Finite Group Databasebeta

← #22563005

Subgroups — #22563005

Order 8576.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524028C211yesmaximal, characteristic
2#16863297C211yesmaximal, characteristic
3#16863291C211yesmaximal, characteristic, complemented
4#16863288C211yesmaximal, characteristic, complemented
5#16863287C211yesmaximal, characteristic, complemented
6#16863285C211yesmaximal, characteristic, complemented
7#16863285C211yesmaximal, characteristic, complemented
8#16863278C211yesmaximal, characteristic, complemented
9#16863274C211yesmaximal, characteristic
10#16863270C211yesmaximal, characteristic, complemented
11#16863255C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic67C211yesmaximal, characteristic, complemented
13C4×(C2×D67)⋊C4C211yesmaximal, characteristic, complemented
14#16863243C211yesmaximal, characteristic, complemented
15#387×C67C211yesmaximal, characteristic, complemented
16C22×D268C2211yescharacteristic
17C22×C4×D67C2211yescharacteristic
18C22×C4×D67C2211yescharacteristic
19#14524028C418yescomplemented
20C2×C4⋊Dic67C2211yescharacteristic
21C2×Dic67⋊C4C2211yescharacteristic
22C2×Dic67⋊C4C2211yescharacteristic
23C2×Dic67⋊C4C2211yescharacteristic
24C2×Dic67⋊C4C2211yescharacteristic
25C2×(C2×D67)⋊C4C2211yescharacteristic
26C2×(C2×D67)⋊C4C2211yescharacteristic
27C2×(C2×D67)⋊C4C2211yescharacteristic
28C2×(C2×D67)⋊C4C2211yescharacteristic
29C2×(C2×D67)⋊C4C2211yescharacteristic
30C2×(C2×D67)⋊C4C2211yescharacteristic
31C2×(C2×D67)⋊C4C2211yescharacteristic
32C2×(C2×D67)⋊C4C2211yescharacteristic
33C2×(C2×D67)⋊C4C2211yescharacteristic
34C2×(C2×D67)⋊C4C2211yescharacteristic
35C2×C4×Dic67C2211yescharacteristic
36C2×C4×Dic67C2211yescharacteristic
37C2×C4×Dic67C2211yescharacteristic
38C2×C4×Dic67C2211yescharacteristic
39C67⋊#95C2211yescharacteristic
40C67⋊#95C2211yescharacteristic
41C67⋊#95C2211yescharacteristic
42C67⋊#95C2211yescharacteristic
43C67⋊#95C2211yescharacteristic
44C67⋊#95C2211yescharacteristic
45C2×C4⋊C4×C67C2211yescharacteristic
46C2×C4⋊C4×C67C2211yescharacteristic
47C2×C42×C67C2211yescharacteristic
48#95×C67C2211yescharacteristic
49#95×C67C2211yescharacteristic
50#95×C67C2211yescharacteristic
51#95×C67C2211yescharacteristic
52C23×D67C2311yescharacteristic
53C23×D67C2311yescharacteristic
54C2×D268—41no—
55C2×D268—41no—
56C2×D268C2×C414yes—
57C22×Dic67C2311yescharacteristic
58C22×Dic67C2311yescharacteristic
59C22×Dic67C2311yescharacteristic
60C22×Dic67C2311yescharacteristic
61C22×Dic67C2311yescharacteristic
62C22×Dic67C2311yescharacteristic
63C2×C4×D67—22no—
64C2×C4×D67—41no—
65C2×C4×D67—41no—
66C2×C4×D67—41no—
67C2×C4×D67—41no—
68C2×C4×D67C2×C414yescomplemented
69C4⋊Dic67—22no—
70Dic67⋊C4—41no—
71Dic67⋊C4—41no—
72Dic67⋊C4—41no—
73Dic67⋊C4C2×C414yescomplemented
74(C2×D67)⋊C4—22no—
75(C2×D67)⋊C4—22no—
76(C2×D67)⋊C4—22no—
77(C2×D67)⋊C4—41no—
78(C2×D67)⋊C4—41no—
79(C2×D67)⋊C4—41no—
80(C2×D67)⋊C4—41no—
81(C2×D67)⋊C4C2×C414yes—
82(C2×D67)⋊C4C2×C414yes—
83(C2×D67)⋊C4C2×C414yescomplemented
84C4×Dic67—22no—
85C4×Dic67—41no—
86C4×Dic67—41no—
87C4×Dic67—41no—
88C22×C4×C67C2311yescharacteristic
89C22×C4×C67C2311yescharacteristic
90C22×C4×C67C2311yescharacteristic
91C22×C4×C67C2311yescharacteristic
92C22×C4×C67C2311yescharacteristic
93C22×C4×C67C2311yescharacteristic
94C22×C4×C67C2311yescharacteristic
95C4⋊C4×C67—41no—
96C4⋊C4×C67C2×C414yescomplemented
97C42×C67—41no—
98C23×C67C2411yescharacteristic
99C22×D67—21no—
100C22×D67—21no—
101C22×D67—21no—
102C22×D67—21no—
103C22×D67—21no—
104C22×D67—21no—
105C22×D67—21no—
106C22×D67—21no—
107C22×D67—21no—
108C22×D67—21no—
109C22×D67—21no—
110C22×D67—21no—
111C22×D67C22×C412yes—
112C22×D67C22×C412yes—
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×C67—21no—
130C2×C4×C67—21no—
131C2×C4×C67C22×C412yes—
132C2×C4×C67C22×C412yes—
133C2×C4×C67C22×C412yes—
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×Dic67C22×C412yes—
149C2×Dic67C22×C412yes—
150C2×Dic67#4112yes—
151C2×Dic67#4112yes—
152D268—44no—
153C4×D67—44no—
154C4×D67—44no—
155C22×C67C23×C411yescharacteristic
156C22×C67C2×#4111yescharacteristic
157C22×C67C2×#4111yescharacteristic
158C22×C67H(5,𝔽2)11yescharacteristic
159C22×C67H(5,𝔽2)11yescharacteristic
160C22×C67H(5,𝔽2)11yescharacteristic
161C22×C672−1+411yescharacteristic
162C2×D67—22no—
163C2×D67—22no—
164C2×D67—22no—
165C2×D67—22no—
166C2×D67—22no—
167C2×D67—22no—
168C2×D67—41no—
169C2×D67—41no—
170C2×D67—41no—
171C2×D67—41no—
172C2×D67—41no—
173C2×D67—41no—
174C2×D67—41no—
175C2×D67—41no—
176C4×C67—22no—
177C4×C67—22no—
178C4×C67—22no—
179C4×C67—41no—
180C4×C67—41no—
181C4×C67—41no—
182C4×C67—41no—
183Dic67—22no—
184Dic67—22no—
185Dic67—22no—
186Dic67—22no—
187Dic67—41no—
188Dic67—41no—
189C2×C67C4×#4111yescharacteristic
190C2×C67#51811yescharacteristic
191C2×C67#52011yescharacteristic
192C2×C67#53811yescharacteristic
193C2×C67#53911yescharacteristic
194C2×C67#54011yescharacteristic
195C2×C67#54211yescharacteristic
196D67—42no—
197D67—42no—
198#2335—671nomaximal, Sylow
199C67#233511yescharacteristic, complemented, Sylow
200C2×#123—671no—
201#390—671no—
202#389—671no—
203#388—671no—
204#388—671no—
205#388—671no—
206#388—671no—
207#387—671no—
208#387D6711yescharacteristic, complemented
209#386—671no—
210#385—671no—
211#380—671no—
212C4×C4⋊C4—671no—
213C4×C22⋊C4—671no—
214C4×C22⋊C4—671no—
215C22×D4—671no—
216C23×C4—671no—
217C23×C4—671no—
218#123—678no—
219C2×C4⋊C4—671no—
220C2×C4⋊C4—671no—
221C2×C4⋊C4—671no—
222C2×C4⋊C4—671no—
223C2×C4⋊C4—671no—
224C2×C4⋊C4C2×D6711yescharacteristic
225C2×C4⋊C4C2×D6711yescharacteristic
226C2×C22⋊C4—671no—
227C2×C22⋊C4—671no—
228C2×C22⋊C4—671no—
229C2×C22⋊C4—671no—
230C2×C22⋊C4—671no—
231C2×C22⋊C4—671no—
232C2×C22⋊C4—671no—
233C2×C22⋊C4—671no—
234C2×C22⋊C4—671no—
235C2×C22⋊C4—671no—
236C2×C42—671no—
237C2×C42—671no—
238C2×C42—671no—
239C2×C42—671no—
240C2×C42C2×D6711yescharacteristic
241#95—671no—
242#95—671no—
243#95—671no—
244#95—671no—
245#95—671no—
246#95—671no—
247#95C2×D6711yescharacteristic
248#95C2×D6711yescharacteristic
249#95C2×D6711yescharacteristic
250#95C2×D6711yescharacteristic
251C24—671no—
252C24—671no—
253C2×D4—2681no—
254C2×D4—2681no—
255C2×D4—674no—
256C22×C4—2681no—
257C22×C4—2681no—
258C22×C4—2681no—
259C22×C4—2681no—
260C22×C4—674no—
261C22×C4—671no—
262C22×C4—671no—
263C22×C4—671no—
264C22×C4—671no—
265C22×C4—671no—
266C22×C4—671no—
267C22×C4—1342no—
268C22×C4C22×D6711yescharacteristic
269C22×C4C22×D6711yescharacteristic
270C22×C4C22×D6711yescharacteristic
271C22×C4C22×D6711yescharacteristic
272C22×C4C22×D6711yescharacteristic
273C22×C4C22×D6711yescharacteristic
274C22×C4C22×D6711yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—2681no—
277C4⋊C4—2681no—
278C4⋊C4—2681no—
279C4⋊C4—674no—
280C4⋊C4—1342no—
281C4⋊C4C4×D6714yescomplemented
282C22⋊C4—2681no—
283C22⋊C4—2681no—
284C22⋊C4—2681no—
285C22⋊C4—2681no—
286C22⋊C4—674no—
287C22⋊C4—674no—
288C22⋊C4—674no—
289C22⋊C4—1342no—
290C22⋊C4—1342no—
291C22⋊C4—1342no—
292C42—41no—
293C42—2681no—
294C42—2681no—
295C42—2681no—
296C42—1342no—
297C23—672no—
298C23—672no—
299C23—1341no—
300C23—1341no—
301C23—1341no—
302C23—1341no—
303C23—1341no—
304C23—1341no—
305C23—1341no—
306C23—1341no—
307C23—1341no—
308C23—1341no—
309C23—1341no—
310C23—1341no—
311C23C23×D6711yescharacteristic
312D4—2684no—
313C2×C4—21no—
314C2×C4—21no—
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—672no—
334C2×C4—672no—
335C2×C4—672no—
336C2×C4—672no—
337C2×C4—1341no—
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×C4C2×C4×D6712yes—
352C2×C4C2×C4×D6712yes—
353C2×C4C2×C4×D6712yes—
354C22—2681no—
355C22—2681no—
356C22—2681no—
357C22—2681no—
358C22—2681no—
359C22—2681no—
360C22—2681no—
361C22—2681no—
362C22—1342no—
363C22—1342no—
364C22—1342no—
365C22—1342no—
366C22—1342no—
367C22—1342no—
368C22C22×C4×D6711yescharacteristic
369C22#41×D6711yescharacteristic
370C22#41×D6711yescharacteristic
371C22#1452408811yescharacteristic
372C22#1452408911yescharacteristic
373C22#1452408911yescharacteristic
374C22#1452409111yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—2681no—
383C4—2681no—
384C4—1342no—
385C4—1342no—
386C4—1342no—
387C4—1342no—
388C2—2682no—
389C2—2682no—
390C2#1686386411yescharacteristic
391C2#1686386811yescharacteristic
392C2#1686387811yescharacteristic
393C2#1686398511yescharacteristic
394C2#1686399411yescharacteristic
395C2#1686399811yescharacteristic
396C2#1686401611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D67
commutatorC22×C67C23×C4
Fitting#387×C67C2
socleC23×C67C24
hypercenter#387D67
hypocenterC67#2335
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #16863291C2semidirect
4. #16863288C2semidirect
5. #16863287C2semidirect
6. #16863285C2semidirect
7. #16863285C2semidirect
8. #16863278C2semidirect
10. #16863270C2semidirect
11. #16863255C2semidirect
12. C4⋊C4×Dic67C2semidirect
13. C4×(C2×D67)⋊C4C2semidirect
14. #16863243C2semidirect
15. #387×C67C2semidirect
19. #14524028C4semidirect
68. C2×C4×D67C2×C4semidirect
73. Dic67⋊C4C2×C4semidirect
83. (C2×D67)⋊C4C2×C4semidirect
96. C4⋊C4×C67C2×C4semidirect
199. C67#2335semidirect
208. #387D67semidirect
281. C4⋊C4C4×D67semidirect