Finite Group Databasebeta

← #27152290

Subgroups — #27152290

Order 12416.

412 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261287C211yesmaximal, characteristic, complemented
2#20261280C211yesmaximal, characteristic
3#20261132C211yesmaximal, characteristic, complemented
4#20261106C211yesmaximal, characteristic, complemented
5#20261084C211yesmaximal, characteristic, complemented
6#20261080C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260962C211yesmaximal, characteristic, complemented
10#20260826C211yesmaximal, characteristic, complemented
11SD16×Dic97C211yesmaximal, characteristic, complemented
12#20260822C211yesmaximal, characteristic, complemented
13#20260820C211yesmaximal, characteristic, complemented
14C2×C8×Dic97C211yesmaximal, characteristic, complemented
15#465×C97C211yesmaximal, characteristic, complemented
16C2×#13095510—21no—
17C2×#13095500C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21#15690809C2211yescharacteristic
22#15690809C2211yescharacteristic
23#15690808C2211yescharacteristic
24#15690806C2211yescharacteristic
25#15690800—21no—
26#15690796—21no—
27#15690789C2211yescharacteristic
28#15690787—21no—
29#15690784C2211yescharacteristic
30#15690782—21no—
31Q8×Dic97C2211yescharacteristic
32#15690775—21no—
33#15690775C2211yescharacteristic
34D4×Dic97C2211yescharacteristic
35#15690772C2211yescharacteristic
36#15690770—21no—
37#15690766—21no—
38C2×C4×Dic97C2211yescharacteristic
39#15690742C2211yescharacteristic
40#15690741C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690735C2211yescharacteristic
43#15690734C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48C8×Dic97C2211yescharacteristic
49C8×Dic97C2211yescharacteristic
50C8×Dic97C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×SD16×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56#106×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59#13095510—44no—
60C2×Dic194—21no—
61C2×Dic194—21no—
62C2×Dic194C2311yescharacteristic
63C2×D388C2311yescharacteristic
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67C22×Dic97—21no—
68C22×Dic97D411yescharacteristic
69C22×Dic97D411yescharacteristic
70C2×C4×D97—21no—
71#13095500—22no—
72#13095499—22no—
73#13095498—22no—
74C2×#11715895—22no—
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C4⋊Dic97—21no—
78C4⋊Dic97C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88#13095488—21no—
89#13095488—21no—
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97D411yescharacteristic
93C4×Dic97D411yescharacteristic
94C4×Dic97C2311yescharacteristic
95C4×Dic97C2311yescharacteristic
96C2×Q8×C97C2311yescharacteristic
97C2×D4×C97—21no—
98C2×D4×C97C2311yescharacteristic
99C22×C4×C97C2311yescharacteristic
100SD16×C97—22no—
101C2×C8×C97—22no—
102C2×C8×C97C2311yescharacteristic
103C2×C8×C97C2311yescharacteristic
104C4⋊C4×C97—21no—
105C4⋊C4×C97C2311yescharacteristic
106C4⋊C4×C97C2311yescharacteristic
107C22⋊C4×C97—21no—
108C22⋊C4×C97—21no—
109C23×C97—21no—
110C23×C97C2×D411yescharacteristic
111C22×D97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114D4×C97—21no—
115D4×C97—21no—
116D4×C97—42no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97C2×D411yescharacteristic
123C2×C4×C97C2411yescharacteristic
124C22⋊D97—42no—
125C22⋊D97—42no—
126C22⋊D97—42no—
127C2×Dic97—22no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—42no—
132C2×Dic97SD1614yescomplemented
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137D388—21no—
138D388—21no—
139C4×D97—42no—
140Dic194—21no—
141Dic194—21no—
142Dic194—42no—
143Dic194—42no—
144C8×C97—21no—
145C8×C97#4112yescomplemented
146#11715895—21no—
147#11715895#4112yescomplemented
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—41no—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97C2×SD1612yes—
154C22×C97C22×D411yescharacteristic
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C4×C97—21no—
159C4×C97—41no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97C2×#4111yescharacteristic
163C4×C97C2×#4111yescharacteristic
164Dic97—21no—
165Dic97—41no—
166Dic97—41no—
167Dic97—41no—
168Dic97C2×SD1612yes—
169Dic97C2×SD1612yes—
170Dic97#13512yes—
171C2×C97—22no—
172C2×C97—81no—
173C2×C97#53211yescharacteristic
174C2×C97C22×SD1611yescharacteristic
175C2×C97C2×#13511yescharacteristic
176D97—81no—
177#3167—971nomaximal, Sylow
178C97#316711yescharacteristic, complemented, Sylow
179#533—971no—
180#532—971no—
181#498—971no—
182#492—971no—
183#487—971no—
184#486—971no—
185#465—971no—
186#465—971no—
187#465—971no—
188#465D9711yescharacteristic, complemented
189C4×SD16—971no—
190C4×SD16—971no—
191C4×SD16—971no—
192C4×SD16—971no—
193C2×C4×C8—971no—
194C2×#41—1941no—
195C2×SD16—971no—
196C2×SD16—971no—
197C2×SD16—971no—
198C2×SD16C2×D9711yescharacteristic
199C22×C8—971no—
200C22×C8C2×D9711yescharacteristic
201C4⋊Q8—971no—
202C4⋊Q8—971no—
203C4⋊Q8—971no—
204#127—971no—
205#125—1941no—
206#124—1941no—
207C22⋊Q8—971no—
208C22⋊Q8—1941no—
209C22⋊Q8C2×D9711yescharacteristic
210#121—971no—
211#121—1941no—
212#121C2×D9711yescharacteristic
213C4×Q8—971no—
214C4×Q8—971no—
215C4×Q8—1941no—
216C4×D4—971no—
217C4×D4—971no—
218C4×D4—1941no—
219#117—1941no—
220C2×C42—971no—
221#106—971no—
222#106—971no—
223#106—971no—
224#106C2×D9711yescharacteristic
225#103—971no—
226#103—971no—
227#103—971no—
228#103C2×D9711yescharacteristic
229#102—971no—
230#102—971no—
231#102—971no—
232#102C2×D9711yescharacteristic
233C4×C8—971no—
234C4×C8—971no—
235C4×C8—971no—
236C4×C8—971no—
237#41—3884no—
238C2×Q8—971no—
239C2×Q8—1941no—
240C2×Q8—1941no—
241C2×Q8C22×D9711yescharacteristic
242C2×D4—21no—
243C2×D4—971no—
244C2×D4—1941no—
245C2×D4—1941no—
246C2×D4—1941no—
247C2×D4C22×D9711yescharacteristic
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—1941no—
251C22×C4—1941no—
252C22×C4C22×D9711yescharacteristic
253SD16—22no—
254SD16—1942no—
255SD16—1942no—
256SD16—1942no—
257C2×C8—22no—
258C2×C8—971no—
259C2×C8—971no—
260C2×C8—1942no—
261C2×C8C22×D9711yescharacteristic
262C2×C8C22×D9711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—971no—
265C4⋊C4—971no—
266C4⋊C4—1941no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22×D9711yescharacteristic
274C4⋊C4C22×D9711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C42—971no—
282C42—971no—
283C42—971no—
284C42—971no—
285C42—1941no—
286C42—1941no—
287C23—21no—
288C23—1941no—
289C23D4×D9711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3882no—
293Q8—3882no—
294Q8—1941no—
295Q8—1941no—
296D4—21no—
297D4—21no—
298D4—42no—
299D4—3882no—
300D4—3882no—
301D4—3882no—
302D4—1941no—
303D4—1941no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—974no—
310C2×C4—971no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—971no—
314C2×C4—3882no—
315C2×C4—3882no—
316C2×C4—1942no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4D4×D9711yescharacteristic
321C2×C4C23×D9711yescharacteristic
322C8—21no—
323C8—972no—
324C8—1941no—
325C8#1309551012yescomplemented
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—3881no—
332C22—3881no—
333C22—3881no—
334C22SD16×D9712yes—
335C22C2×D4×D9711yescharacteristic
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—972no—
341C4—972no—
342C4—972no—
343C4—3881no—
344C4—3881no—
345C4—3881no—
346C4—1941no—
347C4#41×D9711yescharacteristic
348C4C2×#1309551011yescharacteristic
349C2—22no—
350C2—81no—
351C2—7761no—
352C2#2026128011yescharacteristic
353C2C2×SD16×D9711yescharacteristic
354C2#135×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#465×C97C2
socleC22×C97C22×D4
prime radical#20261280C2
hypercenter#465D97
hypocenterC97#3167
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #20261287C2semidirect
3. #20261132C2semidirect
4. #20261106C2semidirect
5. #20261084C2semidirect
6. #20261080C2semidirect
7. #20260966C2semidirect
8. #20260964C2semidirect
9. #20260962C2semidirect
10. #20260826C2semidirect
11. SD16×Dic97C2semidirect
12. #20260822C2semidirect
13. #20260820C2semidirect
14. C2×C8×Dic97C2semidirect
15. #465×C97C2semidirect
132. C2×Dic97SD16semidirect
145. C8×C97#41semidirect
147. #11715895#41semidirect
178. C97#3167semidirect
188. #465D97semidirect
325. C8#13095510semidirect

Maximal subgroups

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

subgroupindex
#202612872
#202612802
#202611322
#202611062
#202610842
#202610802
#202609662
#202609642
#202609622
#202608262
SD16×Dic972
#202608222
#202608202
C2×C8×Dic972
#465×C972
#316797