Finite Group Databasebeta

← #27152102

Subgroups — #27152102

Order 12416.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261365C211yesmaximal, characteristic, complemented
2#20261310C211yesmaximal, characteristic
3#20261102C211yesmaximal, characteristic, complemented
4#20261096C211yesmaximal, characteristic, complemented
5#20261095C211yesmaximal, characteristic, complemented
6#20261092C211yesmaximal, characteristic, complemented
7#20261091C211yesmaximal, characteristic, complemented
8#20261085C211yesmaximal, characteristic, complemented
9#20261084C211yesmaximal, characteristic, complemented
10#20260742C211yesmaximal, characteristic, complemented
11#20260735C211yesmaximal, characteristic
12#20260730C211yesmaximal, characteristic
13#103×D97C211yesmaximal, characteristic, complemented
14#20260692C211yesmaximal, characteristic, complemented
15#491×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×Q8×D97C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690810C2211yescharacteristic
21#15690809—21no—
22#15690809C2211yescharacteristic
23#15690808C2211yescharacteristic
24#15690801C2211yescharacteristic
25#15690800—21no—
26#15690799C2211yescharacteristic
27#15690798C2211yescharacteristic
28#15690795—21no—
29#15690795—21no—
30#15690790—21no—
31#15690788—21no—
32#15690788—21no—
33#15690788—21no—
34#15690768C2211yescharacteristic
35#15690767C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic
37#15690735C2211yescharacteristic
38#15690735C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690734C2211yescharacteristic
43#15690733C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690732C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49#15690720C2211yescharacteristic
50C4×#11715895C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C4⋊Q8×C97C2211yescharacteristic
53#125×C97C2211yescharacteristic
54#103×C97C2211yescharacteristic
55#103×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#97×C97C2211yescharacteristic
59#13095512—22no—
60#13095512—22no—
61C2×Dic194—21no—
62C2×Dic194—21no—
63C2×Dic194—21no—
64C2×Dic194C2311yescharacteristic
65Q8×D97—22no—
66Q8×D97—22no—
67C2×D388—21no—
68C2×D388C2311yescharacteristic
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97C2311yescharacteristic
73#13095494—22no—
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C8×D97—22no—
77C4⋊Dic97—21no—
78C4⋊Dic97—21no—
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490D411yescharacteristic
87#13095490D411yescharacteristic
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488D411yescharacteristic
94#13095488D411yescharacteristic
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98C2×Q8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C4⋊C4×C97—21no—
102C4⋊C4×C97—21no—
103C4⋊C4×C97—21no—
104C4⋊C4×C97—21no—
105C4⋊C4×C97C2311yescharacteristic
106C4⋊C4×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C22×D97—21no—
109C22×D97C2×D411yescharacteristic
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97C2411yescharacteristic
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97C2×D411yescharacteristic
125C2×Dic97C2×D411yescharacteristic
126C2×Dic97C2×D411yescharacteristic
127D388—21no—
128D388—21no—
129D388—42no—
130C4×D97—42no—
131C4×D97—42no—
132C4×D97—81no—
133C4×D97#4112yes—
134C4×D97#4112yes—
135Dic194—21no—
136Dic194—21no—
137Dic194—42no—
138Dic194—81no—
139Dic194—81no—
140C8×C97—21no—
141C8×C97—21no—
142#11715895—21no—
143#11715895—21no—
144C22×C97C22×D411yescharacteristic
145C2×D97—21no—
146C2×D97—21no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97—41no—
150C2×D97#12412yes—
151C4×C97—21no—
152C4×C97—21no—
153C4×C97—41no—
154C4×C97—41no—
155C4×C97—41no—
156C4×C97—41no—
157C4×C97C2×#4111yescharacteristic
158C4×C97C2×#4111yescharacteristic
159Dic97—21no—
160Dic97—21no—
161Dic97—41no—
162Dic97—41no—
163Dic97—41no—
164Dic97#12412yes—
165C2×C97C2×#12411yescharacteristic
166C2×C97#57711yescharacteristic
167C2×C97#57811yescharacteristic
168D97—22no—
169D97—81no—
170#3137—971nomaximal, Sylow
171C97#313711yescharacteristic, complemented, Sylow
172#544—971no—
173#536—971no—
174#491—971no—
175#491D9711yescharacteristic, complemented
176#489—971no—
177#489—971no—
178#488—971no—
179#488—971no—
180#487—971no—
181#487—971no—
182#419—971no—
183#417—971no—
184#416—971no—
185C2×#103—971no—
186#405—971no—
187C2×#41—971no—
188C22×Q8—971no—
189C2×M16—971no—
190C22×C8—971no—
191C4⋊Q8—971no—
192C4⋊Q8—971no—
193C4⋊Q8—971no—
194C4⋊Q8—1941no—
195C4⋊Q8C2×D9711yescharacteristic
196#125—971no—
197#125—1941no—
198#125C2×D9711yescharacteristic
199#124—971no—
200#124—971no—
201#123—1941no—
202#123—1941no—
203C22⋊Q8—1941no—
204C22⋊Q8—1941no—
205C22⋊Q8—1941no—
206C22⋊Q8—1941no—
207#117—971no—
208#117—971no—
209C2×C4⋊C4—971no—
210#103—971no—
211#103—971no—
212#103—971no—
213#103—971no—
214#103—971no—
215#103—971no—
216#103—971no—
217#103—971no—
218#103C2×D9711yescharacteristic
219#103C2×D9711yescharacteristic
220#103C2×D9711yescharacteristic
221#103C2×D9711yescharacteristic
222#102—971no—
223#102—971no—
224#102—971no—
225#102—971no—
226#97—971no—
227#97C2×D9711yescharacteristic
228C4×C8—971no—
229C4×C8—971no—
230#41—1942no—
231#41—1942no—
232C2×Q8—971no—
233C2×Q8—1941no—
234C2×Q8—1941no—
235C2×Q8—1942no—
236C2×Q8—1942no—
237C2×Q8—1941no—
238C2×Q8C22×D9711yescharacteristic
239C2×Q8C22×D9711yescharacteristic
240C2×D4—971no—
241C2×D4—1941no—
242C22×C4—971no—
243C22×C4—1941no—
244C22×C4—1941no—
245C22×C4—1941no—
246M16—1942no—
247C2×C8—971no—
248C2×C8—971no—
249C2×C8—1942no—
250C2×C8C22×D9711yescharacteristic
251C2×C8C22×D9711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—971no—
257C4⋊C4—971no—
258C4⋊C4—971no—
259C4⋊C4—971no—
260C4⋊C4—1941no—
261C4⋊C4—1941no—
262C4⋊C4—1941no—
263C4⋊C4—1941no—
264C4⋊C4—1941no—
265C4⋊C4—1941no—
266C4⋊C4—1941no—
267C4⋊C4C22×D9711yescharacteristic
268C4⋊C4C22×D9711yescharacteristic
269C22⋊C4—971no—
270C22⋊C4—971no—
271C22⋊C4—1941no—
272C22⋊C4—1941no—
273C22⋊C4—1941no—
274C22⋊C4—1941no—
275C22⋊C4—1941no—
276C42—971no—
277C42—1941no—
278C42C22×D9711yescharacteristic
279C23—971no—
280C23—1941no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—7761no—
286Q8—7761no—
287Q8—3882no—
288Q8—1941no—
289Q8—1941no—
290D4—3882no—
291D4—1941no—
292D4—1941no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—7761no—
298C2×C4—971no—
299C2×C4—972no—
300C2×C4—972no—
301C2×C4—971no—
302C2×C4—971no—
303C2×C4—3882no—
304C2×C4—3882no—
305C2×C4—1941no—
306C2×C4—1941no—
307C2×C4—1941no—
308C2×C4D4×D9711yescharacteristic
309C2×C4D4×D9711yescharacteristic
310C2×C4C23×D9711yescharacteristic
311C8—21no—
312C8—21no—
313C8—1941no—
314C8—1941no—
315C22—972no—
316C22—3881no—
317C22—3881no—
318C22—3881no—
319C22—1941no—
320C22—1941no—
321C22C2×D4×D9711yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—972no—
329C4—3881no—
330C4—3881no—
331C4—3881no—
332C4—1941no—
333C4—1941no—
334C4#41×D9711yescharacteristic
335C4#41×D9711yescharacteristic
336C2—7761no—
337C2—1942no—
338C2#124×D9711yescharacteristic
339C2#2026154411yescharacteristic
340C2#2026155111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#491×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095512C22
hypercenter#491D97
hypocenterC97#3137
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #20261365C2semidirect
3. #20261102C2semidirect
4. #20261096C2semidirect
5. #20261095C2semidirect
6. #20261092C2semidirect
7. #20261091C2semidirect
8. #20261085C2semidirect
9. #20261084C2semidirect
10. #20260742C2semidirect
13. #103×D97C2semidirect
14. #20260692C2semidirect
15. #491×C97C2semidirect
171. C97#3137semidirect
175. #491D97semidirect

Maximal subgroups

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

subgroupindex
#202613652
#202613102
#202611022
#202610962
#202610952
#202610922
#202610912
#202610852
#202610842
#202607422
#202607352
#202607302
#103×D972
#202606922
#491×C972
#313797