Finite Group Databasebeta

← #27153164

Subgroups — #27153164

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#20261356C211yesmaximal, characteristic
2#20261347C211yesmaximal, characteristic, complemented
3#20261043C211yesmaximal, characteristic, complemented
4#20261036C211yesmaximal, characteristic, complemented
5#20261031C211yesmaximal, characteristic, complemented
6#20261021C211yesmaximal, characteristic, complemented
7#20260993C211yesmaximal, characteristic, complemented
8#20260981C211yesmaximal, characteristic
9#20260976C211yesmaximal, characteristic, complemented
10#20260964C211yesmaximal, characteristic
11#20260958C211yesmaximal, characteristic, complemented
12#20260958C211yesmaximal, characteristic, complemented
13#20260946C211yesmaximal, characteristic, complemented
14#20260941C211yesmaximal, characteristic, complemented
15#424×C97C211yesmaximal, characteristic, complemented
16C2×Dic388C2211yescharacteristic
17C2×Dic388C2211yescharacteristic
18C2×#13095500C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20#15690810C2211yescharacteristic
21#15690810C2211yescharacteristic
22#15690805—21no—
23#15690803—21no—
24#15690803—21no—
25#15690802C2211yescharacteristic
26#15690800—21no—
27#15690800—21no—
28#15690798C2211yescharacteristic
29#15690796—21no—
30#15690792—21no—
31#15690789C2211yescharacteristic
32#15690789C2211yescharacteristic
33#15690789C2211yescharacteristic
34#15690788—21no—
35#15690784C2211yescharacteristic
36C4×Dic194C2211yescharacteristic
37C4×Dic194C2211yescharacteristic
38C4×Dic194C2211yescharacteristic
39C4×D388C2211yescharacteristic
40#15690744C2211yescharacteristic
41#15690744C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690742C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690735C2211yescharacteristic
49#15690735C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54#117×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56C4⋊C8×C97C2211yescharacteristic
57C4⋊C8×C97C2211yescharacteristic
58C4⋊C8×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194C2311yescharacteristic
61C2×Dic194C2311yescharacteristic
62C2×Dic194C2311yescharacteristic
63C2×D388C2311yescharacteristic
64C2×C22⋊D97—21no—
65C2×C4×D97—21no—
66Dic388—41no—
67Dic388—41no—
68#13095500—41no—
69#13095500—41no—
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97—21no—
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92C4×Dic97—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C22×C4×C97C2311yescharacteristic
96M16×C97—41no—
97C2×C8×C97—41no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C4⋊C4×C97D411yescharacteristic
103C4⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C22⋊C4×C97D411yescharacteristic
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97C2×D411yescharacteristic
113C2×C4×C97C2×D411yescharacteristic
114C2×C4×C97C2×D411yescharacteristic
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2×D411yescharacteristic
117C2×C4×C97C2411yescharacteristic
118C22⋊D97—81no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126D388—21no—
127D388—21no—
128C4×D97—81no—
129Dic194—21no—
130Dic194—21no—
131Dic194—21no—
132Dic194—21no—
133Dic194—21no—
134Dic194—21no—
135Dic194—81no—
136C8×C97—21no—
137C8×C97—21no—
138C8×C97—21no—
139C8×C97—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97—21no—
143C22×C97C22×D411yescharacteristic
144C2×D97—41no—
145C2×D97—41no—
146C2×D97—41no—
147C4×C97—21no—
148C4×C97—21no—
149C4×C97—21no—
150C4×C97—21no—
151C4×C97—21no—
152C4×C97H(5,𝔽2)11yescharacteristic
153C4×C972−1+411yescharacteristic
154Dic97—41no—
155Dic97—41no—
156Dic97—41no—
157Dic97—41no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161C2×C97—41no—
162C2×C97#53711yescharacteristic
163C2×C97#57711yescharacteristic
164C2×C97#57811yescharacteristic
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#462—971no—
181#461—971no—
182#424D9711yescharacteristic, complemented
183C2×Q16—971no—
184C2×Q16—971no—
185C2×SD16—971no—
186C2×SD16—971no—
187C2×M16C2×D9711yescharacteristic
188C22×C8C2×D9711yescharacteristic
189C4⋊Q8—971no—
190C4⋊Q8—971no—
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×Q8—971no—
208C4×D4—971no—
209#117C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#106—971no—
213#106—971no—
214C4⋊C8C2×D9711yescharacteristic
215C4⋊C8C2×D9711yescharacteristic
216C4⋊C8C2×D9711yescharacteristic
217C4⋊C8C2×D9711yescharacteristic
218#103—971no—
219#103—971no—
220#103—971no—
221#103—971no—
222#103—971no—
223#103—971no—
224#102—971no—
225#102—971no—
226C2×Q8—971no—
227C2×Q8—971no—
228C2×Q8—971no—
229C2×Q8—1941no—
230C2×D4—971no—
231C2×D4—1941no—
232C22×C4—1941no—
233C22×C4C22×D9711yescharacteristic
234Q16—3881no—
235Q16—3881no—
236SD16—3881no—
237SD16—3881no—
238M16—41no—
239C2×C8—41no—
240C2×C8C22×D9711yescharacteristic
241C2×C8C22×D9711yescharacteristic
242C2×C8C22×D9711yescharacteristic
243C2×C8C22×D9711yescharacteristic
244C4⋊C4—971no—
245C4⋊C4—971no—
246C4⋊C4—971no—
247C4⋊C4—971no—
248C4⋊C4—1941no—
249C4⋊C4—1941no—
250C4⋊C4—1941no—
251C4⋊C4—1941no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4D38811yescharacteristic
260C4⋊C4D38811yescharacteristic
261C22⋊C4—1941no—
262C22⋊C4—1941no—
263C22⋊C4—1941no—
264C22⋊C4—1941no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4—1941no—
268C22⋊C4D38811yescharacteristic
269C22⋊C4D38811yescharacteristic
270C42—1941no—
271C42—1941no—
272C42—1941no—
273C42C22×D9711yescharacteristic
274C42C22×D9711yescharacteristic
275C23—1941no—
276C23C2×D38811yescharacteristic
277Q8—7761no—
278Q8—1941no—
279Q8—1941no—
280Q8—1941no—
281Q8—1941no—
282Q8—1941no—
283Q8—1941no—
284D4—7761no—
285D4—1941no—
286D4—1941no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—7761no—
290C2×C4—1941no—
291C2×C4—1941no—
292C2×C4—1941no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4C2×D38811yescharacteristic
298C2×C4C2×D38811yescharacteristic
299C2×C4C2×D38811yescharacteristic
300C2×C4C2×D38811yescharacteristic
301C2×C4C2×D38811yescharacteristic
302C2×C4C23×D9711yescharacteristic
303C8—21no—
304C8—21no—
305C8—21no—
306C8—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—3881no—
311C22—3881no—
312C22—3881no—
313C22C22×D38811yescharacteristic
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—3881no—
320C4—3881no—
321C4—3881no—
322C4—3881no—
323C4—3881no—
324C4—3881no—
325C4—3881no—
326C4#1569085511yescharacteristic
327C4#1569085811yescharacteristic
328C2—41no—
329C2—7761no—
330C2#2026131611yescharacteristic
331C2#2026154711yescharacteristic
332C2#2026155211yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#424×C97C2
socleC22×C97C22×D4
prime radical#15690784C22
hypercenter#424D97
hypocenterC97#3258
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. #20261347C2semidirect
3. #20261043C2semidirect
4. #20261036C2semidirect
5. #20261031C2semidirect
6. #20261021C2semidirect
7. #20260993C2semidirect
9. #20260976C2semidirect
11. #20260958C2semidirect
12. #20260958C2semidirect
13. #20260946C2semidirect
14. #20260941C2semidirect
15. #424×C97C2semidirect
167. C97#3258semidirect
182. #424D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202613472
#202610432
#202610362
#202610312
#202610212
#202609932
#202609812
#202609762
#202609642
#202609582
#202609462
#202609412
#424×C972
#325897