Finite Group Databasebeta

← #22563370

Subgroups — #22563370

Order 8576.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524019C211yesmaximal, characteristic
2#16863303C211yesmaximal, characteristic, complemented
3#16863293C211yesmaximal, characteristic, complemented
4#16863291C211yesmaximal, characteristic, complemented
5#16863288C211yesmaximal, characteristic
6#16863287C211yesmaximal, characteristic
7#16863286C211yesmaximal, characteristic
8#16863285C211yesmaximal, characteristic
9#16863281C211yesmaximal, characteristic, complemented
10#16863280C211yesmaximal, characteristic, complemented
11#16863279C211yesmaximal, characteristic, complemented
12#16863278C211yesmaximal, characteristic, complemented
13#16863258C211yesmaximal, characteristic
14#16863255C211yesmaximal, characteristic
15#391×C67C211yesmaximal, characteristic, complemented
16C22×C4×D67C2211yescharacteristic
17#14524019C418yescomplemented
18C2×C4⋊Dic67C2211yescharacteristic
19C2×C4⋊Dic67C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C2×(C2×D67)⋊C4C2211yescharacteristic
28C2×(C2×D67)⋊C4C2211yescharacteristic
29C2×(C2×D67)⋊C4C2211yescharacteristic
30C2×(C2×D67)⋊C4C2211yescharacteristic
31C2×(C2×D67)⋊C4C2211yescharacteristic
32C2×C4×Dic67C2211yescharacteristic
33C2×C4×Dic67C2211yescharacteristic
34C2×C4×Dic67C2211yescharacteristic
35C67⋊#95C2211yescharacteristic
36C67⋊#95C2211yescharacteristic
37C67⋊#95C2211yescharacteristic
38C67⋊#95C2211yescharacteristic
39C67⋊#95C2211yescharacteristic
40C67⋊#95C2211yescharacteristic
41C67⋊#95C2211yescharacteristic
42C67⋊#95C2211yescharacteristic
43C67⋊#95C2211yescharacteristic
44C67⋊#95C2211yescharacteristic
45C22×Q8×C67C2211yescharacteristic
46C2×C4⋊C4×C67C2211yescharacteristic
47C2×C42×C67C2211yescharacteristic
48#95×C67C2211yescharacteristic
49#95×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C23×D67C2311yescharacteristic
53C22×Dic67C2311yescharacteristic
54C22×Dic67C2311yescharacteristic
55C22×Dic67C2311yescharacteristic
56C22×Dic67C2311yescharacteristic
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C2×C4×D67—41no—
61C2×C4×D67—41no—
62C2×C4×D67C2×C414yes—
63C4⋊Dic67—41no—
64C4⋊Dic67C2×C414yes—
65Dic67⋊C4—41no—
66Dic67⋊C4—41no—
67Dic67⋊C4—41no—
68Dic67⋊C4—41no—
69Dic67⋊C4C2×C414yes—
70Dic67⋊C4C2×C414yes—
71(C2×D67)⋊C4—41no—
72(C2×D67)⋊C4—41no—
73(C2×D67)⋊C4—41no—
74(C2×D67)⋊C4—41no—
75(C2×D67)⋊C4C2×C414yes—
76(C2×D67)⋊C4C2×C414yes—
77C4×Dic67—41no—
78C4×Dic67—41no—
79C4×Dic67—41no—
80C2×Q8×C67—41no—
81C2×Q8×C67—41no—
82C2×Q8×C67C2×C414yes—
83C22×C4×C67C2311yescharacteristic
84C22×C4×C67C2311yescharacteristic
85C22×C4×C67C2311yescharacteristic
86C22×C4×C67C2311yescharacteristic
87C22×C4×C67C2311yescharacteristic
88C22×C4×C67C2311yescharacteristic
89C22×C4×C67C2311yescharacteristic
90C4⋊C4×C67—41no—
91C42×C67—41no—
92C23×C67C2411yescharacteristic
93C22×D67—21no—
94C22×D67—21no—
95C22×D67—21no—
96C22×D67—21no—
97C22×D67—21no—
98C22×D67—21no—
99C22×D67C22×C412yes—
100Q8×C67—44no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67C22×C412yes—
120C2×C4×C67C22×C412yes—
121C2×C4×C67C22×C412yes—
122C2×Dic67—21no—
123C2×Dic67—21no—
124C2×Dic67—21no—
125C2×Dic67—21no—
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—21no—
138C2×Dic67—21no—
139C2×Dic67—21no—
140C2×Dic67C22×C412yes—
141C2×Dic67C22×C412yes—
142C2×Dic67C22×C412yes—
143C4×D67—44no—
144C22×C67C23×C411yescharacteristic
145C22×C67H(5,𝔽2)11yescharacteristic
146C22×C67H(5,𝔽2)11yescharacteristic
147C22×C67H(5,𝔽2)11yescharacteristic
148C22×C672−1+411yescharacteristic
149C22×C672−1+411yescharacteristic
150C22×C672−1+411yescharacteristic
151C2×D67—22no—
152C2×D67—22no—
153C2×D67—22no—
154C2×D67—41no—
155C2×D67—41no—
156C2×D67—41no—
157C2×D67—41no—
158C4×C67—22no—
159C4×C67—22no—
160C4×C67—22no—
161C4×C67—41no—
162C4×C67—41no—
163C4×C67—41no—
164C4×C67—41no—
165Dic67—22no—
166Dic67—22no—
167Dic67—22no—
168Dic67—41no—
169Dic67—41no—
170Dic67—41no—
171Dic67—41no—
172C2×C67#51811yescharacteristic
173C2×C67#51911yescharacteristic
174C2×C67#52011yescharacteristic
175C2×C67#56211yescharacteristic
176C2×C67#56211yescharacteristic
177C2×C67#56311yescharacteristic
178C2×C67#56311yescharacteristic
179D67—42no—
180#2378—671nomaximal, Sylow
181C67#237811yescharacteristic, complemented, Sylow
182C2×C22⋊Q8—671no—
183#391—671no—
184#391D6711yescharacteristic, complemented
185#389—671no—
186#389—671no—
187#388—671no—
188#388—671no—
189#388—671no—
190#388—671no—
191#387—671no—
192#387—671no—
193#387—671no—
194#387—671no—
195#381—671no—
196#380—671no—
197C22×Q8C2×D6711yescharacteristic
198C23×C4—671no—
199C22⋊Q8—678no—
200C2×C4⋊C4—671no—
201C2×C4⋊C4—671no—
202C2×C4⋊C4—671no—
203C2×C4⋊C4—671no—
204C2×C4⋊C4—671no—
205C2×C4⋊C4—671no—
206C2×C4⋊C4—671no—
207C2×C4⋊C4—671no—
208C2×C4⋊C4C2×D6711yescharacteristic
209C2×C22⋊C4—671no—
210C2×C22⋊C4—671no—
211C2×C22⋊C4—671no—
212C2×C22⋊C4—671no—
213C2×C22⋊C4—671no—
214C2×C22⋊C4—671no—
215C2×C42—671no—
216C2×C42—671no—
217C2×C42—671no—
218C2×C42C2×D6711yescharacteristic
219#95—671no—
220#95—671no—
221#95—671no—
222#95—671no—
223#95—671no—
224#95—671no—
225#95—671no—
226#95—671no—
227#95—671no—
228#95—671no—
229#95C2×D6711yescharacteristic
230#95C2×D6711yescharacteristic
231#95C2×D6711yescharacteristic
232#95C2×D6711yescharacteristic
233C24—671no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8C4×D6714yes—
237C22×C4—2681no—
238C22×C4—2681no—
239C22×C4—674no—
240C22×C4—671no—
241C22×C4—671no—
242C22×C4—671no—
243C22×C4—671no—
244C22×C4—671no—
245C22×C4—671no—
246C22×C4—671no—
247C22×C4C22×D6711yescharacteristic
248C22×C4C22×D6711yescharacteristic
249C22×C4C22×D6711yescharacteristic
250C22×C4C22×D6711yescharacteristic
251C22×C4C22×D6711yescharacteristic
252C22×C4C22×D6711yescharacteristic
253C22×C4C22×D6711yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—2681no—
256C4⋊C4—2681no—
257C4⋊C4—2681no—
258C4⋊C4—2681no—
259C4⋊C4—2681no—
260C4⋊C4—674no—
261C4⋊C4—674no—
262C4⋊C4—674no—
263C22⋊C4—2681no—
264C22⋊C4—2681no—
265C22⋊C4—2681no—
266C22⋊C4—2681no—
267C22⋊C4—674no—
268C22⋊C4—674no—
269C42—41no—
270C42—2681no—
271C42—2681no—
272C42—2681no—
273C23—672no—
274C23—1341no—
275C23—1341no—
276C23—1341no—
277C23—1341no—
278C23—1341no—
279C23—1341no—
280C23C23×D6711yescharacteristic
281Q8—44no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—2684no—
301C2×C4—672no—
302C2×C4—672no—
303C2×C4—672no—
304C2×C4—1341no—
305C2×C4—1341no—
306C2×C4—1341no—
307C2×C4—1341no—
308C2×C4—1341no—
309C2×C4—1341no—
310C2×C4—1341no—
311C2×C4—1341no—
312C2×C4—1341no—
313C2×C4—1341no—
314C2×C4—1341no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1341no—
318C2×C4—1341no—
319C2×C4—1341no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4C2×C4×D6712yes—
323C2×C4C2×C4×D6712yes—
324C2×C4C2×C4×D6712yes—
325C22—2681no—
326C22—2681no—
327C22—2681no—
328C22—2681no—
329C22—1342no—
330C22—1342no—
331C22—1342no—
332C22C22×C4×D6711yescharacteristic
333C22#1452408811yescharacteristic
334C22#1452408811yescharacteristic
335C22#1452408911yescharacteristic
336C22#1452409011yescharacteristic
337C22#1452409111yescharacteristic
338C22#1452409111yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—2681no—
347C4—2681no—
348C4—2681no—
349C4—2681no—
350C4—1342no—
351C4—1342no—
352C4—1342no—
353C2—2682no—
354C2#1686386911yescharacteristic
355C2#1686387211yescharacteristic
356C2#1686387811yescharacteristic
357C2#1686414011yescharacteristic
358C2#1686414211yescharacteristic
359C2#1686414511yescharacteristic
360C2#1686414711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#391×C67C2
socleC23×C67C24
hypercenter#391D67
hypocenterC67#2378
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. #16863303C2semidirect
3. #16863293C2semidirect
4. #16863291C2semidirect
9. #16863281C2semidirect
10. #16863280C2semidirect
11. #16863279C2semidirect
12. #16863278C2semidirect
15. #391×C67C2semidirect
17. #14524019C4semidirect
181. C67#2378semidirect
184. #391D67semidirect