Finite Group Databasebeta

← #3310×C227 #44397266

Subgroups — #3310×C227 #44397266

Order 29056.

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
1#552×C227C211yesmaximal, characteristic, complemented
2D42×C227C211yesmaximal, characteristic
3#490×C227C211yesmaximal, characteristic, complemented
4#482×C227C211yesmaximal, characteristic, complemented
5#480×C227C211yesmaximal, characteristic, complemented
6#478×C227C211yesmaximal, characteristic, complemented
7#468×C227C211yesmaximal, characteristic, complemented
8#465×C227C211yesmaximal, characteristic, complemented
9#460×C227C211yesmaximal, characteristic, complemented
10#450×C227C211yesmaximal, characteristic, complemented
11#450×C227C211yesmaximal, characteristic, complemented
12#440×C227C211yesmaximal, characteristic, complemented
13#435×C227C211yesmaximal, characteristic, complemented
14#418×C227C211yesmaximal, characteristic, complemented
15C2×#102×C227C211yesmaximal, characteristic, complemented
16C22×D4×C227—21no—
17C22×D4×C227C2211yescharacteristic
18C22×D4×C227C2211yescharacteristic
19C2×SD16×C227C2211yescharacteristic
20C2×SD16×C227C2211yescharacteristic, complemented
21C2×SD16×C227C2211yescharacteristic, complemented
22C2×M16×C227C2211yescharacteristic, complemented
23C22×C8×C227C2211yescharacteristic, complemented
24#127×C227C2211yescharacteristic
25#126×C227—21no—
26#125×C227—21no—
27#125×C227C2211yescharacteristic, complemented
28#123×C227—21no—
29C22⋊Q8×C227C2211yescharacteristic, complemented
30C22⋊Q8×C227C2211yescharacteristic, complemented
31#121×C227—21no—
32#121×C227C2211yescharacteristic
33#121×C227C2211yescharacteristic
34#120×C227—21no—
35#120×C227—21no—
36C4×Q8×C227C2211yescharacteristic, complemented
37C4×D4×C227C2211yescharacteristic
38C4×D4×C227C2211yescharacteristic
39#117×C227—21no—
40#117×C227C2211yescharacteristic, complemented
41C2×C4⋊C4×C227C2211yescharacteristic, complemented
42#107×C227C2211yescharacteristic, complemented
43#107×C227C2211yescharacteristic, complemented
44#106×C227C2211yescharacteristic, complemented
45C4⋊C8×C227C2211yescharacteristic, complemented
46#103×C227C2211yescharacteristic, complemented
47#102×C227C2211yescharacteristic
48#102×C227C2211yescharacteristic
49#102×C227C2211yescharacteristic
50#102×C227C2211yescharacteristic, complemented
51#102×C227C2211yescharacteristic, complemented
52#102×C227C2211yescharacteristic, complemented
53#102×C227C2211yescharacteristic, complemented
54#102×C227C2211yescharacteristic, complemented
55#102×C227C2211yescharacteristic, complemented
56C22⋊C8×C227C2211yescharacteristic, complemented
57C22⋊C8×C227C2211yescharacteristic, complemented
58#97×C227C2211yescharacteristic, complemented
59C24×C227—21no—
60C24×C227—21no—
61C2×Q8×C227C2311yescharacteristic
62C2×D4×C227—22no—
63C2×D4×C227—22no—
64C2×D4×C227—21no—
65C2×D4×C227—21no—
66C2×D4×C227—21no—
67C2×D4×C227—21no—
68C2×D4×C227—21no—
69C2×D4×C227—21no—
70C2×D4×C227—41no—
71C2×D4×C227—41no—
72C2×D4×C227—44no—
73C2×D4×C227D411yescharacteristic
74C2×D4×C227C2311yescharacteristic
75C2×D4×C227C2311yescharacteristic
76C2×D4×C227C2311yescharacteristic
77C22×C4×C227—21no—
78C22×C4×C227—21no—
79C22×C4×C227C2311yescharacteristic
80C22×C4×C227C2311yescharacteristic
81SD16×C227—22no—
82SD16×C227—41no—
83SD16×C227—41no—
84M16×C227—41no—
85C2×C8×C227—41no—
86C2×C8×C227C2311yescharacteristic
87C2×C8×C227C2311yescharacteristic
88C2×C8×C227C2311yescharacteristic
89C2×C8×C227C2311yescharacteristic
90C4⋊C4×C227—21no—
91C4⋊C4×C227—21no—
92C4⋊C4×C227—21no—
93C4⋊C4×C227—21no—
94C4⋊C4×C227—21no—
95C4⋊C4×C227—21no—
96C4⋊C4×C227D411yescharacteristic
97C4⋊C4×C227C2311yescharacteristic
98C4⋊C4×C227C2311yescharacteristic
99C4⋊C4×C227C2311yescharacteristic
100C4⋊C4×C227C2311yescharacteristic
101C22⋊C4×C227—21no—
102C22⋊C4×C227—21no—
103C22⋊C4×C227—21no—
104C22⋊C4×C227—21no—
105C22⋊C4×C227—21no—
106C22⋊C4×C227—21no—
107C22⋊C4×C227D411yescharacteristic
108C22⋊C4×C227D411yescharacteristic
109C42×C227—21no—
110C42×C227—21no—
111C42×C227C2311yescharacteristic
112C23×C227—21no—
113C23×C227—21no—
114C23×C227—21no—
115C23×C227—42no—
116C23×C227—42no—
117C23×C227—42no—
118C23×C227—42no—
119C23×C227—81no—
120C23×C227—81no—
121C23×C227C2×D411yescharacteristic
122C23×C227C2×D411yescharacteristic
123Q8×C227—21no—
124Q8×C227—21no—
125D4×C227—22no—
126D4×C227—21no—
127D4×C227—21no—
128D4×C227—21no—
129D4×C227—21no—
130D4×C227—21no—
131D4×C227—42no—
132D4×C227—42no—
133D4×C227—42no—
134D4×C227—42no—
135D4×C227—42no—
136D4×C227—81no—
137D4×C227#4112yes—
138C2×C4×C227—21no—
139C2×C4×C227—21no—
140C2×C4×C227—21no—
141C2×C4×C227—21no—
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—21no—
145C2×C4×C227—21no—
146C2×C4×C227—21no—
147C2×C4×C227—42no—
148C2×C4×C227—42no—
149C2×C4×C227C2×D411yescharacteristic
150C2×C4×C227C2×D411yescharacteristic
151C2×C4×C227C2×D411yescharacteristic
152C2×C4×C227C2×D411yescharacteristic
153C2×C4×C227C2411yescharacteristic
154C8×C227—21no—
155C8×C227—21no—
156C8×C227—21no—
157C8×C227—21no—
158C22×C227—21no—
159C22×C227—21no—
160C22×C227—21no—
161C22×C227—22no—
162C22×C227—21no—
163C22×C227—21no—
164C22×C227—41no—
165C22×C227—41no—
166C22×C227—41no—
167C22×C227—41no—
168C22×C227—41no—
169C22×C227—41no—
170C22×C227—41no—
171C22×C227—41no—
172C22×C227—84no—
173C22×C227—84no—
174C22×C227AGL(1,ℤ/8)12yes—
175C22×C227C22×D411yescharacteristic
176C4×C227—21no—
177C4×C227—21no—
178C4×C227—21no—
179C4×C227—21no—
180C4×C227—41no—
181C4×C227—41no—
182C4×C227—41no—
183C4×C227—41no—
184C4×C227—41no—
185C4×C227C2×#4111yescharacteristic
186C4×C227H(5,𝔽2)11yescharacteristic
187C2×C227—22no—
188C2×C227—41no—
189C2×C227—42no—
190C2×C227—81no—
191C2×C227—81no—
192C2×C227#54611yescharacteristic
193C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
194C2×C227#57711yescharacteristic
195C227#331011yescharacteristic, direct product factor, Sylow
196#3310C22711yesmaximal, characteristic, direct product factor, Sylow
197#552C2×C22711yescharacteristic, complemented
198D42C2×C22711yescharacteristic
199#490C2×C22711yescharacteristic, complemented
200#482C2×C22711yescharacteristic, complemented
201#480C2×C22711yescharacteristic, complemented
202#478C2×C22711yescharacteristic, complemented
203#468C2×C22711yescharacteristic, complemented
204#465C2×C22711yescharacteristic, complemented
205#460C2×C22711yescharacteristic, complemented
206#450C2×C22711yescharacteristic, complemented
207#450C2×C22711yescharacteristic, complemented
208#440C2×C22711yescharacteristic, complemented
209#435C2×C22711yescharacteristic, complemented
210#418C2×C22711yescharacteristic, complemented
211C2×#102C2×C22711yescharacteristic, complemented
212C22×D4—21no—
213C22×D4C22×C22711yescharacteristic
214C22×D4C22×C22711yescharacteristic
215C2×SD16C22×C22711yescharacteristic
216C2×SD16C22×C22711yescharacteristic, complemented
217C2×SD16C22×C22711yescharacteristic, complemented
218C2×M16C22×C22711yescharacteristic, complemented
219C22×C8C22×C22711yescharacteristic, complemented
220#127C22×C22711yescharacteristic
221#126—21no—
222#125—21no—
223#125C22×C22711yescharacteristic, complemented
224#123—21no—
225C22⋊Q8C22×C22711yescharacteristic, complemented
226C22⋊Q8C22×C22711yescharacteristic, complemented
227#121—21no—
228#121C22×C22711yescharacteristic
229#121C22×C22711yescharacteristic
230#120—21no—
231#120—21no—
232C4×Q8C22×C22711yescharacteristic, complemented
233C4×D4C22×C22711yescharacteristic
234C4×D4C22×C22711yescharacteristic
235#117—21no—
236#117C22×C22711yescharacteristic, complemented
237C2×C4⋊C4C22×C22711yescharacteristic, complemented
238#107C22×C22711yescharacteristic, complemented
239#107C22×C22711yescharacteristic, complemented
240#106C22×C22711yescharacteristic, complemented
241C4⋊C8C22×C22711yescharacteristic, complemented
242#103C22×C22711yescharacteristic, complemented
243#102C22×C22711yescharacteristic
244#102C22×C22711yescharacteristic
245#102C22×C22711yescharacteristic
246#102C22×C22711yescharacteristic, complemented
247#102C22×C22711yescharacteristic, complemented
248#102C22×C22711yescharacteristic, complemented
249#102C22×C22711yescharacteristic, complemented
250#102C22×C22711yescharacteristic, complemented
251#102C22×C22711yescharacteristic, complemented
252C22⋊C8C22×C22711yescharacteristic, complemented
253C22⋊C8C22×C22711yescharacteristic, complemented
254#97C22×C22711yescharacteristic, complemented
255C24—21no—
256C24—21no—
257C2×Q8C23×C22711yescharacteristic
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4D4×C22711yescharacteristic
270C2×D4C23×C22711yescharacteristic
271C2×D4C23×C22711yescharacteristic
272C2×D4C23×C22711yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4C23×C22711yescharacteristic
276C22×C4C23×C22711yescharacteristic
277SD16—22no—
278SD16—41no—
279SD16—41no—
280M16—41no—
281C2×C8—41no—
282C2×C8C23×C22711yescharacteristic
283C2×C8C23×C22711yescharacteristic
284C2×C8C23×C22711yescharacteristic
285C2×C8C23×C22711yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4D4×C22711yescharacteristic
293C4⋊C4C23×C22711yescharacteristic
294C4⋊C4C23×C22711yescharacteristic
295C4⋊C4C23×C22711yescharacteristic
296C4⋊C4C23×C22711yescharacteristic
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4D4×C22711yescharacteristic
304C22⋊C4D4×C22711yescharacteristic
305C42—21no—
306C42—21no—
307C42C23×C22711yescharacteristic
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—81no—
316C23—81no—
317C23C2×D4×C22711yescharacteristic
318C23C2×D4×C22711yescharacteristic
319Q8—21no—
320Q8—21no—
321D4—22no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—42no—
332D4—81no—
333D4#41×C22712yes—
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—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4C2×D4×C22711yescharacteristic
346C2×C4C2×D4×C22711yescharacteristic
347C2×C4C2×D4×C22711yescharacteristic
348C2×C4C2×D4×C22711yescharacteristic
349C2×C4C24×C22711yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—84no—
370C22AGL(1,ℤ/8)×C22712yes—
371C22C22×D4×C22711yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4C2×#41×C22711yescharacteristic
382C4H(5,𝔽2)×C22711yescharacteristic
383C2—22no—
384C2—41no—
385C2—42no—
386C2—81no—
387C2—81no—
388C2#546×C22711yescharacteristic
389C2C2×AGL(1,ℤ/8)×C22711yescharacteristic
390C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
prime radicalD42×C227C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C227 ◁ C4×D4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. #552×C227C2semidirect
3. #490×C227C2semidirect
4. #482×C227C2semidirect
5. #480×C227C2semidirect
6. #478×C227C2semidirect
7. #468×C227C2semidirect
8. #465×C227C2semidirect
9. #460×C227C2semidirect
10. #450×C227C2semidirect
11. #450×C227C2semidirect
12. #440×C227C2semidirect
13. #435×C227C2semidirect
14. #418×C227C2semidirect
15. C2×#102×C227C2semidirect
20. C2×SD16×C227C22semidirect
21. C2×SD16×C227C22semidirect
22. C2×M16×C227C22semidirect
23. C22×C8×C227C22semidirect
27. #125×C227C22semidirect
29. C22⋊Q8×C227C22semidirect
30. C22⋊Q8×C227C22semidirect
36. C4×Q8×C227C22semidirect
40. #117×C227C22semidirect
41. C2×C4⋊C4×C227C22semidirect
42. #107×C227C22semidirect
43. #107×C227C22semidirect
44. #106×C227C22semidirect
45. C4⋊C8×C227C22semidirect
46. #103×C227C22semidirect
50. #102×C227C22semidirect
51. #102×C227C22semidirect
52. #102×C227C22semidirect
53. #102×C227C22semidirect
54. #102×C227C22semidirect
55. #102×C227C22semidirect
56. C22⋊C8×C227C22semidirect
57. C22⋊C8×C227C22semidirect
58. #97×C227C22semidirect
195. C227#3310direct (N × H)
196. #3310C227direct (N × H)
197. #552C2×C227semidirect
199. #490C2×C227semidirect
200. #482C2×C227semidirect
201. #480C2×C227semidirect
202. #478C2×C227semidirect
203. #468C2×C227semidirect
204. #465C2×C227semidirect
205. #460C2×C227semidirect
206. #450C2×C227semidirect
207. #450C2×C227semidirect
208. #440C2×C227semidirect
209. #435C2×C227semidirect
210. #418C2×C227semidirect
211. C2×#102C2×C227semidirect
216. C2×SD16C22×C227semidirect
217. C2×SD16C22×C227semidirect
218. C2×M16C22×C227semidirect
219. C22×C8C22×C227semidirect
223. #125C22×C227semidirect
225. C22⋊Q8C22×C227semidirect
226. C22⋊Q8C22×C227semidirect
232. C4×Q8C22×C227semidirect
236. #117C22×C227semidirect
237. C2×C4⋊C4C22×C227semidirect
238. #107C22×C227semidirect
239. #107C22×C227semidirect
240. #106C22×C227semidirect
241. C4⋊C8C22×C227semidirect
242. #103C22×C227semidirect
246. #102C22×C227semidirect
247. #102C22×C227semidirect
248. #102C22×C227semidirect
249. #102C22×C227semidirect
250. #102C22×C227semidirect
251. #102C22×C227semidirect
252. C22⋊C8C22×C227semidirect
253. C22⋊C8C22×C227semidirect
254. #97C22×C227semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#552×C2272
D42×C2272
#490×C2272
#482×C2272
#480×C2272
#478×C2272
#468×C2272
#465×C2272
#460×C2272
#450×C2272
#440×C2272
#435×C2272
#418×C2272
C2×#102×C2272
#3310227