Finite Group Databasebeta

← #29684213

Subgroups — #29684213

Order 15616.

860 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839823C211yesmaximal, characteristic, complemented
2C2×#16839817C211yesmaximal, characteristic, complemented
3#21448256C211yesmaximal, characteristic
4#21446615C211yesmaximal, characteristic
5#21446611C211yesmaximal, characteristic
6#21446541C211yesmaximal, characteristic, complemented
7#21446536C211yesmaximal, characteristic, complemented
8#21446433C211yesmaximal, characteristic, complemented
9#21446429C211yesmaximal, characteristic, complemented
10#21446426C211yesmaximal, characteristic, complemented
11#21446422C211yesmaximal, characteristic, complemented
12#21445679C211yesmaximal, characteristic, complemented
13C4×#14519689C211yesmaximal, characteristic, complemented
14#21445505C211yesmaximal, characteristic, complemented
15C61×#1750C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊D61C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18#16839823—24no—
19#16839817—24no—
20C2×C4⋊#92418C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839650C411yescharacteristic
39#16839650C411yescharacteristic
40#16839643C411yescharacteristic
41#16839643C411yescharacteristic
42#16839638C411yescharacteristic
43#16839638C411yescharacteristic
44#16839633C411yescharacteristic
45#16839633C411yescharacteristic
46C4×#11758196C2211yescharacteristic
47C4×#11758198C2211yescharacteristic
48C4×C22⋊Dic61C2211yescharacteristic
49#16839608C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#117×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×C22⋊D61C2×C411yescharacteristic
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61C2×C411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C4×C22⋊D61—22no—
70C4×C22⋊D61—22no—
71C2×#11758196C2×C411yescharacteristic
72C2×#11758196C2×C411yescharacteristic
73C2×#11758196C2×C411yescharacteristic
74C2×C22⋊Dic61C2×C411yescharacteristic
75C2×C22⋊Dic61C2×C411yescharacteristic
76C2×C22⋊Dic61C2×C411yescharacteristic
77C2×#11758198C2×C411yescharacteristic
78C2×#11758198C2×C411yescharacteristic
79C2×#11758198C2×C411yescharacteristic
80C2×C4×Dic61C2×C411yescharacteristic
81C2×C4×Dic61C2×C411yescharacteristic
82C2×C4×Dic61C2311yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85C42×D61—22no—
86C42×D61—22no—
87C4⋊#92418—22no—
88C4⋊#92418—22no—
89#14519706—22no—
90#14519706—22no—
91#14519706—22no—
92#14519706—22no—
93C22⋊#92418—22no—
94C22⋊#92418—22no—
95#14519689—22no—
96#14519689—22no—
97#14519689—22no—
98#14519689—22no—
99#14519688D414yescomplemented
100#14519687—22no—
101#14519687—22no—
102C4×#92418—22no—
103C8×Dic61—22no—
104C8×Dic61—22no—
105#14519684C2×C411yescharacteristic
106#14519684C2×C411yescharacteristic
107#14519684C2×C411yescharacteristic
108#14519684C2×C411yescharacteristic
109#14519684C2×C411yescharacteristic
110#14519684C2×C411yescharacteristic
111#14519684C2×C411yescharacteristic
112#14519684C2×C411yescharacteristic
113#14519684C2×C411yescharacteristic
114#14519684C2×C411yescharacteristic
115C23×C4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120#117×C61—22no—
121#117×C61—22no—
122C2×C4⋊C4×C61C2×C411yescharacteristic
123C2×C4⋊C4×C61C2×C411yescharacteristic
124C2×C22⋊C4×C61C2×C411yescharacteristic
125C2×C22⋊C4×C61C2×C411yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C22⋊C8×C61—22no—
129C22⋊C8×C61—22no—
130#97×C61—22no—
131C4×C8×C61—22no—
132C24×C61C22×C411yescharacteristic
133C23×D61C22×C411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—41no—
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C411yescharacteristic
152C22×Dic61C22×C411yescharacteristic
153C22×Dic61C22×C411yescharacteristic
154C22×Dic61C22×C411yescharacteristic
155C22×Dic61C22×C411yescharacteristic
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—22no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61#4112yes—
179C2×C8×C61#4112yes—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61C2×D412yes—
187C42×C61#4112yes—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—41no—
191#11758198—22no—
192#11758198—22no—
193#11758198—41no—
194#11758196—22no—
195#11758196—22no—
196#11758196—41no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61#4112yes—
204C4×Dic61#4112yes—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418C2×D412yes—
216C2×#92418C2×D412yes—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61C23×C411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61#13112yes—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C4×D412yes—
250C2×C4×C61C4×D412yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C22⋊D61—44no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61#11712yes—
277C2×Dic61#11712yes—
278C2×Dic61#13112yes—
279C4×D61—44no—
280C4×D61—44no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284C4×D61—41no—
285C8×C61—22no—
286C8×C61—22no—
287C8×C61—41no—
288C8×C61—41no—
289#92418—22no—
290#92418—22no—
291#92418—41no—
292#92418—41no—
293#21273—611nomaximal, Sylow
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61C2×#11711yescharacteristic
302C22×C61C2×C4×D411yescharacteristic
303C22×C61C4×#4111yescharacteristic
304C22×C61C2×#13111yescharacteristic
305C22×C61C2×#13111yescharacteristic
306C22×C61#56811yescharacteristic
307C22×C61#56811yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C4×C61—22no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61#52911yescharacteristic
321C4×C61#52911yescharacteristic
322C4×C61#53211yescharacteristic
323C4×C61#53211yescharacteristic
324Dic61—22no—
325Dic61—22no—
326Dic61—22no—
327Dic61—41no—
328Dic61—41no—
329Dic61—41no—
330Dic61—41no—
331C2×#433—611no—
332C2×#432—611no—
333#2290—611no—
334#1955—611no—
335#1954—611no—
336#1940—611no—
337#1939—611no—
338#1924—611no—
339#1924—611no—
340#1923—611no—
341#1923—611no—
342#1768—611no—
343#1750D6111yescharacteristic, complemented
344C4×C22⋊C8—611no—
345#1726—611no—
346C2×C61—81no—
347C2×C61#229311yescharacteristic
348C2×C61#292411yescharacteristic
349C2×C61#292511yescharacteristic
350C2×C61#293311yescharacteristic
351C2×C61#293311yescharacteristic
352C2×C61#296611yescharacteristic
353C2×C61#296711yescharacteristic
354D61—42no—
355C2×C4×D4—611no—
356C2×#117C2×D6111yescharacteristic
357C22×C42—611no—
358#433—1224no—
359#432—1224no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8—611no—
366C2×C22⋊C8—611no—
367C2×C22⋊C8—611no—
368C2×C22⋊C8—611no—
369C2×C22⋊C8—611no—
370C2×C22⋊C8—611no—
371C2×C22⋊C8—611no—
372C2×C22⋊C8C2×D6111yescharacteristic
373C2×C22⋊C8C2×D6111yescharacteristic
374C2×#97—611no—
375C2×#97—611no—
376C2×#97—611no—
377C2×#97C2×D6111yescharacteristic
378C2×C4×C8—611no—
379C2×C4×C8—611no—
380C2×C4×C8—611no—
381C2×C4×C8C2×D6111yescharacteristic
382#388—611no—
383#388—611no—
384#387—611no—
385#387—611no—
386#386—611no—
387#386—611no—
388#385—611no—
389#385—611no—
390C4×C4⋊C4—611no—
391C4×C22⋊C4—611no—
392C4×C22⋊C4—611no—
393#376—611no—
394#336—611no—
395#336—611no—
396#336—611no—
397#336—611no—
398#336C2×D6111yescharacteristic
399#336C2×D6111yescharacteristic
400C61#2127311yescharacteristic, complemented, Sylow
401C22×D4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4C22×D6111yescharacteristic
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8—611no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C4×D4—1222no—
415C4×D4—1222no—
416#117—22no—
417#117—22no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4C4×D6111yescharacteristic
422C2×C4⋊C4C4×D6111yescharacteristic
423C2×C22⋊C4—611no—
424C2×C22⋊C4—611no—
425C2×C22⋊C4—611no—
426C2×C22⋊C4—611no—
427C2×C22⋊C4—611no—
428C2×C22⋊C4—611no—
429C2×C22⋊C4C4×D6111yescharacteristic
430C2×C22⋊C4C4×D6111yescharacteristic
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1222no—
437C2×C42—1222no—
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6111yescharacteristic
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442C4⋊C8—1222no—
443C4⋊C8—1222no—
444C4⋊C8—1222no—
445C4⋊C8—1222no—
446C22⋊C8—22no—
447C22⋊C8—22no—
448C22⋊C8—1222no—
449C22⋊C8—1222no—
450C22⋊C8—1222no—
451C22⋊C8—1222no—
452C22⋊C8—1222no—
453C22⋊C8—1222no—
454#97—22no—
455#97—614no—
456#97—1222no—
457#97—1222no—
458C4×C8—22no—
459C4×C8—1222no—
460C4×C8—1222no—
461C4×C8—1222no—
462#95—611no—
463#95—611no—
464#95—611no—
465#95—611no—
466#95—611no—
467#95—611no—
468#95—611no—
469#95—611no—
470#95—611no—
471#95—611no—
472C24—611no—
473C24C2×C4×D6111yescharacteristic
474C2×D4—1222no—
475C2×D4—1222no—
476C2×D4—2441no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1222no—
499C22×C4—1222no—
500C22×C4—2441no—
501C22×C4—2441no—
502C22×C4C2×C4×D6111yescharacteristic
503C22×C4C2×C4×D6111yescharacteristic
504C22×C4C2×C4×D6111yescharacteristic
505C22×C4C2×C4×D6111yescharacteristic
506C22×C4C2×C4×D6111yescharacteristic
507C22×C4C23×D6111yescharacteristic
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—612no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8#1175822212yes—
531C2×C8#1175822212yes—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—2441no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—1222no—
540C22⋊C4—1222no—
541C22⋊C4—1222no—
542C22⋊C4—1222no—
543C22⋊C4—2441no—
544C22⋊C4—2441no—
545C42—21no—
546C42—21no—
547C42—612no—
548C42—612no—
549C42—1221no—
550C42—1221no—
551C42—1221no—
552C42—1221no—
553C42—1222no—
554C42—1222no—
555C42D4×D6112yes—
556C42#1175822412yes—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—612no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23C22×C4×D6111yescharacteristic
572D4—2444no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—2444no—
612C2×C4—2444no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4#1451973712yes—
618C2×C4#1451973712yes—
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822411yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C2×C4C2×#1175822211yescharacteristic
625C8—22no—
626C8—22no—
627C8—41no—
628C8—41no—
629C8—1222no—
630C8—1222no—
631C8—2441no—
632C8—2441no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22#117×D6111yescharacteristic
648C22C2×#1451973711yescharacteristic
649C22C4×#1175822211yescharacteristic
650C22#131×D6111yescharacteristic
651C22#131×D6111yescharacteristic
652C22#1684052711yescharacteristic
653C22#1684052711yescharacteristic
654C4—22no—
655C4—22no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—1222no—
660C4—1222no—
661C4—1222no—
662C4—2441no—
663C4—2441no—
664C4—2441no—
665C4—2441no—
666C4#1684029211yescharacteristic
667C4#1684029211yescharacteristic
668C4#1684030911yescharacteristic
669C4#1684030911yescharacteristic
670C2—81no—
671C2—2442no—
672C2#2144828211yescharacteristic
673C2#2145447511yescharacteristic
674C2#2145448211yescharacteristic
675C2#2145454911yescharacteristic
676C2#2145454911yescharacteristic
677C2#2145479011yescharacteristic
678C2#2145480011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1750C2
prime radicalC22×C22⋊D61C2×C4
cosocleC22×C4×C61C24
hypercenter#1750D61
hypocenterC61#21273
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1750

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16839823C2semidirect
2. C2×#16839817C2semidirect
6. #21446541C2semidirect
7. #21446536C2semidirect
8. #21446433C2semidirect
9. #21446429C2semidirect
10. #21446426C2semidirect
11. #21446422C2semidirect
12. #21445679C2semidirect
13. C4×#14519689C2semidirect
14. #21445505C2semidirect
15. C61×#1750C2semidirect
99. #14519688D4semidirect
343. #1750D61semidirect
400. C61#21273semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168398232
C2×#168398172
#214482562
#214466152
#214466112
#214465412
#214465362
#214464332
#214464292
#214464262
#214464222
#214456792
C4×#145196892
#214455052
C61×#17502
#2127361