Finite Group Databasebeta

← #27151601

Subgroups — #27151601

Order 12416.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261365C211yesmaximal, characteristic, complemented
2#20261217C211yesmaximal, characteristic
3#20261040C211yesmaximal, characteristic, complemented
4#20261038C211yesmaximal, characteristic, complemented
5#20261032C211yesmaximal, characteristic, complemented
6#20261026C211yesmaximal, characteristic, complemented
7#20261024C211yesmaximal, characteristic, complemented
8#20261020C211yesmaximal, characteristic, complemented
9#20261018C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic, complemented
11#106×D97C211yesmaximal, characteristic, complemented
12#20260760C211yesmaximal, characteristic, complemented
13#20260735C211yesmaximal, characteristic
14#20260730C211yesmaximal, characteristic
15#479×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19#15690810C2211yescharacteristic
20#15690809—21no—
21#15690809C2211yescharacteristic
22#15690801C2211yescharacteristic
23#15690800—21no—
24#15690790—21no—
25#15690788—21no—
26Q8×Dic97C2211yescharacteristic
27C4×D388—21no—
28C4×D388C2211yescharacteristic
29#15690772—21no—
30#15690772C2211yescharacteristic
31#15690768C2211yescharacteristic
32#15690767—21no—
33#15690767C2211yescharacteristic
34C4⋊C4×D97—21no—
35C4⋊C4×D97C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690743C2211yescharacteristic
38#15690742C2211yescharacteristic
39#15690742C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690734C2211yescharacteristic
48#15690732C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690730C2211yescharacteristic
51#15690730C2211yescharacteristic
52#125×C97C2211yescharacteristic
53C4×Q8×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic
55#106×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59#13095512—22no—
60#13095512Q814yescomplemented
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×D388—21no—
64C2×D388C2311yescharacteristic
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97C2311yescharacteristic
71#13095494—22no—
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C8×D97—22no—
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490D411yescharacteristic
86#13095490D411yescharacteristic
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488D411yescharacteristic
91#13095488D411yescharacteristic
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97C2311yescharacteristic
95C2×Q8×C97C2311yescharacteristic
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97—21no—
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C42×C97—21no—
105C42×C97C2311yescharacteristic
106C22×D97—21no—
107C22×D97C2×D411yescharacteristic
108Q8×C97—21no—
109Q8×C97C2×Q812yes—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97C2×D411yescharacteristic
121C2×Dic97C2×D411yescharacteristic
122C2×Dic97C2×D411yescharacteristic
123D388—24no—
124D388—21no—
125D388C2×Q812yes—
126C4×D97—24no—
127C4×D97—81no—
128C4×D97—81no—
129C4×D97—81no—
130C4×D97—81no—
131C4×D97C2×Q812yes—
132C4×D97#4112yes—
133Dic194—81no—
134Dic194—81no—
135C8×C97—21no—
136C8×C97—21no—
137#11715895—21no—
138#11715895—21no—
139C22×C97C22×D411yescharacteristic
140C2×D97—21no—
141C2×D97—21no—
142C2×D97—22no—
143C2×D97—41no—
144C2×D97—41no—
145C2×D97C22⋊Q812yes—
146C4×C97—22no—
147C4×C97—21no—
148C4×C97—21no—
149C4×C97—41no—
150C4×C97—41no—
151C4×C97—41no—
152C4×C97C22×Q811yescharacteristic
153C4×C97C2×#4111yescharacteristic
154Dic97—21no—
155Dic97—21no—
156Dic97—41no—
157Dic97—41no—
158Dic97—41no—
159Dic97C22⋊Q812yes—
160C2×C97C2×C22⋊Q811yescharacteristic
161C2×C97#57711yescharacteristic
162C2×C97#57711yescharacteristic
163D97—22no—
164D97—42no—
165#3084—971nomaximal, Sylow
166C97#308411yescharacteristic, complemented, Sylow
167#544—971no—
168#520—971no—
169#479—971no—
170#479D9711yescharacteristic, complemented
171#478—971no—
172#478—971no—
173#476—971no—
174#476—971no—
175#475—971no—
176#475—971no—
177#428—971no—
178C2×#106—971no—
179#424—971no—
180#417—971no—
181#416—971no—
182C2×#41—971no—
183C2×M16—971no—
184C22×C8—971no—
185C4⋊Q8—971no—
186C4⋊Q8—971no—
187C4⋊Q8—1941no—
188#125—971no—
189#125—1941no—
190#125C2×D9711yescharacteristic
191C22⋊Q8—1941no—
192C22⋊Q8—1941no—
193C4×Q8—971no—
194C4×Q8C2×D9711yescharacteristic
195C4×D4—971no—
196C4×D4—971no—
197C4×D4—1941no—
198C4×D4—1941no—
199#117—971no—
200#117—971no—
201#117—1941no—
202C2×C4⋊C4—971no—
203C2×C4⋊C4—971no—
204C2×C4⋊C4—1941no—
205#107—971no—
206#107C2×D9711yescharacteristic
207#106—971no—
208#106—971no—
209#106—971no—
210#106—971no—
211#106—971no—
212#106C2×D9711yescharacteristic
213C4⋊C8—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8C2×D9711yescharacteristic
217#103—971no—
218#103—971no—
219#103C2×D9711yescharacteristic
220#103C2×D9711yescharacteristic
221#102—971no—
222#102—971no—
223#102—971no—
224#102—971no—
225#41—974no—
226#41—1942no—
227C2×Q8—1941no—
228C2×Q8—1941no—
229C2×Q8C22×D9711yescharacteristic
230C2×D4—971no—
231C2×D4—1941no—
232C22×C4—971no—
233C22×C4—1941no—
234C22×C4—1941no—
235C22×C4—1941no—
236C22×C4—1941no—
237C22×C4—1941no—
238M16—1942no—
239C2×C8—971no—
240C2×C8—971no—
241C2×C8—1942no—
242C2×C8C22×D9711yescharacteristic
243C2×C8C22×D9711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—971no—
248C4⋊C4—971no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4C22×D9711yescharacteristic
260C4⋊C4C22×D9711yescharacteristic
261C4⋊C4C22×D9711yescharacteristic
262C22⋊C4—971no—
263C22⋊C4—971no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C42—21no—
268C42—971no—
269C42—1941no—
270C42—1941no—
271C42C22×D9711yescharacteristic
272C23—971no—
273C23—1941no—
274Q8—21no—
275Q8—7761no—
276Q8—7761no—
277Q8Q8×D9712yes—
278D4—972no—
279D4—1944no—
280D4—1941no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—7761no—
286C2×C4—7761no—
287C2×C4—7761no—
288C2×C4—7761no—
289C2×C4—971no—
290C2×C4—972no—
291C2×C4—972no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—1944no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4D4×D9711yescharacteristic
299C2×C4D4×D9711yescharacteristic
300C2×C4C23×D9711yescharacteristic
301C8—21no—
302C8—21no—
303C8—1941no—
304C8—1941no—
305C22—972no—
306C22—3881no—
307C22—3881no—
308C22—1942no—
309C22—1941no—
310C22—1941no—
311C22C2×D4×D9711yescharacteristic
312C4—22no—
313C4—21no—
314C4—21no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—972no—
319C4—3881no—
320C4—3881no—
321C4—3881no—
322C4—1941no—
323C4—1941no—
324C4C2×Q8×D9711yescharacteristic
325C4#41×D9711yescharacteristic
326C2—3882no—
327C2—1942no—
328C2C22⋊Q8×D9711yescharacteristic
329C2#2026154211yescharacteristic
330C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#479×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095512C22
hypercenter#479D97
hypocenterC97#3084
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #479

Complements

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

NH ≅ G/Ntype
1. #20261365C2semidirect
3. #20261040C2semidirect
4. #20261038C2semidirect
5. #20261032C2semidirect
6. #20261026C2semidirect
7. #20261024C2semidirect
8. #20261020C2semidirect
9. #20261018C2semidirect
10. #20260776C2semidirect
11. #106×D97C2semidirect
12. #20260760C2semidirect
15. #479×C97C2semidirect
60. #13095512Q8semidirect
166. C97#3084semidirect
170. #479D97semidirect

Maximal subgroups

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

subgroupindex
#202613652
#202612172
#202610402
#202610382
#202610322
#202610262
#202610242
#202610202
#202610182
#202607762
#106×D972
#202607602
#202607352
#202607302
#479×C972
#308497