Finite Group Databasebeta

← #29834997

Subgroups — #29834997

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2#21448010C212yesmaximal, complemented
3#21447971C211yesmaximal, characteristic, complemented
4#21447966C211yesmaximal, characteristic, complemented
5#21447859C211yesmaximal, characteristic, complemented
6#21447854C211yesmaximal, characteristic, complemented
7#21447742C212yesmaximal, complemented
8C4×#14519720C212yesmaximal, complemented
9C4×#14519718C212yesmaximal, complemented
10#21447618C211yesmaximal, characteristic, complemented
11C4×#135×C61C211yesmaximal, characteristic, complemented
12#16840312—21no—
13C2×#14519720C2212yes—
14C2×#14519718C2212yes—
15#16840167C2211yescharacteristic
16#16840139C2211yescharacteristic
17#16840109—21no—
18#16839998—21no—
19#16839998C2212yes—
20C4×D488—21no—
21C4×D488C2212yes—
22#16839589C2212yes—
23#16839587C2212yes—
24C4⋊#11758185C2211yescharacteristic
25C4⋊#11758185C2211yescharacteristic
26#16839566C2212yes—
27#16839563C2212yes—
28#16839562C2212yes—
29#16839562C2212yes—
30C22⋊#11758185C2212yes—
31C4×#11758185C2212yes—
32C2×#135×C61C2211yescharacteristic
33C4×#41×C61—21no—
34C4×Q16×C61C2211yescharacteristic
35C4×Q16×C61C2211yescharacteristic
36C4×SD16×C61—21no—
37C4×SD16×C61—21no—
38C4×D8×C61C2211yescharacteristic
39C4×D8×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C61×#416—21no—
42C2×C4×C8×C61C2211yescharacteristic
43C2×#11758222—41no—
44C2×D488—21no—
45C2×D488C2312yes—
46#14519777—21no—
47#14519774—21no—
48#14519765—41no—
49#14519751—22no—
50#14519751—41no—
51C4×D244—22no—
52C4×D244—41no—
53#14519720—24no—
54#14519718—24no—
55C2×#11758185C2312yes—
56C2×#11758185C2312yes—
57#14519711—21no—
58#14519711C2312yes—
59#14519699—22no—
60#14519699—22no—
61C2×#41×C61—21no—
62#135×C61—22no—
63#135×C61—22no—
64C2×Q16×C61C2311yescharacteristic
65C2×SD16×C61—21no—
66C2×D8×C61C2311yescharacteristic
67C22×C8×C61D411yescharacteristic
68C22×C8×C61C2311yescharacteristic
69C4×Q8×C61—21no—
70C4×Q8×C61—21no—
71C4×D4×C61—21no—
72C4×D4×C61—21no—
73C4×D4×C61—41no—
74C4×D4×C61—41no—
75#117×C61—21no—
76#117×C61—41no—
77C2×C42×C61—41no—
78C2×C42×C61D411yescharacteristic
79#107×C61C2311yescharacteristic
80#107×C61C2311yescharacteristic
81#106×C61—21no—
82#103×C61—21no—
83#103×C61—21no—
84#102×C61—21no—
85#102×C61—21no—
86C4×C8×C61D411yescharacteristic
87C4×C8×C61D411yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C4×C8×C61C2311yescharacteristic
90#41×C61—24no—
91#41×C61—42no—
92C2×Q8×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61D811yescharacteristic
99C22×C4×C61D811yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—42no—
102#11758222—84no—
103C2×D244—22no—
104C2×D244—41no—
105C2×D244—41no—
106C2×C4×D61—42no—
107C2×Dic122—41no—
108Q16×C61—21no—
109Q16×C61#4112yescomplemented
110SD16×C61—24no—
111D8×C61—21no—
112D8×C61#4112yescomplemented
113C2×C8×C61—21no—
114C2×C8×C61—21no—
115C2×C8×C61—41no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C42×C61—41no—
128C42×C61—41no—
129C42×C61—41no—
130C42×C61D811yescharacteristic
131C42×C61D811yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61C2×D411yescharacteristic
134#11758198—42no—
135#11758198—42no—
136C4⋊Dic61—22no—
137C4⋊Dic61—41no—
138D488—22no—
139D488—22no—
140D488—42no—
141#11758185—22no—
142#11758185—22no—
143C23×C61—41no—
144C23×C61C2×D811yescharacteristic
145C22×D61—42no—
146Q8×C61—22no—
147Q8×C61—41no—
148D4×C61—22no—
149D4×C61—44no—
150D4×C61—41no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—44no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—82no—
160C2×C4×C61—82no—
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61C2×D811yescharacteristic
163C2×C4×C61C2×D811yescharacteristic
164C2×C4×C61C2×D811yescharacteristic
165C2×C4×C61C2×D811yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C22⋊D61—84no—
168C2×Dic61—42no—
169D244—42no—
170D244—42no—
171D244—82no—
172D244—82no—
173C4×D61—84no—
174Dic122—82no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61—21no—
178C8×C61C2×#4111yescharacteristic
179C8×C61C2×#4111yescharacteristic
180#33666—611nomaximal, Sylow
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—42no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61C22×D811yescharacteristic
188C2×D61—82no—
189C2×D61—82no—
190C2×D61—82no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—42no—
195C4×C61—81no—
196C4×C61—81no—
197C4×C61#50812yes—
198C4×C61#50912yes—
199C4×C61#52511yescharacteristic
200C4×C61#57511yescharacteristic
201Dic61—82no—
202#3169—611no—
203C4×#135D6111yescharacteristic, complemented
204#2234—612no—
205#2227—611no—
206#2226—611no—
207#2207—611no—
208#2206—611no—
209#2186—612no—
210C4×SD32—612no—
211C4×D16—612no—
212#2162—611no—
213C2×C61—41no—
214C2×C61—82no—
215C2×C61#310011yescharacteristic
216C2×C61C2×#50811yescharacteristic
217C2×C61C2×#50911yescharacteristic
218D61—162no—
219C2×#135C2×D6111yescharacteristic
220#532—1221no—
221C4×#41—21no—
222C2×SD32—612no—
223C2×D16—612no—
224#500—611no—
225#493—611no—
226#486—1221no—
227#466—612no—
228#466—1221no—
229C4×Q16C2×D6111yescharacteristic
230C4×Q16C2×D6111yescharacteristic
231C4×SD16—21no—
232C4×SD16—21no—
233C4×D8—612no—
234C4×D8—1221no—
235C4×D8C2×D6111yescharacteristic
236C4×D8C2×D6111yescharacteristic
237#427C2×D6111yescharacteristic
238#416—21no—
239C2×C4×C8C2×D6111yescharacteristic
240#367—612no—
241#366—612no—
242C4⋊C16—611no—
243C4⋊C16—611no—
244#358—612no—
245#357—612no—
246#357—612no—
247#357—612no—
248C22⋊C16—612no—
249C4×C16—612no—
250C61#3366611yescharacteristic, complemented, Sylow
251C2×#41—21no—
252C2×#41—2441no—
253#135—22no—
254#135—22no—
255C2×Q16C22×D6111yescharacteristic
256C2×SD16—21no—
257C2×D8—612no—
258C2×D8—1221no—
259C2×D8C22×D6111yescharacteristic
260C22×C8C22⋊D6111yescharacteristic
261C22×C8C22×D6111yescharacteristic
262C4⋊Q8—1221no—
263#127—1221no—
264#124—2441no—
265#121—1222no—
266#121—2441no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—1222no—
274C4×D4—2441no—
275#117—21no—
276#117—41no—
277C2×C42—41no—
278C2×C42C22⋊D6111yescharacteristic
279SD32—1224no—
280D16—1224no—
281C2×C16—612no—
282C2×C16—612no—
283#107—612no—
284#107—1221no—
285#107C22×D6111yescharacteristic
286#107C22×D6111yescharacteristic
287#106—21no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#102—1222no—
293#102—1222no—
294C4×C8C22⋊D6111yescharacteristic
295C4×C8C22⋊D6111yescharacteristic
296C4×C8C22×D6111yescharacteristic
297C4×C8C22×D6111yescharacteristic
298#41—24no—
299#41—42no—
300#41—4884no—
301C2×Q8—21no—
302C2×Q8—2441no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1222no—
306C2×D4—2442no—
307C2×D4—2441no—
308C2×D4—2441no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—2442no—
313C22×C4#1175819911yescharacteristic
314C22×C4#1175819911yescharacteristic
315C22×C4C2×C22⋊D6111yescharacteristic
316Q16—21no—
317Q16#1175822212yescomplemented
318SD16—24no—
319D8—21no—
320D8—1222no—
321D8—1222no—
322D8—2442no—
323D8#1175822212yescomplemented
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—41no—
327C2×C8C2×C22⋊D6111yescharacteristic
328C2×C8C2×C22⋊D6111yescharacteristic
329C2×C8C2×C22⋊D6111yescharacteristic
330C2×C8C23×D6111yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1222no—
336C4⋊C4—2441no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42#1175819911yescharacteristic
346C42#1175819911yescharacteristic
347C42C2×C22⋊D6111yescharacteristic
348C42C2×C22⋊D6111yescharacteristic
349C16—1222no—
350C16—1222no—
351C23—41no—
352C23—2442no—
353C23C2×#1175819911yescharacteristic
354Q8—22no—
355Q8—41no—
356Q8—4882no—
357D4—22no—
358D4—44no—
359D4—41no—
360D4—4882no—
361D4—4884no—
362D4—4882no—
363D4—2442no—
364D4—2442no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—4884no—
376C2×C4—2442no—
377C2×C4C2×#1175819911yescharacteristic
378C2×C4C2×#1175819911yescharacteristic
379C2×C4C2×#1175819911yescharacteristic
380C2×C4C2×#1175819911yescharacteristic
381C2×C4C2×#1175819911yescharacteristic
382C2×C4C22×C22⋊D6111yescharacteristic
383C8—21no—
384C8—21no—
385C8—21no—
386C8C2×#1175822211yescharacteristic
387C8C2×#1175822211yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—81no—
393C22—81no—
394C22—4882no—
395C22—4882no—
396C22—4882no—
397C22C22×#1175819911yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—4882no—
405C4#1684019212yes—
406C4#1684019712yes—
407C4#1684027311yescharacteristic
408C4#1684056011yescharacteristic
409C2—41no—
410C2—82no—
411C2—9762no—
412C2#2145581911yescharacteristic
413C2C2×#1684019211yescharacteristic
414C2C2×#1684019711yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC4×#135×C61C2
socleC22×C61C22×D8
hypercenterC4×#135D61
hypocenterC61#33666
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21448010C2semidirect
3. #21447971C2semidirect
4. #21447966C2semidirect
5. #21447859C2semidirect
6. #21447854C2semidirect
7. #21447742C2semidirect
8. C4×#14519720C2semidirect
9. C4×#14519718C2semidirect
10. #21447618C2semidirect
11. C4×#135×C61C2semidirect
109. Q16×C61#41semidirect
112. D8×C61#41semidirect
203. C4×#135D61semidirect
250. C61#33666semidirect
317. Q16#11758222semidirect
323. D8#11758222semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214480102
#214479712
#214479662
#214478592
#214478542
#214477422
C4×#145197202
C4×#145197182
#214476182
C4×#135×C612
#3366661