Finite Group Databasebeta

← #29657827

Subgroups — #29657827

Order 15616.

1006 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456834C212yesmaximal, complemented
2C2×#16840136C211yesmaximal, characteristic, complemented
3#21444902C212yesmaximal, complemented
4#21444897C212yesmaximal, complemented
5#21444891C214yesmaximal, complemented
6#21444545C211yesmaximal, characteristic, complemented
7#21444279C212yesmaximal, complemented
8C61×#3544C211yesmaximal, characteristic, complemented
9C22×#11758201—22no—
10C2×#14519774C2211yescharacteristic
11#16840152—22no—
12#16840136—22no—
13#16840136C2212yes—
14#16840136C2212yes—
15#16840128—22no—
16#16839962C2214yes—
17#16839957C2214yes—
18#16839912—22no—
19#16839895—22no—
20#16839819C2212yes—
21C2×C4×#92418C2211yescharacteristic
22#16839511C2212yescomplemented
23#16839511C2212yescomplemented
24Q8⋊#92418C2214yescomplemented
25D4⋊#92418C2214yescomplemented
26C61×#563—22no—
27Q82×C61C2211yescharacteristic, complemented
28C61×#555C2211yescharacteristic, complemented
29C61×#554C2212yescomplemented
30C61×#552—22no—
31C61×#547—22no—
32C61×#544—22no—
33C61×#542—22no—
34C61×#536—22no—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C61×#529C2212yes—
37C22×D244—21no—
38C22×D244—22no—
39C2×#11758201—24no—
40C2×#11758201—24no—
41C2×#11758201—24no—
42C2×#11758201—48no—
43C2×#11758199—24no—
44C22×#92418—21no—
45#14519774—22no—
46#14519774—21no—
47#14519774C2312yes—
48C4⋊#92418C2314yes—
49#14519697—24no—
50#14519697—24no—
51C22⋊#92418—22no—
52#14519688—22no—
53C4×#92418—21no—
54C4×#92418C2312yes—
55C2×#41×C61—22no—
56C22×Q8×C61D412yes—
57C4⋊Q8×C61—22no—
58C4⋊Q8×C61—22no—
59C4⋊Q8×C61—22no—
60C4⋊Q8×C61—22no—
61C4⋊Q8×C61D412yes—
62C4⋊Q8×C61C2311yescharacteristic
63C4⋊Q8×C61C2311yescharacteristic
64#126×C61—24no—
65#126×C61—22no—
66#125×C61—24no—
67#125×C61—22no—
68#125×C61D412yes—
69#124×C61—22no—
70#124×C61—22no—
71#124×C61D412yes—
72#123×C61—24no—
73#123×C61—24no—
74#123×C61—22no—
75C22⋊Q8×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61D412yes—
80#121×C61—22no—
81#121×C61D412yes—
82C4×Q8×C61—22no—
83C4×Q8×C61C2312yes—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C4×D4×C61C2312yes—
87#117×C61—22no—
88#117×C61—22no—
89#117×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C42×C61C2311yescharacteristic
93C23×D61—41no—
94#41×C61—84no—
95C2×Q8×C61—22no—
96C2×Q8×C61—22no—
97C2×Q8×C61—44no—
98C2×Q8×C61C2×D412yes—
99C2×Q8×C61C2×D412yes—
100C2×Q8×C61C2×D412yes—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61C2×D412yes—
104C22×C4×C61—22no—
105C22×C4×C61—24no—
106C22×C4×C61—22no—
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2×D411yescharacteristic
109C2×D244—24no—
110C2×D244—22no—
111C2×D244—44no—
112C2×D244—41no—
113C2×D244—42no—
114C2×D244—84no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61C2×D412yes—
126C4⋊C4×C61C2×D412yes—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—24no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C22⋊C4×C61—22no—
134C42×C61—22no—
135C42×C61—22no—
136C42×C61—22no—
137C42×C61C2×D412yes—
138C42×C61C2×D411yescharacteristic
139C42×C61C2411yescharacteristic
140#11758201—48no—
141#11758201—48no—
142#11758201—84no—
143#11758199—84no—
144C2×#92418—24no—
145C2×#92418—22no—
146C2×#92418—24no—
147C23×C61—22no—
148C23×C61#12011yescharacteristic
149C22×D61—42no—
150C22×D61—84no—
151C22×D61—82no—
152Q8×C61—24no—
153Q8×C61—42no—
154Q8×C61—42no—
155Q8×C61—82no—
156Q8×C61—82no—
157D4×C61—24no—
158D4×C61—44no—
159D4×C61—82no—
160C2×C4×C61—22no—
161C2×C4×C61—24no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—44no—
168C2×C4×C61—44no—
169C2×C4×C61—84no—
170C2×C4×C61#12012yes—
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D412yes—
174D244—44no—
175D244—44no—
176D244—44no—
177D244—82no—
178D244—84no—
179#92418—24no—
180#92418—44no—
181#18684—611nomaximal, Sylow
182C22×C61—22no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61#45312yes—
187C22×C61C2×#12011yescharacteristic
188C2×D61—84no—
189C2×D61—82no—
190C2×D61—164no—
191C4×C61—22no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—42no—
195C4×C61—42no—
196C4×C61—42no—
197C4×C61—42no—
198C4×C61—42no—
199C4×C61#57714yes—
200#3544D6111yescharacteristic, complemented
201#3217—612no—
202C2×#492—611no—
203#1624—612no—
204#1623—612no—
205#1622—614no—
206#1558—611no—
207#1494—612no—
208C2×C61—22no—
209C2×C61—82no—
210C2×C61#301012yes—
211C2×C61C2×#45311yescharacteristic
212D61—162no—
213C22×SD16—1222no—
214#563—22no—
215Q82C2×D6111yescharacteristic, complemented
216#555C2×D6111yescharacteristic, complemented
217#554C2×D6112yescomplemented
218#552—22no—
219#547—22no—
220#544—22no—
221#542—22no—
222#536—22no—
223C2×C4⋊Q8C2×D6111yescharacteristic
224C2×#127—611no—
225#529C2×D6112yes—
226#496—1222no—
227#492—612no—
228#492—612no—
229#492—1222no—
230#490—1222no—
231#460—614no—
232#459—614no—
233#450—1222no—
234#447—1222no—
235#432—612no—
236C2×C4×C8—611no—
237#333—612no—
238#333—612no—
239Q8⋊C8—614no—
240D4⋊C8—614no—
241C61#1868411yescharacteristic, complemented, Sylow
242C2×#41—22no—
243C22×Q8C22⋊D6112yes—
244C22×D4—1222no—
245C22×D4—1221no—
246C2×SD16—1224no—
247C2×SD16—1224no—
248C2×SD16—1224no—
249C2×SD16—2448no—
250C2×D8—1224no—
251C22×C8—1221no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8C22⋊D6112yes—
257C4⋊Q8C22×D6111yescharacteristic
258C4⋊Q8C22×D6111yescharacteristic
259#127—612no—
260#127—1222no—
261#127—1221no—
262#126—24no—
263#126—22no—
264#125—24no—
265#125—22no—
266#125C22⋊D6112yes—
267#124—22no—
268#124—22no—
269#124C22⋊D6112yes—
270#123—24no—
271#123—24no—
272#123—22no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—22no—
277C22⋊Q8C22⋊D6112yes—
278#121—22no—
279#121C22⋊D6112yes—
280C4×Q8—22no—
281C4×Q8C22×D6112yes—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4C22×D6112yes—
285#117—22no—
286#117—22no—
287#117—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C42C22×D6111yescharacteristic
291C4⋊C8—614no—
292#102—1224no—
293#102—1224no—
294C22⋊C8—1222no—
295#97—1222no—
296C4×C8—612no—
297C4×C8—1221no—
298C24—2441no—
299#41—84no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—44no—
303C2×Q8C2×C22⋊D6112yes—
304C2×Q8C2×C22⋊D6112yes—
305C2×Q8C2×C22⋊D6112yes—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—1224no—
309C2×D4—1222no—
310C2×D4—4884no—
311C2×D4—2444no—
312C2×D4—2442no—
313C2×D4—2441no—
314C2×D4C2×C22⋊D6112yes—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4C2×C22⋊D6112yes—
319C22×C4C2×C22⋊D6111yescharacteristic
320SD16—4884no—
321SD16—2448no—
322SD16—2448no—
323D8—4884no—
324C2×C8—1224no—
325C2×C8—1222no—
326C2×C8—1224no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4C2×C22⋊D6112yes—
338C4⋊C4C2×C22⋊D6112yes—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42C2×C22⋊D6111yescharacteristic
350C42C2×C22⋊D6112yes—
351C42C23×D6111yescharacteristic
352C23—22no—
353C23—4884no—
354C23—4882no—
355C23—2442no—
356C23#1451974611yescharacteristic
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—82no—
361Q8—82no—
362D4—24no—
363D4—44no—
364D4—82no—
365D4—4884no—
366D4—4882no—
367D4—2444no—
368D4—2444no—
369D4—2444no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—84no—
380C2×C4#1451974612yes—
381C2×C4#1451974611yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C2×C4C22×C22⋊D6112yes—
384C8—1224no—
385C8—2444no—
386C22—22no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—9764no—
391C22—4884no—
392C22—4882no—
393C22#1683993012yes—
394C22C2×#1451974611yescharacteristic
395C4—22no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4#1684057014yes—
404C2—22no—
405C2—82no—
406C2—9762no—
407C2#2145510512yes—
408C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3544C2
socleC22×C61C2×#120
hypercenter#3544D61
hypocenterC61#18684
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

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

NH ≅ G/Ntype
1. #21456834C2semidirect
2. C2×#16840136C2semidirect
3. #21444902C2semidirect
4. #21444897C2semidirect
5. #21444891C2semidirect
6. #21444545C2semidirect
7. #21444279C2semidirect
8. C61×#3544C2semidirect
22. #16839511C22semidirect
23. #16839511C22semidirect
24. Q8⋊#92418C22semidirect
25. D4⋊#92418C22semidirect
27. Q82×C61C22semidirect
28. C61×#555C22semidirect
29. C61×#554C22semidirect
200. #3544D61semidirect
215. Q82C2×D61semidirect
216. #555C2×D61semidirect
217. #554C2×D61semidirect
241. C61#18684semidirect

Maximal subgroups

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

subgroupindex
#214568342
C2×#168401362
#214449022
#214448972
#214448912
#214445452
#214442792
C61×#35442
#1868461