Finite Group Databasebeta

← #27151923

Subgroups — #27151923

Order 12416.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261358C211yesmaximal, characteristic, complemented
2#20261209C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic, complemented
4#20261024C211yesmaximal, characteristic, complemented
5#20261014C211yesmaximal, characteristic, complemented
6#20260953C211yesmaximal, characteristic, complemented
7#20260953C211yesmaximal, characteristic, complemented
8#20260885C211yesmaximal, characteristic, complemented
9#20260871C211yesmaximal, characteristic
10#20260846C211yesmaximal, characteristic, complemented
11#20260838C211yesmaximal, characteristic, complemented
12#20260820C211yesmaximal, characteristic, complemented
13#20260817C211yesmaximal, characteristic, complemented
14#20260793C211yesmaximal, characteristic, complemented
15#485×C97C211yesmaximal, characteristic, complemented
16C22×D388C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095498C2211yescharacteristic
19C2×D776C2211yescharacteristic
20C2×D776C2211yescharacteristic
21#15690809C2211yescharacteristic, complemented
22#15690805—21no—
23#15690799C2211yescharacteristic, complemented
24#15690796—21no—
25#15690792—21no—
26#15690790—21no—
27#15690784C2211yescharacteristic
28Q8×Dic97C2211yescharacteristic, complemented
29C4×D388—21no—
30C4×D388—21no—
31#15690772C2211yescharacteristic
32#15690772C2211yescharacteristic
33#15690772C2211yescharacteristic
34#15690772C2211yescharacteristic
35#15690772C2211yescharacteristic, complemented
36#15690769C2211yescharacteristic
37C22⋊C4×D97—21no—
38C22⋊C4×D97—21no—
39#15690743C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic
44#15690732C2211yescharacteristic
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic, complemented
48#15690730C2211yescharacteristic, complemented
49C22⋊#11715895C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22⋊Q8×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54#107×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56#103×C97C2211yescharacteristic, complemented
57#103×C97C2211yescharacteristic, complemented
58C22⋊C8×C97C2211yescharacteristic
59C23×D97—21no—
60C2×Dic194—21no—
61C2×D388—21no—
62C2×D388—22no—
63C2×D388—22no—
64C2×D388C2311yescharacteristic
65C2×D388C2311yescharacteristic
66C2×D388C2311yescharacteristic
67C2×C22⋊D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97—21no—
72C2×C4×D97—21no—
73#13095498—22no—
74#13095498—22no—
75D776—22no—
76D776—22no—
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C4⋊Dic97—21no—
80C4⋊Dic97D411yescharacteristic
81C4⋊Dic97D411yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84C22⋊Dic97D411yescharacteristic
85C22⋊Dic97D411yescharacteristic
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94C4×Dic97—21no—
95C4×Dic97C2311yescharacteristic
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C2×C8×C97C2311yescharacteristic
101C4⋊C4×C97—21no—
102C4⋊C4×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C42×C97—21no—
108C23×C97C2×D411yescharacteristic
109C22×D97—21no—
110C22×D97—21no—
111C22×D97—24no—
112C22×D97—21no—
113C22×D97—81no—
114C22×D97—81no—
115Q8×C97—21no—
116Q8×C97—21no—
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×C97C2×D411yescharacteristic
124C2×C4×C97C2411yescharacteristic
125C22⋊D97—81no—
126C2×Dic97—21no—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131D388—24no—
132D388—21no—
133D388—21no—
134D388—21no—
135D388—21no—
136D388#4112yes—
137D388#4112yes—
138C4×D97—81no—
139C4×D97—81no—
140C4×D97—81no—
141C4×D97—81no—
142C4×D97—81no—
143Dic194—81no—
144C8×C97—21no—
145C8×C97—21no—
146#11715895—21no—
147#11715895—21no—
148C22×C97—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97C22×D411yescharacteristic
152C2×D97—22no—
153C2×D97—22no—
154C2×D97—41no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—84no—
162C4×C97—21no—
163C4×C97—41no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97C2×#4111yescharacteristic
168C4×C97C2×#4111yescharacteristic
169Dic97—21no—
170Dic97—21no—
171Dic97—21no—
172Dic97—21no—
173Dic97—41no—
174C2×C97—41no—
175C2×C97#52511yescharacteristic
176C2×C97#57611yescharacteristic
177C2×C97#57711yescharacteristic
178D97—42no—
179D97—42no—
180D97—81no—
181#3118—971nomaximal, Sylow
182C97#311811yescharacteristic, complemented, Sylow
183#542—971no—
184#518—971no—
185#485D9711yescharacteristic, complemented
186#482—971no—
187#476—971no—
188#474—971no—
189#463—971no—
190#463—971no—
191#450—971no—
192#447—971no—
193#442—971no—
194#440—971no—
195C4×SD16—971no—
196C4×D8—971no—
197#433—971no—
198C22×D4—971no—
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8—971no—
204#126—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—1941no—
209C22⋊Q8C2×D9711yescharacteristic
210#121—971no—
211C4×Q8—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4—1941no—
218C4×D4—1941no—
219#117—971no—
220#117C2×D9711yescharacteristic
221C2×C22⋊C4—1941no—
222C2×C22⋊C4—1941no—
223#107—971no—
224#107C2×D9711yescharacteristic, complemented
225#106—971no—
226#106C2×D9711yescharacteristic, complemented
227C4⋊C8—971no—
228C4⋊C8—971no—
229#103C2×D9711yescharacteristic, complemented
230#103C2×D9711yescharacteristic, complemented
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237C22⋊C8—971no—
238C22⋊C8C2×D9711yescharacteristic
239#97—971no—
240C4×C8—971no—
241C24—1941no—
242C2×Q8—1941no—
243C2×Q8C22×D9711yescharacteristic
244C2×D4—971no—
245C2×D4—971no—
246C2×D4—971no—
247C2×D4—1942no—
248C2×D4—1942no—
249C2×D4—1941no—
250C2×D4—1941no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4C22×D9711yescharacteristic
257SD16—1942no—
258SD16—1942no—
259D8—1942no—
260D8—1942no—
261C2×C8—971no—
262C2×C8—971no—
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4C22×D9711yescharacteristic
272C4⋊C4C22×D9711yescharacteristic
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—971no—
277C22⋊C4—971no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C42—21no—
287C42—971no—
288C42—971no—
289C42—1941no—
290C23—7761no—
291C23—7761no—
292C23—1944no—
293C23—1941no—
294C23—1941no—
295C23—1941no—
296C23D4×D9711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—7761no—
300D4—7761no—
301D4—972no—
302D4—972no—
303D4—1944no—
304D4—1941no—
305D4—1941no—
306D4—1941no—
307D4—1941no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—7761no—
315C2×C4—7761no—
316C2×C4—7761no—
317C2×C4—7761no—
318C2×C4—7761no—
319C2×C4—971no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—971no—
323C2×C4—1941no—
324C2×C4D4×D9711yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—21no—
327C8—21no—
328C8—1941no—
329C8—1941no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—7764no—
334C22—3881no—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—3881no—
341C22—1942no—
342C22—1942no—
343C22C2×D4×D9711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—3881no—
350C4—1941no—
351C4—1941no—
352C4—1941no—
353C4—1941no—
354C4#41×D9711yescharacteristic
355C4#41×D9711yescharacteristic
356C2—41no—
357C2—7761no—
358C2—3882no—
359C2—3882no—
360C2#2026124311yescharacteristic
361C2#2026153711yescharacteristic
362C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#485×C97C2
socleC22×C97C22×D4
prime radical#20260871C2
hypercenter#485D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #20261358C2semidirect
2. #20261209C2semidirect
3. #20261059C2semidirect
4. #20261024C2semidirect
5. #20261014C2semidirect
6. #20260953C2semidirect
7. #20260953C2semidirect
8. #20260885C2semidirect
10. #20260846C2semidirect
11. #20260838C2semidirect
12. #20260820C2semidirect
13. #20260817C2semidirect
14. #20260793C2semidirect
15. #485×C97C2semidirect
21. #15690809C22semidirect
23. #15690799C22semidirect
28. Q8×Dic97C22semidirect
35. #15690772C22semidirect
39. #15690743C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690739C22semidirect
47. #15690730C22semidirect
48. #15690730C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. #103×C97C22semidirect
57. #103×C97C22semidirect
182. C97#3118semidirect
185. #485D97semidirect
224. #107C2×D97semidirect
226. #106C2×D97semidirect
229. #103C2×D97semidirect
230. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613582
#202612092
#202610592
#202610242
#202610142
#202609532
#202608852
#202608712
#202608462
#202608382
#202608202
#202608172
#202607932
#485×C972
#311897