Finite Group Databasebeta

← #3311×C227 #44397267

Subgroups — #3311×C227 #44397267

Order 29056.

402 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#556×C227C211yesmaximal, characteristic, complemented
2#546×C227C211yesmaximal, characteristic
3#489×C227C211yesmaximal, characteristic, complemented
4#485×C227C211yesmaximal, characteristic, complemented
5#482×C227C211yesmaximal, characteristic, complemented
6#476×C227C211yesmaximal, characteristic, complemented
7#470×C227C211yesmaximal, characteristic, complemented
8#465×C227C211yesmaximal, characteristic, complemented
9#464×C227C211yesmaximal, characteristic, complemented
10#452×C227C211yesmaximal, characteristic, complemented
11#450×C227C211yesmaximal, characteristic, complemented
12#440×C227C211yesmaximal, characteristic, complemented
13#435×C227C211yesmaximal, characteristic, complemented
14#418×C227C211yesmaximal, characteristic, complemented
15#416×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227C2211yescharacteristic
17C22×D4×C227C2211yescharacteristic
18C2×Q16×C227C2211yescharacteristic, complemented
19C2×SD16×C227C2211yescharacteristic
20C2×SD16×C227C2211yescharacteristic
21C2×M16×C227C2211yescharacteristic
22C22×C8×C227C2211yescharacteristic, complemented
23C4⋊Q8×C227C2211yescharacteristic, complemented
24#126×C227—21no—
25#126×C227—21no—
26#124×C227—21no—
27#124×C227C2211yescharacteristic
28#123×C227—21no—
29C22⋊Q8×C227C2211yescharacteristic
30C22⋊Q8×C227C2211yescharacteristic
31C22⋊Q8×C227C2211yescharacteristic, complemented
32#121×C227—21no—
33#121×C227C2211yescharacteristic
34#120×C227—21no—
35C4×Q8×C227—21no—
36C4×Q8×C227C2211yescharacteristic, complemented
37C4×D4×C227C2211yescharacteristic
38C4×D4×C227C2211yescharacteristic
39#117×C227C2211yescharacteristic
40#117×C227C2211yescharacteristic, complemented
41C2×C22⋊C4×C227—21no—
42#107×C227C2211yescharacteristic, complemented
43#106×C227C2211yescharacteristic, complemented
44#106×C227C2211yescharacteristic, complemented
45C4⋊C8×C227C2211yescharacteristic, complemented
46#103×C227C2211yescharacteristic, complemented
47#103×C227C2211yescharacteristic, complemented
48#103×C227C2211yescharacteristic, complemented
49#103×C227C2211yescharacteristic, complemented
50#102×C227C2211yescharacteristic
51#102×C227C2211yescharacteristic
52#102×C227C2211yescharacteristic
53#102×C227C2211yescharacteristic
54#102×C227C2211yescharacteristic
55#102×C227C2211yescharacteristic
56C22⋊C8×C227C2211yescharacteristic
57C22⋊C8×C227C2211yescharacteristic, complemented
58#97×C227C2211yescharacteristic, complemented
59C24×C227—21no—
60#41×C227—22no—
61#41×C227—41no—
62C2×Q8×C227—21no—
63C2×Q8×C227C2311yescharacteristic
64C2×Q8×C227C2311yescharacteristic
65C2×D4×C227—22no—
66C2×D4×C227—21no—
67C2×D4×C227—21no—
68C2×D4×C227—21no—
69C2×D4×C227—41no—
70C2×D4×C227D411yescharacteristic
71C2×D4×C227C2311yescharacteristic
72C2×D4×C227C2311yescharacteristic
73C22×C4×C227—21no—
74C22×C4×C227—21no—
75C22×C4×C227C2311yescharacteristic
76C22×C4×C227C2311yescharacteristic
77Q16×C227—41no—
78SD16×C227—22no—
79SD16×C227—41no—
80M16×C227—41no—
81C2×C8×C227—41no—
82C2×C8×C227C2311yescharacteristic
83C2×C8×C227C2311yescharacteristic
84C2×C8×C227C2311yescharacteristic
85C2×C8×C227C2311yescharacteristic
86C4⋊C4×C227—21no—
87C4⋊C4×C227—21no—
88C4⋊C4×C227—21no—
89C4⋊C4×C227—21no—
90C4⋊C4×C227—21no—
91C4⋊C4×C227D411yescharacteristic
92C4⋊C4×C227C2311yescharacteristic
93C4⋊C4×C227C2311yescharacteristic
94C4⋊C4×C227C2311yescharacteristic
95C4⋊C4×C227C2311yescharacteristic
96C22⋊C4×C227—21no—
97C22⋊C4×C227—21no—
98C22⋊C4×C227—21no—
99C22⋊C4×C227—21no—
100C22⋊C4×C227—21no—
101C22⋊C4×C227—21no—
102C22⋊C4×C227—21no—
103C22⋊C4×C227—21no—
104C22⋊C4×C227—21no—
105C22⋊C4×C227D411yescharacteristic
106C22⋊C4×C227D411yescharacteristic
107C42×C227—21no—
108C42×C227—21no—
109C42×C227—21no—
110C42×C227C2311yescharacteristic
111C23×C227—21no—
112C23×C227—21no—
113C23×C227—42no—
114C23×C227—42no—
115C23×C227—81no—
116C23×C227C2×D411yescharacteristic
117C23×C227C2×D411yescharacteristic
118Q8×C227—21no—
119Q8×C227—21no—
120Q8×C227—21no—
121Q8×C227—21no—
122Q8×C227—42no—
123D4×C227—21no—
124D4×C227—21no—
125D4×C227—21no—
126D4×C227—41no—
127D4×C227—42no—
128D4×C227—42no—
129D4×C227—81no—
130D4×C227#4112yes—
131C2×C4×C227—21no—
132C2×C4×C227—21no—
133C2×C4×C227—21no—
134C2×C4×C227—21no—
135C2×C4×C227—21no—
136C2×C4×C227—21no—
137C2×C4×C227—21no—
138C2×C4×C227—21no—
139C2×C4×C227—21no—
140C2×C4×C227—21no—
141C2×C4×C227—42no—
142C2×C4×C227—81no—
143C2×C4×C227C2×D411yescharacteristic
144C2×C4×C227C2×D411yescharacteristic
145C2×C4×C227C2×D411yescharacteristic
146C2×C4×C227C2×D411yescharacteristic
147C2×C4×C227C2411yescharacteristic
148C8×C227—21no—
149C8×C227—21no—
150C8×C227—21no—
151C8×C227—21no—
152C22×C227—21no—
153C22×C227—21no—
154C22×C227—22no—
155C22×C227—21no—
156C22×C227—21no—
157C22×C227—21no—
158C22×C227—21no—
159C22×C227—41no—
160C22×C227—41no—
161C22×C227—41no—
162C22×C227—41no—
163C22×C227—41no—
164C22×C227—84no—
165C22×C227C22×D411yescharacteristic
166C4×C227—21no—
167C4×C227—21no—
168C4×C227—21no—
169C4×C227—21no—
170C4×C227—41no—
171C4×C227—41no—
172C4×C227—41no—
173C4×C227—41no—
174C4×C227—41no—
175C4×C227—41no—
176C4×C227C2×#4111yescharacteristic
177C4×C227H(5,𝔽2)11yescharacteristic
178C2×C227—41no—
179C2×C227—41no—
180C2×C227—42no—
181C2×C227—81no—
182C2×C227#54611yescharacteristic
183C2×C227#57511yescharacteristic
184C2×C227#57711yescharacteristic
185C227#331111yescharacteristic, direct product factor, Sylow
186#3311C22711yesmaximal, characteristic, direct product factor, Sylow
187#556C2×C22711yescharacteristic, complemented
188#546C2×C22711yescharacteristic
189#489C2×C22711yescharacteristic, complemented
190#485C2×C22711yescharacteristic, complemented
191#482C2×C22711yescharacteristic, complemented
192#476C2×C22711yescharacteristic, complemented
193#470C2×C22711yescharacteristic, complemented
194#465C2×C22711yescharacteristic, complemented
195#464C2×C22711yescharacteristic, complemented
196#452C2×C22711yescharacteristic, complemented
197#450C2×C22711yescharacteristic, complemented
198#440C2×C22711yescharacteristic, complemented
199#435C2×C22711yescharacteristic, complemented
200#418C2×C22711yescharacteristic, complemented
201#416C2×C22711yescharacteristic, complemented
202C2×#41C22×C22711yescharacteristic
203C22×D4C22×C22711yescharacteristic
204C2×Q16C22×C22711yescharacteristic, complemented
205C2×SD16C22×C22711yescharacteristic
206C2×SD16C22×C22711yescharacteristic
207C2×M16C22×C22711yescharacteristic
208C22×C8C22×C22711yescharacteristic, complemented
209C4⋊Q8C22×C22711yescharacteristic, complemented
210#126—21no—
211#126—21no—
212#124—21no—
213#124C22×C22711yescharacteristic
214#123—21no—
215C22⋊Q8C22×C22711yescharacteristic
216C22⋊Q8C22×C22711yescharacteristic
217C22⋊Q8C22×C22711yescharacteristic, complemented
218#121—21no—
219#121C22×C22711yescharacteristic
220#120—21no—
221C4×Q8—21no—
222C4×Q8C22×C22711yescharacteristic, complemented
223C4×D4C22×C22711yescharacteristic
224C4×D4C22×C22711yescharacteristic
225#117C22×C22711yescharacteristic
226#117C22×C22711yescharacteristic, complemented
227C2×C22⋊C4—21no—
228#107C22×C22711yescharacteristic, complemented
229#106C22×C22711yescharacteristic, complemented
230#106C22×C22711yescharacteristic, complemented
231C4⋊C8C22×C22711yescharacteristic, complemented
232#103C22×C22711yescharacteristic, complemented
233#103C22×C22711yescharacteristic, complemented
234#103C22×C22711yescharacteristic, complemented
235#103C22×C22711yescharacteristic, complemented
236#102C22×C22711yescharacteristic
237#102C22×C22711yescharacteristic
238#102C22×C22711yescharacteristic
239#102C22×C22711yescharacteristic
240#102C22×C22711yescharacteristic
241#102C22×C22711yescharacteristic
242C22⋊C8C22×C22711yescharacteristic
243C22⋊C8C22×C22711yescharacteristic, complemented
244#97C22×C22711yescharacteristic, complemented
245C24—21no—
246#41—22no—
247#41—41no—
248C2×Q8—21no—
249C2×Q8C23×C22711yescharacteristic
250C2×Q8C23×C22711yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4D4×C22711yescharacteristic
257C2×D4C23×C22711yescharacteristic
258C2×D4C23×C22711yescharacteristic
259C22×C4—21no—
260C22×C4—21no—
261C22×C4C23×C22711yescharacteristic
262C22×C4C23×C22711yescharacteristic
263Q16—41no—
264SD16—22no—
265SD16—41no—
266M16—41no—
267C2×C8—41no—
268C2×C8C23×C22711yescharacteristic
269C2×C8C23×C22711yescharacteristic
270C2×C8C23×C22711yescharacteristic
271C2×C8C23×C22711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4D4×C22711yescharacteristic
278C4⋊C4C23×C22711yescharacteristic
279C4⋊C4C23×C22711yescharacteristic
280C4⋊C4C23×C22711yescharacteristic
281C4⋊C4C23×C22711yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4D4×C22711yescharacteristic
292C22⋊C4D4×C22711yescharacteristic
293C42—21no—
294C42—21no—
295C42—21no—
296C42C23×C22711yescharacteristic
297C23—21no—
298C23—21no—
299C23—42no—
300C23—42no—
301C23—81no—
302C23C2×D4×C22711yescharacteristic
303C23C2×D4×C22711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—21no—
308Q8—42no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—41no—
313D4—42no—
314D4—42no—
315D4—81no—
316D4#41×C22712yes—
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—42no—
328C2×C4—81no—
329C2×C4C2×D4×C22711yescharacteristic
330C2×C4C2×D4×C22711yescharacteristic
331C2×C4C2×D4×C22711yescharacteristic
332C2×C4C2×D4×C22711yescharacteristic
333C2×C4C24×C22711yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22C22×D4×C22711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4C2×#41×C22711yescharacteristic
363C4H(5,𝔽2)×C22711yescharacteristic
364C2—41no—
365C2—41no—
366C2—42no—
367C2—81no—
368C2#546×C22711yescharacteristic
369C2#575×C22711yescharacteristic
370C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
prime radical#546×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. #556×C227C2semidirect
3. #489×C227C2semidirect
4. #485×C227C2semidirect
5. #482×C227C2semidirect
6. #476×C227C2semidirect
7. #470×C227C2semidirect
8. #465×C227C2semidirect
9. #464×C227C2semidirect
10. #452×C227C2semidirect
11. #450×C227C2semidirect
12. #440×C227C2semidirect
13. #435×C227C2semidirect
14. #418×C227C2semidirect
15. #416×C227C2semidirect
18. C2×Q16×C227C22semidirect
22. C22×C8×C227C22semidirect
23. C4⋊Q8×C227C22semidirect
31. C22⋊Q8×C227C22semidirect
36. C4×Q8×C227C22semidirect
40. #117×C227C22semidirect
42. #107×C227C22semidirect
43. #106×C227C22semidirect
44. #106×C227C22semidirect
45. C4⋊C8×C227C22semidirect
46. #103×C227C22semidirect
47. #103×C227C22semidirect
48. #103×C227C22semidirect
49. #103×C227C22semidirect
57. C22⋊C8×C227C22semidirect
58. #97×C227C22semidirect
185. C227#3311direct (N × H)
186. #3311C227direct (N × H)
187. #556C2×C227semidirect
189. #489C2×C227semidirect
190. #485C2×C227semidirect
191. #482C2×C227semidirect
192. #476C2×C227semidirect
193. #470C2×C227semidirect
194. #465C2×C227semidirect
195. #464C2×C227semidirect
196. #452C2×C227semidirect
197. #450C2×C227semidirect
198. #440C2×C227semidirect
199. #435C2×C227semidirect
200. #418C2×C227semidirect
201. #416C2×C227semidirect
204. C2×Q16C22×C227semidirect
208. C22×C8C22×C227semidirect
209. C4⋊Q8C22×C227semidirect
217. C22⋊Q8C22×C227semidirect
222. C4×Q8C22×C227semidirect
226. #117C22×C227semidirect
228. #107C22×C227semidirect
229. #106C22×C227semidirect
230. #106C22×C227semidirect
231. C4⋊C8C22×C227semidirect
232. #103C22×C227semidirect
233. #103C22×C227semidirect
234. #103C22×C227semidirect
235. #103C22×C227semidirect
243. C22⋊C8C22×C227semidirect
244. #97C22×C227semidirect

Maximal subgroups

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

subgroupindex
#556×C2272
#546×C2272
#489×C2272
#485×C2272
#482×C2272
#476×C2272
#470×C2272
#465×C2272
#464×C2272
#452×C2272
#450×C2272
#440×C2272
#435×C2272
#418×C2272
#416×C2272
#3311227