Finite Group Databasebeta

← #27153625

Subgroups — #27153625

Order 12416.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261425C211yesmaximal, characteristic, complemented
2#20261394C211yesmaximal, characteristic
3#20261090C211yesmaximal, characteristic, complemented
4#20261077C211yesmaximal, characteristic, complemented
5#20261049C211yesmaximal, characteristic, complemented
6#20261043C211yesmaximal, characteristic, complemented
7#20260976C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260958C211yesmaximal, characteristic, complemented
10#20260899C211yesmaximal, characteristic, complemented
11#20260893C211yesmaximal, characteristic, complemented
12C4×Dic388C211yesmaximal, characteristic, complemented
13#20260732C211yesmaximal, characteristic, complemented
14C2×#15690735C211yesmaximal, characteristic, complemented
15C8×D4×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18C2×Dic388C2211yescharacteristic
19C2×Dic388C2211yescharacteristic
20C2×#13095500C2211yescharacteristic
21#15690803—21no—
22#15690802C2211yescharacteristic
23#15690800—21no—
24#15690798C2211yescharacteristic
25#15690796—21no—
26#15690791—21no—
27#15690789C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690789C2211yescharacteristic
30#15690788—21no—
31#15690784C2211yescharacteristic
32#15690782—21no—
33C4×Dic194C2211yescharacteristic
34C4×Dic194C2211yescharacteristic
35#15690770—21no—
36#15690769C2211yescharacteristic
37#15690766—21no—
38C2×C4⋊Dic97C2211yescharacteristic
39#15690744C2211yescharacteristic
40#15690744C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690735C2211yescharacteristic
45#15690735C2211yescharacteristic
46#15690735C2211yescharacteristic
47#15690735C2211yescharacteristic
48#15690735C2211yescharacteristic
49#15690732C2211yescharacteristic
50#15690732C2211yescharacteristic
51#15690732C2211yescharacteristic
52C22×C8×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55C4⋊C8×C97C2211yescharacteristic
56C22⋊C8×C97C2211yescharacteristic
57C22⋊C8×C97C2211yescharacteristic
58C4×C8×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—41no—
61C2×Dic194—22no—
62C2×Dic194—21no—
63C2×Dic194—41no—
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66C2×Dic194C2311yescharacteristic
67C2×D388C2311yescharacteristic
68C2×C22⋊D97—21no—
69C2×C22⋊D97—21no—
70C22×Dic97—21no—
71C22×Dic97—21no—
72C2×C4×D97—21no—
73Dic388—22no—
74Dic388—41no—
75#13095500—41no—
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97—21no—
90C22⋊Dic97—21no—
91C22⋊Dic97—21no—
92#13095488—21no—
93#13095488—21no—
94#13095488—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C4×Dic97—21no—
98C2×D4×C97D411yescharacteristic
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C2×C8×C97—41no—
102C2×C8×C97—41no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C2×C8×C97C2311yescharacteristic
106C2×C8×C97C2311yescharacteristic
107C4⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C22⋊C4×C97D411yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114D4×C97—22no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97C2×D411yescharacteristic
120C2×C4×C97C2×D411yescharacteristic
121C2×C4×C97C2×D411yescharacteristic
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2411yescharacteristic
124C22⋊D97—42no—
125C22⋊D97—42no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—42no—
134C2×Dic97—42no—
135D388—21no—
136D388—21no—
137C4×D97—42no—
138Dic194—21no—
139Dic194—21no—
140Dic194—21no—
141Dic194—21no—
142Dic194—21no—
143Dic194—42no—
144Dic194#4112yes—
145C8×C97—21no—
146C8×C97—21no—
147C8×C97—21no—
148C8×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97#13512yes—
155C22×C97C22×D411yescharacteristic
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97C2×#4111yescharacteristic
164C4×C97H(5,𝔽2)11yescharacteristic
165Dic97—22no—
166Dic97—41no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97—41no—
171Dic97—41no—
172C2×C97—22no—
173C2×C97—41no—
174C2×C97#54611yescharacteristic
175C2×C97C2×#13511yescharacteristic
176C2×C97#57811yescharacteristic
177D97—81no—
178#3300—971nomaximal, Sylow
179C97#330011yescharacteristic, complemented, Sylow
180#552—971no—
181#548—971no—
182#488—971no—
183#485—971no—
184#480—971no—
185#479—971no—
186#467—971no—
187#465—971no—
188#464—971no—
189#452—971no—
190#451—971no—
191C4×Q16—971no—
192C8×D4D9711yescharacteristic, complemented
193#416—971no—
194C2×#103—971no—
195C2×#41—971no—
196C22×Q8—971no—
197C2×Q16—971no—
198C2×Q16—971no—
199C2×SD16—971no—
200C22×C8C2×D9711yescharacteristic
201C22×C8C2×D9711yescharacteristic
202#126—1941no—
203#125—971no—
204#125—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—971no—
209C22⋊Q8—971no—
210C22⋊Q8—971no—
211C22⋊Q8—1941no—
212#121—971no—
213#121—1941no—
214C4×Q8—971no—
215C4×Q8—971no—
216C4×D4—1941no—
217C4×D4C2×D9711yescharacteristic
218#117—971no—
219#117—1941no—
220C2×C4⋊C4—971no—
221#107—971no—
222#107—971no—
223#106—971no—
224C4⋊C8C2×D9711yescharacteristic
225#103—971no—
226#103—971no—
227#103—971no—
228#103—971no—
229#103—971no—
230#103—971no—
231#103—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235C22⋊C8C2×D9711yescharacteristic
236C22⋊C8C2×D9711yescharacteristic
237C4×C8C2×D9711yescharacteristic
238#41—3881no—
239#41—1942no—
240C2×Q8—971no—
241C2×Q8—971no—
242C2×Q8—971no—
243C2×Q8—3881no—
244C2×Q8—1942no—
245C2×Q8—1941no—
246C2×D4—971no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4D38811yescharacteristic
250C22×C4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4C22×D9711yescharacteristic
254C22×C4C22×D9711yescharacteristic
255Q16—3881no—
256Q16—1942no—
257SD16—3881no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8C22×D9711yescharacteristic
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C2×C8C22×D9711yescharacteristic
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4—1941no—
276C4⋊C4D38811yescharacteristic
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4D38811yescharacteristic
285C22⋊C4D38811yescharacteristic
286C42—1941no—
287C42—1941no—
288C42—1941no—
289C42C22×D9711yescharacteristic
290C23—1941no—
291C23C2×D38811yescharacteristic
292C23C2×D38811yescharacteristic
293Q8—972no—
294Q8—3882no—
295Q8—1941no—
296Q8—1941no—
297Q8—1941no—
298Q8—1941no—
299Q8—1941no—
300D4—22no—
301D4—3882no—
302D4—3882no—
303D4—1941no—
304D4—1941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—3882no—
310C2×C4—3882no—
311C2×C4—3882no—
312C2×C4—1941no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4C2×D38811yescharacteristic
320C2×C4C2×D38811yescharacteristic
321C2×C4C2×D38811yescharacteristic
322C2×C4C2×D38811yescharacteristic
323C2×C4C23×D9711yescharacteristic
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—3881no—
334C22—3881no—
335C22—3881no—
336C22#1569083212yes—
337C22C22×D38811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—3881no—
343C4—3881no—
344C4—3881no—
345C4—3881no—
346C4—3881no—
347C4—3881no—
348C4—1942no—
349C4#41×D9711yescharacteristic
350C4#1569085511yescharacteristic
351C2—22no—
352C2—41no—
353C2—7761no—
354C2#2026138011yescharacteristic
355C2C2×#1569083211yescharacteristic
356C2#2026155211yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC8×D4×C97C2
socleC22×C97C22×D4
prime radical#20261394C2
hypercenterC8×D4D97
hypocenterC97#3300
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20261425C2semidirect
3. #20261090C2semidirect
4. #20261077C2semidirect
5. #20261049C2semidirect
6. #20261043C2semidirect
7. #20260976C2semidirect
8. #20260964C2semidirect
9. #20260958C2semidirect
10. #20260899C2semidirect
11. #20260893C2semidirect
12. C4×Dic388C2semidirect
13. #20260732C2semidirect
14. C2×#15690735C2semidirect
15. C8×D4×C97C2semidirect
179. C97#3300semidirect
192. C8×D4D97semidirect

Maximal subgroups

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

subgroupindex
#202614252
#202613942
#202610902
#202610772
#202610492
#202610432
#202609762
#202609642
#202609582
#202608992
#202608932
C4×Dic3882
#202607322
C2×#156907352
C8×D4×C972
#330097