Finite Group Databasebeta

← #29654081

Subgroups — #29654081

Order 15616.

930 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21451585C211yesmaximal, characteristic, complemented
4#21444833C214yesmaximal, complemented
5#21444816C214yesmaximal, complemented
6#21444383C211yesmaximal, characteristic, complemented
7#21444379C211yesmaximal, characteristic, complemented
8#21444133C211yesmaximal, characteristic, complemented
9C61×#3455C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11C2×#14519756—21no—
12#16840165—21no—
13#16840158—21no—
14#16840112—21no—
15#16840108—21no—
16#16840090C2214yes—
17#16840073C2214yes—
18#16840047—22no—
19#16840042—22no—
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22#16839692—21no—
23#16839692C2212yes—
24#16839687—21no—
25#16839687C2212yes—
26#16839668—21no—
27#16839628C2211yescharacteristic
28#16839499C2214yes—
29#16839496C2214yes—
30#16839488C2212yes—
31#16839486C2212yes—
32C61×#542—21no—
33C61×#542—21no—
34C61×#541—21no—
35C61×#540—21no—
36C61×#539—21no—
37C61×#539C2212yes—
38C61×#538—21no—
39C61×#538C2212yes—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#529—22no—
43C61×#529—22no—
44C61×#525—21no—
45C4×#41×C61—21no—
46C2×#117×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519756—44no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61C2312yes—
56C2×#11758196—22no—
57C2×C22⋊Dic61—22no—
58#14519710—24no—
59#14519708—24no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62#14519701—24no—
63#14519698—24no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66C4×#92418—21no—
67C4×#92418—21no—
68#14519684—21no—
69#14519684—22no—
70#14519684—22no—
71#14519684C2312yes—
72C2×#41×C61—21no—
73C23×C4×C61D411yescharacteristic
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61D411yescharacteristic
77#127×C61D411yescharacteristic
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—22no—
81#126×C61—22no—
82#125×C61—22no—
83#125×C61—22no—
84#125×C61—21no—
85#124×C61—22no—
86#124×C61—22no—
87#124×C61—21no—
88#123×C61—22no—
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61C2312yes—
99#121×C61—22no—
100#121×C61—21no—
101#121×C61—21no—
102#121×C61—22no—
103#121×C61—21no—
104#121×C61C2312yes—
105#120×C61—22no—
106C4×Q8×C61—21no—
107C4×Q8×C61—21no—
108C4×Q8×C61—21no—
109C4×Q8×C61D412yes—
110C4×D4×C61—21no—
111C4×D4×C61—21no—
112C4×D4×C61—22no—
113C4×D4×C61—22no—
114C4×D4×C61—22no—
115C4×D4×C61—22no—
116C4×D4×C61—21no—
117C4×D4×C61D412yes—
118#117×C61—22no—
119#117×C61—21no—
120#117×C61—21no—
121#117×C61—21no—
122#117×C61—21no—
123#117×C61—21no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61C2311yescharacteristic
128C2×C42×C61—22no—
129C2×C42×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C24×C61C2×D411yescharacteristic
132#41×C61—42no—
133#41×C61—42no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61C2×D412yes—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—22no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—22no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—22no—
157C22×Dic61—22no—
158C2×Dic122—41no—
159C2×Dic122—41no—
160C2×Dic122—42no—
161C2×Dic122—42no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61C2×D412yes—
177C4⋊C4×C61C2×D412yes—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—22no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—21no—
190C22⋊C4×C61—22no—
191C42×C61—22no—
192C42×C61—22no—
193C42×C61—21no—
194C42×C61—22no—
195C42×C61—21no—
196C42×C61C2×D411yescharacteristic
197C42×C61C2×D411yescharacteristic
198C42×C61C2×D411yescharacteristic
199C42×C61C2×D411yescharacteristic
200C22⋊Dic61—44no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61—42no—
206#11758196—44no—
207C2×#92418—22no—
208C2×#92418—22no—
209C23×C61—21no—
210C23×C61—22no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61C22×D411yescharacteristic
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—81no—
219D4×C61—42no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—42no—
223D4×C61—81no—
224D4×C61—82no—
225D4×C61—82no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—42no—
237C2×C4×C61—42no—
238C2×C4×C61—44no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—82no—
242C2×C4×C61—82no—
243C2×C4×C61#12011yescharacteristic
244C2×C4×C61#12011yescharacteristic
245C2×C4×C61#12011yescharacteristic
246C2×C4×C61#12011yescharacteristic
247C2×C4×C61#13512yes—
248C2×C4×C61AGL(1,ℤ/8)12yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—42no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257Dic122—82no—
258Dic122—82no—
259#92418—42no—
260#92418—42no—
261#18312—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61—42no—
265C22×C61—41no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—41no—
269C22×C61—42no—
270C22×C61—41no—
271C22×C61—81no—
272C22×C61C2×#12011yescharacteristic
273C22×C61C2×#13511yescharacteristic
274C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—41no—
280C4×C61—42no—
281C4×C61—41no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61#44912yes—
286C4×C61#44912yes—
287Dic61—82no—
288Dic61—82no—
289#3455D6111yescharacteristic
290#3229—611no—
291#3226—611no—
292#2621—611no—
293#1612—614no—
294#1609—614no—
295#1518—611no—
296#1517—611no—
297#1458—611no—
298C2×C61—21no—
299C2×C61—21no—
300C2×C61—82no—
301C2×C61—81no—
302C2×C61C2×#44911yescharacteristic
303C2×C61#302211yescharacteristic
304C2×C61#302811yescharacteristic
305#542—21no—
306#542—21no—
307#541—21no—
308#540—21no—
309#539—21no—
310#539C2×D6112yes—
311#538—21no—
312#538C2×D6112yes—
313#533C2×D6111yescharacteristic
314#532C2×D6111yescharacteristic
315C2×C4⋊Q8—611no—
316#529—22no—
317#529—22no—
318#525—21no—
319C2×C22⋊Q8—1221no—
320C4×#41—21no—
321C2×#117C2×D6111yescharacteristic
322#500—1221no—
323#498—1221no—
324#487—1221no—
325#486—1221no—
326#483—614no—
327#480—614no—
328#475—1222no—
329#474—1222no—
330#431—611no—
331#431—611no—
332#398—612no—
333#398—1221no—
334#397—612no—
335#397—1221no—
336#391—1221no—
337#382—611no—
338#328—614no—
339#327—614no—
340#324—612no—
341#323—612no—
342C61#1831211yescharacteristic, complemented, Sylow
343C2×#41—21no—
344C22×Q8—1221no—
345C23×C4C22⋊D6111yescharacteristic
346C4⋊Q8—1221no—
347C4⋊Q8—1221no—
348C4⋊Q8—1221no—
349C4⋊Q8—1221no—
350C4⋊Q8C22⋊D6111yescharacteristic
351C4⋊Q8C22⋊D6111yescharacteristic
352C4⋊Q8C22⋊D6111yescharacteristic
353#127C22⋊D6111yescharacteristic
354#126—22no—
355#126—22no—
356#126—22no—
357#126—22no—
358#125—22no—
359#125—22no—
360#125—21no—
361#124—22no—
362#124—22no—
363#124—21no—
364#123—22no—
365#123—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369C22⋊Q8—21no—
370C22⋊Q8—21no—
371C22⋊Q8—22no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—2444no—
375C22⋊Q8C22×D6112yes—
376#121—21no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121—21no—
381#121C22×D6112yes—
382#120—22no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8C22⋊D6112yes—
387C4×D4—21no—
388C4×D4—21no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—21no—
394C4×D4C22⋊D6112yes—
395#117—22no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—21no—
400#117—21no—
401C2×C4⋊C4—612no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22⋊D6111yescharacteristic
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C22⋊C4—1222no—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22×D6111yescharacteristic
410C2×C42—22no—
411C2×C42C22⋊D6111yescharacteristic
412C2×C42C22×D6111yescharacteristic
413#107—1224no—
414#106—1224no—
415C4⋊C8—1221no—
416C4⋊C8—1221no—
417#103—1224no—
418#102—1224no—
419C22⋊C8—612no—
420C22⋊C8—612no—
421C4×C8—1221no—
422C4×C8—1221no—
423#95—612no—
424#95—1221no—
425#95—1222no—
426#95—1222no—
427C24C2×C22⋊D6111yescharacteristic
428#41—42no—
429#41—42no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—2441no—
433C2×Q8—2441no—
434C2×Q8—2442no—
435C2×Q8—2442no—
436C2×Q8C2×C22⋊D6112yes—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C22⋊D6112yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4C2×C22⋊D6111yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C23×D6111yescharacteristic
458C2×C8—1222no—
459C2×C8—1222no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—2444no—
477C4⋊C4—2441no—
478C4⋊C4—2441no—
479C4⋊C4—2442no—
480C4⋊C4C2×C22⋊D6112yes—
481C4⋊C4C2×C22⋊D6112yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—2444no—
496C42—22no—
497C42—22no—
498C42—21no—
499C42—22no—
500C42—21no—
501C42C2×C22⋊D6111yescharacteristic
502C42C2×C22⋊D6111yescharacteristic
503C42C2×C22⋊D6111yescharacteristic
504C42C2×C22⋊D6111yescharacteristic
505C23—21no—
506C23—22no—
507C23—21no—
508C23—21no—
509C23—41no—
510C23—41no—
511C23C22×C22⋊D6111yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—81no—
515Q8—4882no—
516Q8—4882no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—81no—
522D4—82no—
523D4—82no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—2442no—
542C2×C4—2442no—
543C2×C4—2442no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4#1451974611yescharacteristic
548C2×C4#1451974611yescharacteristic
549C2×C4#1451974611yescharacteristic
550C2×C4#1451974611yescharacteristic
551C2×C4#1451979712yes—
552C2×C4#1451980212yes—
553C2×C4C22×C22⋊D6111yescharacteristic
554C2×C4C22×C22⋊D6111yescharacteristic
555C8—2442no—
556C8—2442no—
557C22—21no—
558C22—21no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—42no—
563C22—41no—
564C22—41no—
565C22—42no—
566C22—81no—
567C22C2×#1451974611yescharacteristic
568C22C2×#1451979711yescharacteristic
569C22C2×#1451980211yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—4882no—
581C4—4882no—
582C4#1683990712yes—
583C4#1683990712yes—
584C2—21no—
585C2—21no—
586C2—81no—
587C2—82no—
588C2C2×#1683990711yescharacteristic
589C2#2145519811yescharacteristic
590C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3455C2
socleC22×C61C2×#120
hypercenter#3455D61
hypocenterC61#18312
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3455

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
2. #21456937C2semidirect
3. #21451585C2semidirect
4. #21444833C2semidirect
5. #21444816C2semidirect
6. #21444383C2semidirect
7. #21444379C2semidirect
8. #21444133C2semidirect
342. C61#18312semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214569372
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