Finite Group Databasebeta

← #21457834

Subgroups — #21457834

Order 7808.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D244C211yesmaximal, characteristic
2#16840128C211yesmaximal, characteristic, complemented
3#16840086C211yesmaximal, characteristic, complemented
4#16840075C211yesmaximal, characteristic, complemented
5#16840059C211yesmaximal, characteristic, complemented
6#16840010C211yesmaximal, characteristic, complemented
7#16839993C211yesmaximal, characteristic, complemented
8#16839962C211yesmaximal, characteristic, complemented
9#16839912C211yesmaximal, characteristic, complemented
10#16839912C211yesmaximal, characteristic, complemented
11#16839860C211yesmaximal, characteristic, complemented
12#16839834C211yesmaximal, characteristic, complemented
13#16839766C211yesmaximal, characteristic, complemented
14C2×#14519697C211yesmaximal, characteristic, complemented
15C61×#552C211yesmaximal, characteristic, complemented
16C22×D244C2211yescharacteristic
17C22×D244C2211yescharacteristic
18C2×D4×D61—21no—
19C2×#11758201C2211yescharacteristic
20C2×#11758201C2211yescharacteristic, complemented
21C2×#11758201C2211yescharacteristic, complemented
22C2×#11758194C2211yescharacteristic, complemented
23C22×#92418C2211yescharacteristic, complemented
24#14519774C2211yescharacteristic
25#14519752—21no—
26#14519751C2211yescharacteristic
27#14519751C2211yescharacteristic
28#14519747—21no—
29#14519747—21no—
30C4×D244C2211yescharacteristic
31#14519710C2211yescharacteristic, complemented
32#14519710C2211yescharacteristic, complemented
33#14519708C2211yescharacteristic, complemented
34C4⋊#92418C2211yescharacteristic, complemented
35#14519701C2211yescharacteristic, complemented
36#14519697C2211yescharacteristic
37#14519697C2211yescharacteristic
38#14519697C2211yescharacteristic
39#14519697C2211yescharacteristic, complemented
40#14519697C2211yescharacteristic, complemented
41#14519697C2211yescharacteristic, complemented
42#14519697C2211yescharacteristic, complemented
43#14519697C2211yescharacteristic, complemented
44#14519697C2211yescharacteristic, complemented
45C22⋊#92418C2211yescharacteristic, complemented
46C22⋊#92418C2211yescharacteristic, complemented
47#14519688C2211yescharacteristic, complemented
48#126×C61—21no—
49#125×C61—21no—
50#125×C61C2211yescharacteristic, complemented
51#123×C61—21no—
52C22⋊Q8×C61C2211yescharacteristic, complemented
53C22⋊Q8×C61C2211yescharacteristic, complemented
54C4×Q8×C61C2211yescharacteristic, complemented
55C4×D4×C61C2211yescharacteristic
56#117×C61—21no—
57#117×C61C2211yescharacteristic, complemented
58C2×C4⋊C4×C61C2211yescharacteristic, complemented
59C23×D61—21no—
60C23×D61—21no—
61C2×Q8×C61C2311yescharacteristic
62C2×D4×C61D411yescharacteristic
63C22×C4×C61—21no—
64C22×C4×C61C2311yescharacteristic
65C22×C4×C61C2311yescharacteristic
66C2×C22⋊D61—21no—
67C2×C22⋊D61—21no—
68D4×D61—44no—
69C2×D244—22no—
70C2×D244—22no—
71C2×D244—21no—
72C2×D244—21no—
73C2×D244—21no—
74C2×D244—21no—
75C2×D244—41no—
76C2×D244—41no—
77C2×D244C2311yescharacteristic
78C2×D244C2311yescharacteristic
79C2×D244C2311yescharacteristic
80C2×C4×D61—21no—
81C4⋊C4×C61—21no—
82C4⋊C4×C61—21no—
83C4⋊C4×C61—21no—
84C4⋊C4×C61—21no—
85C4⋊C4×C61—21no—
86C4⋊C4×C61—21no—
87C4⋊C4×C61D411yescharacteristic
88C4⋊C4×C61C2311yescharacteristic
89C4⋊C4×C61C2311yescharacteristic
90C4⋊C4×C61C2311yescharacteristic
91C22⋊C4×C61—21no—
92C22⋊C4×C61—21no—
93C22⋊C4×C61—21no—
94C22⋊C4×C61D411yescharacteristic
95C22⋊C4×C61D411yescharacteristic
96C42×C61—21no—
97C42×C61—21no—
98C42×C61C2311yescharacteristic
99#11758201—22no—
100#11758201—41no—
101#11758201—41no—
102#11758198—21no—
103#11758198—21no—
104#11758198—21no—
105C4⋊Dic61C2311yescharacteristic
106#11758194—41no—
107C2×#92418—41no—
108C2×#92418C2311yescharacteristic
109C2×#92418C2311yescharacteristic
110C2×#92418C2311yescharacteristic
111C2×#92418C2311yescharacteristic
112C23×C61C2×D411yescharacteristic
113C23×C61C2×D411yescharacteristic
114C22×D61—21no—
115C22×D61—21no—
116C22×D61—21no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—42no—
120C22×D61—42no—
121C22×D61—81no—
122C22×D61—81no—
123Q8×C61—21no—
124Q8×C61—21no—
125D4×C61—22no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—42no—
135C2×C4×C61C2×D411yescharacteristic
136C2×C4×C61C2×D411yescharacteristic
137C2×C4×C61C2×D411yescharacteristic
138C2×C4×C61C2×D411yescharacteristic
139C2×C4×C61C2411yescharacteristic
140C22⋊D61—42no—
141C22⋊D61—42no—
142C2×Dic61—21no—
143D244—21no—
144D244—21no—
145D244—21no—
146D244—21no—
147D244—21no—
148D244—42no—
149D244—42no—
150D244—42no—
151D244—81no—
152D244#4112yes—
153C4×D61—42no—
154#92418—21no—
155#92418—21no—
156#92418—21no—
157#92418—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61AGL(1,ℤ/8)12yes—
164C22×C61C22×D411yescharacteristic
165C2×D61—22no—
166C2×D61—41no—
167C2×D61—41no—
168C2×D61—41no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61—41no—
174C2×D61—84no—
175C2×D61—84no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—41no—
183C4×C61—41no—
184C4×C61C2×#4111yescharacteristic
185C4×C61H(5,𝔽2)11yescharacteristic
186Dic61—41no—
187#3310—611nomaximal, Sylow
188C2×C61—22no—
189C2×C61—41no—
190C2×C61#54611yescharacteristic
191C2×C61C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C61#57711yescharacteristic
193D61—42no—
194D61—81no—
195D61—81no—
196#552D6111yescharacteristic, complemented
197D42—611no—
198#490—611no—
199#482—611no—
200#480—611no—
201#478—611no—
202#468—611no—
203#465—611no—
204#460—611no—
205#450—611no—
206#450—611no—
207#440—611no—
208#435—611no—
209#418—611no—
210C2×#102—611no—
211C61#331011yescharacteristic, complemented, Sylow
212C22×D4—611no—
213C22×D4—611no—
214C22×D4—1221no—
215C2×SD16—611no—
216C2×SD16—611no—
217C2×SD16—611no—
218C2×M16—611no—
219C22×C8—611no—
220#127—611no—
221#126—21no—
222#125—21no—
223#125C2×D6111yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C2×D6111yescharacteristic, complemented
226C22⋊Q8C2×D6111yescharacteristic, complemented
227#121—611no—
228#121—611no—
229#121—1221no—
230#120—1221no—
231#120—1221no—
232C4×Q8C2×D6111yescharacteristic, complemented
233C4×D4—611no—
234C4×D4C2×D6111yescharacteristic
235#117—21no—
236#117C2×D6111yescharacteristic, complemented
237C2×C4⋊C4C2×D6111yescharacteristic, complemented
238#107—611no—
239#107—611no—
240#106—611no—
241C4⋊C8—611no—
242#103—611no—
243#102—611no—
244#102—611no—
245#102—611no—
246#102—611no—
247#102—611no—
248#102—611no—
249#102—611no—
250#102—611no—
251#102—611no—
252C22⋊C8—611no—
253C22⋊C8—611no—
254#97—611no—
255C24—1221no—
256C24—1221no—
257C2×Q8C22×D6111yescharacteristic
258C2×D4—611no—
259C2×D4—611no—
260C2×D4—611no—
261C2×D4—1222no—
262C2×D4—1222no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—1221no—
266C2×D4—1221no—
267C2×D4—1221no—
268C2×D4—1221no—
269C2×D4—2441no—
270C2×D4—2441no—
271C2×D4—2444no—
272C2×D4C22⋊D6111yescharacteristic
273C22×C4—21no—
274C22×C4—1221no—
275C22×C4C22×D6111yescharacteristic
276C22×C4C22×D6111yescharacteristic
277SD16—1222no—
278SD16—2441no—
279SD16—2441no—
280M16—2441no—
281C2×C8—611no—
282C2×C8—611no—
283C2×C8—611no—
284C2×C8—611no—
285C2×C8—2441no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—611no—
293C4⋊C4C22⋊D6111yescharacteristic
294C4⋊C4C22×D6111yescharacteristic
295C4⋊C4C22×D6111yescharacteristic
296C4⋊C4C22×D6111yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—1221no—
301C22⋊C4—1221no—
302C22⋊C4—1221no—
303C22⋊C4C22⋊D6111yescharacteristic
304C22⋊C4C22⋊D6111yescharacteristic
305C42—21no—
306C42—21no—
307C42C22×D6111yescharacteristic
308C23—1221no—
309C23—1221no—
310C23—1221no—
311C23—4881no—
312C23—4881no—
313C23—2442no—
314C23—2442no—
315C23—2442no—
316C23—2442no—
317C23C2×C22⋊D6111yescharacteristic
318C23C2×C22⋊D6111yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—612no—
323D4—1221no—
324D4—1221no—
325D4—1221no—
326D4—1221no—
327D4—1221no—
328D4—4881no—
329D4—2442no—
330D4—2442no—
331D4—2442no—
332D4—2442no—
333D4—2442no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1221no—
344C2×C4—2442no—
345C2×C4C2×C22⋊D6111yescharacteristic
346C2×C4C2×C22⋊D6111yescharacteristic
347C2×C4C2×C22⋊D6111yescharacteristic
348C2×C4C2×C22⋊D6111yescharacteristic
349C2×C4C23×D6111yescharacteristic
350C8—1221no—
351C8—1221no—
352C8—1221no—
353C8—1221no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—1222no—
360C22—4884no—
361C22—4884no—
362C22—2441no—
363C22—2441no—
364C22—2441no—
365C22—2441no—
366C22—2441no—
367C22—2441no—
368C22—2441no—
369C22—2441no—
370C22#1451980212yes—
371C22C22×C22⋊D6111yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—2441no—
381C4#41×D6111yescharacteristic
382C4#1451982311yescharacteristic
383C2—22no—
384C2—41no—
385C2—4881no—
386C2—4881no—
387C2—2442no—
388C2#1684040011yescharacteristic
389C2C2×#1451980211yescharacteristic
390C2#1684057011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D61
commutatorC2×C4×C61C24
FrattiniC2×C4C23×D61
FittingC61×#552C2
socleC22×C61C22×D4
hypercenter#552D61
hypocenterC61#3310
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #16840128C2semidirect
3. #16840086C2semidirect
4. #16840075C2semidirect
5. #16840059C2semidirect
6. #16840010C2semidirect
7. #16839993C2semidirect
8. #16839962C2semidirect
9. #16839912C2semidirect
10. #16839912C2semidirect
11. #16839860C2semidirect
12. #16839834C2semidirect
13. #16839766C2semidirect
14. C2×#14519697C2semidirect
15. C61×#552C2semidirect
20. C2×#11758201C22semidirect
21. C2×#11758201C22semidirect
22. C2×#11758194C22semidirect
23. C22×#92418C22semidirect
31. #14519710C22semidirect
32. #14519710C22semidirect
33. #14519708C22semidirect
34. C4⋊#92418C22semidirect
35. #14519701C22semidirect
39. #14519697C22semidirect
40. #14519697C22semidirect
41. #14519697C22semidirect
42. #14519697C22semidirect
43. #14519697C22semidirect
44. #14519697C22semidirect
45. C22⋊#92418C22semidirect
46. C22⋊#92418C22semidirect
47. #14519688C22semidirect
50. #125×C61C22semidirect
52. C22⋊Q8×C61C22semidirect
53. C22⋊Q8×C61C22semidirect
54. C4×Q8×C61C22semidirect
57. #117×C61C22semidirect
58. C2×C4⋊C4×C61C22semidirect
196. #552D61semidirect
211. C61#3310semidirect
223. #125C2×D61semidirect
225. C22⋊Q8C2×D61semidirect
226. C22⋊Q8C2×D61semidirect
232. C4×Q8C2×D61semidirect
236. #117C2×D61semidirect
237. C2×C4⋊C4C2×D61semidirect