Finite Group Databasebeta

← #22563293

Subgroups — #22563293

Order 8576.

548 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524021C211yesmaximal, characteristic
2#16863302C211yesmaximal, characteristic, complemented
3#16863300C211yesmaximal, characteristic, complemented
4#16863282C211yesmaximal, characteristic, complemented
5#16863280C211yesmaximal, characteristic, complemented
6#16863280C211yesmaximal, characteristic, complemented
7#16863277C211yesmaximal, characteristic, complemented
8#16863270C211yesmaximal, characteristic
9#16863267C211yesmaximal, characteristic
10#16863247C211yesmaximal, characteristic, complemented
11C4×(C2×D67)⋊C4C211yesmaximal, characteristic
12C4×(C2×D67)⋊C4C211yesmaximal, characteristic
13C4×(C2×D67)⋊C4C211yesmaximal, characteristic
14#95×D67C211yesmaximal, characteristic
15C4×C4⋊C4×C67C211yesmaximal, characteristic, complemented
16C22×Dic134C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C22×C4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20#14524021C418yescomplemented
21C2×C4⋊Dic67C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C2×(C2×D67)⋊C4C2211yescharacteristic
28C2×(C2×D67)⋊C4C2211yescharacteristic
29C2×(C2×D67)⋊C4C2211yescharacteristic
30C2×C4×Dic67C2211yescharacteristic
31C2×C4×Dic67C2211yescharacteristic
32C2×C4×Dic67C2211yescharacteristic
33C67⋊#95C2211yescharacteristic
34C67⋊#95C2211yescharacteristic
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×C42×C67C2211yescharacteristic
48C2×C42×C67C2211yescharacteristic
49C2×C42×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C23×D67C2311yescharacteristic
53C2×Dic134—41no—
54C2×Dic134—41no—
55C2×Dic134C2×C414yescomplemented
56C22×Dic67C2311yescharacteristic
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C22×Dic67C2311yescharacteristic
61C22×Dic67C2311yescharacteristic
62C22×Dic67C2311yescharacteristic
63C2×C4×D67—22no—
64C2×C4×D67—22no—
65C2×C4×D67—22no—
66C2×C4×D67—22no—
67C2×C4×D67—41no—
68C2×C4×D67—41no—
69C2×C4×D67—41no—
70C2×C4×D67—41no—
71C2×C4×D67C2×C414yes—
72C4⋊Dic67—41no—
73Dic67⋊C4—22no—
74Dic67⋊C4—41no—
75Dic67⋊C4C2×C414yescomplemented
76Dic67⋊C4C2×C414yescomplemented
77(C2×D67)⋊C4—22no—
78(C2×D67)⋊C4C2×C414yes—
79(C2×D67)⋊C4C2×C414yes—
80(C2×D67)⋊C4Q814yescomplemented
81C4×Dic67—22no—
82C4×Dic67—41no—
83C4×Dic67—41no—
84C22×C4×C67C2311yescharacteristic
85C22×C4×C67C2311yescharacteristic
86C22×C4×C67C2311yescharacteristic
87C22×C4×C67C2311yescharacteristic
88C22×C4×C67C2311yescharacteristic
89C22×C4×C67C2311yescharacteristic
90C22×C4×C67C2311yescharacteristic
91C4⋊C4×C67—22no—
92C4⋊C4×C67C2×C414yescomplemented
93C42×C67—22no—
94C42×C67—41no—
95C42×C67—41no—
96C23×C67C2411yescharacteristic
97C22×D67—21no—
98C22×D67—21no—
99C22×D67—21no—
100C22×D67—21no—
101C22×D67C22×C412yes—
102C22×D67C2×Q812yes—
103C22×D67#4112yes—
104C2×C4×C67—21no—
105C2×C4×C67—21no—
106C2×C4×C67—21no—
107C2×C4×C67—21no—
108C2×C4×C67—21no—
109C2×C4×C67—21no—
110C2×C4×C67—21no—
111C2×C4×C67—21no—
112C2×C4×C67—21no—
113C2×C4×C67—21no—
114C2×C4×C67—21no—
115C2×C4×C67—21no—
116C2×C4×C67—21no—
117C2×C4×C67—21no—
118C2×C4×C67—21no—
119C2×C4×C67C22×C412yes—
120C2×C4×C67C22×C412yes—
121C2×C4×C67C22×C412yes—
122C2×C4×C67C2×Q812yes—
123C2×C4×C67#4112yes—
124C2×C4×C67#4112yes—
125C2×Dic67—21no—
126C2×Dic67—21no—
127C2×Dic67—21no—
128C2×Dic67—21no—
129C2×Dic67—21no—
130C2×Dic67—21no—
131C2×Dic67—21no—
132C2×Dic67—21no—
133C2×Dic67—21no—
134C2×Dic67—21no—
135C2×Dic67—21no—
136C2×Dic67—21no—
137C2×Dic67—21no—
138C2×Dic67—21no—
139C2×Dic67—21no—
140C2×Dic67—21no—
141C2×Dic67—21no—
142C2×Dic67C22×C412yes—
143C2×Dic67C22×C412yes—
144C2×Dic67C22×C412yes—
145C2×Dic67C2×Q812yes—
146C4×D67—44no—
147C4×D67—44no—
148C4×D67—44no—
149Dic134—44no—
150C22×C67C23×C411yescharacteristic
151C22×C67C22×Q811yescharacteristic
152C22×C67C2×#4111yescharacteristic
153C22×C67C2×#4111yescharacteristic
154C22×C67C2×#4111yescharacteristic
155C22×C67H(5,𝔽2)11yescharacteristic
156C22×C67H(5,𝔽2)11yescharacteristic
157C2×D67—22no—
158C2×D67—22no—
159C2×D67—22no—
160C2×D67—22no—
161C2×D67—22no—
162C2×D67—22no—
163C2×D67C4×Q814yes—
164C4×C67—22no—
165C4×C67—22no—
166C4×C67—22no—
167C4×C67—22no—
168C4×C67—22no—
169C4×C67—22no—
170C4×C67—41no—
171Dic67—22no—
172Dic67—22no—
173Dic67—22no—
174Dic67—22no—
175Dic67—41no—
176Dic67—41no—
177Dic67—41no—
178C2×C67C2×C4×Q811yescharacteristic
179C2×C67C4×#4111yescharacteristic
180C2×C67#51811yescharacteristic
181C2×C67#55111yescharacteristic
182C2×C67#55111yescharacteristic
183C2×C67#55411yescharacteristic
184C2×C67#55411yescharacteristic
185D67—24no—
186#2368—671nomaximal, Sylow
187C67#236811yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—671no—
189#391—671no—
190#391—671no—
191#387—671no—
192#387—671no—
193#387—671no—
194#387—671no—
195#385—671no—
196#385—671no—
197C4×C4⋊C4—671no—
198C4×C4⋊C4D6711yescharacteristic, complemented
199C4×C22⋊C4—671no—
200C4×C22⋊C4—671no—
201C4×C22⋊C4—671no—
202C2×#95—671no—
203C22×Q8—671no—
204C23×C4—671no—
205C23×C4—671no—
206C23×C4—671no—
207C22⋊Q8—678no—
208C2×C4⋊C4—671no—
209C2×C4⋊C4—671no—
210C2×C4⋊C4—671no—
211C2×C4⋊C4—671no—
212C2×C4⋊C4—671no—
213C2×C4⋊C4C2×D6711yescharacteristic
214C2×C4⋊C4C2×D6711yescharacteristic
215C2×C22⋊C4—671no—
216C2×C22⋊C4—671no—
217C2×C22⋊C4—671no—
218C2×C22⋊C4—671no—
219C2×C42—671no—
220C2×C42—671no—
221C2×C42—671no—
222C2×C42C2×D6711yescharacteristic
223C2×C42C2×D6711yescharacteristic
224C2×C42C2×D6711yescharacteristic
225#95—671no—
226#95—671no—
227#95—671no—
228#95—671no—
229#95—671no—
230#95—671no—
231#95—671no—
232#95—671no—
233#95—671no—
234#95—671no—
235#95—671no—
236#95—671no—
237#95C2×D6711yescharacteristic
238#95C2×D6711yescharacteristic
239C24—671no—
240C2×Q8—2681no—
241C2×Q8—2681no—
242C2×Q8—674no—
243C22×C4—2681no—
244C22×C4—2681no—
245C22×C4—2681no—
246C22×C4—2681no—
247C22×C4—674no—
248C22×C4—671no—
249C22×C4—671no—
250C22×C4—671no—
251C22×C4—671no—
252C22×C4—671no—
253C22×C4—671no—
254C22×C4—671no—
255C22×C4—1342no—
256C22×C4—1342no—
257C22×C4—1342no—
258C22×C4—1342no—
259C22×C4C22×D6711yescharacteristic
260C22×C4C22×D6711yescharacteristic
261C22×C4C22×D6711yescharacteristic
262C22×C4C22×D6711yescharacteristic
263C22×C4C22×D6711yescharacteristic
264C22×C4C22×D6711yescharacteristic
265C22×C4C22×D6711yescharacteristic
266C4⋊C4—22no—
267C4⋊C4—2681no—
268C4⋊C4—2681no—
269C4⋊C4—674no—
270C4⋊C4—674no—
271C4⋊C4—1342no—
272C4⋊C4C4×D6714yescomplemented
273C22⋊C4—674no—
274C22⋊C4—674no—
275C22⋊C4—674no—
276C22⋊C4—1342no—
277C42—22no—
278C42—41no—
279C42—41no—
280C42—2681no—
281C42—2681no—
282C42—1342no—
283C23—672no—
284C23—672no—
285C23—672no—
286C23—1341no—
287C23—1341no—
288C23—1341no—
289C23—1341no—
290C23C23×D6711yescharacteristic
291Q8—2684no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—2684no—
308C2×C4—2684no—
309C2×C4—2684no—
310C2×C4—672no—
311C2×C4—672no—
312C2×C4—672no—
313C2×C4—672no—
314C2×C4—1341no—
315C2×C4—1341no—
316C2×C4—1341no—
317C2×C4—1341no—
318C2×C4—1341no—
319C2×C4—1341no—
320C2×C4—1341no—
321C2×C4—1341no—
322C2×C4—1341no—
323C2×C4—1341no—
324C2×C4—1341no—
325C2×C4—1341no—
326C2×C4—1341no—
327C2×C4—1341no—
328C2×C4—1341no—
329C2×C4—1341no—
330C2×C4—1341no—
331C2×C4C2×C4×D6712yes—
332C2×C4C2×C4×D6712yes—
333C2×C4C2×C4×D6712yes—
334C2×C4Q8×D6712yes—
335C2×C4#1176162812yes—
336C2×C4#1176162812yes—
337C22—674no—
338C22—1342no—
339C22—1342no—
340C22—1342no—
341C22—1342no—
342C22—1342no—
343C22—1342no—
344C22C22×C4×D6711yescharacteristic
345C22C2×Q8×D6711yescharacteristic
346C22#41×D6711yescharacteristic
347C22C2×#1176162811yescharacteristic
348C22C2×#1176162811yescharacteristic
349C22#1452408811yescharacteristic
350C22#1452408911yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—41no—
358C4—2681no—
359C4—2681no—
360C4—2681no—
361C4—1342no—
362C4—1342no—
363C4—1342no—
364C4—1342no—
365C2—1344no—
366C2C4×Q8×D6711yescharacteristic
367C2C4×#1176162811yescharacteristic
368C2#1686386811yescharacteristic
369C2#1686407011yescharacteristic
370C2#1686407511yescharacteristic
371C2#1686410111yescharacteristic
372C2#1686410611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
FittingC4×C4⋊C4×C67C2
socleC23×C67C24
hypercenterC4×C4⋊C4D67
hypocenterC67#2368
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #16863302C2semidirect
3. #16863300C2semidirect
4. #16863282C2semidirect
5. #16863280C2semidirect
6. #16863280C2semidirect
7. #16863277C2semidirect
10. #16863247C2semidirect
15. C4×C4⋊C4×C67C2semidirect
20. #14524021C4semidirect
55. C2×Dic134C2×C4semidirect
75. Dic67⋊C4C2×C4semidirect
76. Dic67⋊C4C2×C4semidirect
80. (C2×D67)⋊C4Q8semidirect
92. C4⋊C4×C67C2×C4semidirect
187. C67#2368semidirect
198. C4×C4⋊C4D67semidirect
272. C4⋊C4C4×D67semidirect