Finite Group Databasebeta

← #29652720

Subgroups — #29652720

Order 15616.

1042 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2C2×#16839692C211yesmaximal, characteristic, complemented
3#21444827C212yesmaximal, complemented
4#21444822C212yesmaximal, complemented
5#21444816C214yesmaximal, complemented
6#21444323C212yesmaximal, complemented
7#21444149C211yesmaximal, characteristic, complemented
8C61×#3492C211yesmaximal, characteristic
9C22×C4⋊Dic61C2211yescharacteristic
10C22×#11758196—21no—
11#16840095—22no—
12#16840090C2214yes—
13#16840078—22no—
14#16840073C2214yes—
15#16839805—22no—
16#16839763—22no—
17#16839726C2212yes—
18#16839692—21no—
19#16839692—21no—
20#16839692C2212yes—
21#16839692C2212yes—
22C2×#14519684—21no—
23C2×#14519684C2211yescharacteristic
24#16839499C2214yes—
25#16839496C2214yes—
26#16839488C2212yes—
27#16839488C2212yes—
28C61×#554—22no—
29C61×#553—22no—
30C61×#552—22no—
31C61×#547—22no—
32C61×#544C2211yescharacteristic
33C61×#539—21no—
34C61×#539C2212yes—
35C61×#537C2211yescharacteristic
36C61×#525—21no—
37C61×#525—21no—
38C61×#525C2212yes—
39C61×#520—21no—
40C22×C4⋊C4×C61C2211yescharacteristic
41C23×Dic61—21no—
42C23×Dic61—21no—
43C2×#11758194—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61C2312yes—
48C2×#11758196—22no—
49C2×#11758196—22no—
50C2×#11758196—44no—
51#14519710—24no—
52#14519708—24no—
53#14519701—24no—
54#14519698—24no—
55C22⋊#92418—22no—
56C22⋊#92418C2314yes—
57#14519684—21no—
58#14519684—22no—
59#14519684—22no—
60#14519684C2312yes—
61C2×#41×C61—21no—
62C2×#41×C61—21no—
63C2×#41×C61D412yes—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C4⋊Q8×C61—22no—
68#126×C61—22no—
69#126×C61—22no—
70#125×C61—22no—
71#125×C61—22no—
72#125×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77#123×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—21no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61D412yes—
86C22⋊Q8×C61C2312yes—
87#121×C61—21no—
88#121×C61—21no—
89#121×C61—21no—
90#121×C61—22no—
91#121×C61—21no—
92#121×C61D412yes—
93#121×C61C2312yes—
94#120×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103#117×C61—21no—
104#117×C61—22no—
105#117×C61—21no—
106#117×C61D412yes—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C24×C61C2×D411yescharacteristic
115#41×C61—42no—
116#41×C61—44no—
117#41×C61—42no—
118#41×C61—82no—
119C2×Q8×C61—21no—
120C2×Q8×C61—21no—
121C2×Q8×C61C2×D412yes—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—22no—
126C2×D4×C61—21no—
127C2×D4×C61C2×D412yes—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—22no—
136C22×C4×C61—41no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61—42no—
154C22×Dic61—41no—
155C22×Dic61—41no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61—22no—
183C42×C61—22no—
184C4⋊Dic61—24no—
185C4⋊Dic61—42no—
186#11758196—44no—
187#11758196—44no—
188#11758194—44no—
189C2×#92418—24no—
190C23×C61—21no—
191C23×C61—21no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61#12011yescharacteristic
196C23×C61#12011yescharacteristic
197C23×C61C22×D411yescharacteristic
198Q8×C61—42no—
199Q8×C61—42no—
200Q8×C61—81no—
201D4×C61—42no—
202D4×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—44no—
206D4×C61—44no—
207D4×C61—81no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—24no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—42no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—44no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—82no—
229C2×C4×C61—81no—
230C2×C4×C61—82no—
231C2×C4×C61#12012yes—
232C2×C4×C61AGL(1,ℤ/8)12yes—
233C2×C4×C61C22×D412yes—
234C2×Dic61—24no—
235C2×Dic61—48no—
236C2×Dic61—42no—
237C2×Dic61—42no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—82no—
242C2×Dic61—82no—
243#92418—44no—
244#18171—611nomaximal, Sylow
245C22×C61—24no—
246C22×C61—24no—
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—21no—
252C22×C61—21no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61#45612yes—
257C22×C61#45812yes—
258C22×C61C2×#12011yescharacteristic
259C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C61#57511yescharacteristic
261C4×C61—22no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—41no—
265C4×C61—42no—
266C4×C61—42no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—42no—
270Dic61—44no—
271Dic61—82no—
272#3492D6111yescharacteristic
273#3092—612no—
274C2×#398—611no—
275#1611—612no—
276#1610—612no—
277#1609—614no—
278#1504—612no—
279#1463—611no—
280C2×C61—21no—
281C2×C61—21no—
282C2×C61—22no—
283C2×C61—22no—
284C2×C61—82no—
285C2×C61#300611yescharacteristic
286C2×C61C2×#45611yescharacteristic
287C2×C61C2×#45811yescharacteristic
288#554—22no—
289#553—22no—
290#552—22no—
291#547—22no—
292#544C2×D6111yescharacteristic
293#539—21no—
294#539C2×D6112yes—
295#537C2×D6111yescharacteristic
296#525—21no—
297#525—21no—
298#525C2×D6112yes—
299#520—21no—
300C22×C4⋊C4—611no—
301C22×C4⋊C4—1221no—
302C22×C4⋊C4C2×D6111yescharacteristic
303#484—1222no—
304#483—614no—
305#481—1222no—
306#480—614no—
307#428—1222no—
308#417—1222no—
309#407—612no—
310#398—612no—
311#398—612no—
312#398—1221no—
313#398—1221no—
314C2×#95—611no—
315C2×#95—1221no—
316#328—614no—
317#327—614no—
318#324—612no—
319#324—612no—
320C61#1817111yescharacteristic, complemented, Sylow
321C2×#41—21no—
322C2×#41—21no—
323C2×#41C22⋊D6112yes—
324C23×C4—1221no—
325C23×C4—1221no—
326C23×C4C22⋊D6111yescharacteristic
327C23×C4C22⋊D6111yescharacteristic
328C23×C4C22×D6111yescharacteristic
329C2×M16—1222no—
330C4⋊Q8—22no—
331#126—22no—
332#126—22no—
333#125—22no—
334#125—22no—
335#125—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—21no—
347C22⋊Q8—22no—
348C22⋊Q8C22⋊D6112yes—
349C22⋊Q8C22×D6112yes—
350#121—21no—
351#121—21no—
352#121—21no—
353#121—22no—
354#121—21no—
355#121C22⋊D6112yes—
356#121C22×D6112yes—
357#120—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366#117—22no—
367#117—21no—
368#117—21no—
369#117C22⋊D6112yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—612no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—2444no—
380C2×C4⋊C4C22⋊D6111yescharacteristic
381C2×C4⋊C4C22⋊D6111yescharacteristic
382C2×C4⋊C4C22×D6111yescharacteristic
383C2×C4⋊C4C22×D6111yescharacteristic
384#107—1224no—
385#106—1224no—
386#103—1224no—
387#102—1224no—
388C22⋊C8—614no—
389C22⋊C8—1222no—
390#95—612no—
391#95—1221no—
392#95—1222no—
393#95—1222no—
394C24C2×C22⋊D6111yescharacteristic
395#41—42no—
396#41—44no—
397#41—42no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×C22⋊D6112yes—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4C2×C22⋊D6112yes—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—1224no—
420C22×C4—1222no—
421C22×C4—1222no—
422C22×C4—2442no—
423C22×C4—2442no—
424C22×C4—2441no—
425C22×C4—2441no—
426C22×C4—2442no—
427C22×C4—2442no—
428C22×C4C2×C22⋊D6111yescharacteristic
429C22×C4C2×C22⋊D6112yes—
430C22×C4C2×C22⋊D6111yescharacteristic
431C22×C4C2×C22⋊D6111yescharacteristic
432C22×C4C2×C22⋊D6111yescharacteristic
433C22×C4C2×C22⋊D6112yes—
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C23×D6111yescharacteristic
436M16—2444no—
437C2×C8—1224no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
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⋊C4—22no—
452C4⋊C4—1224no—
453C4⋊C4—2444no—
454C4⋊C4—2442no—
455C4⋊C4—2444no—
456C4⋊C4C2×C22⋊D6112yes—
457C4⋊C4C2×C22⋊D6112yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C23—22no—
471C23—22no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23#1451974611yescharacteristic
476C23#1451974611yescharacteristic
477C23C22×C22⋊D6111yescharacteristic
478Q8—42no—
479Q8—42no—
480Q8—81no—
481D4—42no—
482D4—42no—
483D4—44no—
484D4—44no—
485D4—42no—
486D4—42no—
487D4—81no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—1224no—
512C2×C4—4882no—
513C2×C4—4882no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2442no—
517C2×C4—2448no—
518C2×C4—2442no—
519C2×C4—2442no—
520C2×C4#1451974612yes—
521C2×C4#1451980212yes—
522C2×C4C22×C22⋊D6112yes—
523C8—2444no—
524C22—24no—
525C22—24no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22#1683994512yes—
536C22#1683995212yes—
537C22C2×#1451974611yescharacteristic
538C22C2×#1451980211yescharacteristic
539C22#1684056011yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—4882no—
550C4—2444no—
551C2—21no—
552C2—22no—
553C2—22no—
554C2—21no—
555C2—82no—
556C2#2145507111yescharacteristic
557C2C2×#1683994511yescharacteristic
558C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3492C2
socleC22×C61C2×#120
hypercenter#3492D61
hypocenterC61#18171
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. #21455739C2semidirect
2. C2×#16839692C2semidirect
3. #21444827C2semidirect
4. #21444822C2semidirect
5. #21444816C2semidirect
6. #21444323C2semidirect
7. #21444149C2semidirect
320. C61#18171semidirect

Maximal subgroups

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

subgroupindex
#214557392
C2×#168396922
#214448272
#214448222
#214448162
#214443232
#214441492
C61×#34922
#1817161