Finite Group Databasebeta

← #29663630

Subgroups — #29663630

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457164C211yesmaximal, characteristic, complemented
2#21456762C211yesmaximal, characteristic
3#21456194C211yesmaximal, characteristic
4#21454766C211yesmaximal, characteristic, complemented
5#21445388C212yesmaximal, complemented
6#21445360C212yesmaximal, complemented
7#21445264C212yesmaximal, complemented
8#21445236C212yesmaximal, complemented
9#21444474C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic
11C61×#1565C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840319C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840135—21no—
16#16840127—22no—
17#16840109—21no—
18#16840109C2212yes—
19#16840109C2212yes—
20#16840058—21no—
21#16840058C2212yescomplemented
22#16840003—22no—
23#16839973—21no—
24#16839973C2212yescomplemented
25#16839963C2212yes—
26#16839963C2212yes—
27#16839915—22no—
28#16839892C2212yescomplemented
29#16839890—22no—
30C8×Dic122C2212yescomplemented
31#16839790—21no—
32C2×C4⋊#92418C2211yescharacteristic
33#16839768—21no—
34#16839768—21no—
35#16839720—21no—
36#16839510C2211yescharacteristic, complemented
37#16839510C2211yescharacteristic, complemented
38#16839509C2211yescharacteristic, complemented
39#16839509C2211yescharacteristic, complemented
40#16839505C2212yes—
41#16839505C2212yes—
42C61×#431C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#335C2212yescomplemented
46C61×#334C2212yescomplemented
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C2×Dic244—22no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C22×#92418—21no—
54#14519777D411yescharacteristic
55#14519777D411yescharacteristic
56#14519777C2311yescharacteristic
57#14519777C2311yescharacteristic
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774C2312yes—
62#14519769—21no—
63#14519769—21no—
64#14519756—21no—
65#14519756—21no—
66#14519756—21no—
67#14519756D412yes—
68C4×Dic122—21no—
69C4×Dic122—21no—
70C4×Dic122—21no—
71C4×Dic122C2312yes—
72#14519736—21no—
73#14519736—21no—
74#14519711—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2311yescharacteristic
77C4⋊#92418C2311yescharacteristic
78#14519706—22no—
79#14519706—22no—
80#14519702—22no—
81#14519702—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85#14519699—22no—
86#14519699—22no—
87#14519699—22no—
88#14519687—22no—
89C8×Dic61—22no—
90C2×M16×C61D411yescharacteristic
91C2×M16×C61D411yescharacteristic
92C22×C8×C61—21no—
93C2×C42×C61C2311yescharacteristic
94C4⋊C8×C61—21no—
95C4⋊C8×C61C2311yescharacteristic
96C4⋊C8×C61C2311yescharacteristic
97C4⋊C8×C61C2312yes—
98C22⋊C8×C61—21no—
99#97×C61D412yes—
100C4×C8×C61—21no—
101C4×C8×C61C2312yes—
102C23×D61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—42no—
110C2×D244—42no—
111C2×D244—41no—
112C2×D244—41no—
113C2×D244—41no—
114C2×D244—81no—
115C2×D244—81no—
116C2×D244—82no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122C2×D412yes—
121M16×C61—22no—
122M16×C61—22no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—22no—
126C2×C8×C61—81no—
127C2×C8×C61C2×D412yes—
128C2×C8×C61C2×D412yes—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—22no—
134C22⋊Dic61—41no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61C2×D412yes—
143#11758196—22no—
144#11758196—22no—
145#11758196—41no—
146#11758196—41no—
147C4×Dic61—22no—
148C4×Dic61—41no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—42no—
152Dic244—82no—
153#11758190—82no—
154#11758190—82no—
155C23×C61#12111yescharacteristic
156C22×D61—42no—
157C22×D61—82no—
158C22×D61—82no—
159C22×D61—82no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#13512yes—
167C2×C4×C61#13712yes—
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C2×Dic61—22no—
172C2×Dic61—22no—
173C2×Dic61—41no—
174C2×Dic61—41no—
175D244—42no—
176D244—42no—
177D244—42no—
178D244—42no—
179D244—82no—
180D244—82no—
181D244—82no—
182D244—82no—
183Dic122—22no—
184Dic122—22no—
185Dic122—41no—
186Dic122—41no—
187Dic122—81no—
188Dic122—81no—
189C8×C61—22no—
190C8×C61—22no—
191C8×C61—41no—
192C8×C61—42no—
193C8×C61—41no—
194#92418—41no—
195#92418—41no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#46412yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61#46412yes—
209C4×C61C2×#13511yescharacteristic
210C4×C61C2×#13711yescharacteristic
211C4×C61#57611yescharacteristic
212C4×C61#57711yescharacteristic
213Dic61—42no—
214Dic61—42no—
215Dic61—81no—
216Dic61—81no—
217#3249—611no—
218#3208—611no—
219#3138—611no—
220#2962—611no—
221#1702—612no—
222#1698—612no—
223#1682—612no—
224#1678—612no—
225#1565D6111yescharacteristic, complemented
226#1542—611no—
227C2×#331—611no—
228C2×C61—22no—
229C2×C61C2×#46411yescharacteristic
230C2×C61#306211yescharacteristic
231C2×C61#306811yescharacteristic
232D61—162no—
233#533—611no—
234#533—611no—
235C2×#127—611no—
236#492—1221no—
237#490—1222no—
238#486—612no—
239#486—612no—
240#486—1221no—
241#477—612no—
242#477—1221no—
243#467—1222no—
244#462—612no—
245#462—1221no—
246#460—612no—
247#460—612no—
248#450—1222no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431C2×D6111yescharacteristic
253#424—1221no—
254C2×C4⋊C8—611no—
255C2×C4⋊C8C2×D6111yescharacteristic
256#418—1221no—
257#418—1221no—
258#405—1221no—
259C4×M16C2×D6111yescharacteristic
260#335C2×D6112yescomplemented
261#334C2×D6112yescomplemented
262#333—611no—
263#333—611no—
264#332—611no—
265#332—611no—
266#331—612no—
267#331—612no—
268C61#1924211yescharacteristic, complemented, Sylow
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4—1221no—
272C2×Q16—1222no—
273C2×SD16—1222no—
274C2×SD16—1222no—
275C2×M16D24411yescharacteristic
276C2×M16D24411yescharacteristic
277C22×C8—21no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8—611no—
282C4⋊Q8—611no—
283#127—612no—
284#127—1221no—
285#127—1221no—
286#127—1221no—
287#125—1221no—
288#125—1221no—
289C22⋊Q8—612no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292C22⋊Q8—1221no—
293C4×Q8—612no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×Q8—1221no—
297#117—1221no—
298#117—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301C4⋊C8—21no—
302C4⋊C8—611no—
303C4⋊C8—611no—
304C4⋊C8—1222no—
305C4⋊C8—1222no—
306C4⋊C8—1221no—
307C4⋊C8C22×D6112yes—
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6111yescharacteristic
310#103—1222no—
311#103—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—21no—
319#97—1222no—
320#97D24412yes—
321C4×C8—21no—
322C4×C8—1222no—
323C4×C8C22×D6112yes—
324C24—2441no—
325C2×Q8—612no—
326C2×Q8—1221no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—4881no—
333C2×D4—4881no—
334C2×D4—4882no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C22×C4C2×D24411yescharacteristic
341C22×C4D4×D6111yescharacteristic
342C22×C4#1175822611yescharacteristic
343Q16—4882no—
344SD16—4882no—
345SD16—4882no—
346M16—22no—
347M16—22no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—81no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—2442no—
355C2×C8C2×D24412yes—
356C2×C8C2×D24412yes—
357C4⋊C4—612no—
358C4⋊C4—1222no—
359C4⋊C4—1222no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C22⋊C4—1222no—
370C22⋊C4—2441no—
371C42—1222no—
372C42—2441no—
373C42C2×D24411yescharacteristic
374C42D4×D6111yescharacteristic
375C42#1175822611yescharacteristic
376C42C23×D6111yescharacteristic
377C23—4882no—
378C23—4882no—
379C23—4882no—
380C23—2442no—
381C23#1451975211yescharacteristic
382Q8—1222no—
383Q8—1222no—
384Q8—4881no—
385Q8—4881no—
386Q8—2441no—
387Q8—2441no—
388D4—4882no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395D4—2442no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—1222no—
400C2×C4—1222no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451975211yescharacteristic
404C2×C4#1451975211yescharacteristic
405C2×C4#1451975211yescharacteristic
406C2×C4#1451979912yes—
407C2×C4#1451981012yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C22×D24411yescharacteristic
410C2×C4C2×#1175822611yescharacteristic
411C8—22no—
412C8—22no—
413C8—42no—
414C8—41no—
415C8—41no—
416C8—2441no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—9764no—
421C22—4882no—
422C22—4882no—
423C22—4882no—
424C22#1683998512yes—
425C22C2×#1451975211yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—4881no—
430C4—4881no—
431C4—2442no—
432C4—2442no—
433C4#1683998512yes—
434C4C2×#1451979911yescharacteristic
435C4C2×#1451981011yescharacteristic
436C4#1684056611yescharacteristic
437C4#1684057111yescharacteristic
438C2—22no—
439C2—9762no—
440C2C2×#1683998511yescharacteristic
441C2#2145550011yescharacteristic
442C2#2145554811yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#1565D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
1. #21457164C2semidirect
4. #21454766C2semidirect
5. #21445388C2semidirect
6. #21445360C2semidirect
7. #21445264C2semidirect
8. #21445236C2semidirect
9. #21444474C2semidirect
11. C61×#1565C2semidirect
21. #16840058C22semidirect
24. #16839973C22semidirect
28. #16839892C22semidirect
30. C8×Dic122C22semidirect
36. #16839510C22semidirect
37. #16839510C22semidirect
38. #16839509C22semidirect
39. #16839509C22semidirect
45. C61×#335C22semidirect
46. C61×#334C22semidirect
225. #1565D61semidirect
260. #335C2×D61semidirect
261. #334C2×D61semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571642
#214567622
#214561942
#214547662
#214453882
#214453602
#214452642
#214452362
#214444742
C2×#168395052
C61×#15652
#1924261