Finite Group Databasebeta

← #29737147

Subgroups — #29737147

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840045C211yesmaximal, characteristic, complemented
2C22×#14519701C211yesmaximal, characteristic
3#21449522C211yesmaximal, characteristic
4Q8×C22⋊Dic61C211yesmaximal, characteristic
5#21447360C211yesmaximal, characteristic, complemented
6#21447198C211yesmaximal, characteristic, complemented
7#21447196C211yesmaximal, characteristic, complemented
8#21446958C211yesmaximal, characteristic, complemented
9#21446089C211yesmaximal, characteristic, complemented
10#21446085C211yesmaximal, characteristic, complemented
11#21446014C211yesmaximal, characteristic
12#21445996C211yesmaximal, characteristic
13#21445842C211yesmaximal, characteristic
14#21445822C211yesmaximal, characteristic, complemented
15C61×#1787C211yesmaximal, characteristic
16C23×#92418C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22#16840045—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519701—24no—
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34#16839696—21no—
35#16839683—21no—
36#16839669—21no—
37#16839669—21no—
38#16839669C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839634—21no—
43#16839631C2211yescharacteristic
44#16839622—21no—
45#16839622C2211yescharacteristic
46C4×C22⋊Dic61—21no—
47C4×C22⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C23×Q8×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C61×#380C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C22×#92418—22no—
64C22×#92418—22no—
65C22×#92418—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519754—44no—
71Q8×Dic61—22no—
72Q8×Dic61—22no—
73Q8×Dic61—41no—
74Q8×Dic61—41no—
75C2×C4⋊Dic61—22no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×C22⋊Dic61D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519709—22no—
97#14519709—41no—
98#14519708—22no—
99#14519708—41no—
100#14519706—22no—
101#14519706—41no—
102#14519701—22no—
103#14519701—22no—
104#14519701—22no—
105#14519701D414yes—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115C22×Q8×C61—21no—
116C22×Q8×C61—22no—
117C22×Q8×C61—22no—
118C22×Q8×C61—22no—
119C22×Q8×C61—41no—
120C22×Q8×C61—41no—
121C22×Q8×C61Q814yescomplemented
122C22×Q8×C61C2311yescharacteristic
123C22×Q8×C61C2311yescharacteristic
124C23×C4×C61—21no—
125C23×C4×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131#103×C61—22no—
132#103×C61—22no—
133#103×C61—41no—
134#103×C61—41no—
135C22⋊C8×C61—22no—
136#95×C61—21no—
137C24×C61C2×D411yescharacteristic
138C2×Q8×C61—24no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—28no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—42no—
149C2×Q8×C61—42no—
150C2×Q8×C61—44no—
151C2×Q8×C61—44no—
152C2×Q8×C61—44no—
153C2×Q8×C61C2×D412yes—
154C2×Q8×C61C2×Q812yes—
155C2×Q8×C61C2×Q812yes—
156C2×Q8×C61#4112yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—21no—
162C22×C4×C61—24no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—42no—
167C22×C4×C61—42no—
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×Q812yes—
170C22×C4×C61#4112yes—
171C22×C4×C61C2411yescharacteristic
172C22×Dic61—21no—
173C22×Dic61—21no—
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—42no—
177C22×Dic61—42no—
178C22×Dic61—81no—
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D411yescharacteristic
183C2×Dic122—42no—
184C2×Dic122—42no—
185C2×Dic122—81no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C22⋊C4×C61—42no—
197C22⋊Dic61—22no—
198C22⋊Dic61—42no—
199C22⋊Dic61SD1614yes—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210C4⋊Dic61—42no—
211C4⋊Dic61C2×D412yes—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—81no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—42no—
221C4×Dic61—42no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—44no—
228C2×#92418C2×D412yes—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C22⋊Q812yes—
234C23×C61C2×SD1612yes—
235C23×C61#13712yes—
236C23×C61C22×D411yescharacteristic
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—48no—
243Q8×C61—41no—
244Q8×C61—41no—
245Q8×C61C22⋊Q814yescomplemented
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—48no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61C22⋊Q812yes—
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×Q811yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—84no—
295C2×Dic61C2×SD1612yes—
296C2×Dic61C2×SD1612yes—
297Dic122—84no—
298C8×C61—41no—
299C8×C61—41no—
300#92418—22no—
301#92418—41no—
302#25955—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61#47514yescomplemented
310C22×C61C2×C22⋊Q811yescharacteristic
311C22×C61D4211yescharacteristic
312C22×C61#54611yescharacteristic
313C22×C61C22×SD1611yescharacteristic
314C22×C61C2×#13711yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C4×C61—22no—
318C4×C61—24no—
319C4×C61—42no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61C2×C22⋊Q811yescharacteristic
323C4×C61#54311yescharacteristic
324C4×C61#54611yescharacteristic
325C4×C61#55111yescharacteristic
326Dic61—22no—
327Dic61—22no—
328Dic61—41no—
329Dic61—42no—
330Dic61—41no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#475—611no—
335C22×#103—611no—
336#2440—611no—
337Q8×C22⋊C4—611no—
338#2096—611no—
339#2066—611no—
340#2066—611no—
341#2024—611no—
342#1865—611no—
343#1865—611no—
344#1853—611no—
345#1850—611no—
346#1807—611no—
347#1803—611no—
348#1787D6111yescharacteristic
349C2×C61—24no—
350C2×C61#243711yescharacteristic
351C2×C61C2×#47511yescharacteristic
352C2×C61#308211yescharacteristic
353C2×C61#327011yescharacteristic
354C2×C61D4×SD1611yescharacteristic
355C2×C61#329811yescharacteristic
356C2×C61#331611yescharacteristic
357C23×Q8C2×D6111yescharacteristic
358C23×C8—611no—
359C2×C4⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×Q8—611no—
362C2×C4×Q8—611no—
363C22×C4⋊C4—611no—
364#475—1224no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—1224no—
376C2×#103C2×D6111yescharacteristic
377C2×#103C2×D6111yescharacteristic
378C2×#103C2×D6111yescharacteristic
379C2×#103C2×D6111yescharacteristic
380C2×C22⋊C8C2×D6111yescharacteristic
381#398—1221no—
382#395—1221no—
383#391—611no—
384#391—611no—
385#391—1221no—
386#391—1221no—
387#389—611no—
388#389—611no—
389#385—1221no—
390#384—611no—
391#380—611no—
392#380—1221no—
393#380C2×D6111yescharacteristic
394C4×C22⋊C4—611no—
395C4×C22⋊C4—1221no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#336—611no—
401#336—611no—
402C61#2595511yescharacteristic, complemented, Sylow
403C22×Q8—21no—
404C22×Q8—22no—
405C22×Q8—22no—
406C22×Q8—22no—
407C22×Q8—41no—
408C22×Q8—41no—
409C22×Q8—1221no—
410C22×Q8Dic12214yescomplemented
411C22×Q8C22×D6111yescharacteristic
412C22×Q8C22×D6111yescharacteristic
413C23×C4—21no—
414C23×C4—1221no—
415C23×C4C22×D6111yescharacteristic
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—1222no—
419C22×C8—1222no—
420C22×C8—2441no—
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423C4⋊Q8—1222no—
424C4⋊Q8—2441no—
425C22⋊Q8—2444no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×Q8—2441no—
429C4×Q8—2441no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1222no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—2441no—
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—611no—
449C2×C22⋊C4—1221no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42—1221no—
454#106—1222no—
455#106—1222no—
456#106—2441no—
457#106—2441no—
458C4⋊C8—1222no—
459C4⋊C8—2441no—
460#103—22no—
461#103—22no—
462#103—41no—
463#103—41no—
464#103—614no—
465#103—1222no—
466#103—1222no—
467#103—1222no—
468C22⋊C8—22no—
469#95—21no—
470#95—611no—
471#95—611no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24D4×D6111yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—24no—
485C2×Q8—28no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—4881no—
496C2×Q8—2442no—
497C2×Q8—2442no—
498C2×Q8C2×Dic12212yes—
499C2×Q8C2×Dic12212yes—
500C2×Q8D4×D6112yes—
501C2×Q8#1175822412yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—4881no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4C2×Dic12212yes—
525C22×C4D4×D6111yescharacteristic
526C22×C4#1175822412yes—
527C22×C4C23×D6111yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—612no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—2444no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—612no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—4881no—
554C4⋊C4—2442no—
555C4⋊C4—2442no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C22⋊C4—42no—
562C22⋊C4—614no—
563C22⋊C4—1222no—
564C22⋊C4—2442no—
565C42—1221no—
566C42—1221no—
567C42—1221no—
568C42—1221no—
569C42—2442no—
570C42—2442no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1451975412yes—
576C23SD16×D6112yes—
577C23#1451980912yes—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—48no—
585Q8—41no—
586Q8—41no—
587Q8—4884no—
588Q8#1451975414yescomplemented
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—612no—
609C2×C4—612no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—4884no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4#1451975412yes—
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C22×Dic12211yescharacteristic
636C2×C4C2×#1175822411yescharacteristic
637C2×C4C2×#1175822411yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982211yescharacteristic
640C8—41no—
641C8—41no—
642C8—1222no—
643C8—2441no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1684004514yescomplemented
651C22C2×#1451975411yescharacteristic
652C22#1684039411yescharacteristic
653C22#1684039911yescharacteristic
654C22C2×SD16×D6111yescharacteristic
655C22C2×#1451980911yescharacteristic
656C22#1684057111yescharacteristic
657C22#1684057811yescharacteristic
658C4—22no—
659C4—24no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—1222no—
664C4—1222no—
665C4—4881no—
666C4—4881no—
667C4—4881no—
668C4—2441no—
669C4—2442no—
670C4—2441no—
671C4C2×#1451975411yescharacteristic
672C4#1684038811yescharacteristic
673C4#1684039911yescharacteristic
674C4#1684044311yescharacteristic
675C2—24no—
676C2#2144948511yescharacteristic
677C2C2×#1684004511yescharacteristic
678C2#2145566311yescharacteristic
679C2#2145739311yescharacteristic
680C2#2145750811yescharacteristic
681C2#2145767711yescharacteristic
682C2#2145793411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#25955
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C2×#16840045C2semidirect
5. #21447360C2semidirect
6. #21447198C2semidirect
7. #21447196C2semidirect
8. #21446958C2semidirect
9. #21446089C2semidirect
10. #21446085C2semidirect
14. #21445822C2semidirect
121. C22×Q8×C61Q8semidirect
245. Q8×C61C22⋊Q8semidirect
309. C22×C61#475semidirect
402. C61#25955semidirect
410. C22×Q8Dic122semidirect
588. Q8#14519754semidirect
650. C22#16840045semidirect

Maximal subgroups

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

subgroupindex
C2×#168400452
C22×#145197012
#214495222
Q8×C22⋊Dic612
#214473602
#214471982
#214471962
#214469582
#214460892
#214460852
#214460142
#214459962
#214458422
#214458222
C61×#17872
#2595561