Finite Group Databasebeta

← #22562947

Subgroups — #22562947

Order 8576.

444 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524033C211yesmaximal, characteristic
2#16863293C211yesmaximal, characteristic
3#16863292C211yesmaximal, characteristic
4#16863291C211yesmaximal, characteristic
5#16863291C211yesmaximal, characteristic
6#16863291C211yesmaximal, characteristic
7#16863282C211yesmaximal, characteristic
8#16863280C211yesmaximal, characteristic
9#16863279C211yesmaximal, characteristic
10#16863277C211yesmaximal, characteristic
11#16863265C211yesmaximal, characteristic
12C4×Dic67⋊C4C211yesmaximal, characteristic
13#16863247C211yesmaximal, characteristic
14#16863241C211yesmaximal, characteristic
15#389×C67C211yesmaximal, characteristic
16#14524033C418yescomplemented
17C2×C4⋊Dic67C2211yescharacteristic
18C2×C4⋊Dic67C2211yescharacteristic
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×Dic67⋊C4C2211yescharacteristic
27C2×Dic67⋊C4C2211yescharacteristic
28C2×Dic67⋊C4C2211yescharacteristic
29C2×C4×Dic67C2211yescharacteristic
30C2×C4×Dic67C2211yescharacteristic
31C2×C4×Dic67C2211yescharacteristic
32C2×C4×Dic67C2211yescharacteristic
33C2×C4×Dic67C2211yescharacteristic
34C2×C4×Dic67C2211yescharacteristic
35C67⋊#95C2211yescharacteristic
36C67⋊#95C2211yescharacteristic
37C67⋊#95C2211yescharacteristic
38C67⋊#95C2211yescharacteristic
39C67⋊#95C2211yescharacteristic
40C67⋊#95C2211yescharacteristic
41C67⋊#95C2211yescharacteristic
42C67⋊#95C2211yescharacteristic
43C67⋊#95C2211yescharacteristic
44C67⋊#95C2211yescharacteristic
45C2×C4⋊C4×C67C2211yescharacteristic
46C2×C4⋊C4×C67C2211yescharacteristic
47C2×C4⋊C4×C67C2211yescharacteristic
48C2×C4⋊C4×C67C2211yescharacteristic
49C2×C42×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C22×Dic67C2311yescharacteristic
53C22×Dic67C2311yescharacteristic
54C22×Dic67C2311yescharacteristic
55C22×Dic67C2311yescharacteristic
56C22×Dic67C2311yescharacteristic
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C4⋊Dic67—41no—
61C4⋊Dic67C2×C414yes—
62Dic67⋊C4—22no—
63Dic67⋊C4—22no—
64Dic67⋊C4—41no—
65Dic67⋊C4—41no—
66Dic67⋊C4—41no—
67Dic67⋊C4—41no—
68Dic67⋊C4C2×C414yes—
69Dic67⋊C4C2×C414yes—
70Dic67⋊C4C2×C414yes—
71Dic67⋊C4C2×C414yes—
72C4×Dic67—22no—
73C4×Dic67—41no—
74C4×Dic67—41no—
75C4×Dic67—41no—
76C4×Dic67C2×C414yes—
77C4×Dic67Q814yes—
78C22×C4×C67C2311yescharacteristic
79C22×C4×C67C2311yescharacteristic
80C22×C4×C67C2311yescharacteristic
81C22×C4×C67C2311yescharacteristic
82C22×C4×C67C2311yescharacteristic
83C22×C4×C67C2311yescharacteristic
84C22×C4×C67C2311yescharacteristic
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C4⋊C4×C67C2×C414yes—
89C42×C67—41no—
90C23×C67C2411yescharacteristic
91C2×C4×C67—21no—
92C2×C4×C67—21no—
93C2×C4×C67—21no—
94C2×C4×C67—21no—
95C2×C4×C67—21no—
96C2×C4×C67—21no—
97C2×C4×C67—21no—
98C2×C4×C67—21no—
99C2×C4×C67—21no—
100C2×C4×C67—21no—
101C2×C4×C67—21no—
102C2×C4×C67—21no—
103C2×C4×C67—21no—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67C22×C412yes—
109C2×C4×C67C22×C412yes—
110C2×C4×C67C22×C412yes—
111C2×C4×C67C2×Q812yes—
112C2×Dic67—21no—
113C2×Dic67—21no—
114C2×Dic67—21no—
115C2×Dic67—21no—
116C2×Dic67—21no—
117C2×Dic67—21no—
118C2×Dic67—21no—
119C2×Dic67—21no—
120C2×Dic67—21no—
121C2×Dic67—21no—
122C2×Dic67—21no—
123C2×Dic67—21no—
124C2×Dic67—21no—
125C2×Dic67—21no—
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67C22×C412yes—
130C2×Dic67C22×C412yes—
131C2×Dic67C22×C412yes—
132C2×Dic67C22×C412yes—
133C2×Dic67C2×Q812yes—
134C2×Dic67C2×Q812yes—
135C2×Dic67#4112yes—
136C22×C67C23×C411yescharacteristic
137C22×C67C22×Q811yescharacteristic
138C22×C67C2×#4111yescharacteristic
139C22×C67H(5,𝔽2)11yescharacteristic
140C22×C67H(5,𝔽2)11yescharacteristic
141C22×C672−1+411yescharacteristic
142C22×C672−1+411yescharacteristic
143C4×C67—22no—
144C4×C67—22no—
145C4×C67—22no—
146C4×C67—22no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150Dic67—22no—
151Dic67—22no—
152Dic67—22no—
153Dic67—22no—
154Dic67—41no—
155Dic67—41no—
156Dic67—41no—
157Dic67C4×Q814yes—
158C2×C67C2×C4×Q811yescharacteristic
159C2×C67#52011yescharacteristic
160C2×C67#52011yescharacteristic
161C2×C67#54011yescharacteristic
162C2×C67#54111yescharacteristic
163C2×C67#54411yescharacteristic
164C2×C67#54411yescharacteristic
165#2330—671nomaximal, Sylow
166C67#233011yescharacteristic, complemented, Sylow
167C2×#125—671no—
168#389—671no—
169#389—671no—
170#389—671no—
171#389—671no—
172#389—671no—
173#389D6711yescharacteristic
174#387—671no—
175#387—671no—
176#387—671no—
177#387—671no—
178#384—671no—
179C4×C4⋊C4—671no—
180C4×C4⋊C4—671no—
181#376—671no—
182#125—678no—
183C2×C4⋊C4—671no—
184C2×C4⋊C4—671no—
185C2×C4⋊C4—671no—
186C2×C4⋊C4—671no—
187C2×C4⋊C4—671no—
188C2×C4⋊C4—671no—
189C2×C4⋊C4—671no—
190C2×C4⋊C4—671no—
191C2×C4⋊C4—671no—
192C2×C4⋊C4—671no—
193C2×C4⋊C4—671no—
194C2×C4⋊C4—671no—
195C2×C4⋊C4C2×D6711yescharacteristic
196C2×C4⋊C4C2×D6711yescharacteristic
197C2×C4⋊C4C2×D6711yescharacteristic
198C2×C4⋊C4C2×D6711yescharacteristic
199C2×C42—671no—
200C2×C42—671no—
201C2×C42—671no—
202C2×C42—671no—
203C2×C42—671no—
204C2×C42—671no—
205C2×C42C2×D6711yescharacteristic
206#95—671no—
207#95—671no—
208#95—671no—
209#95—671no—
210#95—671no—
211#95—671no—
212#95—671no—
213#95—671no—
214#95—671no—
215#95—671no—
216#95C2×D6711yescharacteristic
217#95C2×D6711yescharacteristic
218C22×C4—671no—
219C22×C4—671no—
220C22×C4—671no—
221C22×C4—671no—
222C22×C4—671no—
223C22×C4—671no—
224C22×C4—671no—
225C22×C4—671no—
226C22×C4C22×D6711yescharacteristic
227C22×C4C22×D6711yescharacteristic
228C22×C4C22×D6711yescharacteristic
229C22×C4C22×D6711yescharacteristic
230C22×C4C22×D6711yescharacteristic
231C22×C4C22×D6711yescharacteristic
232C22×C4C22×D6711yescharacteristic
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—2681no—
237C4⋊C4—2681no—
238C4⋊C4—2681no—
239C4⋊C4—2681no—
240C4⋊C4—2681no—
241C4⋊C4—674no—
242C4⋊C4—674no—
243C4⋊C4—674no—
244C4⋊C4—674no—
245C4⋊C4—674no—
246C4⋊C4—1342no—
247C4⋊C4—1342no—
248C4⋊C4C4×D6714yes—
249C42—41no—
250C42—2681no—
251C42—2681no—
252C42—2681no—
253C42—674no—
254C42—674no—
255C42—1342no—
256C23C23×D6711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—672no—
275C2×C4—672no—
276C2×C4—672no—
277C2×C4—672no—
278C2×C4—672no—
279C2×C4—672no—
280C2×C4—672no—
281C2×C4—1341no—
282C2×C4—1341no—
283C2×C4—1341no—
284C2×C4—1341no—
285C2×C4—1341no—
286C2×C4—1341no—
287C2×C4—1341no—
288C2×C4—1341no—
289C2×C4—1341no—
290C2×C4—1341no—
291C2×C4—1341no—
292C2×C4—1341no—
293C2×C4—1341no—
294C2×C4—1341no—
295C2×C4—1341no—
296C2×C4—1341no—
297C2×C4—1341no—
298C2×C4C2×C4×D6712yes—
299C2×C4C2×C4×D6712yes—
300C2×C4C2×C4×D6712yes—
301C2×C4Q8×D6712yes—
302C22C22×C4×D6711yescharacteristic
303C22C2×Q8×D6711yescharacteristic
304C22#41×D6711yescharacteristic
305C22#1452408811yescharacteristic
306C22#1452408811yescharacteristic
307C22#1452409011yescharacteristic
308C22#1452409111yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—2681no—
317C4—2681no—
318C4—2681no—
319C4—674no—
320C4—1342no—
321C4—1342no—
322C4—1342no—
323C4—1342no—
324C2C4×Q8×D6711yescharacteristic
325C2#1686387511yescharacteristic
326C2#1686387711yescharacteristic
327C2#1686399611yescharacteristic
328C2#1686400511yescharacteristic
329C2#1686402311yescharacteristic
330C2#1686402511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#389×C67C2
socleC23×C67C24
hypercenter#389D67
hypocenterC67#2330
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
16. #14524033C4semidirect
166. C67#2330semidirect