Finite Group Databasebeta

← #29571529

Subgroups — #29571529

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446263C212yesmaximal, complemented
2#21444323C212yesmaximal, complemented
3#21444149C211yesmaximal, characteristic, complemented
4#21444051C211yesmaximal, characteristic, complemented
5C61×#3492C211yesmaximal, characteristic
6C22×C4×Dic61C2211yescharacteristic
7C2×#14519704—22no—
8#16839726C2212yes—
9C2×#14519684—21no—
10C2×#14519684—21no—
11C2×#14519684C2211yescharacteristic
12#16839521—22no—
13#16839499—22no—
14#16839496—22no—
15#16839488—21no—
16#16839488—21no—
17C61×#554—22no—
18C61×#553—22no—
19C61×#552—22no—
20C61×#547—22no—
21C61×#544C411yescharacteristic, complemented
22C61×#539—21no—
23C61×#539C412yescomplemented
24C61×#537C411yescharacteristic, complemented
25C61×#525—21no—
26C61×#525—21no—
27C61×#525C2212yes—
28C61×#520—21no—
29C22×C4⋊C4×C61C2211yescharacteristic
30C23×Dic61—21no—
31C23×Dic61—21no—
32C2×#11758194—22no—
33C2×C4×Dic61—22no—
34C2×C4×Dic61—42no—
35#14519704—44no—
36C22⋊#92418—22no—
37C22⋊#92418—22no—
38#14519684—24no—
39#14519684—22no—
40#14519684—42no—
41C2×#41×C61—21no—
42C2×#41×C61—21no—
43C2×#41×C61D412yes—
44C23×C4×C61D411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C23×C4×C61C2311yescharacteristic
47C4⋊Q8×C61—22no—
48#126×C61—22no—
49#126×C61—22no—
50#125×C61—22no—
51#125×C61—22no—
52#125×C61—22no—
53#123×C61—22no—
54#123×C61—22no—
55#123×C61—22no—
56#123×C61—22no—
57#123×C61—22no—
58C22⋊Q8×C61—22no—
59C22⋊Q8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—22no—
63C22⋊Q8×C61—21no—
64C22⋊Q8×C61—22no—
65C22⋊Q8×C61C2×C412yes—
66C22⋊Q8×C61D412yes—
67#121×C61—21no—
68#121×C61—21no—
69#121×C61—21no—
70#121×C61—22no—
71#121×C61—21no—
72#121×C61C2×C412yes—
73#121×C61D412yes—
74#120×C61—22no—
75C4×Q8×C61—22no—
76C4×D4×C61—22no—
77C4×D4×C61—22no—
78C4×D4×C61—22no—
79C4×D4×C61—22no—
80C4×D4×C61—22no—
81C4×D4×C61—22no—
82C4×D4×C61—22no—
83#117×C61—21no—
84#117×C61—22no—
85#117×C61—21no—
86#117×C61D412yes—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61C2×C411yescharacteristic
91C2×C4⋊C4×C61C2×C411yescharacteristic
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D411yescharacteristic
94C24×C61C2×D411yescharacteristic
95#41×C61—42no—
96#41×C61—42no—
97#41×C61—42no—
98#41×C61—42no—
99#41×C61—81no—
100#41×C61—81no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61C22⋊C412yes—
104C2×D4×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—21no—
109C2×D4×C61C22⋊C412yes—
110C22×C4×C61—22no—
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—41no—
119C22×C4×C61—41no—
120C22×C4×C61—41no—
121C22×C4×C61—41no—
122C22×C4×C61—41no—
123C22×C4×C61C22⋊C411yescharacteristic
124C22×C4×C61C22⋊C411yescharacteristic
125C22×C4×C61C22⋊C411yescharacteristic
126C22×C4×C61C22⋊C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—41no—
136C22×Dic61—41no—
137C22×Dic61—41no—
138C22×Dic61—41no—
139C22×Dic61—41no—
140C22×Dic61—41no—
141C22×Dic61—41no—
142C22×Dic61—41no—
143C22×Dic61—41no—
144C22×Dic61—41no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C22⋊C412yes—
160C4⋊C4×C61C22⋊C412yes—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C22⋊C4×C61—22no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—22no—
171C42×C61—22no—
172C42×C61—22no—
173C4×Dic61—24no—
174C4×Dic61—42no—
175#11758194—42no—
176#11758194—42no—
177C2×#92418—42no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—21no—
181C23×C61—21no—
182C23×C61—21no—
183C23×C61—21no—
184C23×C61—22no—
185C23×C61C2×C22⋊C411yescharacteristic
186C23×C61#12011yescharacteristic
187C23×C61#12011yescharacteristic
188Q8×C61—42no—
189Q8×C61—42no—
190Q8×C61—81no—
191D4×C61—42no—
192D4×C61—42no—
193D4×C61—42no—
194D4×C61—42no—
195D4×C61—42no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—81no—
200C2×C4×C61—21no—
201C2×C4×C61—21no—
202C2×C4×C61—22no—
203C2×C4×C61—22no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—21no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—82no—
223C2×C4×C61—81no—
224C2×C4×C61—82no—
225C2×C4×C61#10412yescomplemented
226C2×C4×C61C2×C22⋊C412yes—
227C2×C4×C61#12012yes—
228C2×Dic61—24no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—42no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—81no—
239C2×Dic61—81no—
240C2×Dic61—81no—
241C2×Dic61—81no—
242#92418—82no—
243#8347—611nomaximal, Sylow
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—21no—
256C22×C61—21no—
257C22×C61—21no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61#37911yescharacteristic
262C22×C61C2×#10411yescharacteristic
263C22×C61#42111yescharacteristic
264C22×C61#45411yescharacteristic
265C22×C61#45611yescharacteristic
266C22×C61B(4,𝔽2)11yescharacteristic
267C22×C61#45811yescharacteristic
268C4×C61—22no—
269C4×C61—22no—
270C4×C61—22no—
271C4×C61—42no—
272C4×C61—41no—
273C4×C61—42no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—42no—
277Dic61—44no—
278Dic61—82no—
279#3492D6111yescharacteristic
280#1896—612no—
281#1504—612no—
282#1463—611no—
283#1436—611no—
284C2×C61—21no—
285C2×C61—21no—
286C2×C61—21no—
287C2×C61—21no—
288C2×C61—21no—
289C2×C61—21no—
290C2×C61—82no—
291C2×C61#178911yescharacteristic
292C2×C61#179011yescharacteristic
293C2×C61#179311yescharacteristic
294#554—22no—
295#553—22no—
296#552—22no—
297#547—22no—
298#544Dic6111yescharacteristic, complemented
299#539—21no—
300#539Dic6112yescomplemented
301#537Dic6111yescharacteristic, complemented
302#525—21no—
303#525—21no—
304#525C2×D6112yes—
305#520—21no—
306C22×C4⋊C4C2×D6111yescharacteristic
307C22×C42—611no—
308C2×#104—1222no—
309#407—612no—
310C2×#95—611no—
311C2×#95—1221no—
312C2×#95—1221no—
313#339—1222no—
314#328—1222no—
315#327—1222no—
316#324—1221no—
317#324—1221no—
318C61#834711yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C2×#41—21no—
321C2×#41C22⋊D6112yes—
322C23×C4—1221no—
323C23×C4—1221no—
324C23×C4C22⋊D6111yescharacteristic
325C23×C4C22⋊D6111yescharacteristic
326C23×C4C22×D6111yescharacteristic
327C2×M16—1222no—
328C4⋊Q8—22no—
329#126—22no—
330#126—22no—
331#125—22no—
332#125—22no—
333#125—22no—
334#123—22no—
335#123—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8—22no—
344C22⋊Q8—21no—
345C22⋊Q8—22no—
346C22⋊Q8C2×Dic6112yes—
347C22⋊Q8C22⋊D6112yes—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—22no—
352#121—21no—
353#121C2×Dic6112yes—
354#121C22⋊D6112yes—
355#120—22no—
356C4×Q8—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364#117—22no—
365#117—21no—
366#117—21no—
367#117C22⋊D6112yes—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4C2×Dic6111yescharacteristic
372C2×C4⋊C4C2×Dic6111yescharacteristic
373C2×C4⋊C4C22⋊D6111yescharacteristic
374C2×C4⋊C4C22⋊D6111yescharacteristic
375C2×C42—1222no—
376C2×C42—2442no—
377#104—2444no—
378C22⋊C8—1222no—
379C22⋊C8—1222no—
380#95—1224no—
381#95—1222no—
382#95—2442no—
383C24C2×C22⋊D6111yescharacteristic
384#41—42no—
385#41—42no—
386#41—42no—
387#41—42no—
388#41—81no—
389#41—81no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8C22⋊Dic6112yes—
393C2×D4—21no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—21no—
398C2×D4C22⋊Dic6112yes—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4—2441no—
417C22×C4—2441no—
418C22×C4—2441no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4—2441no—
422C22×C4—2441no—
423C22×C4—2441no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4C22⋊Dic6111yescharacteristic
427C22×C4C22⋊Dic6111yescharacteristic
428C22×C4C22⋊Dic6111yescharacteristic
429C22×C4C22⋊Dic6111yescharacteristic
430C22×C4C22×Dic6111yescharacteristic
431C22×C4C2×C22⋊D6112yes—
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6111yescharacteristic
434M16—2442no—
435M16—2442no—
436C2×C8—2442no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4C22⋊Dic6112yes—
452C4⋊C4C22⋊Dic6112yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—1224no—
466C42—2442no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23C2×C22⋊Dic6111yescharacteristic
475C23#1451974611yescharacteristic
476C23#1451974611yescharacteristic
477Q8—42no—
478Q8—42no—
479Q8—81no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—81no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—1224no—
515C2×C4—4881no—
516C2×C4—4881no—
517C2×C4—4881no—
518C2×C4—4881no—
519C2×C4—2442no—
520C2×C4—2442no—
521C2×C4—2442no—
522C2×C4—2442no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2442no—
528C2×C4#1451970412yescomplemented
529C2×C4C2×C22⋊Dic6112yes—
530C2×C4#1451974612yes—
531C8—4882no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22#1683961811yescharacteristic
550C22C2×#1451970411yescharacteristic
551C22#1683977911yescharacteristic
552C22#1683993511yescharacteristic
553C22#1683994511yescharacteristic
554C22#1683994911yescharacteristic
555C22#1683995211yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—4882no—
566C4—2444no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—82no—
574C2#2144576311yescharacteristic
575C2#2144576711yescharacteristic
576C2#2144577811yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3492C2
socleC22×C61#379
cosocleC23×C4×C61C23
hypercenter#3492D61
hypocenterC61#8347
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
1. #21446263C2semidirect
2. #21444323C2semidirect
3. #21444149C2semidirect
4. #21444051C2semidirect
21. C61×#544C4semidirect
23. C61×#539C4semidirect
24. C61×#537C4semidirect
225. C2×C4×C61#104semidirect
298. #544Dic61semidirect
300. #539Dic61semidirect
301. #537Dic61semidirect
318. C61#8347semidirect
528. C2×C4#14519704semidirect

Maximal subgroups

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

subgroupindex
#214462632
#214443232
#214441492
#214440512
C61×#34922
#834761