Finite Group Databasebeta

← #22565622

Subgroups — #22565622

Order 8576.

412 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16863327C211yesmaximal, characteristic
2#16863316C211yesmaximal, characteristic
3#16863316C211yesmaximal, characteristic
4#16863316C211yesmaximal, characteristic
5#16863301C211yesmaximal, characteristic
6#16863300C211yesmaximal, characteristic
7#16863300C211yesmaximal, characteristic
8#16863291C211yesmaximal, characteristic
9#16863280C211yesmaximal, characteristic
10#16863278C211yesmaximal, characteristic
11#16863277C211yesmaximal, characteristic
12#16863277C211yesmaximal, characteristic
13C4×Dic67⋊C4C211yesmaximal, characteristic
14#16863247C211yesmaximal, characteristic
15#389×C67C211yesmaximal, characteristic
16C22×Dic134C2211yescharacteristic
17C22×Dic134C2211yescharacteristic
18C2×Dic67⋊C4C2211yescharacteristic
19C2×Dic67⋊C4C2211yescharacteristic
20C2×Dic67⋊C4C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×C4×Dic67C2211yescharacteristic
27C2×C4×Dic67C2211yescharacteristic
28C2×C4×Dic67C2211yescharacteristic
29C2×C4×Dic67C2211yescharacteristic
30C67⋊#95C2211yescharacteristic
31C67⋊#95C2211yescharacteristic
32C67⋊#95C2211yescharacteristic
33C67⋊#95C2211yescharacteristic
34C67⋊#95C2211yescharacteristic
35C67⋊#95C2211yescharacteristic
36C67⋊#95C2211yescharacteristic
37C67⋊#95C2211yescharacteristic
38C67⋊#95C2211yescharacteristic
39C67⋊#95C2211yescharacteristic
40C67⋊#95C2211yescharacteristic
41C67⋊#95C2211yescharacteristic
42C67⋊#95C2211yescharacteristic
43C67⋊#95C2211yescharacteristic
44C2×C4⋊C4×C67C2211yescharacteristic
45C2×C4⋊C4×C67C2211yescharacteristic
46C2×C4⋊C4×C67C2211yescharacteristic
47C2×C4⋊C4×C67C2211yescharacteristic
48C2×C42×C67C2211yescharacteristic
49#95×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51C2×Dic134—22no—
52C2×Dic134—22no—
53C2×Dic134—22no—
54C2×Dic134—41no—
55C2×Dic134—41no—
56C2×Dic134—41no—
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C22×Dic67C2311yescharacteristic
61C22×Dic67C2311yescharacteristic
62C22×Dic67C2311yescharacteristic
63C22×Dic67C2311yescharacteristic
64C22×Dic67C2311yescharacteristic
65Dic67⋊C4—22no—
66Dic67⋊C4—22no—
67Dic67⋊C4—22no—
68Dic67⋊C4—41no—
69Dic67⋊C4—41no—
70Dic67⋊C4—41no—
71Dic67⋊C4Q814yescomplemented
72Dic67⋊C4Q814yescomplemented
73C4×Dic67—22no—
74C4×Dic67—22no—
75C4×Dic67—41no—
76C4×Dic67—41no—
77C22×C4×C67C2311yescharacteristic
78C22×C4×C67C2311yescharacteristic
79C22×C4×C67C2311yescharacteristic
80C22×C4×C67C2311yescharacteristic
81C22×C4×C67C2311yescharacteristic
82C22×C4×C67C2311yescharacteristic
83C22×C4×C67C2311yescharacteristic
84C4⋊C4×C67—22no—
85C4⋊C4×C67—41no—
86C4⋊C4×C67—41no—
87C4⋊C4×C67—41no—
88C42×C67—41no—
89C23×C67C2411yescharacteristic
90C2×C4×C67—21no—
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×C67C2×Q812yes—
109C2×C4×C67C2×Q812yes—
110C2×C4×C67#4112yes—
111C2×Dic67—21no—
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×Dic67C2×Q812yes—
130C2×Dic67C2×Q812yes—
131C2×Dic67C2×Q812yes—
132C2×Dic67C2×Q812yes—
133C2×Dic67#4112yes—
134C2×Dic67#4112yes—
135Dic134—44no—
136Dic134—44no—
137C22×C67C22×Q811yescharacteristic
138C22×C67C22×Q811yescharacteristic
139C22×C67C2×#4111yescharacteristic
140C22×C67C2×#4111yescharacteristic
141C22×C67C2×#4111yescharacteristic
142C22×C67H(5,𝔽2)11yescharacteristic
143C22×C67H(5,𝔽2)11yescharacteristic
144C4×C67—22no—
145C4×C67—22no—
146C4×C67—22no—
147C4×C67—41no—
148C4×C67—41no—
149C4×C67—41no—
150C4×C67—41no—
151Dic67—22no—
152Dic67—22no—
153Dic67—22no—
154Dic67—22no—
155Dic67—22no—
156Dic67—22no—
157Dic67—41no—
158Dic67—41no—
159C2×C67#53311yescharacteristic
160C2×C67#53311yescharacteristic
161C2×C67#53811yescharacteristic
162C2×C67#55111yescharacteristic
163C2×C67#55411yescharacteristic
164C2×C67#55411yescharacteristic
165C2×C67Q8211yescharacteristic
166#2586—671nomaximal, Sylow
167C67#258611yescharacteristic, complemented, Sylow
168#398—671no—
169#395—671no—
170#395—671no—
171#395—671no—
172#391—671no—
173#391—671no—
174#391—671no—
175#389—671no—
176#389D6711yescharacteristic
177#387—671no—
178#387—671no—
179#387—671no—
180#387—671no—
181C4×C4⋊C4—671no—
182C4×C4⋊C4—671no—
183C22×Q8—671no—
184C22×Q8—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⋊C4C2×D6711yescharacteristic
194C2×C4⋊C4C2×D6711yescharacteristic
195C2×C4⋊C4C2×D6711yescharacteristic
196C2×C4⋊C4C2×D6711yescharacteristic
197C2×C42—671no—
198C2×C42—671no—
199C2×C42—671no—
200C2×C42—671no—
201C2×C42C2×D6711yescharacteristic
202#95—671no—
203#95—671no—
204#95—671no—
205#95—671no—
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
218C2×Q8—2681no—
219C2×Q8—2681no—
220C2×Q8—2681no—
221C2×Q8—1342no—
222C2×Q8—1342no—
223C2×Q8—1342no—
224C22×C4—671no—
225C22×C4—671no—
226C22×C4—671no—
227C22×C4—671no—
228C22×C4—671no—
229C22×C4—671no—
230C22×C4—671no—
231C22×C4—671no—
232C22×C4C22×D6711yescharacteristic
233C22×C4C22×D6711yescharacteristic
234C22×C4C22×D6711yescharacteristic
235C22×C4C22×D6711yescharacteristic
236C22×C4C22×D6711yescharacteristic
237C22×C4C22×D6711yescharacteristic
238C22×C4C22×D6711yescharacteristic
239C4⋊C4—22no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—2681no—
244C4⋊C4—2681no—
245C4⋊C4—2681no—
246C4⋊C4—674no—
247C4⋊C4—674no—
248C4⋊C4—1342no—
249C4⋊C4—1342no—
250C4⋊C4—1342no—
251C42—41no—
252C42—2681no—
253C42—2681no—
254C42—1342no—
255C42—1342no—
256C23C23×D6711yescharacteristic
257Q8—2684no—
258Q8—2684no—
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—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—672no—
278C2×C4—672no—
279C2×C4—672no—
280C2×C4—672no—
281C2×C4—672no—
282C2×C4—672no—
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×C4—1341no—
299C2×C4—1341no—
300C2×C4—1341no—
301C2×C4Q8×D6712yes—
302C2×C4Q8×D6712yes—
303C2×C4#1176162812yes—
304C22C2×Q8×D6711yescharacteristic
305C22C2×Q8×D6711yescharacteristic
306C22#41×D6711yescharacteristic
307C22#41×D6711yescharacteristic
308C22C2×#1176162811yescharacteristic
309C22#1452408811yescharacteristic
310C22#1452408811yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—2681no—
319C4—2681no—
320C4—1342no—
321C4—1342no—
322C4—1342no—
323C4—1342no—
324C4—1342no—
325C4—1342no—
326C2#1686395011yescharacteristic
327C2#1686395211yescharacteristic
328C2#1686397811yescharacteristic
329C2#1686407511yescharacteristic
330C2#1686410011yescharacteristic
331C2#1686410111yescharacteristic
332C2#1686412611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutator, socleC23×C67C24
Fitting#389×C67C2
hypercenter#389D67
hypocenterC67#2586
LCSG ▷ C23×C67 ▷ C67
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
71. Dic67⋊C4Q8semidirect
72. Dic67⋊C4Q8semidirect
167. C67#2586semidirect