Finite Group Databasebeta

← #29655154

Subgroups — #29655154

Order 15616.

872 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic
2#21456764C211yesmaximal, characteristic, complemented
3#21451940C211yesmaximal, characteristic, complemented
4#21444793C214yesmaximal, complemented
5#21444782C214yesmaximal
6#21444331C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3563C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840157—21no—
12#16840141—21no—
13#16840132—21no—
14#16840114—21no—
15#16839972—22no—
16#16839972—21no—
17#16839967—21no—
18#16839917C2214yes—
19#16839917C2212yes—
20#16839900C2212yes—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839692—22no—
24#16839682C2214yes—
25#16839631—21no—
26#16839627—21no—
27#16839608C2211yescharacteristic
28#16839499C2214yes—
29#16839499C2212yes—
30#16839496C2212yes—
31#16839488C2214yes—
32C61×#564—21no—
33C61×#563—21no—
34C61×#557—21no—
35C61×#554—21no—
36C61×#552—21no—
37C61×#552—22no—
38C61×#552—21no—
39C61×#544—21no—
40C61×#544C2212yes—
41C61×#541—21no—
42C61×#541—22no—
43C61×#539C2212yes—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122C2312yes—
48C2×#11758202—24no—
49C2×#11758202—22no—
50C2×#11758200—22no—
51#14519777—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×#11758196—24no—
58C2×C4×Dic61—21no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519700—22no—
62#14519700—24no—
63#14519700—22no—
64C22⋊#92418C2312yes—
65C22⋊#92418C2312yes—
66#14519688—21no—
67C4×#92418—21no—
68#14519684—24no—
69#14519684—21no—
70#14519684C2312yes—
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—22no—
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D411yescharacteristic
75C4⋊Q8×C61D411yescharacteristic
76#126×C61—22no—
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—22no—
80#126×C61—21no—
81#125×C61—22no—
82#125×C61—22no—
83#125×C61—22no—
84#125×C61—22no—
85#125×C61—21no—
86#125×C61—22no—
87#125×C61—22no—
88#125×C61—22no—
89#125×C61—22no—
90#125×C61—21no—
91#125×C61—21no—
92#125×C61—21no—
93#125×C61—21no—
94#125×C61—21no—
95#125×C61D411yescharacteristic
96#124×C61D411yescharacteristic
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—21no—
100C22⋊Q8×C61—21no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—22no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—22no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2312yes—
108C22⋊Q8×C61C2311yescharacteristic
109#121×C61C2311yescharacteristic
110C4×Q8×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61D412yes—
115C4×Q8×C61D411yescharacteristic
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61D411yescharacteristic
119#117×C61—22no—
120#117×C61—21no—
121#117×C61—21no—
122#117×C61—22no—
123#117×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130C2×Q8×C61—22no—
131C2×Q8×C61—21no—
132C2×Q8×C61C2×D412yes—
133C2×Q8×C61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—22no—
144C22×Dic61—22no—
145C2×Dic122—22no—
146C2×Dic122—22no—
147C2×Dic122—82no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
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—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—22no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D412yes—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—22no—
189C42×C61—21no—
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194#11758202—82no—
195#11758202—84no—
196#11758200—82no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201#11758196—84no—
202C4×Dic61—24no—
203C2×#92418—22no—
204C2×#92418—22no—
205C23×C61—21no—
206C23×C61C22×D411yescharacteristic
207Q8×C61—41no—
208Q8×C61—42no—
209Q8×C61—42no—
210Q8×C61—82no—
211D4×C61—41no—
212D4×C61—42no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—24no—
223C2×C4×C61—41no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—82no—
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61#12011yescharacteristic
230C2×C4×C61#12011yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×Dic61—24no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—44no—
237C2×Dic61—42no—
238Dic122—42no—
239Dic122—42no—
240Dic122—44no—
241#92418—42no—
242#92418—42no—
243#18433—611nomaximal, Sylow
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—41no—
247C22×C61—41no—
248C22×C61—41no—
249C22×C61C2×#12011yescharacteristic
250C22×C61#57812yes—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—21no—
254C4×C61—21no—
255C4×C61—21no—
256C4×C61—42no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—42no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61#45612yes—
264Dic61—44no—
265Dic61—82no—
266#3563D6111yescharacteristic
267#3218—611no—
268#3209—611no—
269#2660—611no—
270#1605—614no—
271#1603—614no—
272#1506—611no—
273#1500—611no—
274#1470—611no—
275C2×C61—22no—
276C2×C61—81no—
277C2×C61#301111yescharacteristic
278C2×C61C2×#45611yescharacteristic
279C2×C61#302711yescharacteristic
280#564—21no—
281#563—21no—
282#557—21no—
283#554—21no—
284#552—21no—
285#552—21no—
286#552—22no—
287#544—21no—
288#544C2×D6112yes—
289#541—21no—
290#541—22no—
291#539C2×D6112yes—
292#533C2×D6111yescharacteristic
293C2×C4⋊Q8—611no—
294#529C2×D6111yescharacteristic
295C2×#125C2×D6111yescharacteristic
296#497—1221no—
297#494—1221no—
298#491—1221no—
299#487—1221no—
300#462—1222no—
301#462—1221no—
302#461—1221no—
303#451—614no—
304#451—612no—
305#448—612no—
306#432—611no—
307#431—611no—
308#398—1222no—
309#395—614no—
310#384—1221no—
311#382—1221no—
312#376—611no—
313#328—614no—
314#328—612no—
315#327—612no—
316#324—614no—
317C61#1843311yescharacteristic, complemented, Sylow
318C22×Q8—612no—
319C2×Q16—1224no—
320C2×Q16—1222no—
321C2×SD16—1222no—
322C4⋊Q8—21no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8C22⋊D6111yescharacteristic
330C4⋊Q8C22⋊D6111yescharacteristic
331#126—22no—
332#126—22no—
333#126—22no—
334#126—22no—
335#126—21no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125—21no—
341#125—22no—
342#125—22no—
343#125—22no—
344#125—22no—
345#125—21no—
346#125—21no—
347#125—21no—
348#125—21no—
349#125—21no—
350#125C22⋊D6111yescharacteristic
351#124C22⋊D6111yescharacteristic
352#123—22no—
353#123—21no—
354#123—22no—
355C22⋊Q8—21no—
356C22⋊Q8—21no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—21no—
361C22⋊Q8—22no—
362C22⋊Q8C22×D6111yescharacteristic
363C22⋊Q8C22×D6112yes—
364#121C22×D6111yescharacteristic
365C4×Q8—21no—
366C4×Q8—22no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8C22⋊D6111yescharacteristic
370C4×Q8C22⋊D6112yes—
371C4×D4—21no—
372C4×D4—22no—
373C4×D4C22⋊D6111yescharacteristic
374#117—22no—
375#117—21no—
376#117—21no—
377#117—22no—
378#117—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1221no—
382C2×C4⋊C4—1221no—
383C2×C4⋊C4C22⋊D6111yescharacteristic
384C2×C4⋊C4C22⋊D6112yes—
385C2×C4⋊C4C22⋊D6111yescharacteristic
386C2×C4⋊C4C22×D6112yes—
387C2×C42—1221no—
388C2×C42C22×D6111yescharacteristic
389C4⋊C8—1221no—
390C4⋊C8—1221no—
391#103—1222no—
392#103—1222no—
393#103—1224no—
394C22⋊C8—612no—
395C22⋊C8—612no—
396#97—1221no—
397C4×C8—1221no—
398#95—612no—
399#95—1221no—
400#95—1224no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—1222no—
404C2×Q8—1222no—
405C2×Q8—4882no—
406C2×Q8C2×C22⋊D6112yes—
407C2×Q8C2×C22⋊D6111yescharacteristic
408C2×D4—21no—
409C2×D4C2×C22⋊D6111yescharacteristic
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—1222no—
413C22×C4—1222no—
414C22×C4C2×C22⋊D6112yes—
415C22×C4C2×C22⋊D6111yescharacteristic
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6112yes—
418C22×C4C23×D6111yescharacteristic
419Q16—4882no—
420Q16—4884no—
421SD16—4882no—
422C2×C8—1222no—
423C2×C8—1222no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—4884no—
450C4⋊C4—2441no—
451C4⋊C4—2441no—
452C4⋊C4—2441no—
453C4⋊C4—2441no—
454C4⋊C4C2×C22⋊D6111yescharacteristic
455C4⋊C4C2×C22⋊D6111yescharacteristic
456C4⋊C4C2×C22⋊D6112yes—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—1224no—
472C42C2×C22⋊D6111yescharacteristic
473C42C2×C22⋊D6111yescharacteristic
474C42C2×C22⋊D6111yescharacteristic
475C42C2×C22⋊D6111yescharacteristic
476C23—21no—
477C23C22×C22⋊D6111yescharacteristic
478Q8—41no—
479Q8—42no—
480Q8—42no—
481Q8—82no—
482Q8—2442no—
483Q8—2442no—
484Q8—2444no—
485D4—41no—
486D4—42no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—1224no—
502C2×C4—2444no—
503C2×C4—2442no—
504C2×C4—2442no—
505C2×C4—2442no—
506C2×C4#1451974611yescharacteristic
507C2×C4#1451974611yescharacteristic
508C2×C4#1451974611yescharacteristic
509C2×C4#1451974611yescharacteristic
510C2×C4C22×C22⋊D6111yescharacteristic
511C2×C4C22×C22⋊D6111yescharacteristic
512C8—2442no—
513C8—2442no—
514C22—21no—
515C22—21no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22C2×#1451974611yescharacteristic
520C22#1684057712yes—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—42no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—41no—
533C4—4882no—
534C4—2444no—
535C4#1683994512yes—
536C2—22no—
537C2—81no—
538C2#2145511411yescharacteristic
539C2C2×#1683994511yescharacteristic
540C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61C23
hypercenter#3563D61
hypocenterC61#18433
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. #21456764C2semidirect
3. #21451940C2semidirect
4. #21444793C2semidirect
6. #21444331C2semidirect
8. #21444175C2semidirect
317. C61#18433semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214567642
#214519402
#214447932
#214447822
#214443312
#214443052
#214441752
C61×#35632
#1843361