Finite Group Databasebeta

← #22572809

Subgroups — #22572809

Order 8576.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16864092C211yesmaximal, characteristic, complemented
2#16863795C211yesmaximal, characteristic, complemented
3#16863720C211yesmaximal, characteristic, complemented
4#16863717C211yesmaximal, characteristic, complemented
5#16863709C211yesmaximal, characteristic, complemented
6#16863707C211yesmaximal, characteristic, complemented
7#16863696C211yesmaximal, characteristic, complemented
8#16863693C211yesmaximal, characteristic, complemented
9#16863647C211yesmaximal, characteristic, complemented
10#16863647C211yesmaximal, characteristic, complemented
11C4×#11761613C211yesmaximal, characteristic, complemented
12#16863474C211yesmaximal, characteristic, complemented
13#16863439C211yesmaximal, characteristic, complemented
14#16863436C211yesmaximal, characteristic, complemented
15#553×C67C211yesmaximal, characteristic, complemented
16C2×#11761613C2211yescharacteristic
17C2×#11761612C2211yescharacteristic
18C2×#11761612C2211yescharacteristic
19#14524038—21no—
20#14524035C2211yescharacteristic
21#14524025—21no—
22#14524017C2211yescharacteristic
23#14524017C2211yescharacteristic
24#14524015—21no—
25C4×D268C2211yescharacteristic
26(C22⋊D67)⋊C4—21no—
27#14524002C2211yescharacteristic
28C2×C4⋊Dic67C2211yescharacteristic
29#14523976C2211yescharacteristic
30#14523976C2211yescharacteristic
31#14523976C2211yescharacteristic
32#14523976C2211yescharacteristic
33#14523976C2211yescharacteristic
34#14523974C2211yescharacteristic
35#14523974C2211yescharacteristic
36#14523974C2211yescharacteristic
37#14523974C2211yescharacteristic
38C4⋊(C67⋊C8)C2211yescharacteristic
39#14523964C2211yescharacteristic
40#14523964C2211yescharacteristic
41#14523964C2211yescharacteristic
42#14523963C2211yescharacteristic
43#14523963C2211yescharacteristic
44#14523963C2211yescharacteristic
45C22⋊(C67⋊C8)C2211yescharacteristic
46C22⋊(C67⋊C8)C2211yescharacteristic
47C4×C67⋊C8C2211yescharacteristic
48#126×C67—21no—
49#125×C67C2211yescharacteristic
50#123×C67—21no—
51#121×C67—21no—
52#121×C67C2211yescharacteristic
53#121×C67C2211yescharacteristic
54C4×D4×C67—21no—
55C4×D4×C67C2211yescharacteristic
56C4×D4×C67C2211yescharacteristic
57#117×C67C2211yescharacteristic
58C2×C4⋊C4×C67C2211yescharacteristic
59C2×D268C2311yescharacteristic
60C2×C22⋊D67—21no—
61C2×C22⋊D67—21no—
62C22×Dic67—21no—
63C2×C4×D67—21no—
64#11761613—22no—
65#11761612—22no—
66#11761612—22no—
67C2×C67⋊C8C2311yescharacteristic
68C2×C67⋊C8C2311yescharacteristic
69C2×C67⋊C8C2311yescharacteristic
70C2×C67⋊C8C2311yescharacteristic
71C4⋊Dic67—21no—
72C4⋊Dic67C2311yescharacteristic
73C4⋊Dic67C2311yescharacteristic
74C4⋊Dic67C2311yescharacteristic
75Dic67⋊C4—21no—
76Dic67⋊C4—21no—
77C22⋊Dic67—21no—
78(C2×D67)⋊C4—21no—
79(C2×D67)⋊C4—21no—
80(C2×D67)⋊C4—21no—
81C4×Dic67—21no—
82C2×D4×C67—21no—
83C2×D4×C67—21no—
84C2×D4×C67D411yescharacteristic
85C2×D4×C67C2311yescharacteristic
86C22×C4×C67—21no—
87C22×C4×C67—21no—
88C22×C4×C67C2311yescharacteristic
89C22×C4×C67C2311yescharacteristic
90C4⋊C4×C67—21no—
91C4⋊C4×C67—21no—
92C4⋊C4×C67—21no—
93C4⋊C4×C67D411yescharacteristic
94C4⋊C4×C67C2311yescharacteristic
95C4⋊C4×C67C2311yescharacteristic
96C4⋊C4×C67C2311yescharacteristic
97C22⋊C4×C67—21no—
98C22⋊C4×C67—21no—
99C22⋊C4×C67—21no—
100C22⋊C4×C67—21no—
101C22⋊C4×C67D411yescharacteristic
102C22⋊C4×C67D411yescharacteristic
103C42×C67—21no—
104C42×C67C2311yescharacteristic
105C23×C67—21no—
106C23×C67C2×D411yescharacteristic
107C23×C67C2×D411yescharacteristic
108C22×D67—21no—
109D4×C67—21no—
110D4×C67—21no—
111D4×C67—41no—
112D4×C67—42no—
113D4×C67—81no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—81no—
122C2×C4×C67—81no—
123C2×C4×C67C2×D411yescharacteristic
124C2×C4×C67C2×D411yescharacteristic
125C2×C4×C67C2×D411yescharacteristic
126C2×C4×C67C2×D411yescharacteristic
127C2×C4×C67C2411yescharacteristic
128C22⋊D67—42no—
129C22⋊D67—81no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—81no—
134D268—21no—
135D268—21no—
136C4×D67—81no—
137C67⋊C8—21no—
138C67⋊C8—21no—
139C67⋊C8—21no—
140C67⋊C8#4112yes—
141C22×C67—21no—
142C22×C67—21no—
143C22×C67—21no—
144C22×C67—21no—
145C22×C67—21no—
146C22×C67—21no—
147C22×C67—41no—
148C22×C67—41no—
149C22×C67—41no—
150C22×C67C22×D411yescharacteristic
151C2×D67—41no—
152C2×D67—41no—
153C2×D67—41no—
154C4×C67—21no—
155C4×C67—21no—
156C4×C67—21no—
157C4×C67—21no—
158C4×C67—41no—
159C4×C67—41no—
160C4×C67—41no—
161C4×C67C2×#4111yescharacteristic
162C4×C672−1+411yescharacteristic
163Dic67—41no—
164Dic67—41no—
165Dic67—41no—
166C2×C67—41no—
167C2×C67—41no—
168C2×C67—81no—
169C2×C67#54711yescharacteristic
170C2×C67#57511yescharacteristic
171C2×C67#57611yescharacteristic
172D67—81no—
173#3355—671nomaximal, Sylow
174C67#335511yescharacteristic, complemented, Sylow
175#553—671no—
176#553D6711yescharacteristic, complemented
177#499—671no—
178#482—671no—
179#482—671no—
180#480—671no—
181#480—671no—
182#478—671no—
183#478—671no—
184#469—671no—
185#469—671no—
186C4×D8—671no—
187#436—671no—
188#428—671no—
189#428—671no—
190C2×D8—671no—
191C2×M16—671no—
192C2×M16—671no—
193#126—21no—
194#126—1341no—
195#125—671no—
196#125C2×D6711yescharacteristic
197#123—21no—
198#123—1341no—
199#121—21no—
200#121—671no—
201#121—671no—
202#121—1341no—
203#121C2×D6711yescharacteristic
204#121C2×D6711yescharacteristic
205C4×D4—21no—
206C4×D4—671no—
207C4×D4—1341no—
208C4×D4C2×D6711yescharacteristic
209C4×D4C2×D6711yescharacteristic
210#117—671no—
211#117C2×D6711yescharacteristic
212C2×C4⋊C4—671no—
213C2×C4⋊C4C2×D6711yescharacteristic
214#107—671no—
215#107—671no—
216#107—671no—
217#107—671no—
218#107—671no—
219#106—671no—
220#106—671no—
221#106—671no—
222#106—671no—
223C4⋊C8—671no—
224#102—671no—
225#102—671no—
226#102—671no—
227#102—671no—
228#102—671no—
229#102—671no—
230C22⋊C8—671no—
231C22⋊C8—671no—
232C4×C8—671no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—671no—
236C2×D4—1341no—
237C2×D4—1341no—
238C2×D4C22⋊D6711yescharacteristic
239C2×D4C22×D6711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1341no—
243C22×C4—1341no—
244C22×C4C22×D6711yescharacteristic
245C22×C4C22×D6711yescharacteristic
246D8—1342no—
247M16—1342no—
248M16—1342no—
249C2×C8—671no—
250C2×C8—671no—
251C2×C8—671no—
252C2×C8—671no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—671no—
257C4⋊C4—671no—
258C4⋊C4—671no—
259C4⋊C4—1341no—
260C4⋊C4—1341no—
261C4⋊C4—1341no—
262C4⋊C4C22⋊D6711yescharacteristic
263C4⋊C4C22×D6711yescharacteristic
264C4⋊C4C22×D6711yescharacteristic
265C4⋊C4C22×D6711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—1341no—
271C22⋊C4—1341no—
272C22⋊C4—1341no—
273C22⋊C4—1341no—
274C22⋊C4C22⋊D6711yescharacteristic
275C22⋊C4C22⋊D6711yescharacteristic
276C42—21no—
277C42—1341no—
278C42C22×D6711yescharacteristic
279C23—21no—
280C23—1341no—
281C23C2×C22⋊D6711yescharacteristic
282C23C2×C22⋊D6711yescharacteristic
283D4—21no—
284D4—21no—
285D4—41no—
286D4—42no—
287D4—81no—
288D4—2682no—
289D4—5361no—
290D4—1341no—
291D4—1341no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—5361no—
302C2×C4—5361no—
303C2×C4—1341no—
304C2×C4—1341no—
305C2×C4—1341no—
306C2×C4C2×C22⋊D6711yescharacteristic
307C2×C4C2×C22⋊D6711yescharacteristic
308C2×C4C2×C22⋊D6711yescharacteristic
309C2×C4C2×C22⋊D6711yescharacteristic
310C2×C4C23×D6711yescharacteristic
311C8—672no—
312C8—1341no—
313C8—1341no—
314C8—1341no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—2681no—
325C22—2681no—
326C22—2681no—
327C22C22×C22⋊D6711yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—2681no—
336C4—2681no—
337C4—2681no—
338C4#41×D6711yescharacteristic
339C4#1452409011yescharacteristic
340C2—41no—
341C2—41no—
342C2—81no—
343C2—5361no—
344C2#1686404511yescharacteristic
345C2#1686419411yescharacteristic
346C2#1686419911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D67
commutatorC2×C4×C67C24
FrattiniC2×C4C23×D67
Fitting#553×C67C2
socleC22×C67C22×D4
hypercenter#553D67
hypocenterC67#3355
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #16864092C2semidirect
2. #16863795C2semidirect
3. #16863720C2semidirect
4. #16863717C2semidirect
5. #16863709C2semidirect
6. #16863707C2semidirect
7. #16863696C2semidirect
8. #16863693C2semidirect
9. #16863647C2semidirect
10. #16863647C2semidirect
11. C4×#11761613C2semidirect
12. #16863474C2semidirect
13. #16863439C2semidirect
14. #16863436C2semidirect
15. #553×C67C2semidirect
174. C67#3355semidirect
176. #553D67semidirect