Finite Group Databasebeta

← #29737150

Subgroups — #29737150

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×#16840048C211yesmaximal, characteristic, complemented
2C22×#14519702C211yesmaximal, characteristic
3#21449521C211yesmaximal, characteristic
4C22⋊C4×Dic122C211yesmaximal, characteristic
5#21447361C211yesmaximal, characteristic, complemented
6#21447199C211yesmaximal, characteristic, complemented
7#21447199C211yesmaximal, characteristic, complemented
8#21446961C211yesmaximal, characteristic, complemented
9#21446090C211yesmaximal, characteristic, complemented
10#21446090C211yesmaximal, characteristic, complemented
11#21446014C211yesmaximal, characteristic
12#21445844C211yesmaximal, characteristic
13#21445825C211yesmaximal, characteristic, complemented
14#21445755C211yesmaximal, characteristic
15C61×#1850C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519754—21no—
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22#16840048—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519702—24no—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36#16839694—21no—
37#16839683—21no—
38#16839668C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839666—21no—
41#16839666—21no—
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839637—21no—
45#16839632C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839620—21no—
49C22⋊C4×Dic61—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C23×C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—21no—
64C22×Dic122—22no—
65C22×Dic122—21no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122Q814yescomplemented
69C22×Dic122C2311yescharacteristic
70C22×Dic122C2311yescharacteristic
71C23×Dic61—21no—
72C23×Dic61—21no—
73#14519777—22no—
74#14519777—41no—
75#14519754—44no—
76C4×Dic122—22no—
77C4×Dic122—22no—
78C4×Dic122—41no—
79C4×Dic122—41no—
80C2×C4⋊Dic61—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—41no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100#14519709—22no—
101#14519709—22no—
102#14519709—41no—
103#14519709—41no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519702—22no—
108#14519702—22no—
109#14519702—41no—
110#14519702—41no—
111#14519702D414yes—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61—22no—
122C22×C8×C61—22no—
123C22×C8×C61—41no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—22no—
133C4⋊C8×C61—41no—
134C22⋊C8×C61—22no—
135#95×C61D411yescharacteristic
136#95×C61D411yescharacteristic
137C24×C61C2×D411yescharacteristic
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×Q812yes—
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—24no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—28no—
172C2×Dic122—24no—
173C2×Dic122—21no—
174C2×Dic122—44no—
175C2×Dic122—44no—
176C2×Dic122—44no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—81no—
182C2×Dic122C2×D412yes—
183C2×Dic122C2×Q812yes—
184C2×Dic122C2×Q812yes—
185C2×Dic122#4112yes—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—44no—
198C2×C8×C61C2×D412yes—
199C22⋊C4×C61—22no—
200C22⋊C4×C61SD1614yes—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C22⋊Dic61—42no—
206C22⋊Dic61—42no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—42no—
219C4⋊Dic61—42no—
220C4⋊Dic61—42no—
221C4⋊Dic61—81no—
222C4⋊Dic61C2×D412yes—
223#11758196—42no—
224#11758196—42no—
225#11758196—42no—
226#11758196—42no—
227C4×Dic61—42no—
228C4×Dic61—42no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C22⋊Q812yes—
234C23×C61C2×SD1612yes—
235C23×C61#13712yes—
236C23×C61C22×D411yescharacteristic
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—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61C22⋊Q812yes—
254C2×C4×C61C2×SD1612yes—
255C2×C4×C61C2×SD1612yes—
256C2×C4×C61C22×D411yescharacteristic
257C2×C4×C61C22×Q811yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—48no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—84no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—22no—
293Dic122—48no—
294Dic122—41no—
295Dic122—41no—
296Dic122—84no—
297Dic122C22⋊Q814yescomplemented
298C8×C61—22no—
299C8×C61—41no—
300C8×C61—41no—
301C8×C61—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—22no—
319C4×C61—22no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61C2×C22⋊Q811yescharacteristic
323C4×C61#54311yescharacteristic
324C4×C61#54611yescharacteristic
325C4×C61#55111yescharacteristic
326Dic61—24no—
327Dic61—42no—
328Dic61—42no—
329Dic61—81no—
330Dic61—81no—
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#1850D6111yescharacteristic
346#1807—611no—
347#1803—611no—
348#1787—611no—
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×Q8—611no—
358C23×C8C2×D6111yescharacteristic
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⋊C8C2×D6111yescharacteristic
370C2×C4⋊C8C2×D6111yescharacteristic
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—1224no—
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—611no—
393#380—1221no—
394C4×C22⋊C4—1221no—
395C4×C22⋊C4C2×D6111yescharacteristic
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#336C2×D6111yescharacteristic
401#336C2×D6111yescharacteristic
402C61#2595511yescharacteristic, complemented, Sylow
403C22×Q8—614no—
404C22×Q8—611no—
405C22×Q8—611no—
406C22×Q8—1222no—
407C22×Q8—1222no—
408C22×Q8—1221no—
409C22×Q8—1221no—
410C22×Q8—1222no—
411C22×Q8—2441no—
412C22×Q8—2441no—
413C23×C4—1221no—
414C23×C4—1221no—
415C23×C4C22×D6111yescharacteristic
416C22×C8—22no—
417C22×C8—22no—
418C22×C8—41no—
419C22×C8C22×D6111yescharacteristic
420C22×C8C22×D6111yescharacteristic
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—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1222no—
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—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—2441no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4D24411yescharacteristic
449C2×C22⋊C4D24411yescharacteristic
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453C2×C42C22×D6111yescharacteristic
454#106—1222no—
455#106—1222no—
456#106—2441no—
457#106—2441no—
458C4⋊C8—22no—
459C4⋊C8—41no—
460#103—614no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—2441no—
467#103—2441no—
468C22⋊C8—22no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95D24411yescharacteristic
478#95D24411yescharacteristic
479C24C2×D24411yescharacteristic
480C2×Q8—612no—
481C2×Q8—612no—
482C2×Q8—612no—
483C2×Q8—612no—
484C2×Q8—1228no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1224no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—4881no—
495C2×Q8—2442no—
496C2×Q8—2442no—
497C2×Q8—2442no—
498C2×Q8—2444no—
499C2×Q8—2444no—
500C2×Q8—2444no—
501C2×Q8—2442no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1224no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—4881no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4C2×D24411yescharacteristic
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24411yescharacteristic
523C22×C4C2×D24411yescharacteristic
524C22×C4C2×D24411yescharacteristic
525C22×C4#1175822412yes—
526C22×C4Q8×D6112yes—
527C22×C4C23×D6111yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—44no—
540C2×C8D4×D6112yes—
541C4⋊C4—612no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
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—22no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C22⋊C4#1175819014yes—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—2442no—
570C42—2442no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1451975712yes—
576C23C2×#1175819012yes—
577C23#1451981012yes—
578C23C22×D24411yescharacteristic
579Q8—614no—
580Q8—1222no—
581Q8—1222no—
582Q8—1222no—
583Q8—1222no—
584Q8—1222no—
585Q8—4884no—
586Q8—2448no—
587Q8—2441no—
588Q8—2441no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—4884no—
611C2×C4—2448no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
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#1451975712yes—
632C2×C4C2×#1175819012yes—
633C2×C4C2×#1175819012yes—
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C2×Q8×D6111yescharacteristic
636C2×C4#41×D6111yescharacteristic
637C2×C4C2×#1175822411yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982311yescharacteristic
640C8—22no—
641C8—41no—
642C8—41no—
643C8—41no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1684004814yescomplemented
651C22C2×#1451975711yescharacteristic
652C22D4×D24411yescharacteristic
653C22#1684040711yescharacteristic
654C22C22×#1175819011yescharacteristic
655C22C2×#1451981011yescharacteristic
656C22#1684057411yescharacteristic
657C22#1684057911yescharacteristic
658C4—22no—
659C4—22no—
660C4—22no—
661C4—41no—
662C4—41no—
663C4—1224no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4—2442no—
670C4—2442no—
671C4C22⋊Q8×D6111yescharacteristic
672C4#1684038711yescharacteristic
673C4#1684039911yescharacteristic
674C4#1684043511yescharacteristic
675C2—24no—
676C2#2144948311yescharacteristic
677C2C2×#1684004811yescharacteristic
678C2#2145566211yescharacteristic
679C2#2145739411yescharacteristic
680C2D4×#1175819011yescharacteristic
681C2#2145768511yescharacteristic
682C2#2145793211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1850C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1850D61
hypocenterC61#25955
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
1. C2×#16840048C2semidirect
5. #21447361C2semidirect
6. #21447199C2semidirect
7. #21447199C2semidirect
8. #21446961C2semidirect
9. #21446090C2semidirect
10. #21446090C2semidirect
13. #21445825C2semidirect
68. C22×Dic122Q8semidirect
297. Dic122C22⋊Q8semidirect
309. C22×C61#475semidirect
402. C61#25955semidirect
650. C22#16840048semidirect

Maximal subgroups

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

subgroupindex
C2×#168400482
C22×#145197022
#214495212
C22⋊C4×Dic1222
#214473612
#214471992
#214469612
#214460902
#214460142
#214458442
#214458252
#214457552
C61×#18502
#2595561