Finite Group Databasebeta

← #22562839

Subgroups — #22562839

Order 8576.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D67)⋊C4C211yesmaximal, characteristic
2#16863289C211yesmaximal, characteristic, complemented
3#16863280C211yesmaximal, characteristic, complemented
4#16863278C211yesmaximal, characteristic, complemented
5#16863272C211yesmaximal, characteristic
6#16863272C211yesmaximal, characteristic
7#16863270C211yesmaximal, characteristic, complemented
8#16863267C211yesmaximal, characteristic, complemented
9#16863263C211yesmaximal, characteristic, complemented
10#16863261C211yesmaximal, characteristic, complemented
11#16863257C211yesmaximal, characteristic, complemented
12#16863254C211yesmaximal, characteristic, complemented
13C4×(C2×D67)⋊C4C211yesmaximal, characteristic, complemented
14#95×D67C211yesmaximal, characteristic, complemented
15#388×C67C211yesmaximal, characteristic, complemented
16C22×C22⋊D67C2211yescharacteristic
17C23×Dic67C2211yescharacteristic
18C22×C4×D67C2211yescharacteristic
19C22×C4×D67C2211yescharacteristic
20C22×C4×D67C2211yescharacteristic
21(C22⋊D67)⋊C4C418yescomplemented
22C2×C4⋊Dic67C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×Dic67⋊C4C2211yescharacteristic
26C2×C22⋊Dic67C2211yescharacteristic
27C2×C22⋊Dic67C2211yescharacteristic
28C2×(C2×D67)⋊C4C2211yescharacteristic
29C2×(C2×D67)⋊C4C2211yescharacteristic
30C2×(C2×D67)⋊C4C2211yescharacteristic
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×C22⋊C4×C67C2211yescharacteristic
47C2×C22⋊C4×C67C2211yescharacteristic
48C2×C22⋊C4×C67C2211yescharacteristic
49C2×C42×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C23×D67C2311yescharacteristic
53C2×C22⋊D67—22no—
54C2×C22⋊D67—22no—
55C2×C22⋊D67C2×C414yes—
56C22×Dic67—41no—
57C22×Dic67—41no—
58C22×Dic67C2×C414yescomplemented
59C22×Dic67C2311yescharacteristic
60C22×Dic67C2311yescharacteristic
61C22×Dic67C2311yescharacteristic
62C22×Dic67C2311yescharacteristic
63C22×Dic67C2311yescharacteristic
64C22×Dic67C2311yescharacteristic
65C22×Dic67C2311yescharacteristic
66C2×C4×D67—22no—
67C2×C4×D67—22no—
68C2×C4×D67—41no—
69C2×C4×D67—41no—
70C2×C4×D67—41no—
71C2×C4×D67—41no—
72C2×C4×D67—41no—
73C2×C4×D67—41no—
74C2×C4×D67C2×C414yes—
75C4⋊Dic67—41no—
76Dic67⋊C4—41no—
77Dic67⋊C4—41no—
78Dic67⋊C4C2×C414yescomplemented
79C22⋊Dic67—41no—
80C22⋊Dic67—41no—
81(C2×D67)⋊C4—22no—
82(C2×D67)⋊C4—22no—
83(C2×D67)⋊C4C2×C414yes—
84C4×Dic67—41no—
85C4×Dic67C2×C414yescomplemented
86C24×C67C2311yescharacteristic
87C22×C4×C67C2311yescharacteristic
88C22×C4×C67C2311yescharacteristic
89C22×C4×C67C2311yescharacteristic
90C22×C4×C67C2311yescharacteristic
91C22×C4×C67C2311yescharacteristic
92C22×C4×C67C2311yescharacteristic
93C4⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—41no—
96C22⋊C4×C67C2×C414yescomplemented
97C42×C67—41no—
98C23×C67—21no—
99C23×C67—21no—
100C23×C67—21no—
101C23×C67—21no—
102C23×C67—21no—
103C23×C67—21no—
104C23×C67C22×C412yes—
105C23×C67C2411yescharacteristic
106C22×D67—21no—
107C22×D67—21no—
108C22×D67—21no—
109C22×D67—21no—
110C22×D67C22×C412yes—
111C22×D67#4112yes—
112C22×D67#4112yes—
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×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—21no—
122C2×C4×C67—21no—
123C2×C4×C67—21no—
124C2×C4×C67—21no—
125C2×C4×C67—21no—
126C2×C4×C67—21no—
127C2×C4×C67—21no—
128C2×C4×C67—21no—
129C2×C4×C67C22×C412yes—
130C2×C4×C67C22×C412yes—
131C22⋊D67—28no—
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×Dic67—21no—
143C2×Dic67—21no—
144C2×Dic67—21no—
145C2×Dic67—21no—
146C2×Dic67—21no—
147C2×Dic67—21no—
148C2×Dic67—21no—
149C2×Dic67—21no—
150C2×Dic67—44no—
151C2×Dic67C22×C412yes—
152C2×Dic67C22×C412yes—
153C2×Dic67C22×C412yes—
154C4×D67—44no—
155C4×D67—44no—
156C4×D67—44no—
157C22×C67—22no—
158C22×C67—22no—
159C22×C67—22no—
160C22×C67—41no—
161C22×C67—41no—
162C22×C67—41no—
163C22×C67—41no—
164C22×C67C23×C411yescharacteristic
165C22×C67C2×#4111yescharacteristic
166C22×C67C2×#4111yescharacteristic
167C22×C67H(5,𝔽2)11yescharacteristic
168C22×C67H(5,𝔽2)11yescharacteristic
169C22×C67H(5,𝔽2)11yescharacteristic
170C22×C672−1+411yescharacteristic
171C2×D67—22no—
172C2×D67—22no—
173C2×D67—22no—
174C2×D67—22no—
175C2×D67—22no—
176C2×D67—22no—
177C2×D67#11714yes—
178C4×C67—22no—
179C4×C67—22no—
180C4×C67—41no—
181C4×C67—41no—
182C4×C67—41no—
183C4×C67—41no—
184Dic67—22no—
185Dic67—22no—
186Dic67—22no—
187Dic67—41no—
188Dic67—41no—
189Dic67—41no—
190Dic67—41no—
191C2×C67—42no—
192C2×C67C2×#11711yescharacteristic
193C2×C67#51811yescharacteristic
194C2×C67#52011yescharacteristic
195C2×C67#53811yescharacteristic
196C2×C67#53811yescharacteristic
197C2×C67#53911yescharacteristic
198C2×C67#53911yescharacteristic
199D67—24no—
200#2318—671nomaximal, Sylow
201C67#231811yescharacteristic, complemented, Sylow
202C2×C4×D4—671no—
203#388—671no—
204#388D6711yescharacteristic, complemented
205#387—671no—
206#387—671no—
207#386—671no—
208#386—671no—
209#385—671no—
210#385—671no—
211#383—671no—
212#382—671no—
213#381—671no—
214#380—671no—
215C4×C22⋊C4—671no—
216C2×#95—671no—
217C22×D4—671no—
218C23×C4—671no—
219C23×C4—671no—
220C23×C4—671no—
221C23×C4—671no—
222C4×D4—678no—
223C2×C4⋊C4—671no—
224C2×C4⋊C4—671no—
225C2×C4⋊C4—671no—
226C2×C4⋊C4—671no—
227C2×C4⋊C4C2×D6711yescharacteristic
228C2×C22⋊C4—671no—
229C2×C22⋊C4—671no—
230C2×C22⋊C4—671no—
231C2×C22⋊C4—671no—
232C2×C22⋊C4—671no—
233C2×C22⋊C4C2×D6711yescharacteristic
234C2×C22⋊C4C2×D6711yescharacteristic
235C2×C22⋊C4C2×D6711yescharacteristic
236C2×C42—671no—
237C2×C42—671no—
238C2×C42C2×D6711yescharacteristic
239#95—671no—
240#95—671no—
241#95—671no—
242#95—671no—
243#95—671no—
244#95—671no—
245#95—671no—
246#95—671no—
247#95—671no—
248#95—671no—
249#95—671no—
250#95—671no—
251#95C2×D6711yescharacteristic
252#95C2×D6711yescharacteristic
253C24—671no—
254C24C22×D6711yescharacteristic
255C2×D4—674no—
256C2×D4—1342no—
257C2×D4—1342no—
258C22×C4—2681no—
259C22×C4—2681no—
260C22×C4—2681no—
261C22×C4—2681no—
262C22×C4—2681no—
263C22×C4—2681no—
264C22×C4—2681no—
265C22×C4—2681no—
266C22×C4—674no—
267C22×C4—674no—
268C22×C4—671no—
269C22×C4—671no—
270C22×C4—671no—
271C22×C4—671no—
272C22×C4—671no—
273C22×C4—671no—
274C22×C4—671no—
275C22×C4—1342no—
276C22×C4—1342no—
277C22×C4C22×D6711yescharacteristic
278C22×C4C22×D6711yescharacteristic
279C22×C4C22×D6711yescharacteristic
280C22×C4C22×D6711yescharacteristic
281C22×C4C22×D6711yescharacteristic
282C22×C4C22×D6711yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—2681no—
285C4⋊C4—2681no—
286C4⋊C4—2681no—
287C4⋊C4—674no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—2681no—
291C22⋊C4—2681no—
292C22⋊C4—674no—
293C22⋊C4—1342no—
294C22⋊C4—1342no—
295C22⋊C4C4×D6714yescomplemented
296C42—41no—
297C42—2681no—
298C42—674no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—672no—
306C23—672no—
307C23—672no—
308C23—1341no—
309C23—1341no—
310C23—1341no—
311C23—1341no—
312C23C2×C4×D6712yes—
313C23C23×D6711yescharacteristic
314D4—1348no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—2684no—
332C2×C4—2684no—
333C2×C4—2684no—
334C2×C4—2684no—
335C2×C4—672no—
336C2×C4—672no—
337C2×C4—672no—
338C2×C4—1341no—
339C2×C4—1341no—
340C2×C4—1341no—
341C2×C4—1341no—
342C2×C4—1341no—
343C2×C4—1341no—
344C2×C4—1341no—
345C2×C4—1341no—
346C2×C4—1341no—
347C2×C4—1341no—
348C2×C4—1341no—
349C2×C4—1341no—
350C2×C4—1341no—
351C2×C4—1341no—
352C2×C4—1341no—
353C2×C4—1341no—
354C2×C4—1341no—
355C2×C4—1341no—
356C2×C4C2×C4×D6712yes—
357C2×C4C2×C4×D6712yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—674no—
366C22—1342no—
367C22—1342no—
368C22—1342no—
369C22—1342no—
370C22—1342no—
371C22—1342no—
372C22C22×C4×D6711yescharacteristic
373C22#41×D6711yescharacteristic
374C22#41×D6711yescharacteristic
375C22#1452408811yescharacteristic
376C22#1452408811yescharacteristic
377C22#1452408911yescharacteristic
378C22#1452409011yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—2681no—
386C4—2681no—
387C4—2681no—
388C4—2681no—
389C4—1342no—
390C4—1342no—
391C4—1342no—
392C2—42no—
393C2—1344no—
394C2#117×D6711yescharacteristic
395C2#1686386811yescharacteristic
396C2#1686387511yescharacteristic
397C2#1686397811yescharacteristic
398C2#1686398011yescharacteristic
399C2#1686398711yescharacteristic
400C2#1686398911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#388×C67C2
socleC23×C67C24
hypercenter#388D67
hypocenterC67#2318
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #16863289C2semidirect
3. #16863280C2semidirect
4. #16863278C2semidirect
7. #16863270C2semidirect
8. #16863267C2semidirect
9. #16863263C2semidirect
10. #16863261C2semidirect
11. #16863257C2semidirect
12. #16863254C2semidirect
13. C4×(C2×D67)⋊C4C2semidirect
14. #95×D67C2semidirect
15. #388×C67C2semidirect
21. (C22⋊D67)⋊C4C4semidirect
58. C22×Dic67C2×C4semidirect
78. Dic67⋊C4C2×C4semidirect
85. C4×Dic67C2×C4semidirect
96. C22⋊C4×C67C2×C4semidirect
201. C67#2318semidirect
204. #388D67semidirect
295. C22⋊C4C4×D67semidirect