Finite Group Databasebeta

← #22572290

Subgroups — #22572290

Order 8576.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864103C211yesmaximal, characteristic, complemented
2#16863755C211yesmaximal, characteristic, complemented
3#16863729C211yesmaximal, characteristic, complemented
4#16863707C211yesmaximal, characteristic, complemented
5#16863676C211yesmaximal, characteristic, complemented
6#16863640C211yesmaximal, characteristic, complemented
7#16863635C211yesmaximal, characteristic, complemented
8#16863615C211yesmaximal, characteristic, complemented
9#16863558C211yesmaximal, characteristic, complemented
10#16863547C211yesmaximal, characteristic, complemented
11#16863496C211yesmaximal, characteristic, complemented
12#16863468C211yesmaximal, characteristic, complemented
13#16863397C211yesmaximal, characteristic, complemented
14C2×#14523964C211yesmaximal, characteristic, complemented
15#546×C67C211yesmaximal, characteristic
16C2×#11761618C2211yescharacteristic, complemented
17C2×#11761616C2211yescharacteristic
18C2×#11761616C2211yescharacteristic
19C2×#11761612C2211yescharacteristic, complemented
20C22×C67⋊C8C2211yescharacteristic
21#14524043C2211yescharacteristic, complemented
22#14524034—21no—
23#14524022C2211yescharacteristic
24#14524022C2211yescharacteristic, complemented
25#14524020—21no—
26#14524020—21no—
27C4×Dic134C2211yescharacteristic, complemented
28D4×Dic67—21no—
29C2×C4⋊Dic67C2211yescharacteristic
30C2×C4⋊Dic67C2211yescharacteristic, complemented
31#14523976C2211yescharacteristic, complemented
32#14523976C2211yescharacteristic, complemented
33#14523976C2211yescharacteristic, complemented
34C4⋊(C67⋊C8)C2211yescharacteristic, complemented
35#14523967C2211yescharacteristic, complemented
36#14523967C2211yescharacteristic, complemented
37#14523967C2211yescharacteristic, complemented
38#14523966C2211yescharacteristic, complemented
39#14523964C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523964C2211yescharacteristic
43#14523964C2211yescharacteristic
44#14523964C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic, complemented
47#14523954C2211yescharacteristic, complemented
48C2×#41×C67C2211yescharacteristic
49C22×D4×C67C2211yescharacteristic
50#124×C67C2211yescharacteristic
51#123×C67—21no—
52C22⋊Q8×C67C2211yescharacteristic
53#121×C67—21no—
54#121×C67C2211yescharacteristic
55#120×C67—21no—
56C4×D4×C67C2211yescharacteristic
57C4×D4×C67C2211yescharacteristic
58C2×C22⋊C4×C67—21no—
59C2×Dic134—21no—
60C2×Dic134C2311yescharacteristic
61C22×Dic67—21no—
62C22×Dic67—21no—
63#11761618—41no—
64#11761616—22no—
65#11761616—41no—
66#11761612—41no—
67C2×C67⋊C8—41no—
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C2×C67⋊C8C2311yescharacteristic
72C4⋊Dic67—21no—
73C4⋊Dic67—21no—
74C4⋊Dic67—21no—
75C4⋊Dic67C2311yescharacteristic
76C4⋊Dic67C2311yescharacteristic
77C4⋊Dic67C2311yescharacteristic
78Dic67⋊C4—21no—
79Dic67⋊C4—21no—
80Dic67⋊C4—21no—
81C22⋊Dic67—21no—
82C22⋊Dic67—21no—
83C4×Dic67—21no—
84C24×C67—21no—
85#41×C67—22no—
86#41×C67—41no—
87C2×Q8×C67C2311yescharacteristic
88C2×D4×C67—22no—
89C2×D4×C67—21no—
90C2×D4×C67—21no—
91C2×D4×C67—21no—
92C2×D4×C67—41no—
93C2×D4×C67D411yescharacteristic
94C2×D4×C67C2311yescharacteristic
95C2×D4×C67C2311yescharacteristic
96C22×C4×C67—21no—
97C22×C4×C67—21no—
98C22×C4×C67C2311yescharacteristic
99C22×C4×C67C2311yescharacteristic
100C4⋊C4×C67—21no—
101C4⋊C4×C67D411yescharacteristic
102C4⋊C4×C67C2311yescharacteristic
103C22⋊C4×C67—21no—
104C22⋊C4×C67—21no—
105C22⋊C4×C67—21no—
106C22⋊C4×C67—21no—
107C22⋊C4×C67—21no—
108C22⋊C4×C67D411yescharacteristic
109C22⋊C4×C67D411yescharacteristic
110C42×C67C2311yescharacteristic
111C23×C67—21no—
112C23×C67—21no—
113C23×C67—42no—
114C23×C67—42no—
115C23×C67—81no—
116C23×C67C2×D411yescharacteristic
117C23×C67C2×D411yescharacteristic
118Q8×C67—21no—
119Q8×C67—21no—
120D4×C67—22no—
121D4×C67—21no—
122D4×C67—21no—
123D4×C67—21no—
124D4×C67—42no—
125D4×C67—42no—
126D4×C67—81no—
127D4×C67#4112yes—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—21no—
132C2×C4×C67—21no—
133C2×C4×C67—21no—
134C2×C4×C67—42no—
135C2×C4×C67—81no—
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2×D411yescharacteristic
138C2×C4×C67C2×D411yescharacteristic
139C2×C4×C67C2×D411yescharacteristic
140C2×C4×C67C2411yescharacteristic
141C2×Dic67—21no—
142C2×Dic67—21no—
143C2×Dic67—21no—
144C2×Dic67—21no—
145C2×Dic67—81no—
146C2×Dic67—81no—
147Dic134—21no—
148Dic134—21no—
149Dic134—81no—
150C67⋊C8—21no—
151C67⋊C8—21no—
152C67⋊C8—21no—
153C67⋊C8—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—21no—
157C22×C67—22no—
158C22×C67—21no—
159C22×C67—21no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67—41no—
165C22×C67—84no—
166C22×C67#13712yes—
167C22×C67C22×D411yescharacteristic
168C4×C67—21no—
169C4×C67—21no—
170C4×C67—21no—
171C4×C67—21no—
172C4×C67—41no—
173C4×C67—41no—
174C4×C67C2×#4111yescharacteristic
175C4×C67H(5,𝔽2)11yescharacteristic
176Dic67—41no—
177Dic67—41no—
178Dic67—41no—
179Dic67—41no—
180C2×C67—22no—
181C2×C67—41no—
182C2×C67—42no—
183C2×C67—81no—
184C2×C67#54611yescharacteristic
185C2×C67C2×#13711yescharacteristic
186C2×C67#57611yescharacteristic
187#3314—671nomaximal, Sylow
188C67#331411yescharacteristic, complemented, Sylow
189#554—671no—
190#546D6711yescharacteristic
191#489—671no—
192#484—671no—
193#480—671no—
194#474—671no—
195#468—671no—
196#467—671no—
197#464—671no—
198#452—671no—
199#450—671no—
200#440—671no—
201#435—671no—
202#417—671no—
203C2×#102—671no—
204C2×#41C2×D6711yescharacteristic
205C22×D4C2×D6711yescharacteristic
206C2×Q16—671no—
207C2×SD16—671no—
208C2×SD16—671no—
209C2×M16—671no—
210C22×C8—671no—
211C4⋊Q8—671no—
212#125—1341no—
213#124C2×D6711yescharacteristic
214#123—21no—
215C22⋊Q8—671no—
216C22⋊Q8—671no—
217C22⋊Q8—1341no—
218C22⋊Q8—1341no—
219C22⋊Q8C2×D6711yescharacteristic
220#121—21no—
221#121C2×D6711yescharacteristic
222#120—21no—
223C4×Q8—671no—
224C4×D4—1341no—
225C4×D4C2×D6711yescharacteristic
226C4×D4C2×D6711yescharacteristic
227C2×C4⋊C4—671no—
228C2×C4⋊C4—671no—
229C2×C22⋊C4—21no—
230#107—671no—
231#107—671no—
232#107—671no—
233C4⋊C8—671no—
234#103—671no—
235#103—671no—
236#103—671no—
237#103—671no—
238#102—671no—
239#102—671no—
240#102—671no—
241#102—671no—
242#102—671no—
243#102—671no—
244C22⋊C8—671no—
245C22⋊C8—671no—
246#97—671no—
247C24—21no—
248#41—22no—
249#41—41no—
250C2×Q8—671no—
251C2×Q8—1341no—
252C2×Q8C22×D6711yescharacteristic
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—41no—
258C2×D4C22⋊D6711yescharacteristic
259C2×D4C22×D6711yescharacteristic
260C2×D4C22×D6711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—1341no—
264C22×C4—1341no—
265C22×C4C22×D6711yescharacteristic
266C22×C4C22×D6711yescharacteristic
267Q16—2681no—
268SD16—2681no—
269SD16—1342no—
270M16—2681no—
271C2×C8—2681no—
272C2×C8—671no—
273C2×C8—671no—
274C2×C8—671no—
275C2×C8—671no—
276C4⋊C4—21no—
277C4⋊C4—671no—
278C4⋊C4—671no—
279C4⋊C4—671no—
280C4⋊C4—1341no—
281C4⋊C4—1341no—
282C4⋊C4—1341no—
283C4⋊C4—1341no—
284C4⋊C4—1341no—
285C4⋊C4—1341no—
286C4⋊C4C22⋊D6711yescharacteristic
287C4⋊C4C22×D6711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—1341no—
294C22⋊C4—1341no—
295C22⋊C4C22⋊D6711yescharacteristic
296C22⋊C4C22⋊D6711yescharacteristic
297C42—1341no—
298C42C22×D6711yescharacteristic
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23C2×C22⋊D6711yescharacteristic
305C23C2×C22⋊D6711yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—5361no—
309Q8—1341no—
310Q8—1341no—
311D4—22no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—42no—
317D4—81no—
318D4#1176162712yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—81no—
327C2×C4—5361no—
328C2×C4—5361no—
329C2×C4—1341no—
330C2×C4—1341no—
331C2×C4—1341no—
332C2×C4—1341no—
333C2×C4C2×C22⋊D6711yescharacteristic
334C2×C4C2×C22⋊D6711yescharacteristic
335C2×C4C2×C22⋊D6711yescharacteristic
336C2×C4C2×C22⋊D6711yescharacteristic
337C2×C4C23×D6711yescharacteristic
338C8—1341no—
339C8—1341no—
340C8—1341no—
341C8—1341no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—84no—
354C22#1452407312yes—
355C22C22×C22⋊D6711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—2681no—
363C4—2681no—
364C4—2681no—
365C4—2681no—
366C4C2×#1176162711yescharacteristic
367C4#1452408811yescharacteristic
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#1686404011yescharacteristic
373C2C2×#1452407311yescharacteristic
374C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#546×C67C2
socleC22×C67C22×D4
hypercenter#546D67
hypocenterC67#3314
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #16864103C2semidirect
2. #16863755C2semidirect
3. #16863729C2semidirect
4. #16863707C2semidirect
5. #16863676C2semidirect
6. #16863640C2semidirect
7. #16863635C2semidirect
8. #16863615C2semidirect
9. #16863558C2semidirect
10. #16863547C2semidirect
11. #16863496C2semidirect
12. #16863468C2semidirect
13. #16863397C2semidirect
14. C2×#14523964C2semidirect
16. C2×#11761618C22semidirect
19. C2×#11761612C22semidirect
21. #14524043C22semidirect
24. #14524022C22semidirect
27. C4×Dic134C22semidirect
30. C2×C4⋊Dic67C22semidirect
31. #14523976C22semidirect
32. #14523976C22semidirect
33. #14523976C22semidirect
34. C4⋊(C67⋊C8)C22semidirect
35. #14523967C22semidirect
36. #14523967C22semidirect
37. #14523967C22semidirect
38. #14523966C22semidirect
46. C22⋊(C67⋊C8)C22semidirect
47. #14523954C22semidirect
188. C67#3314semidirect