Finite Group Databasebeta

← #29663986

Subgroups — #29663986

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#21455307C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4C8×#11758222C211yesmaximal, characteristic, complemented
5#21445264C214yesmaximal, complemented
6#21445257C214yesmaximal, complemented
7#21444492C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic, complemented
9C61×#1567C211yesmaximal, characteristic, complemented
10C2×#14519810—21no—
11C2×#14519805—21no—
12C2×#14519777C2211yescharacteristic
13C2×#14519774C2211yescharacteristic
14C4×#11758222C2211yescharacteristic
15#16840109—21no—
16#16840109—21no—
17#16840109C2214yes—
18#16839973—21no—
19#16839973C2212yescomplemented
20#16839968—21no—
21#16839968C2212yescomplemented
22#16839963—21no—
23#16839963C2212yes—
24#16839958—21no—
25#16839958C2212yes—
26C8×Dic122C2212yescomplemented
27C8×D244C2212yescomplemented
28C8×C22⋊D61—22no—
29#16839815C2211yescharacteristic
30#16839813—21no—
31#16839772—21no—
32#16839768—21no—
33C2×#14519699—22no—
34C4×C8×D61—21no—
35#16839509C2214yescomplemented
36#16839505C2214yes—
37C61×#423C2211yescharacteristic
38C61×#423C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C61×#335C2214yescomplemented
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43C22×D244—21no—
44C22×D244—21no—
45C22×D244—21no—
46#14519810—44no—
47#14519805—44no—
48C2×Dic244—22no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×D488—22no—
52C2×C8×D61—22no—
53#14519777—21no—
54#14519777—21no—
55#14519777—21no—
56#14519777C2312yes—
57#14519774—21no—
58#14519774—21no—
59#14519774—21no—
60#14519774C2312yes—
61C4×Dic122D411yescharacteristic
62C4×Dic122C2311yescharacteristic
63C4×D244D411yescharacteristic
64C4×D244C2311yescharacteristic
65C4×C22⋊D61—21no—
66C4×C22⋊D61—21no—
67#14519736D411yescharacteristic
68#14519734—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C42×D61—21no—
72C4⋊#92418C2312yes—
73#14519706—22no—
74#14519702—22no—
75#14519702—22no—
76#14519699—24no—
77#14519699—24no—
78#14519699—22no—
79#14519699—22no—
80C22⋊#92418—21no—
81#14519689—22no—
82C4×#92418—21no—
83C8×Dic61—22no—
84C2×M16×C61—21no—
85C2×M16×C61—21no—
86C22×C8×C61D412yes—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61—21no—
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C4×C8×C61D411yescharacteristic
93C4×C8×C61D411yescharacteristic
94C4×C8×C61C2312yes—
95C23×D61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61#4111yescharacteristic
99C2×C22⋊D61—21no—
100C22×Dic61—41no—
101#11758222—22no—
102#11758222D814yes—
103C2×D244—22no—
104C2×D244—22no—
105C2×D244—22no—
106C2×D244—44no—
107C2×D244—42no—
108C2×D244—42no—
109C2×D244—41no—
110C2×D244—41no—
111C2×D244—41no—
112C2×D244—82no—
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×Dic122—22no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122—41no—
121C2×Dic122C2×D411yescharacteristic
122M16×C61—42no—
123M16×C61—42no—
124C2×C8×C61—22no—
125C2×C8×C61—22no—
126C2×C8×C61—42no—
127C2×C8×C61C2×D412yes—
128C2×C8×C61C2×D412yes—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—21no—
134#11758198—21no—
135#11758198—21no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61—41no—
139C4⋊Dic61—41no—
140C4⋊Dic61C2×D411yescharacteristic
141C4⋊Dic61C2×D411yescharacteristic
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C2×#92418—22no—
148Dic244—44no—
149D488—44no—
150#11758190—44no—
151#11758190—44no—
152C8×D61—44no—
153C23×C61#12111yescharacteristic
154C22×D61—21no—
155C22×D61—42no—
156C22×D61—82no—
157C22×D61—82no—
158C22×D61—82no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61C2×D812yes—
165C2×C4×C61#13512yes—
166C2×C4×C61C22×D411yescharacteristic
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C2×#4111yescharacteristic
169C22⋊D61—24no—
170C2×Dic61—21no—
171C2×Dic61—21no—
172C2×Dic61—21no—
173C2×Dic61—42no—
174C2×Dic61—82no—
175D244—21no—
176D244—42no—
177D244—42no—
178D244—44no—
179D244—82no—
180D244—82no—
181D244—82no—
182D244—82no—
183D244C2×D812yes—
184C4×D61—24no—
185C4×D61—42no—
186C4×D61—42no—
187Dic122—21no—
188Dic122—42no—
189Dic122—42no—
190Dic122—82no—
191Dic122C2×D812yes—
192C8×C61—22no—
193C8×C61—22no—
194C8×C61—42no—
195C8×C61—42no—
196#92418—42no—
197#19276—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#46612yes—
201C22×C61C2×#12111yescharacteristic
202C2×D61—22no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—164no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61#46612yes—
212C4×C61#47112yes—
213C4×C61C22×D811yescharacteristic
214C4×C61C2×#13511yescharacteristic
215C4×C61#57512yes—
216Dic61—22no—
217Dic61—41no—
218Dic61—41no—
219Dic61—82no—
220C2×#486—611no—
221#3039—611no—
222#3038—611no—
223C8×#41—611no—
224#1682—614no—
225#1681—614no—
226#1567D6111yescharacteristic, complemented
227#1546—611no—
228#1545—611no—
229C2×C61—22no—
230C2×C61#304011yescharacteristic
231C2×C61C2×#46611yescharacteristic
232C2×C61C2×#47111yescharacteristic
233D61—42no—
234D61—162no—
235C2×#137—1221no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×C4⋊Q8—611no—
238C2×#127—611no—
239C4×#41—611no—
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×Q8—612no—
252C8×D4—612no—
253C8×D4—1222no—
254#431—611no—
255#431—1221no—
256#423C2×D6111yescharacteristic
257#423C2×D6111yescharacteristic
258#419—1221no—
259#418—1221no—
260C2×#102—1222no—
261C2×C4×C8—1221no—
262C2×C4×C8C2×D6111yescharacteristic
263#335C2×D6114yescomplemented
264#332—614no—
265#331—614no—
266C61#1927611yescharacteristic, complemented, Sylow
267C2×#41—611no—
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—21no—
279C2×M16—21no—
280C22×C8—1222no—
281C22×C8D24412yes—
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×Q8—611no—
291C4×Q8—611no—
292C4×D4—611no—
293C4×D4—611no—
294C4×D4—1221no—
295C4×D4—1221no—
296#117—611no—
297#117—1221no—
298C2×C4⋊C4—1221no—
299C2×C4⋊C4—1221no—
300C2×C42—1221no—
301C2×C42C22×D6111yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—612no—
305C4⋊C8—1222no—
306C4⋊C8C22×D6112yes—
307C4⋊C8C22×D6112yes—
308#103—1222no—
309#103—1222no—
310#102—1222no—
311#102—1222no—
312#102—1224no—
313#102—1224no—
314C22⋊C8—1221no—
315C22⋊C8—1222no—
316C4×C8—1222no—
317C4×C8—1221no—
318C4×C8D24411yescharacteristic
319C4×C8D24411yescharacteristic
320C4×C8C22×D6112yes—
321C24—2441no—
322#41—614no—
323#41—1222no—
324C2×Q8—611no—
325C2×Q8—1222no—
326C2×Q8—2442no—
327C2×Q8—2442no—
328C2×Q8—2441no—
329C2×D4—611no—
330C2×D4—1221no—
331C2×D4—1222no—
332C2×D4—1222no—
333C2×D4—1222no—
334C2×D4—4882no—
335C2×D4—2444no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C2×D4—2442no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2441no—
345C22×C4C2×D24411yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4#1175822611yescharacteristic
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2444no—
352M16—42no—
353M16—42no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—42no—
357C2×C8—1222no—
358C2×C8—2444no—
359C2×C8C2×D24412yes—
360C2×C8C2×D24412yes—
361C4⋊C4—611no—
362C4⋊C4—611no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1222no—
366C4⋊C4—1222no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—1221no—
373C42—1221no—
374C42—1221no—
375C42C2×D24411yescharacteristic
376C42D4×D6111yescharacteristic
377C42#1175822611yescharacteristic
378C42C23×D6111yescharacteristic
379C23—1221no—
380C23—4882no—
381C23—4882no—
382C23—4882no—
383C23—2442no—
384C23#1451975211yescharacteristic
385Q8—612no—
386Q8—1221no—
387Q8—4882no—
388Q8—2442no—
389Q8—2442no—
390D4—612no—
391D4—1224no—
392D4—1221no—
393D4—4882no—
394D4—4882no—
395D4—4882no—
396D4—4882no—
397D4—2442no—
398D4—2442no—
399D4—2444no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—1221no—
403C2×C4—1224no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—4882no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4#1451975211yescharacteristic
411C2×C4#1451975211yescharacteristic
412C2×C4#1451975211yescharacteristic
413C2×C4D8×D6112yes—
414C2×C4#1451980012yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C22×D24411yescharacteristic
417C2×C4C2×#1175822611yescharacteristic
418C8—22no—
419C8—22no—
420C8—42no—
421C8—42no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—1222no—
426C22—9764no—
427C22—4882no—
428C22—4882no—
429C22—4882no—
430C22—2441no—
431C22—2441no—
432C22#1683999912yes—
433C22C2×#1451975211yescharacteristic
434C4—21no—
435C4—21no—
436C4—1222no—
437C4—4882no—
438C4—2441no—
439C4—2441no—
440C4#1683999912yes—
441C4#1684002612yes—
442C4C2×D8×D6111yescharacteristic
443C4C2×#1451980011yescharacteristic
444C4#1684056212yes—
445C2—22no—
446C2—9762no—
447C2—2442no—
448C2#2145531411yescharacteristic
449C2C2×#1683999911yescharacteristic
450C2C2×#1684002611yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1567C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1567D61
hypocenterC61#19276
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1567

Complements

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

NH ≅ G/Ntype
1. C2×#16840109C2semidirect
2. #21455307C2semidirect
4. C8×#11758222C2semidirect
5. #21445264C2semidirect
6. #21445257C2semidirect
7. #21444492C2semidirect
8. #21444488C2semidirect
9. C61×#1567C2semidirect
19. #16839973C22semidirect
21. #16839968C22semidirect
26. C8×Dic122C22semidirect
27. C8×D244C22semidirect
35. #16839509C22semidirect
40. C61×#335C22semidirect
226. #1567D61semidirect
263. #335C2×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
#214553072
#214553002
C8×#117582222
#214452642
#214452572
#214444922
#214444882
C61×#15672
#1927661