Finite Group Databasebeta

← #22570141

Subgroups — #22570141

Order 8576.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863909C211yesmaximal, characteristic
2#117×D67C211yesmaximal, characteristic, complemented
3#16863736C211yesmaximal, characteristic, complemented
4#16863735C211yesmaximal, characteristic, complemented
5#16863734C211yesmaximal, characteristic, complemented
6#16863723C211yesmaximal, characteristic, complemented
7#16863722C211yesmaximal, characteristic, complemented
8#16863721C211yesmaximal, characteristic, complemented
9#16863720C211yesmaximal, characteristic, complemented
10#16863437C211yesmaximal, characteristic, complemented
11#16863435C211yesmaximal, characteristic, complemented
12#16863396C211yesmaximal, characteristic, complemented
13#16863396C211yesmaximal, characteristic, complemented
14#16863358C211yesmaximal, characteristic, complemented
15#485×C67C211yesmaximal, characteristic, complemented
16C2×#11761629C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×#11761611C2211yescharacteristic
20#14524028—21no—
21#14524019C2211yescharacteristic
22#14524018C2211yescharacteristic
23#14524017C2211yescharacteristic
24#14524013—21no—
25#14524005—21no—
26(C22⋊D67)⋊C4—21no—
27#14524002C2211yescharacteristic
28#14524001C2211yescharacteristic
29#14524001C2211yescharacteristic
30#14524001C2211yescharacteristic
31#14524000—21no—
32#14523999—21no—
33C4⋊C4×D67C2211yescharacteristic
34C4⋊C4×D67C2211yescharacteristic
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#14523967C2211yescharacteristic, complemented
44#14523967C2211yescharacteristic, complemented
45#14523965C2211yescharacteristic, complemented
46#14523965C2211yescharacteristic, complemented
47#14523963C2211yescharacteristic, complemented
48#14523963C2211yescharacteristic, 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
60#11761629—22no—
61#11761629—22no—
62C2×D268—21no—
63C2×D268—21no—
64C2×D268C2311yescharacteristic
65C2×C22⋊D67—21no—
66C22×Dic67D411yescharacteristic
67C2×C4×D67—22no—
68C2×C4×D67—22no—
69C2×C4×D67—21no—
70C2×C4×D67—21no—
71C2×C4×D67—21no—
72C2×C4×D67D411yescharacteristic
73C2×C4×D67D411yescharacteristic
74C2×C4×D67C2311yescharacteristic
75C2×C4×D67C2311yescharacteristic
76#11761611—22no—
77#11761611—22no—
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C4⋊Dic67C2311yescharacteristic
81C4⋊Dic67C2311yescharacteristic
82C4⋊Dic67C2311yescharacteristic
83Dic67⋊C4—21no—
84Dic67⋊C4—21no—
85Dic67⋊C4—21no—
86C22⋊Dic67—21no—
87(C2×D67)⋊C4—21no—
88(C2×D67)⋊C4—21no—
89(C2×D67)⋊C4—21no—
90(C2×D67)⋊C4—21no—
91(C2×D67)⋊C4—21no—
92(C2×D67)⋊C4—21no—
93C4×Dic67—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—21no—
111C22×D67—41no—
112C22×D67—41no—
113C22×D67C2×D411yescharacteristic
114C22×D67C2×D411yescharacteristic
115Q8×C67—21no—
116Q8×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2411yescharacteristic
125C22⋊D67—42no—
126C2×Dic67—22no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67C2×D411yescharacteristic
131C2×Dic67C2×D411yescharacteristic
132D268—21no—
133D268—21no—
134D268—42no—
135D268—42no—
136C4×D67—22no—
137C4×D67—21no—
138C4×D67—21no—
139C4×D67—22no—
140C4×D67—42no—
141C4×D67—42no—
142C4×D67—42no—
143C4×D67#4112yes—
144C4×D67#4112yes—
145C8×C67—21no—
146C8×C67—21no—
147C67⋊C8—21no—
148C67⋊C8—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67C22×D411yescharacteristic
153C2×D67—21no—
154C2×D67—21no—
155C2×D67—21no—
156C2×D67—21no—
157C2×D67—21no—
158C2×D67—44no—
159C2×D67—41no—
160C2×D67—41no—
161C2×D67—41no—
162C2×D67#12312yes—
163C4×C67—21no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67C2×#4111yescharacteristic
169C4×C67C2×#4111yescharacteristic
170Dic67—21no—
171Dic67—41no—
172Dic67—41no—
173Dic67—41no—
174Dic67#12312yes—
175C2×C67—41no—
176C2×C67C2×#12311yescharacteristic
177C2×C67#57511yescharacteristic
178C2×C67#57511yescharacteristic
179D67—22no—
180D67—41no—
181D67—81no—
182#3093—671nomaximal, Sylow
183C67#309311yescharacteristic, complemented, Sylow
184#525—671no—
185C2×#117—671no—
186#485—671no—
187#485—671no—
188#485—671no—
189#485D6711yescharacteristic, complemented
190#482—671no—
191#482—671no—
192#482—671no—
193#482—671no—
194#428—671no—
195#428—671no—
196#417—671no—
197#417—671no—
198#407—671no—
199C2×#41—671no—
200C23×C4—671no—
201C2×M16—671no—
202C2×M16—671no—
203#123—1341no—
204C22⋊Q8—671no—
205C22⋊Q8C2×D6711yescharacteristic
206#121—671no—
207#121—671no—
208#120—1341no—
209C4×D4—1341no—
210C4×D4—1341no—
211#117—671no—
212#117—671no—
213#117—671no—
214#117—671no—
215#117—1341no—
216#117—1341no—
217#117C2×D6711yescharacteristic
218C2×C4⋊C4—671no—
219C2×C4⋊C4—671no—
220C2×C22⋊C4—1341no—
221C2×C42—1341no—
222#107—671no—
223#107—671no—
224#107—671no—
225#107C2×D6711yescharacteristic, complemented
226#106—671no—
227#106—671no—
228#106—671no—
229#106C2×D6711yescharacteristic, complemented
230#103—671no—
231#103—671no—
232#103C2×D6711yescharacteristic, complemented
233#103C2×D6711yescharacteristic, complemented
234#102—671no—
235#102—671no—
236#102—671no—
237#102—671no—
238C22⋊C8—671no—
239C22⋊C8—671no—
240C22⋊C8—671no—
241C22⋊C8C2×D6711yescharacteristic
242C24—671no—
243#41—1342no—
244#41—1342no—
245C2×Q8C22×D6711yescharacteristic
246C2×D4—671no—
247C2×D4—1341no—
248C2×D4—1341no—
249C2×D4—1341no—
250C22×C4—671no—
251C22×C4—671no—
252C22×C4—671no—
253C22×C4—671no—
254C22×C4—671no—
255C22×C4—1341no—
256C22×C4—1341no—
257C22×C4—1341no—
258C22×C4—1342no—
259C22×C4—1342no—
260C22×C4C22×D6711yescharacteristic
261M16—1342no—
262M16—1342no—
263C2×C8—671no—
264C2×C8—671no—
265C2×C8C22×D6711yescharacteristic
266C2×C8C22×D6711yescharacteristic
267C4⋊C4—21no—
268C4⋊C4—671no—
269C4⋊C4—671no—
270C4⋊C4—671no—
271C4⋊C4—1341no—
272C4⋊C4—1341no—
273C4⋊C4—1341no—
274C4⋊C4C22×D6711yescharacteristic
275C4⋊C4C22×D6711yescharacteristic
276C4⋊C4C22×D6711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1341no—
280C22⋊C4—1341no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4—1341no—
285C22⋊C4—1341no—
286C42—21no—
287C42—1341no—
288C42—1341no—
289C42—1341no—
290C42—1341no—
291C23—2681no—
292C23—2681no—
293C23—671no—
294C23—671no—
295C23—1342no—
296C23—1341no—
297C23D4×D6711yescharacteristic
298Q8—21no—
299Q8—21no—
300D4—2682no—
301D4—2682no—
302D4—2682no—
303D4—1341no—
304D4—1341no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—2682no—
312C2×C4—2682no—
313C2×C4—2682no—
314C2×C4—671no—
315C2×C4—672no—
316C2×C4—672no—
317C2×C4—671no—
318C2×C4—1342no—
319C2×C4—1342no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4—1341no—
323C2×C4—1342no—
324C2×C4—1341no—
325C2×C4—1341no—
326C2×C4D4×D6711yescharacteristic
327C2×C4C23×D6711yescharacteristic
328C8—21no—
329C8—21no—
330C8—1341no—
331C8—1341no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—2684no—
336C22—2681no—
337C22—2681no—
338C22—2681no—
339C22—672no—
340C22—1341no—
341C22—1341no—
342C22—1341no—
343C22—1341no—
344C22—1341no—
345C22C2×D4×D6711yescharacteristic
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—2681no—
352C4—2681no—
353C4—2681no—
354C4—672no—
355C4—1341no—
356C4#41×D6711yescharacteristic
357C4#41×D6711yescharacteristic
358C2—41no—
359C2—2681no—
360C2—5361no—
361C2—1342no—
362C2#123×D6711yescharacteristic
363C2#1686419311yescharacteristic
364C2#1686419511yescharacteristic
subgroupG/H
centerC22C2×D4×D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#485×C67C2
socleC22×C67C22×D4
hypercenter#485D67
hypocenterC67#3093
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
2. #117×D67C2semidirect
3. #16863736C2semidirect
4. #16863735C2semidirect
5. #16863734C2semidirect
6. #16863723C2semidirect
7. #16863722C2semidirect
8. #16863721C2semidirect
9. #16863720C2semidirect
10. #16863437C2semidirect
11. #16863435C2semidirect
12. #16863396C2semidirect
13. #16863396C2semidirect
14. #16863358C2semidirect
15. #485×C67C2semidirect
37. #14523977C22semidirect
38. #14523976C22semidirect
39. #14523976C22semidirect
40. #14523975C22semidirect
41. #14523974C22semidirect
42. #14523974C22semidirect
43. #14523967C22semidirect
44. #14523967C22semidirect
45. #14523965C22semidirect
46. #14523965C22semidirect
47. #14523963C22semidirect
48. #14523963C22semidirect
54. #107×C67C22semidirect
55. #106×C67C22semidirect
56. #103×C67C22semidirect
57. #103×C67C22semidirect
183. C67#3093semidirect
189. #485D67semidirect
225. #107C2×D67semidirect
229. #106C2×D67semidirect
232. #103C2×D67semidirect
233. #103C2×D67semidirect