Finite Group Databasebeta

← #27153160

Subgroups — #27153160

Order 12416.

334 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261357C211yesmaximal, characteristic
2#20261346C211yesmaximal, characteristic, complemented
3#20261042C211yesmaximal, characteristic, complemented
4#20261036C211yesmaximal, characteristic, complemented
5#20261028C211yesmaximal, characteristic, complemented
6#20261018C211yesmaximal, characteristic, complemented
7#20260990C211yesmaximal, characteristic, complemented
8#20260984C211yesmaximal, characteristic
9#20260973C211yesmaximal, characteristic, complemented
10#20260967C211yesmaximal, characteristic
11#20260959C211yesmaximal, characteristic, complemented
12#20260959C211yesmaximal, characteristic, complemented
13#20260937C211yesmaximal, characteristic, complemented
14#20260760C211yesmaximal, characteristic, complemented
15#462×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690809C2211yescharacteristic
21#15690805—21no—
22#15690805—21no—
23#15690804—21no—
24#15690802C2211yescharacteristic
25#15690800—21no—
26#15690800—21no—
27#15690799C2211yescharacteristic
28#15690798—21no—
29#15690795—21no—
30#15690788—21no—
31#15690786C2211yescharacteristic
32#15690786C2211yescharacteristic
33#15690786C2211yescharacteristic
34#15690785C2211yescharacteristic
35Q8×Dic97C2211yescharacteristic
36Q8×Dic97C2211yescharacteristic
37C4×D388C2211yescharacteristic
38#15690767C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690741C2211yescharacteristic
43C4⋊#11715895C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690739C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690734C2211yescharacteristic
48#15690734C2211yescharacteristic
49#15690734C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52C2×Q16×C97C2211yescharacteristic
53C2×Q16×C97C2211yescharacteristic
54C4⋊Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×C4×D97—21no—
63C2×C4×D97C2311yescharacteristic
64#13095498—41no—
65#13095498—41no—
66#13095494—41no—
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C8×D97—41no—
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82#13095490D411yescharacteristic
83#13095488—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
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×Q8×C97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98Q16×C97—41no—
99Q16×C97—41no—
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97—21no—
103C4⋊C4×C97—21no—
104C4⋊C4×C97—21no—
105C4⋊C4×C97C2311yescharacteristic
106C42×C97—21no—
107C42×C97C2311yescharacteristic
108C22×D97—21no—
109C22×D97C2×D411yescharacteristic
110Q8×C97—21no—
111Q8×C97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114Q8×C97—21no—
115Q8×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2×D411yescharacteristic
122C2×C4×C97C2411yescharacteristic
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97C2×D411yescharacteristic
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129D388—21no—
130D388—21no—
131D388—81no—
132C4×D97—21no—
133C4×D97—21no—
134C4×D97—81no—
135Dic194—81no—
136C8×C97—21no—
137C8×C97—21no—
138#11715895—21no—
139#11715895—21no—
140C22×C97C22×D411yescharacteristic
141C2×D97—21no—
142C2×D97—21no—
143C2×D97—21no—
144C2×D97—41no—
145C2×D97—41no—
146C2×D97—41no—
147C4×C97—21no—
148C4×C97—21no—
149C4×C97—41no—
150C4×C97—41no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97H(5,𝔽2)11yescharacteristic
154C4×C972−1+411yescharacteristic
155Dic97—21no—
156Dic97—21no—
157Dic97—21no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161C2×C97#53711yescharacteristic
162C2×C97#57711yescharacteristic
163C2×C97#57811yescharacteristic
164D97—41no—
165D97—81no—
166#3258—971nomaximal, Sylow
167C97#325811yescharacteristic, complemented, Sylow
168#542—971no—
169#541—971no—
170#479—971no—
171#478—971no—
172#477—971no—
173#475—971no—
174#470—971no—
175#468—971no—
176#467—971no—
177#465—971no—
178#464—971no—
179#464—971no—
180#462D9711yescharacteristic, complemented
181#461—971no—
182#424—971no—
183C2×Q16C2×D9711yescharacteristic
184C2×Q16C2×D9711yescharacteristic
185C2×SD16—971no—
186C2×SD16—971no—
187C2×M16—971no—
188C22×C8—971no—
189C4⋊Q8—971no—
190C4⋊Q8C2×D9711yescharacteristic
191#126—1941no—
192#126—1941no—
193#126—1941no—
194#125—971no—
195#125—1941no—
196#125—1941no—
197#124—971no—
198#124—1941no—
199#123—1941no—
200C22⋊Q8—971no—
201C22⋊Q8—971no—
202C22⋊Q8—971no—
203C22⋊Q8—1941no—
204#121—971no—
205C4×Q8—971no—
206C4×Q8—971no—
207C4×Q8C2×D9711yescharacteristic
208C4×D4—971no—
209#117—971no—
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8—971no—
215C4⋊C8—971no—
216C4⋊C8—971no—
217C4⋊C8C2×D9711yescharacteristic
218#103—971no—
219#103—971no—
220#103—971no—
221#103—971no—
222#103C2×D9711yescharacteristic
223#103C2×D9711yescharacteristic
224#102—971no—
225#102—971no—
226C2×Q8—1941no—
227C2×Q8C22×D9711yescharacteristic
228C2×Q8C22×D9711yescharacteristic
229C2×Q8C22×D9711yescharacteristic
230C2×D4—971no—
231C2×D4—1941no—
232C22×C4—971no—
233C22×C4—1941no—
234Q16—41no—
235Q16—41no—
236SD16—3881no—
237SD16—3881no—
238M16—3881no—
239C2×C8—971no—
240C2×C8—971no—
241C2×C8—3881no—
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⋊C4—1941no—
260C4⋊C4C22×D9711yescharacteristic
261C22⋊C4—971no—
262C22⋊C4—971no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4—1941no—
268C22⋊C4—1941no—
269C22⋊C4—1941no—
270C42—21no—
271C42—971no—
272C42—1941no—
273C42—1941no—
274C42C22×D9711yescharacteristic
275C23—971no—
276C23—1941no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—7761no—
284D4—7761no—
285D4—1941no—
286D4—1941no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—7761no—
292C2×C4—971no—
293C2×C4—971no—
294C2×C4—971no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4D4×D9711yescharacteristic
301C2×C4D4×D9711yescharacteristic
302C2×C4C23×D9711yescharacteristic
303C8—21no—
304C8—21no—
305C8—1941no—
306C8—1941no—
307C22—3881no—
308C22—3881no—
309C22—3881no—
310C22—1941no—
311C22—1941no—
312C22—1941no—
313C22C2×D4×D9711yescharacteristic
314C4—21no—
315C4—21no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—3881no—
321C4—3881no—
322C4—3881no—
323C4—1941no—
324C4—1941no—
325C4—1941no—
326C4#1569085511yescharacteristic
327C4#1569085711yescharacteristic
328C2—7761no—
329C2—3881no—
330C2#2026131411yescharacteristic
331C2#2026154411yescharacteristic
332C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#462×C97C2
socleC22×C97C22×D4
prime radical#15690785C22
hypercenter#462D97
hypocenterC97#3258
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #20261346C2semidirect
3. #20261042C2semidirect
4. #20261036C2semidirect
5. #20261028C2semidirect
6. #20261018C2semidirect
7. #20260990C2semidirect
9. #20260973C2semidirect
11. #20260959C2semidirect
12. #20260959C2semidirect
13. #20260937C2semidirect
14. #20260760C2semidirect
15. #462×C97C2semidirect
167. C97#3258semidirect
180. #462D97semidirect

Maximal subgroups

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

subgroupindex
#202613572
#202613462
#202610422
#202610362
#202610282
#202610182
#202609902
#202609842
#202609732
#202609672
#202609592
#202609372
#202607602
#462×C972
#325897