Finite Group Databasebeta

← #29663985

Subgroups — #29663985

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic, complemented
2#21455306C211yesmaximal, characteristic, complemented
3#21455299C211yesmaximal, characteristic
4#21445263C214yesmaximal, complemented
5#21445256C214yesmaximal, complemented
6#21444578C211yesmaximal, characteristic, complemented
7#21444490C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C8×#41×C61C211yesmaximal, characteristic, complemented
10C2×#14519807—21no—
11C2×#14519802—21no—
12C2×#14519777C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14#16840109—21no—
15#16840109—21no—
16#16840109C2214yes—
17#16839972—21no—
18#16839972C2212yescomplemented
19#16839967—21no—
20#16839967C2212yescomplemented
21#16839962—21no—
22#16839962C2212yes—
23#16839957—21no—
24#16839957C2212yes—
25#16839786C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839770—21no—
28#16839766—21no—
29C2×#14519699—22no—
30#16839515C2214yescomplemented
31#16839507C2214yescomplemented
32#16839505C2214yes—
33C4×#41×C61C2211yescharacteristic
34C8×Q8×C61C2212yescomplemented
35C8×D4×C61—22no—
36C8×D4×C61C2212yescomplemented
37C61×#431—21no—
38C61×#431C2211yescharacteristic
39C2×C4×C8×C61—21no—
40C2×C4×C8×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45#14519807—44no—
46#14519802—44no—
47C2×#11758202—22no—
48C2×#11758200—22no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51C2×#11758194—21no—
52C2×#11758194—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774C2312yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C4⋊#92418—21no—
64C4⋊#92418—21no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2312yes—
67#14519700—22no—
68#14519700—22no—
69#14519699—24no—
70#14519699—24no—
71#14519697—22no—
72#14519697—22no—
73C2×#41×C61D411yescharacteristic
74C22×C8×C61—22no—
75C22×C8×C61D412yes—
76C4×Q8×C61D411yescharacteristic
77C4×Q8×C61C2311yescharacteristic
78C4×D4×C61—21no—
79C4×D4×C61—21no—
80C4×D4×C61D411yescharacteristic
81C4×D4×C61C2311yescharacteristic
82#117×C61—21no—
83#117×C61D411yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61C2311yescharacteristic
86C4⋊C8×C61—22no—
87C4⋊C8×C61C2312yes—
88C22⋊C8×C61—21no—
89C22⋊C8×C61—22no—
90C4×C8×C61—22no—
91C4×C8×C61—21no—
92C4×C8×C61D411yescharacteristic
93C4×C8×C61D411yescharacteristic
94C4×C8×C61C2312yes—
95C23×D61—41no—
96#41×C61—22no—
97#41×C61D814yes—
98C2×Q8×C61C2×D411yescharacteristic
99C2×D4×C61—21no—
100C2×D4×C61C2×D411yescharacteristic
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61#4111yescharacteristic
107C22×Dic61—41no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—44no—
112C2×D244—42no—
113C2×D244—42no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—82no—
118C2×Dic122—22no—
119C2×Dic122—42no—
120C2×Dic122—42no—
121C2×Dic122—41no—
122C2×C8×C61—22no—
123C2×C8×C61—42no—
124C2×C8×C61—44no—
125C2×C8×C61C2×D412yes—
126C2×C8×C61C2×D412yes—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61C2×D411yescharacteristic
130C4⋊C4×C61C2×D411yescharacteristic
131C22⋊C4×C61—21no—
132C22⋊C4×C61—21no—
133C22⋊C4×C61—21no—
134C42×C61—21no—
135C42×C61—21no—
136C42×C61—21no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141#11758202—44no—
142#11758201—44no—
143#11758200—44no—
144#11758199—44no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149#11758194—42no—
150#11758194—42no—
151C2×#92418—22no—
152C2×#92418—22no—
153C23×C61—21no—
154C23×C61#12111yescharacteristic
155C22×D61—42no—
156C22×D61—82no—
157C22×D61—82no—
158C22×D61—82no—
159Q8×C61—21no—
160Q8×C61C2×D812yes—
161D4×C61—24no—
162D4×C61—21no—
163D4×C61C2×D812yes—
164C2×C4×C61—21no—
165C2×C4×C61—22no—
166C2×C4×C61—24no—
167C2×C4×C61—22no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C2×D812yes—
176C2×C4×C61#13512yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C2×Dic61—42no—
181C2×Dic61—82no—
182D244—42no—
183D244—42no—
184D244—44no—
185D244—82no—
186D244—82no—
187D244—82no—
188D244—82no—
189Dic122—42no—
190Dic122—42no—
191Dic122—82no—
192C8×C61—22no—
193C8×C61—22no—
194C8×C61—42no—
195#92418—42no—
196#92418—42no—
197#19276—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—22no—
201C22×C61—41no—
202C22×C61—41no—
203C22×C61#46612yes—
204C22×C61C2×#12111yescharacteristic
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—164no—
209C4×C61—21no—
210C4×C61—22no—
211C4×C61—21no—
212C4×C61—41no—
213C4×C61—41no—
214C4×C61#46612yes—
215C4×C61#47112yes—
216C4×C61C22×D811yescharacteristic
217C4×C61C2×#13511yescharacteristic
218C4×C61#57512yes—
219Dic61—82no—
220C2×#486—611no—
221#3039—611no—
222#3038—611no—
223C8×#41D6111yescharacteristic, complemented
224#1682—614no—
225#1681—614no—
226#1567—611no—
227#1546—611no—
228#1545—611no—
229C2×C61—22no—
230C2×C61—42no—
231C2×C61#304011yescharacteristic
232C2×C61C2×#46611yescharacteristic
233C2×C61C2×#47111yescharacteristic
234D61—162no—
235C2×#137—1221no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×C4⋊Q8—611no—
238C2×#127—611no—
239C4×#41C2×D6111yescharacteristic
240#486—614no—
241#486—1221no—
242#486—1221no—
243#462—612no—
244#462—1221no—
245#461—612no—
246#461—1221no—
247#460—612no—
248#460—1221no—
249#459—612no—
250#459—1221no—
251C8×Q8C2×D6112yescomplemented
252C8×D4—22no—
253C8×D4C2×D6112yescomplemented
254#431—21no—
255#431C2×D6111yescharacteristic
256#423—611no—
257#423—611no—
258#419—1221no—
259#418—1221no—
260C2×#102—1222no—
261C2×C4×C8—21no—
262C2×C4×C8C2×D6111yescharacteristic
263#335—614no—
264#332—614no—
265#331—614no—
266C61#1927611yescharacteristic, complemented, Sylow
267C2×#41D24411yescharacteristic
268C22×Q8—1221no—
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4—1221no—
272#137—2444no—
273AGL(1,ℤ/8)—2444no—
274C2×Q16—1222no—
275C2×SD16—1222no—
276C2×SD16—1222no—
277C2×D8—1222no—
278C2×M16—1221no—
279C2×M16—1221no—
280C22×C8—22no—
281C22×C8C22⋊D6112yes—
282C4⋊Q8—612no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8—1221no—
286#127—612no—
287#127—1221no—
288#127—1221no—
289#127—1221no—
290C4×Q8D24411yescharacteristic
291C4×Q8C22×D6111yescharacteristic
292C4×D4—21no—
293C4×D4—21no—
294C4×D4D24411yescharacteristic
295C4×D4C22×D6111yescharacteristic
296#117—21no—
297#117D24411yescharacteristic
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C42—21no—
301C2×C42C22×D6111yescharacteristic
302C4⋊C8—22no—
303C4⋊C8—612no—
304C4⋊C8—612no—
305C4⋊C8—1221no—
306C4⋊C8—1221no—
307C4⋊C8C22×D6112yes—
308#103—1222no—
309#103—1222no—
310#102—1224no—
311#102—1224no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—21no—
315C22⋊C8—22no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8C22⋊D6111yescharacteristic
319C4×C8C22⋊D6111yescharacteristic
320C4×C8C22×D6112yes—
321C24—2441no—
322#41—22no—
323#41D48814yes—
324C2×Q8—1222no—
325C2×Q8—2442no—
326C2×Q8—2442no—
327C2×Q8—2441no—
328C2×Q8C2×D24411yescharacteristic
329C2×D4—21no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—1222no—
333C2×D4—4882no—
334C2×D4—2444no—
335C2×D4—2441no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2442no—
339C2×D4—2442no—
340C2×D4C2×D24411yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—2441no—
345C22×C4C2×D24411yescharacteristic
346C22×C4#1175822211yescharacteristic
347C22×C4C2×C22⋊D6111yescharacteristic
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2444no—
352M16—2442no—
353M16—2442no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8—44no—
357C2×C8—1222no—
358C2×C8—1222no—
359C2×C8C2×C22⋊D6112yes—
360C2×C8C2×C22⋊D6112yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—1222no—
364C4⋊C4—1222no—
365C4⋊C4—2441no—
366C4⋊C4—2441no—
367C4⋊C4C2×D24411yescharacteristic
368C4⋊C4C2×D24411yescharacteristic
369C22⋊C4—21no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42C2×D24411yescharacteristic
376C42#1175822211yescharacteristic
377C42C2×C22⋊D6111yescharacteristic
378C42C23×D6111yescharacteristic
379C23—21no—
380C23—4882no—
381C23—4882no—
382C23—4882no—
383C23—2442no—
384C23#1451975111yescharacteristic
385Q8—21no—
386Q8—4882no—
387Q8—2442no—
388Q8—2442no—
389Q8C2×D48812yes—
390D4—24no—
391D4—21no—
392D4—4882no—
393D4—4882no—
394D4—4882no—
395D4—4882no—
396D4—2442no—
397D4—2442no—
398D4—2444no—
399D4C2×D48812yes—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—4882no—
409C2×C4—2442no—
410C2×C4#1451975111yescharacteristic
411C2×C4#1451975111yescharacteristic
412C2×C4#1451975111yescharacteristic
413C2×C4C2×D48812yes—
414C2×C4#1451979912yes—
415C2×C4C22×C22⋊D6111yescharacteristic
416C2×C4C22×D24411yescharacteristic
417C2×C4C2×#1175822211yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—2442no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—9764no—
429C22—4882no—
430C22—4882no—
431C22—4882no—
432C22#1683999812yes—
433C22C2×#1451975111yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—41no—
438C4—41no—
439C4—4882no—
440C4#1683999812yes—
441C4#1684002512yes—
442C4C22×D48811yescharacteristic
443C4C2×#1451979911yescharacteristic
444C4#1684056012yes—
445C2—22no—
446C2—42no—
447C2—9762no—
448C2#2145531311yescharacteristic
449C2C2×#1683999811yescharacteristic
450C2C2×#1684002511yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×#41×C61C2
socleC22×C61C2×#121
prime radical#21455299C2
hypercenterC8×#41D61
hypocenterC61#19276
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840109C2semidirect
2. #21455306C2semidirect
4. #21445263C2semidirect
5. #21445256C2semidirect
6. #21444578C2semidirect
7. #21444490C2semidirect
8. #21444486C2semidirect
9. C8×#41×C61C2semidirect
18. #16839972C22semidirect
20. #16839967C22semidirect
30. #16839515C22semidirect
31. #16839507C22semidirect
34. C8×Q8×C61C22semidirect
36. C8×D4×C61C22semidirect
223. C8×#41D61semidirect
251. C8×Q8C2×D61semidirect
253. C8×D4C2×D61semidirect
266. C61#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
#214553062
#214552992
#214452632
#214452562
#214445782
#214444902
#214444862
C8×#41×C612
#1927661