Finite Group Databasebeta

← #29661625

Subgroups — #29661625

Order 15616.

836 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456843C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3C2×#16840172C211yesmaximal, characteristic, complemented
4#21445147C214yesmaximal, complemented
5#21445130C214yesmaximal, complemented
6#21444434C211yesmaximal, characteristic, complemented
7#21444421C211yesmaximal, characteristic, complemented
8#21444407C211yesmaximal, characteristic
9C61×#3563C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—22no—
12#16840172C2214yes—
13#16840172C2214yes—
14#16840165—21no—
15#16840158—21no—
16#16840157C2212yes—
17#16840132C2212yes—
18#16839972—21no—
19#16839967—21no—
20#16839917—21no—
21#16839900—21no—
22#16839819C2211yescharacteristic
23#16839805—22no—
24C2×#14519710—21no—
25C2×#14519708—21no—
26C4×#11758194C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28#16839504C2214yes—
29#16839504C2214yes—
30#16839504C2212yes—
31#16839502C2212yes—
32C61×#564C2212yes—
33C61×#563C2212yes—
34C61×#557—21no—
35C61×#554—21no—
36C61×#552—21no—
37C61×#552—22no—
38C61×#552—21no—
39C61×#544—21no—
40C61×#544—21no—
41C61×#541—21no—
42C61×#541—22no—
43C61×#539—21no—
44C61×#533C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48C2×#11758202—22no—
49C2×#11758200—22no—
50C2×#11758194—21no—
51C22×#92418—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58#14519710—24no—
59#14519710—24no—
60#14519708—24no—
61#14519708—24no—
62C4⋊#92418—21no—
63#14519700—22no—
64#14519700—22no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688C2312yes—
68#14519688C2312yes—
69#14519688C2312yes—
70C4×#92418—21no—
71C4⋊Q8×C61—22no—
72C4⋊Q8×C61—21no—
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—21no—
84#125×C61—22no—
85#125×C61—22no—
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×C61D412yes—
93#125×C61C2312yes—
94#125×C61C2312yes—
95#125×C61C2311yescharacteristic
96#124×C61C2311yescharacteristic
97#123×C61—22no—
98#123×C61—22no—
99#123×C61—21no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—21no—
102C22⋊Q8×C61—21no—
103C22⋊Q8×C61—22no—
104C22⋊Q8×C61—21no—
105C22⋊Q8×C61—21no—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61D411yescharacteristic
109#121×C61D411yescharacteristic
110C4×Q8×C61—21no—
111C4×Q8×C61—22no—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61—21no—
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—22no—
122#117×C61—21no—
123#117×C61D412yes—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×Q8×C61—22no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61C2×D411yescharacteristic
134C2×D4×C61—21no—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—21no—
137C22×C4×C61—22no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×Dic61—41no—
144C2×Dic122—22no—
145C2×Dic122—41no—
146C2×Dic122—82no—
147C4⋊C4×C61—22no—
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—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C4⋊C4×C61C2×D412yes—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—21no—
182C22⋊C4×C61—21no—
183C22⋊C4×C61—21no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—22no—
187C42×C61—22no—
188C42×C61—21no—
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2411yescharacteristic
193#11758202—82no—
194#11758200—82no—
195C4⋊Dic61—22no—
196C4⋊Dic61—22no—
197C4⋊Dic61—41no—
198C4⋊Dic61—41no—
199#11758194—24no—
200C2×#92418—24no—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61#12011yescharacteristic
206Q8×C61—41no—
207Q8×C61—42no—
208Q8×C61—81no—
209Q8×C61—82no—
210D4×C61—41no—
211D4×C61—42no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—41no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—81no—
225C2×C4×C61—82no—
226C2×C4×C61#12011yescharacteristic
227C2×C4×C61#12011yescharacteristic
228C2×C4×C61#12011yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×Dic61—42no—
233C2×Dic61—82no—
234Dic122—42no—
235Dic122—42no—
236Dic122—82no—
237#92418—24no—
238#92418—24no—
239#92418—42no—
240#19034—611nomaximal, Sylow
241C22×C61—21no—
242C22×C61—21no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61#45612yes—
247C22×C61C2×#12011yescharacteristic
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—21no—
253C4×C61—42no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61—42no—
257C4×C61—42no—
258C4×C61—42no—
259C4×C61—41no—
260C4×C61#45612yes—
261Dic61—82no—
262#3563D6111yescharacteristic
263#3218—611no—
264#3165—611no—
265C2×#501—611no—
266#1663—614no—
267#1660—614no—
268#1531—611no—
269#1528—611no—
270#1524—611no—
271C2×C61—22no—
272C2×C61—81no—
273C2×C61C2×#45611yescharacteristic
274C2×C61C2×#45611yescharacteristic
275C2×C61#302111yescharacteristic
276#564C2×D6112yes—
277#563C2×D6112yes—
278#557—21no—
279#554—21no—
280#552—21no—
281#552—22no—
282#552—21no—
283#544—21no—
284#544—21no—
285#541—21no—
286#541—22no—
287#539—21no—
288#533C2×D6111yescharacteristic
289C2×C4⋊Q8—611no—
290#529C2×D6111yescharacteristic
291C2×#125C2×D6111yescharacteristic
292#501—614no—
293#501—614no—
294#501—1222no—
295#500—1221no—
296#498—1221no—
297#497—612no—
298#491—612no—
299#462—1221no—
300#461—1221no—
301#451—1221no—
302#448—1221no—
303#432—611no—
304#428—1222no—
305C2×#107—1221no—
306C2×#106—1221no—
307C4×M16—611no—
308C2×#97—611no—
309#330—614no—
310#330—614no—
311#330—612no—
312#329—612no—
313C61#1903411yescharacteristic, complemented, Sylow
314C22×Q8—1221no—
315C2×Q16—1222no—
316C2×SD16—1222no—
317C2×M16—1221no—
318C22×C8—1221no—
319C4⋊Q8—22no—
320C4⋊Q8—21no—
321C4⋊Q8—22no—
322C4⋊Q8—612no—
323C4⋊Q8—1221no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8C22⋊D6111yescharacteristic
327C4⋊Q8C22⋊D6111yescharacteristic
328#126—22no—
329#126—22no—
330#126—22no—
331#126—22no—
332#126—21no—
333#125—22no—
334#125—22no—
335#125—21no—
336#125—22no—
337#125—22no—
338#125—22no—
339#125—22no—
340#125—22no—
341#125—22no—
342#125—21no—
343#125—21no—
344#125C22⋊D6112yes—
345#125C22×D6112yes—
346#125C22×D6111yescharacteristic
347#125C22×D6112yes—
348#124C22×D6111yescharacteristic
349#123—22no—
350#123—22no—
351#123—21no—
352C22⋊Q8—22no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8C22⋊D6111yescharacteristic
361#121C22⋊D6111yescharacteristic
362C4×Q8—21no—
363C4×Q8—22no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C22⋊D6111yescharacteristic
368C4×D4—22no—
369C4×D4—21no—
370C4×D4C22⋊D6111yescharacteristic
371#117—22no—
372#117—21no—
373#117—22no—
374#117—21no—
375#117C22⋊D6112yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4—1221no—
381C2×C4⋊C4C22⋊D6111yescharacteristic
382C2×C4⋊C4C22⋊D6111yescharacteristic
383C2×C42C22×D6111yescharacteristic
384#107—1224no—
385#107—1224no—
386#106—1224no—
387#106—1224no—
388C4⋊C8—1221no—
389#103—1222no—
390#103—1222no—
391C22⋊C8—1221no—
392#97—612no—
393#97—612no—
394#97—612no—
395#97—1221no—
396C4×C8—1221no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—1222no—
401C2×Q8—4882no—
402C2×Q8—2441no—
403C2×Q8C2×C22⋊D6111yescharacteristic
404C2×D4—21no—
405C2×D4C2×C22⋊D6111yescharacteristic
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—2441no—
411C22×C4C2×C22⋊D6111yescharacteristic
412C22×C4C2×C22⋊D6111yescharacteristic
413C22×C4C2×C22⋊D6111yescharacteristic
414Q16—4882no—
415SD16—4882no—
416M16—1224no—
417C2×C8—1224no—
418C2×C8—1222no—
419C2×C8—1222no—
420C2×C8—1222no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—1222no—
444C4⋊C4—1222no—
445C4⋊C4—2441no—
446C4⋊C4—2441no—
447C4⋊C4C2×C22⋊D6111yescharacteristic
448C4⋊C4C2×C22⋊D6111yescharacteristic
449C4⋊C4C2×C22⋊D6112yes—
450C4⋊C4C2×C22⋊D6112yes—
451C4⋊C4C2×C22⋊D6112yes—
452C4⋊C4C2×C22⋊D6112yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—21no—
457C22⋊C4—22no—
458C22⋊C4—21no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—21no—
467C42C2×C22⋊D6111yescharacteristic
468C42C2×C22⋊D6111yescharacteristic
469C42C2×C22⋊D6111yescharacteristic
470C42C23×D6111yescharacteristic
471C23—21no—
472C23#1451974611yescharacteristic
473Q8—41no—
474Q8—42no—
475Q8—81no—
476Q8—82no—
477Q8—4882no—
478Q8—2442no—
479Q8—2442no—
480D4—41no—
481D4—42no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—4882no—
497C2×C4—2442no—
498C2×C4#1451974611yescharacteristic
499C2×C4#1451974611yescharacteristic
500C2×C4#1451974611yescharacteristic
501C2×C4C22×C22⋊D6111yescharacteristic
502C2×C4C22×C22⋊D6111yescharacteristic
503C2×C4C22×C22⋊D6111yescharacteristic
504C8—1224no—
505C8—1224no—
506C8—2442no—
507C22—21no—
508C22—21no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22#1683994512yes—
513C22C2×#1451974611yescharacteristic
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—41no—
526C4—4882no—
527C4#1683994512yes—
528C2—22no—
529C2—81no—
530C2C2×#1683994511yescharacteristic
531C2C2×#1683994511yescharacteristic
532C2#2145519111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3563C2
socleC22×C61C2×#120
prime radical#121×C61D4
hypercenter#3563D61
hypocenterC61#19034
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3563

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. C2×#16840172C2semidirect
4. #21445147C2semidirect
5. #21445130C2semidirect
6. #21444434C2semidirect
7. #21444421C2semidirect
313. C61#19034semidirect

Maximal subgroups

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

subgroupindex
#214568432
#214563662
C2×#168401722
#214451472
#214451302
#214444342
#214444212
#214444072
C61×#35632
#1903461