Finite Group Databasebeta

← #22570156

Subgroups — #22570156

Order 8576.

410 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×D67C211yesmaximal, characteristic
2#117×D67C211yesmaximal, characteristic
3#16863738C211yesmaximal, characteristic, complemented
4#16863737C211yesmaximal, characteristic, complemented
5#16863734C211yesmaximal, characteristic, complemented
6#16863733C211yesmaximal, characteristic, complemented
7#16863730C211yesmaximal, characteristic, complemented
8#16863728C211yesmaximal, characteristic, complemented
9#16863725C211yesmaximal, characteristic, complemented
10#16863437C211yesmaximal, characteristic, complemented
11#16863435C211yesmaximal, characteristic, complemented
12#16863403C211yesmaximal, characteristic, complemented
13#16863403C211yesmaximal, characteristic, complemented
14#16863358C211yesmaximal, characteristic
15#485×C67C211yesmaximal, characteristic, complemented
16C2×Q8×D67C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×#11761611C2211yescharacteristic
20#14524023C2211yescharacteristic
21#14524022C2211yescharacteristic
22#14524021—21no—
23#14524020—21no—
24#14524019C2211yescharacteristic
25#14524002C2211yescharacteristic
26#14524001C2211yescharacteristic
27#14524001C2211yescharacteristic
28#14524000—21no—
29#14523999—21no—
30C4⋊C4×D67—21no—
31C4⋊C4×D67C2211yescharacteristic
32C4⋊C4×D67C2211yescharacteristic
33C4⋊C4×D67C2211yescharacteristic
34C22⋊C4×D67—21no—
35C22⋊C4×D67—21no—
36C42×D67—21no—
37#14523977C2211yescharacteristic, complemented
38#14523976C2211yescharacteristic, complemented
39#14523976C2211yescharacteristic, complemented
40#14523975C2211yescharacteristic, complemented
41#14523974C2211yescharacteristic, complemented
42#14523974C2211yescharacteristic, complemented
43#14523968C2211yescharacteristic, complemented
44#14523968C2211yescharacteristic, complemented
45#14523967C2211yescharacteristic, complemented
46#14523967C2211yescharacteristic, complemented
47#14523966C2211yescharacteristic, complemented
48#14523966C2211yescharacteristic, complemented
49C22⋊(C67⋊C8)C2211yescharacteristic
50(C2×D67)⋊C8C2211yescharacteristic
51(C2×D67)⋊C8C2211yescharacteristic
52C22⋊Q8×C67C2211yescharacteristic
53#117×C67C2211yescharacteristic
54#107×C67C2211yescharacteristic, complemented
55#106×C67C2211yescharacteristic, complemented
56#103×C67C2211yescharacteristic, complemented
57#103×C67C2211yescharacteristic, complemented
58C22⋊C8×C67C2211yescharacteristic
59C23×D67D411yescharacteristic
60C2×Dic134—21no—
61C2×Dic134C2311yescharacteristic
62Q8×D67—22no—
63Q8×D67—22no—
64C22×Dic67D411yescharacteristic
65C2×C4×D67—22no—
66C2×C4×D67—22no—
67C2×C4×D67—21no—
68C2×C4×D67—21no—
69C2×C4×D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67D411yescharacteristic
72C2×C4×D67D411yescharacteristic
73C2×C4×D67C2311yescharacteristic
74C2×C4×D67C2311yescharacteristic
75#11761611—22no—
76#11761611—22no—
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C4⋊Dic67—21no—
80C4⋊Dic67C2311yescharacteristic
81C4⋊Dic67C2311yescharacteristic
82C4⋊Dic67C2311yescharacteristic
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86Dic67⋊C4—21no—
87Dic67⋊C4—21no—
88C22⋊Dic67—21no—
89C22⋊Dic67—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93(C2×D67)⋊C4—21no—
94C4×Dic67—21no—
95C4×Dic67—21no—
96C4×Dic67—21no—
97C2×Q8×C67C2311yescharacteristic
98C22×C4×C67C2311yescharacteristic
99C2×C8×C67C2311yescharacteristic
100C2×C8×C67C2311yescharacteristic
101C4⋊C4×C67—21no—
102C4⋊C4×C67C2311yescharacteristic
103C4⋊C4×C67C2311yescharacteristic
104C4⋊C4×C67C2311yescharacteristic
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C42×C67—21no—
108C23×C67C2×D411yescharacteristic
109C22×D67—22no—
110C22×D67—22no—
111C22×D67—41no—
112C22×D67C2×D411yescharacteristic
113C22×D67C2×D411yescharacteristic
114Q8×C67—21no—
115Q8×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67C2×D411yescharacteristic
123C2×C4×C67C2411yescharacteristic
124C2×Dic67—22no—
125C2×Dic67—21no—
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67C2×D411yescharacteristic
130C2×Dic67C2×D411yescharacteristic
131C4×D67—22no—
132C4×D67—21no—
133C4×D67—21no—
134C4×D67—22no—
135C4×D67—42no—
136C4×D67—42no—
137C4×D67—42no—
138C4×D67—42no—
139C4×D67#4112yes—
140C4×D67#4112yes—
141Dic134—21no—
142Dic134—21no—
143Dic134—42no—
144C8×C67—21no—
145C8×C67—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C22×C67—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67C22×D411yescharacteristic
152C2×D67—21no—
153C2×D67—21no—
154C2×D67—21no—
155C2×D67—21no—
156C2×D67—44no—
157C2×D67#12312yes—
158C2×D67#13712yes—
159C4×C67—21no—
160C4×C67—41no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—41no—
164C4×C67C2×#4111yescharacteristic
165C4×C67C2×#4111yescharacteristic
166Dic67—21no—
167Dic67—41no—
168Dic67—41no—
169Dic67—41no—
170Dic67—41no—
171Dic67#12312yes—
172C2×C67—41no—
173C2×C67C2×#12311yescharacteristic
174C2×C67C2×#13711yescharacteristic
175C2×C67#57511yescharacteristic
176D67—22no—
177D67—22no—
178#3095—671nomaximal, Sylow
179C67#309511yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—671no—
181C2×#117—671no—
182#485—671no—
183#485—671no—
184#485—671no—
185#485D6711yescharacteristic, complemented
186#484—671no—
187#484—671no—
188#483—671no—
189#483—671no—
190#428—671no—
191#428—671no—
192#419—671no—
193#419—671no—
194#407—671no—
195C22×Q8—671no—
196C23×C4—671no—
197C2×M16—671no—
198C2×M16—671no—
199C22⋊Q8—671no—
200C22⋊Q8—671no—
201C22⋊Q8—671no—
202C22⋊Q8—1341no—
203C22⋊Q8—1341no—
204C22⋊Q8C2×D6711yescharacteristic
205#117—671no—
206#117—671no—
207#117—671no—
208#117—1341no—
209#117—1341no—
210#117C2×D6711yescharacteristic
211C2×C4⋊C4—671no—
212C2×C4⋊C4—671no—
213C2×C4⋊C4—671no—
214C2×C4⋊C4—1341no—
215C2×C22⋊C4—1341no—
216C2×C22⋊C4—1341no—
217C2×C42—1341no—
218#107—671no—
219#107—671no—
220#107—671no—
221#107C2×D6711yescharacteristic, complemented
222#106—671no—
223#106—671no—
224#106—671no—
225#106C2×D6711yescharacteristic, complemented
226#103—671no—
227#103—671no—
228#103—671no—
229#103—671no—
230#103—671no—
231#103—671no—
232#103C2×D6711yescharacteristic, complemented
233#103C2×D6711yescharacteristic, complemented
234C22⋊C8—671no—
235C22⋊C8—671no—
236C22⋊C8—671no—
237C22⋊C8C2×D6711yescharacteristic
238C24—671no—
239C2×Q8—671no—
240C2×Q8—1341no—
241C2×Q8—1342no—
242C2×Q8—1342no—
243C2×Q8C22×D6711yescharacteristic
244C22×C4—671no—
245C22×C4—671no—
246C22×C4—671no—
247C22×C4—671no—
248C22×C4—671no—
249C22×C4—1341no—
250C22×C4—1341no—
251C22×C4—1341no—
252C22×C4—1341no—
253C22×C4—1342no—
254C22×C4—1342no—
255C22×C4C22×D6711yescharacteristic
256M16—1342no—
257M16—1342no—
258C2×C8—671no—
259C2×C8—671no—
260C2×C8C22×D6711yescharacteristic
261C2×C8C22×D6711yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—671no—
264C4⋊C4—671no—
265C4⋊C4—671no—
266C4⋊C4—1341no—
267C4⋊C4—1341no—
268C4⋊C4—1341no—
269C4⋊C4—1341no—
270C4⋊C4—1341no—
271C4⋊C4—1341no—
272C4⋊C4C22×D6711yescharacteristic
273C4⋊C4C22×D6711yescharacteristic
274C4⋊C4C22×D6711yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—1341no—
278C22⋊C4—1341no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C42—21no—
284C42—1341no—
285C42—1341no—
286C42—1341no—
287C23—2681no—
288C23—671no—
289C23—671no—
290C23—1342no—
291C23—1342no—
292C23D4×D6711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2682no—
296Q8—1341no—
297Q8—1341no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—2682no—
305C2×C4—2682no—
306C2×C4—2682no—
307C2×C4—2682no—
308C2×C4—671no—
309C2×C4—672no—
310C2×C4—672no—
311C2×C4—671no—
312C2×C4—1342no—
313C2×C4—1342no—
314C2×C4—1341no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1341no—
318C2×C4—1342no—
319C2×C4—1341no—
320C2×C4—1341no—
321C2×C4D4×D6711yescharacteristic
322C2×C4C23×D6711yescharacteristic
323C8—21no—
324C8—21no—
325C8—1341no—
326C8—1341no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—2684no—
331C22—672no—
332C22—672no—
333C22—1341no—
334C22—1341no—
335C22—1341no—
336C22—1341no—
337C22C2×D4×D6711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—2681no—
344C4—2681no—
345C4—2681no—
346C4—2681no—
347C4—672no—
348C4—1341no—
349C4#41×D6711yescharacteristic
350C4#41×D6711yescharacteristic
351C2—41no—
352C2—1342no—
353C2—1342no—
354C2#123×D6711yescharacteristic
355C2#137×D6711yescharacteristic
356C2#1686419311yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#485×C67C2
socleC22×C67C22×D4
hypercenter#485D67
hypocenterC67#3095
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
3. #16863738C2semidirect
4. #16863737C2semidirect
5. #16863734C2semidirect
6. #16863733C2semidirect
7. #16863730C2semidirect
8. #16863728C2semidirect
9. #16863725C2semidirect
10. #16863437C2semidirect
11. #16863435C2semidirect
12. #16863403C2semidirect
13. #16863403C2semidirect
15. #485×C67C2semidirect
37. #14523977C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523974C22semidirect
42. #14523974C22semidirect
43. #14523968C22semidirect
44. #14523968C22semidirect
45. #14523967C22semidirect
46. #14523967C22semidirect
47. #14523966C22semidirect
48. #14523966C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. #103×C67C22semidirect
57. #103×C67C22semidirect
179. C67#3095semidirect
185. #485D67semidirect
221. #107C2×D67semidirect
225. #106C2×D67semidirect
232. #103C2×D67semidirect
233. #103C2×D67semidirect