Finite Group Databasebeta

← #22573888

Subgroups — #22573888

Order 8576.

1522 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14524091C211yesmaximal, characteristic, complemented
2C2×#41×D67C211yesmaximal, characteristic, complemented
3Q8×D268C211yesmaximal, characteristic, complemented
4Q8×C22⋊D67C211yesmaximal, characteristic, complemented
5#16864053C211yesmaximal, characteristic, complemented
6#16864051C213yesmaximal, complemented
7#16864048C213yesmaximal, complemented
8#16864043C213yesmaximal, complemented
9#16864039C213yesmaximal, complemented
10#16863975C213yesmaximal, complemented
11#16863966C213yesmaximal, complemented
12#16863905C213yesmaximal, complemented
13C22⋊Q8×D67C213yesmaximal, complemented
14#16863878C211yesmaximal, characteristic, complemented
15#548×C67C211yesmaximal, characteristic, complemented
16#14524091—28no—
17C2×#11761629C2211yescharacteristic
18C2×#11761629C2213yescomplemented
19C2×#11761628C2213yes—
20C2×#11761628C2213yescomplemented
21C2×#11761628C2213yescomplemented
22C2×#11761627C2213yes—
23#41×D67—28no—
24C2×Q8×D67C2211yescharacteristic
25C2×Q8×D67C2211yescharacteristic, complemented
26C2×Q8×D67C2213yescomplemented
27C2×D4×D67C2213yes—
28C22×C4×D67C2213yes—
29#14524043C2213yescomplemented
30#14524041C2213yescomplemented
31#14524030C2213yescomplemented
32#14524028C2213yescomplemented
33#14524025C2216yescomplemented
34#14524025C2216yescomplemented
35#14524025C2213yescomplemented
36#14524023C2213yes—
37#14524023C2213yescomplemented
38#14524022C2213yescomplemented
39#14524022C2213yescomplemented
40#14524021C2213yes—
41#14524020C2216yescomplemented
42#14524020C2213yescomplemented
43#14524019C2213yes—
44#14524016C2211yescharacteristic
45#14524016C2216yescomplemented
46#14524015C2213yescomplemented
47#14524014C2213yescomplemented
48#14524014C2216yescomplemented
49#14524013C2213yes—
50#14524011C2213yes—
51Q8×Dic67C2211yescharacteristic, complemented
52C4×D268C2213yes—
53C4×D268C2213yescomplemented
54#14524005C2213yescomplemented
55C4×C22⋊D67C2213yes—
56C4×C22⋊D67C2213yescomplemented
57#14524000C2213yescomplemented
58C4⋊C4×D67C2216yescomplemented
59C4⋊C4×D67C2213yescomplemented
60C22⋊C4×D67C2216yes—
61C2×#41×C67C2211yescharacteristic
62C22×Q8×C67C2211yescharacteristic, complemented
63#124×C67C2213yescomplemented
64C22⋊Q8×C67C2213yescomplemented
65#121×C67C2213yescomplemented
66C4×Q8×C67C2211yescharacteristic, complemented
67C4×D4×C67C2213yescomplemented
68C23×D67C2313yes—
69#11761629—212no—
70#11761629D418yescomplemented
71#11761628—212no—
72#11761628—212no—
73#11761628—212no—
74#11761627—212no—
75C2×Dic134C2313yes—
76C2×Dic134C2313yes—
77C2×Dic134C2313yes—
78Q8×D67—24no—
79Q8×D67—24no—
80Q8×D67—212no—
81C2×D268C2311yescharacteristic
82C2×D268C2313yes—
83C2×D268C2313yes—
84C2×C22⋊D67C2311yescharacteristic
85C2×C22⋊D67C2313yes—
86C2×C22⋊D67C2316yes—
87C2×C22⋊D67C2311yescharacteristic
88C2×C22⋊D67C2313yes—
89D4×D67—212no—
90C22×Dic67C2313yes—
91C2×C4×D67—212no—
92C2×C4×D67C2316yes—
93C2×C4×D67C2313yes—
94C2×C4×D67C2313yes—
95C2×C4×D67C2313yes—
96C2×C4×D67C2311yescharacteristic
97C2×C4×D67C2316yes—
98C2×C4×D67C2313yes—
99C4⋊Dic67C2313yes—
100C4⋊Dic67C2316yes—
101C4⋊Dic67C2313yes—
102Dic67⋊C4C2313yes—
103Dic67⋊C4C2313yes—
104Dic67⋊C4C2316yes—
105Dic67⋊C4C2311yescharacteristic
106Dic67⋊C4C2313yes—
107C22⋊Dic67C2316yes—
108C22⋊Dic67C2313yes—
109C22⋊Dic67C2313yes—
110(C2×D67)⋊C4C2313yes—
111(C2×D67)⋊C4C2313yes—
112(C2×D67)⋊C4C2313yes—
113(C2×D67)⋊C4C2313yes—
114(C2×D67)⋊C4C2311yescharacteristic
115(C2×D67)⋊C4C2313yes—
116C4×Dic67C2313yes—
117#41×C67—24no—
118C2×Q8×C67—24no—
119C2×Q8×C67C2313yes—
120C2×Q8×C67C2311yescharacteristic
121C2×D4×C67C2313yes—
122C2×D4×C67C2313yes—
123C22×C4×C67C2313yes—
124C22×C4×C67C2313yes—
125C4⋊C4×C67C2313yes—
126C4⋊C4×C67C2313yes—
127C22⋊C4×C67C2311yescharacteristic
128C22⋊C4×C67C2313yes—
129C22⋊C4×C67C2316yes—
130C42×C67C2313yes—
131C23×C67C2413yes—
132C23×C67C2411yescharacteristic
133C22×D67—26no—
134C22×D67—26no—
135C22×D67—43no—
136C22×D67C2411yescharacteristic
137C22×D67C2413yes—
138C22×D67C2411yescharacteristic
139Q8×C67—26no—
140Q8×C67C2×D414yes—
141D4×C67—26no—
142D4×C67—43no—
143C2×C4×C67—26no—
144C2×C4×C67—26no—
145C2×C4×C67C2411yescharacteristic
146C2×C4×C67C2413yes—
147C2×C4×C67C2413yes—
148C2×C4×C67C2411yescharacteristic
149C2×C4×C67C2413yes—
150C22⋊D67—212no—
151C22⋊D67—26no—
152C22⋊D67—22no—
153C22⋊D67—22no—
154C22⋊D67—43no—
155C2×Dic67—26no—
156C2×Dic67C2411yescharacteristic
157C2×Dic67C2413yes—
158C2×Dic67C2413yes—
159C2×Dic67C2413yes—
160C2×Dic67C2411yescharacteristic
161D268—26no—
162D268—22no—
163D268C2×D4112yes—
164C4×D67—26no—
165C4×D67—212no—
166C4×D67—26no—
167C4×D67—26no—
168C4×D67—22no—
169C4×D67—46no—
170C4×D67C2×D4112yes—
171Dic134—26no—
172Dic134—26no—
173Dic134—26no—
174C22×C67—23no—
175C22×C67—21no—
176C22×C67—21no—
177C22×C67—23no—
178C22×C67—23no—
179C22×C67—21no—
180C22×C67C2511yescharacteristic
181C2×D67—21no—
182C2×D67—23no—
183C2×D67—23no—
184C2×D67—21no—
185C2×D67—21no—
186C2×D67—21no—
187C2×D67—21no—
188C2×D67—412no—
189C2×D67C22×D416yes—
190C2×D672−1+412yescomplemented
191C4×C67—23no—
192C4×C67—21no—
193C4×C67—21no—
194C4×C67—23no—
195C4×C67C22×D416yes—
196Dic67—23no—
197Dic67—23no—
198Dic67—21no—
199Dic67—23no—
200Dic67C22×D412yes—
201C2×C67—43no—
202C2×C67—41no—
203C2×C67C23×D411yescharacteristic
204C2×C67C2×2−1+411yescharacteristic
205C2×C67#58511yescharacteristic
206D67—22no—
207D67—26no—
208D67—41no—
209#3489—671nomaximal, Sylow
210C67#348911yescharacteristic, complemented, Sylow
211C2×2−1+4—671no—
212C22×#41—671no—
213D4×Q8—671no—
214D4×Q8—671no—
215#548—671no—
216#548D6711yescharacteristic, complemented
217#547—673no—
218#547—673no—
219#546—673no—
220#546—673no—
221#537—673no—
222#536—673no—
223#525—673no—
224C2×C22⋊Q8—673no—
225#520—671no—
2262−1+4—1348no—
227C2×#41—671no—
228C2×#41—673no—
229C2×#41—673no—
230C2×#41—673no—
231C2×#41—673no—
232C2×#41—673no—
233C2×#41—1348no—
234C2×#41C2×D6711yescharacteristic
235C22×Q8—673no—
236C22×Q8—671no—
237C22×Q8—671no—
238C22×Q8C2×D6711yescharacteristic, complemented
239C22×D4—673no—
240C23×C4—673no—
241C4⋊Q8—673no—
242C4⋊Q8—673no—
243#124—673no—
244#124C2×D6713yescomplemented
245#123—673no—
246#123—673no—
247#123—676no—
248#123—676no—
249C22⋊Q8—673no—
250C22⋊Q8—673no—
251C22⋊Q8—673no—
252C22⋊Q8—673no—
253C22⋊Q8—673no—
254C22⋊Q8—673no—
255C22⋊Q8—676no—
256C22⋊Q8—673no—
257C22⋊Q8C2×D6713yescomplemented
258#121—673no—
259#121—676no—
260#121—673no—
261#121—671no—
262#121—676no—
263#121C2×D6713yescomplemented
264#120—673no—
265#120—673no—
266C4×Q8—671no—
267C4×Q8C2×D6711yescharacteristic, complemented
268C4×D4—673no—
269C4×D4—673no—
270C4×D4—673no—
271C4×D4—673no—
272C4×D4—673no—
273C4×D4C2×D6713yescomplemented
274#117—673no—
275C2×C4⋊C4—673no—
276C2×C4⋊C4—676no—
277C2×C22⋊C4—676no—
278C24—673no—
279#41—24no—
280#41—678no—
281#41—13412no—
282#41—13412no—
283#41—13412no—
284#41—13412no—
285#41—13412no—
286C2×Q8—24no—
287C2×Q8—673no—
288C2×Q8—673no—
289C2×Q8—673no—
290C2×Q8—13412no—
291C2×Q8—1344no—
292C2×Q8—1344no—
293C2×Q8C22×D6713yes—
294C2×Q8C22×D6711yescharacteristic
295C2×D4—673no—
296C2×D4—671no—
297C2×D4—673no—
298C2×D4—671no—
299C2×D4—676no—
300C2×D4—673no—
301C2×D4—671no—
302C2×D4—673no—
303C2×D4—13412no—
304C2×D4C22×D6713yes—
305C2×D4C22×D6713yes—
306C22×C4—676no—
307C22×C4—673no—
308C22×C4—673no—
309C22×C4—676no—
310C22×C4—673no—
311C22×C4—673no—
312C22×C4—673no—
313C22×C4—671no—
314C22×C4—13412no—
315C22×C4C22×D6713yes—
316C22×C4C22×D6713yes—
317C4⋊C4—673no—
318C4⋊C4—673no—
319C4⋊C4—673no—
320C4⋊C4—676no—
321C4⋊C4—676no—
322C4⋊C4—671no—
323C4⋊C4—673no—
324C4⋊C4—673no—
325C4⋊C4C22×D6713yes—
326C4⋊C4C22×D6713yes—
327C22⋊C4—676no—
328C22⋊C4—673no—
329C22⋊C4—673no—
330C22⋊C4—673no—
331C22⋊C4—673no—
332C22⋊C4—673no—
333C22⋊C4—673no—
334C22⋊C4—671no—
335C22⋊C4—673no—
336C22⋊C4C22×D6716yes—
337C22⋊C4C22×D6711yescharacteristic
338C22⋊C4C22×D6713yes—
339C42—673no—
340C42C22×D6713yes—
341C23—2683no—
342C23—671no—
343C23—673no—
344C23—671no—
345C23—1346no—
346C23—1346no—
347C23C23×D6713yes—
348C23C23×D6711yescharacteristic
349Q8—26no—
350Q8—1346no—
351Q8—1346no—
352Q8—1346no—
353Q8D4×D6714yes—
354D4—26no—
355D4—43no—
356D4—2683no—
357D4—6712no—
358D4—1346no—
359D4—1342no—
360D4—1342no—
361D4—13412no—
362D4—1346no—
363D4—1342no—
364C2×C4—26no—
365C2×C4—26no—
366C2×C4—2686no—
367C2×C4—671no—
368C2×C4—6712no—
369C2×C4—671no—
370C2×C4—673no—
371C2×C4—673no—
372C2×C4—673no—
373C2×C4—1342no—
374C2×C4—1346no—
375C2×C4—13412no—
376C2×C4—1346no—
377C2×C4—1346no—
378C2×C4—1346no—
379C2×C4C23×D6711yescharacteristic
380C2×C4C23×D6713yes—
381C2×C4C23×D6713yes—
382C2×C4C23×D6711yescharacteristic
383C2×C4C23×D6713yes—
384C22—21no—
385C22—23no—
386C22—21no—
387C22—23no—
388C22—21no—
389C22—23no—
390C22—26812no—
391C22—676no—
392C22—672no—
393C22—1341no—
394C22—1341no—
395C22—1341no—
396C22—1341no—
397C22—1343no—
398C22—1343no—
399C22—1341no—
400C22C24×D6711yescharacteristic
401C4—21no—
402C4—23no—
403C4—21no—
404C4—23no—
405C4—672no—
406C4—1343no—
407C4—1343no—
408C4—1341no—
409C4—1343no—
410C4C2×D4×D6716yes—
411C2—43no—
412C2—41no—
413C2—2681no—
414C2—1342no—
415C2—1346no—
416C2C22×D4×D6711yescharacteristic
417C22−1+4×D6711yescharacteristic
418C2#1686423211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D67
commutator, socleC22×C67C25
Fitting#548×C67C2
hypercenter#548D67
hypocenterC67#3489
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#14524091C2semidirect
2. C2×#41×D67C2semidirect
3. Q8×D268C2semidirect
4. Q8×C22⋊D67C2semidirect
5. #16864053C2semidirect
6. #16864051C2semidirect
7. #16864048C2semidirect
8. #16864043C2semidirect
9. #16864039C2semidirect
10. #16863975C2semidirect
11. #16863966C2semidirect
12. #16863905C2semidirect
13. C22⋊Q8×D67C2semidirect
14. #16863878C2semidirect
15. #548×C67C2semidirect
18. C2×#11761629C22semidirect
20. C2×#11761628C22semidirect
21. C2×#11761628C22semidirect
25. C2×Q8×D67C22semidirect
26. C2×Q8×D67C22semidirect
29. #14524043C22semidirect
30. #14524041C22semidirect
31. #14524030C22semidirect
32. #14524028C22semidirect
33. #14524025C22semidirect
34. #14524025C22semidirect
35. #14524025C22semidirect
37. #14524023C22semidirect
38. #14524022C22semidirect
39. #14524022C22semidirect
41. #14524020C22semidirect
42. #14524020C22semidirect
45. #14524016C22semidirect
46. #14524015C22semidirect
47. #14524014C22semidirect
48. #14524014C22semidirect
51. Q8×Dic67C22semidirect
53. C4×D268C22semidirect
54. #14524005C22semidirect
56. C4×C22⋊D67C22semidirect
57. #14524000C22semidirect
58. C4⋊C4×D67C22semidirect
59. C4⋊C4×D67C22semidirect
62. C22×Q8×C67C22semidirect
63. #124×C67C22semidirect
64. C22⋊Q8×C67C22semidirect
65. #121×C67C22semidirect
66. C4×Q8×C67C22semidirect
67. C4×D4×C67C22semidirect
70. #11761629D4semidirect
190. C2×D672−1+4semidirect
210. C67#3489semidirect
216. #548D67semidirect
238. C22×Q8C2×D67semidirect
244. #124C2×D67semidirect
257. C22⋊Q8C2×D67semidirect
263. #121C2×D67semidirect
267. C4×Q8C2×D67semidirect
273. C4×D4C2×D67semidirect