Finite Group Databasebeta

← #22572320

Subgroups — #22572320

Order 8576.

388 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
1#16864103C211yesmaximal, characteristic, complemented
2#16863755C211yesmaximal, characteristic, complemented
3#16863724C211yesmaximal, characteristic, complemented
4#16863712C211yesmaximal, characteristic, complemented
5#16863681C211yesmaximal, characteristic, complemented
6#16863652C211yesmaximal, characteristic, complemented
7#16863623C211yesmaximal, characteristic, complemented
8#16863615C211yesmaximal, characteristic, complemented
9#16863558C211yesmaximal, characteristic, complemented
10#16863552C211yesmaximal, characteristic, complemented
11#16863501C211yesmaximal, characteristic, complemented
12#16863468C211yesmaximal, characteristic, complemented
13#16863397C211yesmaximal, characteristic, complemented
14C2×#14523967C211yesmaximal, characteristic, complemented
15#548×C67C211yesmaximal, characteristic
16C2×#11761618C2211yescharacteristic
17C2×#11761618C2211yescharacteristic
18C2×#11761616C2211yescharacteristic
19C2×#11761612C2211yescharacteristic
20C22×C67⋊C8C2211yescharacteristic
21#14524043C2211yescharacteristic
22#14524034—21no—
23#14524022C2211yescharacteristic
24#14524022C2211yescharacteristic
25#14524020—21no—
26#14524020—21no—
27C4×Dic134C2211yescharacteristic
28D4×Dic67—21no—
29C2×C4⋊Dic67C2211yescharacteristic
30C2×C4⋊Dic67C2211yescharacteristic
31#14523974C2211yescharacteristic
32#14523974C2211yescharacteristic
33#14523974C2211yescharacteristic
34C4⋊(C67⋊C8)C2211yescharacteristic
35#14523967C2211yescharacteristic
36#14523967C2211yescharacteristic
37#14523967C2211yescharacteristic
38#14523967C2211yescharacteristic
39#14523967C2211yescharacteristic
40#14523967C2211yescharacteristic
41#14523966C2211yescharacteristic
42#14523964C2211yescharacteristic
43#14523964C2211yescharacteristic
44#14523964C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic
47#14523954C2211yescharacteristic
48C2×#41×C67C2211yescharacteristic
49C22×Q8×C67C2211yescharacteristic
50#124×C67—21no—
51#124×C67C2211yescharacteristic
52C22⋊Q8×C67—21no—
53C22⋊Q8×C67C2211yescharacteristic
54#121×C67—21no—
55#121×C67C2211yescharacteristic
56C4×Q8×C67C2211yescharacteristic
57C4×D4×C67—21no—
58C4×D4×C67C2211yescharacteristic
59C2×Dic134—21no—
60C2×Dic134C2311yescharacteristic
61C22×Dic67—21no—
62C22×Dic67—21no—
63#11761618—22no—
64#11761618—41no—
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—
84#41×C67—22no—
85#41×C67—41no—
86C2×Q8×C67—21no—
87C2×Q8×C67—22no—
88C2×Q8×C67—41no—
89C2×Q8×C67C2311yescharacteristic
90C2×Q8×C67C2311yescharacteristic
91C2×D4×C67—21no—
92C2×D4×C67—21no—
93C2×D4×C67D411yescharacteristic
94C2×D4×C67C2311yescharacteristic
95C22×C4×C67—21no—
96C22×C4×C67—21no—
97C22×C4×C67C2311yescharacteristic
98C22×C4×C67C2311yescharacteristic
99C4⋊C4×C67—21no—
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×C67D411yescharacteristic
108C22⋊C4×C67D411yescharacteristic
109C42×C67—21no—
110C42×C67C2311yescharacteristic
111C23×C67—21no—
112C23×C67C2×D411yescharacteristic
113C23×C67C2×D411yescharacteristic
114Q8×C67—21no—
115Q8×C67—21no—
116Q8×C67—21no—
117Q8×C67—42no—
118Q8×C67#4112yes—
119D4×C67—22no—
120D4×C67—21no—
121D4×C67—21no—
122D4×C67—42no—
123D4×C67—42no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67—42no—
132C2×C4×C67—42no—
133C2×C4×C67C2×D411yescharacteristic
134C2×C4×C67C2×D411yescharacteristic
135C2×C4×C67C2×D411yescharacteristic
136C2×C4×C67C2×D411yescharacteristic
137C2×C4×C67C2411yescharacteristic
138C2×Dic67—21no—
139C2×Dic67—21no—
140C2×Dic67—21no—
141C2×Dic67—21no—
142C2×Dic67—81no—
143C2×Dic67—81no—
144Dic134—21no—
145Dic134—21no—
146Dic134—81no—
147C67⋊C8—21no—
148C67⋊C8—21no—
149C67⋊C8—21no—
150C67⋊C8—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—21no—
155C22×C67—21no—
156C22×C67—41no—
157C22×C67—41no—
158C22×C67—41no—
159C22×C67#13712yes—
160C22×C67C22×D411yescharacteristic
161C4×C67—21no—
162C4×C67—21no—
163C4×C67—22no—
164C4×C67—21no—
165C4×C67—21no—
166C4×C67—41no—
167C4×C67—41no—
168C4×C67C2×#4111yescharacteristic
169C4×C67H(5,𝔽2)11yescharacteristic
170Dic67—41no—
171Dic67—41no—
172Dic67—41no—
173Dic67—41no—
174C2×C67—22no—
175C2×C67—41no—
176C2×C67—81no—
177C2×C67#54611yescharacteristic
178C2×C67C2×#13711yescharacteristic
179C2×C67#57711yescharacteristic
180#3316—671nomaximal, Sylow
181C67#331611yescharacteristic, complemented, Sylow
182#554—671no—
183#548D6711yescharacteristic
184#489—671no—
185#483—671no—
186#481—671no—
187#475—671no—
188#470—671no—
189#465—671no—
190#464—671no—
191#452—671no—
192#451—671no—
193#441—671no—
194#435—671no—
195#417—671no—
196C2×#103—671no—
197C2×#41C2×D6711yescharacteristic
198C22×Q8C2×D6711yescharacteristic
199C2×Q16—671no—
200C2×Q16—671no—
201C2×SD16—671no—
202C2×M16—671no—
203C22×C8—671no—
204C4⋊Q8—671no—
205#125—1341no—
206#124—21no—
207#124C2×D6711yescharacteristic
208C22⋊Q8—21no—
209C22⋊Q8—671no—
210C22⋊Q8—671no—
211C22⋊Q8—1341no—
212C22⋊Q8—1341no—
213C22⋊Q8C2×D6711yescharacteristic
214#121—21no—
215#121C2×D6711yescharacteristic
216C4×Q8—671no—
217C4×Q8C2×D6711yescharacteristic
218C4×D4—21no—
219C4×D4—1341no—
220C4×D4C2×D6711yescharacteristic
221C2×C4⋊C4—671no—
222C2×C4⋊C4—671no—
223#106—671no—
224#106—671no—
225#106—671no—
226C4⋊C8—671no—
227#103—671no—
228#103—671no—
229#103—671no—
230#103—671no—
231#103—671no—
232#103—671no—
233#103—671no—
234#102—671no—
235#102—671no—
236#102—671no—
237C22⋊C8—671no—
238C22⋊C8—671no—
239#97—671no—
240#41—22no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—41no—
245C2×Q8—671no—
246C2×Q8—1341no—
247C2×Q8C22×D6711yescharacteristic
248C2×Q8C22×D6711yescharacteristic
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊D6711yescharacteristic
252C2×D4C22×D6711yescharacteristic
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1341no—
256C22×C4—1341no—
257C22×C4C22×D6711yescharacteristic
258C22×C4C22×D6711yescharacteristic
259Q16—2681no—
260Q16—1342no—
261SD16—2681no—
262M16—2681no—
263C2×C8—2681no—
264C2×C8—671no—
265C2×C8—671no—
266C2×C8—671no—
267C2×C8—671no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—671no—
271C4⋊C4—671no—
272C4⋊C4—671no—
273C4⋊C4—1341no—
274C4⋊C4—1341no—
275C4⋊C4—1341no—
276C4⋊C4—1341no—
277C4⋊C4—1341no—
278C4⋊C4—1341no—
279C4⋊C4C22⋊D6711yescharacteristic
280C4⋊C4C22×D6711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—1341no—
286C22⋊C4—1341no—
287C22⋊C4C22⋊D6711yescharacteristic
288C22⋊C4C22⋊D6711yescharacteristic
289C42—21no—
290C42—1341no—
291C42C22×D6711yescharacteristic
292C23—21no—
293C23C2×C22⋊D6711yescharacteristic
294C23C2×C22⋊D6711yescharacteristic
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—42no—
299Q8—5361no—
300Q8—1341no—
301Q8—1341no—
302Q8#1176162712yes—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—42no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—5361no—
318C2×C4—5361no—
319C2×C4—1341no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4—1341no—
323C2×C4C2×C22⋊D6711yescharacteristic
324C2×C4C2×C22⋊D6711yescharacteristic
325C2×C4C2×C22⋊D6711yescharacteristic
326C2×C4C2×C22⋊D6711yescharacteristic
327C2×C4C23×D6711yescharacteristic
328C8—1341no—
329C8—1341no—
330C8—1341no—
331C8—1341no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1452407312yes—
341C22C22×C22⋊D6711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—2681no—
350C4—2681no—
351C4—2681no—
352C4—2681no—
353C4C2×#1176162711yescharacteristic
354C4#1452408811yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2#1686404011yescharacteristic
359C2C2×#1452407311yescharacteristic
360C2#1686420411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#548×C67C2
socleC22×C67C22×D4
hypercenter#548D67
hypocenterC67#3316
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #16864103C2semidirect
2. #16863755C2semidirect
3. #16863724C2semidirect
4. #16863712C2semidirect
5. #16863681C2semidirect
6. #16863652C2semidirect
7. #16863623C2semidirect
8. #16863615C2semidirect
9. #16863558C2semidirect
10. #16863552C2semidirect
11. #16863501C2semidirect
12. #16863468C2semidirect
13. #16863397C2semidirect
14. C2×#14523967C2semidirect
181. C67#3316semidirect