Finite Group Databasebeta

← #27153617

Subgroups — #27153617

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#20261394C211yesmaximal, characteristic
2#20261092C211yesmaximal, characteristic, complemented
3#20261076C211yesmaximal, characteristic, complemented
4#20261048C211yesmaximal, characteristic, complemented
5#20261039C211yesmaximal, characteristic, complemented
6#20260974C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260957C211yesmaximal, characteristic, complemented
9#20260895C211yesmaximal, characteristic, complemented
10#20260890C211yesmaximal, characteristic, complemented
11C4×#13095501C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260729C211yesmaximal, characteristic, complemented
14C2×#15690733C211yesmaximal, characteristic, complemented
15#552×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18C2×#13095501C2211yescharacteristic
19C2×#13095501C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22C22×#11715895C2211yescharacteristic
23#15690798C2211yescharacteristic
24#15690796—21no—
25#15690789C2211yescharacteristic
26#15690788—21no—
27#15690784C2211yescharacteristic
28#15690782—21no—
29C4×Dic194C2211yescharacteristic
30#15690770—21no—
31#15690743C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690734C2211yescharacteristic
36#15690733C2211yescharacteristic
37#15690733C2211yescharacteristic
38#15690733C2211yescharacteristic
39#15690733C2211yescharacteristic
40#15690733C2211yescharacteristic
41#15690733C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#126×C97—21no—
49#125×C97—21no—
50#125×C97C2211yescharacteristic
51#123×C97—21no—
52C22⋊Q8×C97C2211yescharacteristic
53C22⋊Q8×C97C2211yescharacteristic
54C4×Q8×C97C2211yescharacteristic
55C4×D4×C97C2211yescharacteristic
56#117×C97—21no—
57#117×C97C2211yescharacteristic
58C2×C4⋊C4×C97C2211yescharacteristic
59#13095511—22no—
60#13095511—41no—
61C2×Dic194—22no—
62C2×Dic194—21no—
63C2×Dic194—41no—
64C2×Dic194C2311yescharacteristic
65C2×Dic194C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C22×Dic97—21no—
70C2×C4×D97—21no—
71#13095501—22no—
72#13095501—41no—
73#13095498—41no—
74C2×#11715895—41no—
75C2×#11715895—41no—
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C4⋊Dic97C2311yescharacteristic
81#13095490—21no—
82#13095490—21no—
83C22⋊Dic97—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C2×Q8×C97C2311yescharacteristic
89C2×D4×C97D411yescharacteristic
90C22×C4×C97—21no—
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97—21no—
97C4⋊C4×C97—21no—
98C4⋊C4×C97—21no—
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97D411yescharacteristic
107C22⋊C4×C97D411yescharacteristic
108C42×C97—21no—
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C23×C97C2×D411yescharacteristic
113C22×D97—21no—
114Q8×C97—21no—
115Q8×C97—21no—
116D4×C97—22no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—42no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2411yescharacteristic
131C22⋊D97—42no—
132C22⋊D97—42no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—42no—
137D388—21no—
138D388—21no—
139C4×D97—42no—
140Dic194—21no—
141Dic194—21no—
142Dic194—21no—
143Dic194—42no—
144Dic194#4112yes—
145#11715895—21no—
146#11715895—21no—
147#11715895—21no—
148#11715895—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×C97—41no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97C2×#4111yescharacteristic
168C4×C97H(5,𝔽2)11yescharacteristic
169Dic97—22no—
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#552D9711yescharacteristic, complemented
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×D4—971no—
193#416—971no—
194C2×#103—971no—
195C2×#41—971no—
196C22×Q8—971no—
197C2×Q16—971no—
198C2×Q16—971no—
199C2×SD16—971no—
200C22×C8—971no—
201C22×C8—971no—
202#126—21no—
203#125—21no—
204#125C2×D9711yescharacteristic
205#124—971no—
206#124—1941no—
207#123—21no—
208C22⋊Q8—971no—
209C22⋊Q8—1941no—
210C22⋊Q8C2×D9711yescharacteristic
211C22⋊Q8C2×D9711yescharacteristic
212#121—971no—
213#121—1941no—
214C4×Q8—971no—
215C4×Q8C2×D9711yescharacteristic
216C4×D4—1941no—
217C4×D4C2×D9711yescharacteristic
218#117—21no—
219#117C2×D9711yescharacteristic
220C2×C4⋊C4C2×D9711yescharacteristic
221#107—971no—
222#107—971no—
223#106—971no—
224C4⋊C8—971no—
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⋊C8—971no—
236C22⋊C8—971no—
237C4×C8—971no—
238#41—3881no—
239#41—1942no—
240C2×Q8—971no—
241C2×Q8—971no—
242C2×Q8—3881no—
243C2×Q8—1942no—
244C2×Q8—1941no—
245C2×Q8C22×D9711yescharacteristic
246C2×D4—971no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4C22⋊D9711yescharacteristic
250C22×C4—21no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4C22×D9711yescharacteristic
254C22×C4C22×D9711yescharacteristic
255Q16—3881no—
256Q16—1942no—
257SD16—3881no—
258C2×C8—971no—
259C2×C8—971no—
260C2×C8—971no—
261C2×C8—971no—
262C2×C8—3881no—
263C2×C8—3881no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—971no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22⋊D9711yescharacteristic
274C4⋊C4C22×D9711yescharacteristic
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4C22⋊D9711yescharacteristic
285C22⋊C4C22⋊D9711yescharacteristic
286C42—21no—
287C42—21no—
288C42—1941no—
289C42C22×D9711yescharacteristic
290C23—1941no—
291C23C2×C22⋊D9711yescharacteristic
292C23C2×C22⋊D9711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—972no—
296Q8—3882no—
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—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—3882no—
315C2×C4—3882no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4C2×C22⋊D9711yescharacteristic
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C2×C22⋊D9711yescharacteristic
322C2×C4C2×C22⋊D9711yescharacteristic
323C2×C4C23×D9711yescharacteristic
324C8—1941no—
325C8—1941no—
326C8—1941no—
327C8—1941no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—3881no—
334C22—3881no—
335C22—3881no—
336C22#1569083012yes—
337C22C22×C22⋊D9711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3881no—
347C4—3881no—
348C4—1942no—
349C4#41×D9711yescharacteristic
350C4#1569085611yescharacteristic
351C2—22no—
352C2—41no—
353C2—7761no—
354C2#2026137311yescharacteristic
355C2C2×#1569083011yescharacteristic
356C2#2026155011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#552×C97C2
socleC22×C97C22×D4
prime radical#20261394C2
hypercenter#552D97
hypocenterC97#3300
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20261092C2semidirect
3. #20261076C2semidirect
4. #20261048C2semidirect
5. #20261039C2semidirect
6. #20260974C2semidirect
7. #20260966C2semidirect
8. #20260957C2semidirect
9. #20260895C2semidirect
10. #20260890C2semidirect
11. C4×#13095501C2semidirect
12. D4×#11715895C2semidirect
13. #20260729C2semidirect
14. C2×#15690733C2semidirect
15. #552×C97C2semidirect
179. C97#3300semidirect
180. #552D97semidirect

Maximal subgroups

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

subgroupindex
#202613942
#202610922
#202610762
#202610482
#202610392
#202609742
#202609662
#202609572
#202608952
#202608902
C4×#130955012
D4×#117158952
#202607292
C2×#156907332
#552×C972
#330097