Finite Group Databasebeta

← #485×D97 #27151624

Subgroups — #485×D97 #27151624

Order 12416.

416 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
1C22⋊Q8×D97C211yesmaximal, characteristic
2#117×D97C211yesmaximal, characteristic
3#20261079C211yesmaximal, characteristic, complemented
4#20261078C211yesmaximal, characteristic, complemented
5#20261077C211yesmaximal, characteristic, complemented
6#20261076C211yesmaximal, characteristic, complemented
7#20261075C211yesmaximal, characteristic, complemented
8#20261074C211yesmaximal, characteristic, complemented
9#20261073C211yesmaximal, characteristic, complemented
10#107×D97C211yesmaximal, characteristic, complemented
11#106×D97C211yesmaximal, characteristic, complemented
12#103×D97C211yesmaximal, characteristic, complemented
13#103×D97C211yesmaximal, characteristic, complemented
14C22⋊C8×D97C211yesmaximal, characteristic
15#485×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2211yescharacteristic
17C22×C4×D97C2211yescharacteristic
18C2×C8×D97C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690790C2211yescharacteristic
21#15690789C2211yescharacteristic
22#15690788—21no—
23#15690787—21no—
24#15690786C2211yescharacteristic
25#15690769C2211yescharacteristic
26#15690768C2211yescharacteristic
27#15690768C2211yescharacteristic
28#15690767—21no—
29#15690766—21no—
30C4⋊C4×D97—21no—
31C4⋊C4×D97C2211yescharacteristic
32C4⋊C4×D97C2211yescharacteristic
33C4⋊C4×D97C2211yescharacteristic
34C22⋊C4×D97—21no—
35C22⋊C4×D97—21no—
36C42×D97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690742C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690735C2211yescharacteristic, complemented
45#15690734C2211yescharacteristic, complemented
46#15690734C2211yescharacteristic, complemented
47#15690733C2211yescharacteristic, complemented
48#15690733C2211yescharacteristic, complemented
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51#15690722C2211yescharacteristic
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×D97D411yescharacteristic
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62Q8×D97—22no—
63Q8×D97—22no—
64C22×Dic97D411yescharacteristic
65C2×C4×D97—22no—
66C2×C4×D97—22no—
67C2×C4×D97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97D411yescharacteristic
72C2×C4×D97D411yescharacteristic
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C8×D97—22no—
78C8×D97—22no—
79C4⋊Dic97—21no—
80C4⋊Dic97C2311yescharacteristic
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83#13095490—21no—
84#13095490—21no—
85#13095490—21no—
86#13095490—21no—
87#13095490—21no—
88C22⋊Dic97—21no—
89C22⋊Dic97—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93#13095488—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
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—22no—
110C22×D97—22no—
111C22×D97—22no—
112C22×D97C2×D411yescharacteristic
113C22×D97C2×D411yescharacteristic
114Q8×C97—21no—
115Q8×C97—21no—
116C2×C4×C97—21no—
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
124C2×Dic97—22no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131C4×D97—22no—
132C4×D97—21no—
133C4×D97—21no—
134C4×D97—22no—
135C4×D97—42no—
136C4×D97—42no—
137C4×D97—42no—
138C4×D97—42no—
139C4×D97#4112yes—
140C4×D97#4112yes—
141Dic194—21no—
142Dic194—21no—
143Dic194—42no—
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—21no—
153C2×D97—21no—
154C2×D97—21no—
155C2×D97—44no—
156C2×D97#12312yes—
157C2×D97#13512yes—
158C2×D97#13712yes—
159C4×C97—21no—
160C4×C97—41no—
161C4×C97—41no—
162C4×C97—41no—
163C4×C97—41no—
164C4×C97C2×#4111yescharacteristic
165C4×C97C2×#4111yescharacteristic
166Dic97—21no—
167Dic97—41no—
168Dic97—41no—
169Dic97—41no—
170Dic97—41no—
171Dic97#12312yes—
172C2×C97—41no—
173C2×C97C2×#12311yescharacteristic
174C2×C97C2×#13511yescharacteristic
175C2×C97C2×#13711yescharacteristic
176D97—41no—
177D97#48514yesdirect product factor
178C2×#485—971nomaximal, Sylow
179C97C2×#48511yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—971no—
181C2×#117—971no—
182#485—971no—
183#485—971no—
184#485—971no—
185#485—971no—
186#485—971no—
187#485—971no—
188#485—971no—
189#485D9711yescharacteristic, direct product factor
190C2×#107—971no—
191C2×#106—971no—
192C2×#103—971no—
193C2×#103—971no—
194C2×C22⋊C8—971no—
195C22×Q8—971no—
196C23×C4—971no—
197C22×C8—971no—
198C22×C8—971no—
199C22⋊Q8—971no—
200C22⋊Q8—971no—
201C22⋊Q8—971no—
202C22⋊Q8—1941no—
203C22⋊Q8—1941no—
204C22⋊Q8C2×D9711yescharacteristic
205#117—971no—
206#117—971no—
207#117—971no—
208#117—1941no—
209#117—1941no—
210#117C2×D9711yescharacteristic
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—1941no—
215C2×C22⋊C4—1941no—
216C2×C22⋊C4—1941no—
217C2×C42—1941no—
218#107—971no—
219#107—971no—
220#107—971no—
221#107C2×D9711yescharacteristic, complemented
222#106—971no—
223#106—971no—
224#106—971no—
225#106C2×D9711yescharacteristic, complemented
226#103—971no—
227#103—971no—
228#103—971no—
229#103—971no—
230#103—971no—
231#103—971no—
232#103C2×D9711yescharacteristic, complemented
233#103C2×D9711yescharacteristic, complemented
234C22⋊C8—971no—
235C22⋊C8—971no—
236C22⋊C8—971no—
237C22⋊C8C2×D9711yescharacteristic
238C24—971no—
239C2×Q8—971no—
240C2×Q8—1941no—
241C2×Q8—1942no—
242C2×Q8—1942no—
243C2×Q8C22×D9711yescharacteristic
244C22×C4—971no—
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—1942no—
250C22×C4—1942no—
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4C22×D9711yescharacteristic
256C2×C8—971no—
257C2×C8—971no—
258C2×C8—1942no—
259C2×C8—1942no—
260C2×C8C22×D9711yescharacteristic
261C2×C8C22×D9711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—971no—
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⋊C4C22×D9711yescharacteristic
273C4⋊C4C22×D9711yescharacteristic
274C4⋊C4C22×D9711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C42—21no—
284C42—1941no—
285C42—1941no—
286C42—1941no—
287C23—971no—
288C23—971no—
289C23—1942no—
290C23—1942no—
291C23—1942no—
292C23D4×D9711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—3882no—
296Q8—1941no—
297Q8—1941no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—971no—
305C2×C4—972no—
306C2×C4—972no—
307C2×C4—971no—
308C2×C4—3882no—
309C2×C4—3882no—
310C2×C4—3882no—
311C2×C4—3882no—
312C2×C4—1942no—
313C2×C4—1942no—
314C2×C4—1942no—
315C2×C4—1941no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4D4×D9711yescharacteristic
322C2×C4C23×D9711yescharacteristic
323C8—21no—
324C8—21no—
325C8—1941no—
326C8—1941no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—972no—
331C22—972no—
332C22—972no—
333C22—3884no—
334C22—1941no—
335C22—1941no—
336C22—1941no—
337C22C2×D4×D9711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—972no—
344C4—3881no—
345C4—3881no—
346C4—3881no—
347C4—3881no—
348C4—1941no—
349C4#41×D9711yescharacteristic
350C4#41×D9711yescharacteristic
351C2—41no—
352C2—974no—
353C2—3881no—
354C2#123×D9711yescharacteristic
355C2#135×D9711yescharacteristic
356C2#137×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#485×C97C2
socleC22×C97C22×D4
prime radicalC23×D97D4
hypercenter#485D97
hypocenterC97C2×#485
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
3. #20261079C2semidirect
4. #20261078C2semidirect
5. #20261077C2semidirect
6. #20261076C2semidirect
7. #20261075C2semidirect
8. #20261074C2semidirect
9. #20261073C2semidirect
10. #107×D97C2semidirect
11. #106×D97C2semidirect
12. #103×D97C2semidirect
13. #103×D97C2semidirect
15. #485×C97C2semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690743C22semidirect
40. #15690742C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
43. #15690735C22semidirect
44. #15690735C22semidirect
45. #15690734C22semidirect
46. #15690734C22semidirect
47. #15690733C22semidirect
48. #15690733C22semidirect
54. #107×C97C22semidirect
55. #106×C97C22semidirect
56. #103×C97C22semidirect
57. #103×C97C22semidirect
177. D97#485direct (N × H)
179. C97C2×#485semidirect
189. #485D97direct (N × H)
221. #107C2×D97semidirect
225. #106C2×D97semidirect
232. #103C2×D97semidirect
233. #103C2×D97semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D972
#117×D972
#202610792
#202610782
#202610772
#202610762
#202610752
#202610742
#202610732
#107×D972
#106×D972
#103×D972
C22⋊C8×D972
#485×C972
C2×#48597