Finite Group Databasebeta

← #29834942

Subgroups — #29834942

Order 15616.

610 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456400C211yesmaximal, characteristic, complemented
2C2×#16839844C211yesmaximal, characteristic
3#21448016C211yesmaximal, characteristic, complemented
4#21447965C212yesmaximal, complemented
5#21447863C212yesmaximal, complemented
6#21447750C211yesmaximal, characteristic
7#21447747C211yesmaximal, characteristic
8#21447670C212yesmaximal, complemented
9#21447668C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2235C211yesmaximal, characteristic, complemented
12C22×D488C2211yescharacteristic
13#16840314—21no—
14C2×#14519739—21no—
15C2×#14519722C2212yes—
16C2×#14519720C2212yes—
17#16840166C2211yescharacteristic, complemented
18#16840140C2211yescharacteristic, complemented
19#16840111—21no—
20#16840003C2211yescharacteristic
21#16840001—21no—
22Q16×Dic61C2211yescharacteristic, complemented
23#16839856C2211yescharacteristic, complemented
24#16839854—21no—
25#16839844—21no—
26#16839844—21no—
27#16839844C2212yes—
28#16839844C2212yes—
29C2×#14519697—21no—
30C2×C8×Dic61C2211yescharacteristic
31#16839589C2212yescomplemented
32#16839580C2212yescomplemented
33#16839565C2212yescomplemented
34#16839564C2212yes—
35#16839562C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yescomplemented
38C61×#467C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yescomplemented
41C61×#358C2212yescomplemented
42C61×C22⋊C16C2211yescharacteristic
43C22×D244—21no—
44C22×C4×D61—41no—
45C2×Dic244C2311yescharacteristic
46C2×#11758202—21no—
47C2×D488—21no—
48C2×D488—24no—
49C2×D488C2312yes—
50C22×#92418D411yescharacteristic
51#14519776—21no—
52#14519775—21no—
53#14519767—41no—
54#14519756—21no—
55#14519754—41no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519742—41no—
59#14519739—22no—
60#14519739—22no—
61#14519739—41no—
62#14519739—41no—
63#14519733—41no—
64C2×#11758198—41no—
65C2×C4×Dic61D411yescharacteristic
66#14519722—24no—
67#14519720—24no—
68C2×#11758185C2312yes—
69#14519711C2311yescharacteristic
70#14519710—21no—
71#14519702—21no—
72#14519701—21no—
73#14519700—21no—
74#14519697—22no—
75#14519697—22no—
76C8×Dic61D411yescharacteristic
77C8×Dic61D411yescharacteristic
78C8×Dic61C2311yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82C22⋊Q8×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C16×C61C2312yes—
85#107×C61—21no—
86#107×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C23×D61—41no—
90C2×Q8×C61—21no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C22×Dic61D811yescharacteristic
94C22×Dic61D811yescharacteristic
95C2×D244—24no—
96C2×D244—22no—
97C2×D244—42no—
98C2×D244—41no—
99C2×C4×D61—42no—
100C2×C4×D61—42no—
101C2×C4×D61—84no—
102C2×Dic122—21no—
103C2×Dic122—41no—
104Q16×C61—22no—
105C2×C8×C61—22no—
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2411yescharacteristic
108C4⋊C4×C61—21no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C22⋊C4×C61—41no—
113C22⋊Dic61—41no—
114#11758202—42no—
115#11758198—42no—
116#11758198—42no—
117C4⋊Dic61—21no—
118C4⋊Dic61—41no—
119#11758196—41no—
120#11758196—41no—
121#11758196—41no—
122#11758196—41no—
123#11758196—41no—
124C4×Dic61—41no—
125C4×Dic61—41no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C4×Dic61C2×D411yescharacteristic
130C2×#92418—41no—
131C2×#92418C2×D411yescharacteristic
132C2×#92418C2×D411yescharacteristic
133Dic244—22no—
134D488—24no—
135D488—22no—
136D488#4114yes—
137C16×C61—22no—
138#11758185—22no—
139C23×C61C2×D811yescharacteristic
140C22×D61—44no—
141C22×D61—42no—
142C22×D61—82no—
143C22×D61—82no—
144Q8×C61—41no—
145Q8×C61—41no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—82no—
152C2×C4×C61C2×D811yescharacteristic
153C2×C4×C61C22×D411yescharacteristic
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157C2×Dic61SD3214yescomplemented
158C2×Dic61C2×D811yescharacteristic
159C2×Dic61C2×D811yescharacteristic
160C2×Dic61C2×D811yescharacteristic
161C2×Dic61C2×D811yescharacteristic
162D244—24no—
163D244—44no—
164D244—42no—
165C4×D61—84no—
166C4×D61—84no—
167Dic122—41no—
168Dic122—41no—
169Dic122—82no—
170C8×C61—21no—
171C8×C61C2×#4112yes—
172#92418—21no—
173#92418—21no—
174#33661—611nomaximal, Sylow
175C22×C61—21no—
176C22×C61—21no—
177C22×C61C2×SD3212yes—
178C22×C61C22×D811yescharacteristic
179C2×D61—44no—
180C2×D61—88no—
181C2×D61—82no—
182C2×D61—82no—
183C4×C61—21no—
184C4×C61—81no—
185C4×C61—82no—
186C4×C61—81no—
187C4×C61#52511yescharacteristic
188C4×C61#57511yescharacteristic
189Dic61—21no—
190Dic61—21no—
191Dic61—81no—
192Dic61—81no—
193Dic61C2×SD3212yes—
194Dic61C2×SD3212yes—
195#3170—611no—
196C2×C4×D8—611no—
197#2235—611no—
198#2235D6111yescharacteristic, complemented
199#2226—612no—
200#2208—612no—
201#2187—611no—
202#2187—611no—
203C4×SD32—612no—
204C4×SD32—612no—
205#2162—611no—
206C2×C61—22no—
207C2×C61#310011yescharacteristic
208C2×C61C22×SD3211yescharacteristic
209C2×C61#341411yescharacteristic
210D61—84no—
211C22×D8—611no—
212#533—1221no—
213C2×C4×D4—1221no—
214C2×SD32—612no—
215C2×SD32—612no—
216#500—611no—
217#494—611no—
218#487—1221no—
219#467—611no—
220#467—1221no—
221#467C2×D6111yescharacteristic
222C4×Q16—611no—
223C4×Q16—611no—
224C4×Q16—1221no—
225C4×D8—612no—
226C4×D8—612no—
227C4×D8—1221no—
228C4×D8—1221no—
229C2×#107C2×D6111yescharacteristic
230C2×#102—1221no—
231C2×C4×C8—611no—
232#367—612no—
233#367C2×D6112yescomplemented
234C4⋊C16—612no—
235#358—612no—
236#358C2×D6112yescomplemented
237#357—612no—
238#357—612no—
239C22⋊C16—611no—
240C22⋊C16C2×D6111yescharacteristic
241C4×C16—612no—
242C61#3366111yescharacteristic, complemented, Sylow
243C22×D4—1221no—
244C23×C4—2441no—
245C2×Q16—611no—
246C2×Q16—1221no—
247C2×Q16C22×D6111yescharacteristic
248C2×D8—612no—
249C2×D8—1224no—
250C2×D8—1221no—
251C22×C8—611no—
252C22×C8C22×D6111yescharacteristic
253C4⋊Q8—1221no—
254C4⋊Q8—1221no—
255#125—2441no—
256C22⋊Q8—21no—
257C22⋊Q8—1221no—
258C22⋊Q8—2441no—
259C4×Q8—1221no—
260C4×Q8—1221no—
261C4×Q8—2441no—
262C4×D4—1222no—
263C4×D4—1222no—
264C4×D4—2441no—
265C4×D4—2441no—
266#117—2441no—
267C2×C4⋊C4—21no—
268C2×C22⋊C4—2441no—
269C2×C42—611no—
270SD32—1224no—
271SD32—1224no—
272C2×C16—612no—
273C2×C16C22×D6112yes—
274#107—21no—
275#107—611no—
276#107—1221no—
277#107C22×D6111yescharacteristic
278#107C22×D6112yes—
279#103—21no—
280#103—1221no—
281#103—1221no—
282#103—1221no—
283#102—1222no—
284#102—1222no—
285C4×C8—611no—
286C4×C8—611no—
287C4×C8—611no—
288C4×C8—611no—
289C24—2441no—
290C2×Q8—21no—
291C2×Q8—1221no—
292C2×Q8—2441no—
293C2×D4—1224no—
294C2×D4—1222no—
295C2×D4—2442no—
296C2×D4—2441no—
297C22×C4—41no—
298C22×C4—611no—
299C22×C4—611no—
300C22×C4—4884no—
301C22×C4—2442no—
302C22×C4—2442no—
303C22×C4D4×D6111yescharacteristic
304Q16—22no—
305Q16—1222no—
306Q16—2442no—
307D8—614no—
308D8—1224no—
309D8—1222no—
310C2×C8—22no—
311C2×C8—611no—
312C2×C8—611no—
313C2×C8—2441no—
314C2×C8D4×D6111yescharacteristic
315C2×C8C23×D6111yescharacteristic
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—1221no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C4⋊C4—2441no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—41no—
328C22⋊C4—2442no—
329C22⋊C4—2442no—
330C22⋊C4—2441no—
331C42—611no—
332C42—611no—
333C42—611no—
334C42—611no—
335C42—2441no—
336C42—2441no—
337C16—22no—
338C16—1222no—
339C23—4882no—
340C23—4882no—
341C23—2444no—
342C23—2442no—
343C23D8×D6111yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—4882no—
347Q8—2441no—
348Q8—2441no—
349D4—1224no—
350D4—2444no—
351D4—2442no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—611no—
359C2×C4—611no—
360C2×C4—614no—
361C2×C4—611no—
362C2×C4—611no—
363C2×C4—4884no—
364C2×C4—4884no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4D8×D6111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—21no—
371C8—1221no—
372C8—1221no—
373C8#41×D6112yes—
374C22—21no—
375C22—21no—
376C22—4888no—
377C22—4882no—
378C22—4882no—
379C22—2444no—
380C22SD32×D6112yes—
381C22C2×D8×D6111yescharacteristic
382C4—21no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4—612no—
387C4—612no—
388C4—1221no—
389C4—1221no—
390C4—4881no—
391C4—4881no—
392C4#1684027011yescharacteristic
393C4#1684056111yescharacteristic
394C2—22no—
395C2—4884no—
396C2#2145581611yescharacteristic
397C2C2×SD32×D6111yescharacteristic
398C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2235C2
socleC22×C61C22×D8
prime radicalC22×D488C22
hypercenter#2235D61
hypocenterC61#33661
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2235

Complements

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

NH ≅ G/Ntype
1. #21456400C2semidirect
3. #21448016C2semidirect
4. #21447965C2semidirect
5. #21447863C2semidirect
8. #21447670C2semidirect
9. #21447668C2semidirect
10. #21447615C2semidirect
11. C61×#2235C2semidirect
17. #16840166C22semidirect
18. #16840140C22semidirect
22. Q16×Dic61C22semidirect
23. #16839856C22semidirect
31. #16839589C22semidirect
32. #16839580C22semidirect
33. #16839565C22semidirect
37. C16×Dic61C22semidirect
40. C61×#367C22semidirect
41. C61×#358C22semidirect
157. C2×Dic61SD32semidirect
198. #2235D61semidirect
233. #367C2×D61semidirect
236. #358C2×D61semidirect
242. C61#33661semidirect

Maximal subgroups

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

subgroupindex
#214564002
C2×#168398442
#214480162
#214479652
#214478632
#214477502
#214477472
#214476702
#214476682
#214476152
C61×#22352
#3366161