Finite Group Databasebeta

← #22569282

Subgroups — #22569282

Order 8576.

402 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863942C211yesmaximal, characteristic
2#16863472C211yesmaximal, characteristic, complemented
3#16863471C211yesmaximal, characteristic, complemented
4#16863470C211yesmaximal, characteristic, complemented
5#16863467C211yesmaximal, characteristic, complemented
6#16863465C211yesmaximal, characteristic, complemented
7#16863464C211yesmaximal, characteristic, complemented
8C8×C22⋊D67C211yesmaximal, characteristic, complemented
9(C22⋊D67)⋊C8C211yesmaximal, characteristic, complemented
10#16863450C211yesmaximal, characteristic, complemented
11#16863446C211yesmaximal, characteristic, complemented
12#16863363C211yesmaximal, characteristic, complemented
13#16863361C211yesmaximal, characteristic, complemented
14#16863361C211yesmaximal, characteristic, complemented
15#408×C67C211yesmaximal, characteristic, complemented
16C2×#11761628C2211yescharacteristic
17C2×#11761612C2211yescharacteristic
18C2×#11761611C2211yescharacteristic
19C2×#11761611C2211yescharacteristic
20C22×C67⋊C8C2211yescharacteristic
21C2×C8×D67C2211yescharacteristic
22C2×C8×D67C2211yescharacteristic
23#14524041C411yescharacteristic
24#14524039C411yescharacteristic
25#14524032C411yescharacteristic
26#14524030C411yescharacteristic
27#14524015C411yescharacteristic
28#14524015C411yescharacteristic
29#14524015C411yescharacteristic
30#14524015C411yescharacteristic
31C4×C22⋊D67C2211yescharacteristic
32C4×C22⋊D67C2211yescharacteristic
33C4×C22⋊D67C2211yescharacteristic
34C4×C22⋊D67C2211yescharacteristic
35C2×C4×Dic67C2211yescharacteristic
36Dic67⋊C8C2211yescharacteristic
37Dic67⋊C8C2211yescharacteristic
38Dic67⋊C8C2211yescharacteristic
39Dic67⋊C8C2211yescharacteristic
40C22⋊(C67⋊C8)C2211yescharacteristic
41C22⋊(C67⋊C8)C2211yescharacteristic
42C22⋊(C67⋊C8)C2211yescharacteristic
43C22⋊(C67⋊C8)C2211yescharacteristic
44(C2×D67)⋊C8C2211yescharacteristic
45(C2×D67)⋊C8C2211yescharacteristic
46(C2×D67)⋊C8C2211yescharacteristic
47(C2×D67)⋊C8C2211yescharacteristic
48#14523953C2211yescharacteristic
49#14523953C2211yescharacteristic
50C8×Dic67C2211yescharacteristic
51C8×Dic67C2211yescharacteristic
52C2×#41×C67C2211yescharacteristic
53C2×M16×C67C2211yescharacteristic
54C22×C8×C67C2211yescharacteristic
55C22⋊C8×C67C2211yescharacteristic
56C22⋊C8×C67C2211yescharacteristic
57C22⋊C8×C67C2211yescharacteristic
58C22⋊C8×C67C2211yescharacteristic
59#11761628—41no—
60#11761628—41no—
61C2×Dic134C2×C411yescharacteristic
62C2×D268C2×C411yescharacteristic
63C2×C22⋊D67C2×C411yescharacteristic
64C2×C22⋊D67C2×C411yescharacteristic
65C2×C22⋊D67C2×C411yescharacteristic
66C2×C22⋊D67C2×C411yescharacteristic
67C2×C22⋊D67C2×C411yescharacteristic
68C2×C22⋊D67C2×C411yescharacteristic
69C22×Dic67C2×C411yescharacteristic
70C22×Dic67C2×C411yescharacteristic
71C2×C4×D67C2311yescharacteristic
72C2×C4×D67C2311yescharacteristic
73#11761612—41no—
74#11761611—41no—
75#11761611—41no—
76C2×C67⋊C8—41no—
77C2×C67⋊C8C2311yescharacteristic
78C2×C67⋊C8C2311yescharacteristic
79C2×C67⋊C8C2311yescharacteristic
80C2×C67⋊C8C2311yescharacteristic
81C8×D67—41no—
82C8×D67—41no—
83Dic67⋊C4C2×C411yescharacteristic
84Dic67⋊C4C2×C411yescharacteristic
85Dic67⋊C4C2×C411yescharacteristic
86Dic67⋊C4C2×C411yescharacteristic
87C22⋊Dic67C2×C411yescharacteristic
88C22⋊Dic67C2×C411yescharacteristic
89C22⋊Dic67C2×C411yescharacteristic
90C22⋊Dic67C2×C411yescharacteristic
91(C2×D67)⋊C4C2×C411yescharacteristic
92(C2×D67)⋊C4C2×C411yescharacteristic
93(C2×D67)⋊C4C2×C411yescharacteristic
94(C2×D67)⋊C4C2×C411yescharacteristic
95C4×Dic67C2×C411yescharacteristic
96C4×Dic67C2×C411yescharacteristic
97C4×Dic67C2311yescharacteristic
98C4×Dic67C2311yescharacteristic
99#41×C67—41no—
100#41×C67—41no—
101C2×Q8×C67C2×C411yescharacteristic
102C2×D4×C67C2×C411yescharacteristic
103C2×D4×C67C2×C411yescharacteristic
104C2×D4×C67C2×C411yescharacteristic
105C22×C4×C67C2311yescharacteristic
106C22×C4×C67C2311yescharacteristic
107C22×C4×C67C2311yescharacteristic
108M16×C67—41no—
109C2×C8×C67—41no—
110C2×C8×C67C2311yescharacteristic
111C2×C8×C67C2311yescharacteristic
112C2×C8×C67C2311yescharacteristic
113C2×C8×C67C2311yescharacteristic
114C23×C67C22×C411yescharacteristic
115C23×C67C22×C411yescharacteristic
116C23×C67C22×C411yescharacteristic
117C22×D67C22×C411yescharacteristic
118C22×D67C22×C411yescharacteristic
119Q8×C67—41no—
120D4×C67—41no—
121D4×C67—41no—
122D4×C67—41no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67C22×C411yescharacteristic
130C2×C4×C67C22×C411yescharacteristic
131C2×C4×C67C22×C411yescharacteristic
132C2×C4×C67C2411yescharacteristic
133C22⋊D67—22no—
134C22⋊D67—22no—
135C22⋊D67—41no—
136C22⋊D67—41no—
137C22⋊D67—41no—
138C22⋊D67—41no—
139C2×Dic67—22no—
140C2×Dic67—41no—
141C2×Dic67C22×C411yescharacteristic
142C2×Dic67C22×C411yescharacteristic
143C2×Dic67C22×C411yescharacteristic
144C2×Dic67C22×C411yescharacteristic
145C2×Dic67C22×C411yescharacteristic
146C2×Dic67C22×C411yescharacteristic
147D268—41no—
148C4×D67—21no—
149C4×D67—21no—
150C4×D67—21no—
151C4×D67—21no—
152Dic134—41no—
153C8×C67—21no—
154C8×C67—21no—
155C8×C67—21no—
156C8×C67—21no—
157C67⋊C8—21no—
158C67⋊C8—21no—
159C67⋊C8—21no—
160C67⋊C8—21no—
161C22×C67—21no—
162C22×C67—21no—
163C22×C67—21no—
164C22×C67—21no—
165C22×C67—21no—
166C22×C67—21no—
167C22×C67—21no—
168C22×C67—21no—
169C22×C67—21no—
170C22×C67C23×C411yescharacteristic
171C2×D67—21no—
172C2×D67—21no—
173C2×D67—21no—
174C2×D67—21no—
175C2×D67—21no—
176C2×D67—21no—
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67H(5,𝔽2)11yescharacteristic
181C4×C67H(5,𝔽2)11yescharacteristic
182Dic67—21no—
183Dic67—21no—
184Dic67—21no—
185Dic67—21no—
186Dic67—21no—
187Dic67#13112yes—
188C2×C67—41no—
189C2×C67—41no—
190C2×C67—41no—
191C2×C67#51811yescharacteristic
192C2×C67C2×#13111yescharacteristic
193C2×C67#56811yescharacteristic
194D67—41no—
195D67—41no—
196#2981—671nomaximal, Sylow
197C67#298111yescharacteristic, complemented, Sylow
198#532—671no—
199#436—671no—
200#436—671no—
201#435—671no—
202#435—671no—
203#435—671no—
204#435—671no—
205C8×D4—671no—
206C8×D4—671no—
207#432—671no—
208#431—671no—
209#408—671no—
210#408—671no—
211#408—671no—
212#408D6711yescharacteristic, complemented
213C2×#41—671no—
214C2×#41C2×D6711yescharacteristic
215C2×M16—671no—
216C2×M16—671no—
217C2×M16—671no—
218C2×M16C2×D6711yescharacteristic
219C22×C8—671no—
220C22×C8—671no—
221C22×C8—671no—
222C22×C8C2×D6711yescharacteristic
223C4⋊Q8—671no—
224#127—671no—
225#124—671no—
226#124—671no—
227#121—671no—
228#121—671no—
229#121—671no—
230#121—671no—
231C4×D4—671no—
232C4×D4—671no—
233C4×D4—671no—
234C4×D4—671no—
235C2×C42—671no—
236C4⋊C8—671no—
237C4⋊C8—671no—
238C4⋊C8—671no—
239C4⋊C8—671no—
240C22⋊C8—671no—
241C22⋊C8—671no—
242C22⋊C8—671no—
243C22⋊C8—671no—
244C22⋊C8—671no—
245C22⋊C8—671no—
246C22⋊C8—671no—
247C22⋊C8—671no—
248C22⋊C8C2×D6711yescharacteristic
249C22⋊C8C2×D6711yescharacteristic
250C22⋊C8C2×D6711yescharacteristic
251C22⋊C8C2×D6711yescharacteristic
252#97—671no—
253#97—671no—
254C4×C8—671no—
255C4×C8—671no—
256#41—41no—
257#41—41no—
258#41—2681no—
259#41—2681no—
260C2×Q8—671no—
261C2×Q8C4×D6711yescharacteristic
262C2×D4—671no—
263C2×D4—671no—
264C2×D4—671no—
265C2×D4—671no—
266C2×D4—671no—
267C2×D4—671no—
268C2×D4—671no—
269C2×D4C4×D6711yescharacteristic
270C2×D4C4×D6711yescharacteristic
271C2×D4C4×D6711yescharacteristic
272C22×C4—671no—
273C22×C4—671no—
274C22×C4—671no—
275C22×C4—671no—
276C22×C4C22×D6711yescharacteristic
277C22×C4C22×D6711yescharacteristic
278C22×C4C22×D6711yescharacteristic
279M16—41no—
280M16—2681no—
281M16—2681no—
282M16—2681no—
283C2×C8—41no—
284C2×C8—2681no—
285C2×C8—2681no—
286C2×C8—2681no—
287C2×C8—671no—
288C2×C8—671no—
289C2×C8—671no—
290C2×C8—671no—
291C2×C8C22×D6711yescharacteristic
292C2×C8C22×D6711yescharacteristic
293C2×C8C22×D6711yescharacteristic
294C2×C8C22×D6711yescharacteristic
295C4⋊C4—671no—
296C4⋊C4—671no—
297C4⋊C4—671no—
298C4⋊C4—671no—
299C22⋊C4—671no—
300C22⋊C4—671no—
301C22⋊C4—671no—
302C22⋊C4—671no—
303C22⋊C4—671no—
304C22⋊C4—671no—
305C22⋊C4—671no—
306C22⋊C4—671no—
307C42—671no—
308C42—671no—
309C42—671no—
310C42—671no—
311C23—671no—
312C23—671no—
313C23C2×C4×D6711yescharacteristic
314C23C2×C4×D6711yescharacteristic
315C23C2×C4×D6711yescharacteristic
316Q8—41no—
317Q8—2681no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—2681no—
322D4—2681no—
323D4—2681no—
324D4—2681no—
325D4—2681no—
326D4—1342no—
327D4—1342no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—2681no—
335C2×C4—671no—
336C2×C4—671no—
337C2×C4—671no—
338C2×C4—671no—
339C2×C4—671no—
340C2×C4—671no—
341C2×C4—1342no—
342C2×C4—1341no—
343C2×C4—1341no—
344C2×C4—1341no—
345C2×C4—1341no—
346C2×C4C2×C4×D6711yescharacteristic
347C2×C4C2×C4×D6711yescharacteristic
348C2×C4C2×C4×D6711yescharacteristic
349C2×C4C23×D6711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—1341no—
355C8—1341no—
356C8—1341no—
357C8—1341no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—1341no—
368C22—1341no—
369C22—1341no—
370C22—1341no—
371C22—1341no—
372C22—1341no—
373C22C22×C4×D6711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—672no—
378C4—1341no—
379C4—1341no—
380C4—1341no—
381C4—1341no—
382C4—1341no—
383C4#1452408811yescharacteristic
384C4#1452408811yescharacteristic
385C2—41no—
386C2—41no—
387C2—41no—
388C2—2681no—
389C2—2681no—
390C2#1686386911yescharacteristic
391C2#131×D6711yescharacteristic
392C2#1686416111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D67
commutator, socleC22×C67C23×C4
Fitting#408×C67C2
hypercenter#408D67
hypocenterC67#2981
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #16863472C2semidirect
3. #16863471C2semidirect
4. #16863470C2semidirect
5. #16863467C2semidirect
6. #16863465C2semidirect
7. #16863464C2semidirect
8. C8×C22⋊D67C2semidirect
9. (C22⋊D67)⋊C8C2semidirect
10. #16863450C2semidirect
11. #16863446C2semidirect
12. #16863363C2semidirect
13. #16863361C2semidirect
14. #16863361C2semidirect
15. #408×C67C2semidirect
197. C67#2981semidirect
212. #408D67semidirect