Finite Group Databasebeta

← #22572096

Subgroups — #22572096

Order 8576.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864086C211yesmaximal, characteristic, complemented
2#16863750C211yesmaximal, characteristic, complemented
3#16863734C211yesmaximal, characteristic, complemented
4#16863707C211yesmaximal, characteristic, complemented
5#16863703C211yesmaximal, characteristic, complemented
6#16863635C211yesmaximal, characteristic, complemented
7#16863623C211yesmaximal, characteristic, complemented
8#16863615C211yesmaximal, characteristic, complemented
9#16863558C211yesmaximal, characteristic, complemented
10#16863552C211yesmaximal, characteristic, complemented
11C4×#11761618C211yesmaximal, characteristic, complemented
12D4×C67⋊C8C211yesmaximal, characteristic, complemented
13#16863392C211yesmaximal, characteristic, complemented
14C2×#14523967C211yesmaximal, characteristic, complemented
15#548×C67C211yesmaximal, characteristic
16C2×#11761618C2211yescharacteristic
17C2×#11761618C2211yescharacteristic
18C2×#11761616C2211yescharacteristic
19C22×C67⋊C8C2211yescharacteristic
20C22×C67⋊C8C2211yescharacteristic
21#14524036—21no—
22#14524035C2211yescharacteristic
23#14524033—21no—
24#14524024—21no—
25#14524022C2211yescharacteristic
26#14524022C2211yescharacteristic
27C4×Dic134C2211yescharacteristic
28#14524002C2211yescharacteristic
29#14523999—21no—
30C2×C4⋊Dic67C2211yescharacteristic
31#14523976C2211yescharacteristic
32#14523976C2211yescharacteristic
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
47C4×C67⋊C8C2211yescharacteristic
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×Dic134C2311yescharacteristic
60C22×Dic67—21no—
61#11761618—22no—
62#11761618—41no—
63#11761616—41no—
64C2×C67⋊C8—41no—
65C2×C67⋊C8—41no—
66C2×C67⋊C8C2311yescharacteristic
67C2×C67⋊C8C2311yescharacteristic
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C4⋊Dic67—21no—
71C4⋊Dic67C2311yescharacteristic
72C4⋊Dic67C2311yescharacteristic
73C4⋊Dic67C2311yescharacteristic
74Dic67⋊C4—21no—
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77Dic67⋊C4—21no—
78Dic67⋊C4—21no—
79C22⋊Dic67—21no—
80C22⋊Dic67—21no—
81C22⋊Dic67—21no—
82C4×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—42no—
143Dic134—21no—
144Dic134—21no—
145C67⋊C8—21no—
146C67⋊C8—21no—
147C67⋊C8—21no—
148C67⋊C8—21no—
149C22×C67—21no—
150C22×C67—21no—
151C22×C67—21no—
152C22×C67—21no—
153C22×C67—21no—
154C22×C67—41no—
155C22×C67—41no—
156C22×C67—41no—
157C22×C67#13512yes—
158C22×C67C22×D411yescharacteristic
159C4×C67—21no—
160C4×C67—21no—
161C4×C67—22no—
162C4×C67—21no—
163C4×C67—21no—
164C4×C67—41no—
165C4×C67—41no—
166C4×C67C2×#4111yescharacteristic
167C4×C67H(5,𝔽2)11yescharacteristic
168Dic67—41no—
169Dic67—41no—
170Dic67—41no—
171Dic67—41no—
172C2×C67—22no—
173C2×C67—41no—
174C2×C67—81no—
175C2×C67#54611yescharacteristic
176C2×C67C2×#13511yescharacteristic
177C2×C67#57811yescharacteristic
178#3300—671nomaximal, Sylow
179C67#330011yescharacteristic, complemented, Sylow
180#552—671no—
181#548D6711yescharacteristic
182#488—671no—
183#485—671no—
184#480—671no—
185#479—671no—
186#467—671no—
187#465—671no—
188#464—671no—
189#452—671no—
190#451—671no—
191C4×Q16—671no—
192C8×D4—671no—
193#416—671no—
194C2×#103—671no—
195C2×#41C2×D6711yescharacteristic
196C22×Q8C2×D6711yescharacteristic
197C2×Q16—671no—
198C2×Q16—671no—
199C2×SD16—671no—
200C22×C8—671no—
201C22×C8—671no—
202#126—1341no—
203#125—671no—
204#125—1341no—
205#124—21no—
206#124C2×D6711yescharacteristic
207#123—1341no—
208C22⋊Q8—21no—
209C22⋊Q8—671no—
210C22⋊Q8—671no—
211C22⋊Q8C2×D6711yescharacteristic
212#121—21no—
213#121C2×D6711yescharacteristic
214C4×Q8—671no—
215C4×Q8C2×D6711yescharacteristic
216C4×D4—21no—
217C4×D4C2×D6711yescharacteristic
218#117—671no—
219#117—1341no—
220C2×C4⋊C4—671no—
221#107—671no—
222#107—671no—
223#106—671no—
224C4⋊C8—671no—
225#103—671no—
226#103—671no—
227#103—671no—
228#103—671no—
229#103—671no—
230#103—671no—
231#103—671no—
232#102—671no—
233#102—671no—
234#102—671no—
235C22⋊C8—671no—
236C22⋊C8—671no—
237C4×C8—671no—
238#41—22no—
239#41—41no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—671no—
244C2×Q8C22×D6711yescharacteristic
245C2×Q8C22×D6711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22⋊D6711yescharacteristic
249C2×D4C22×D6711yescharacteristic
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—1341no—
253C22×C4C22×D6711yescharacteristic
254C22×C4C22×D6711yescharacteristic
255Q16—2681no—
256Q16—1342no—
257SD16—2681no—
258C2×C8—2681no—
259C2×C8—2681no—
260C2×C8—671no—
261C2×C8—671no—
262C2×C8—671no—
263C2×C8—671no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—671no—
267C4⋊C4—671no—
268C4⋊C4—671no—
269C4⋊C4—1341no—
270C4⋊C4—1341no—
271C4⋊C4—1341no—
272C4⋊C4—1341no—
273C4⋊C4—1341no—
274C4⋊C4—1341no—
275C4⋊C4C22⋊D6711yescharacteristic
276C4⋊C4C22×D6711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—1341no—
282C22⋊C4—1341no—
283C22⋊C4—1341no—
284C22⋊C4C22⋊D6711yescharacteristic
285C22⋊C4C22⋊D6711yescharacteristic
286C42—21no—
287C42—1341no—
288C42—1341no—
289C42C22×D6711yescharacteristic
290C23—21no—
291C23C2×C22⋊D6711yescharacteristic
292C23C2×C22⋊D6711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—1341no—
298Q8—1341no—
299Q8#1176162712yes—
300D4—22no—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—42no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—2682no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1341no—
318C2×C4—1341no—
319C2×C4C2×C22⋊D6711yescharacteristic
320C2×C4C2×C22⋊D6711yescharacteristic
321C2×C4C2×C22⋊D6711yescharacteristic
322C2×C4C2×C22⋊D6711yescharacteristic
323C2×C4C23×D6711yescharacteristic
324C8—1341no—
325C8—1341no—
326C8—1341no—
327C8—1341no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1452406312yes—
337C22C22×C22⋊D6711yescharacteristic
338C4—21no—
339C4—21no—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—2681no—
346C4—2681no—
347C4—2681no—
348C4—2681no—
349C4C2×#1176162711yescharacteristic
350C4#1452408811yescharacteristic
351C2—22no—
352C2—41no—
353C2—81no—
354C2#1686404011yescharacteristic
355C2C2×#1452406311yescharacteristic
356C2#1686421111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#548×C67C2
socleC22×C67C22×D4
hypercenter#548D67
hypocenterC67#3300
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. #16864086C2semidirect
2. #16863750C2semidirect
3. #16863734C2semidirect
4. #16863707C2semidirect
5. #16863703C2semidirect
6. #16863635C2semidirect
7. #16863623C2semidirect
8. #16863615C2semidirect
9. #16863558C2semidirect
10. #16863552C2semidirect
11. C4×#11761618C2semidirect
12. D4×C67⋊C8C2semidirect
13. #16863392C2semidirect
14. C2×#14523967C2semidirect
179. C67#3300semidirect