Finite Group Databasebeta

← #3306×C227 #44397262

Subgroups — #3306×C227 #44397262

Order 29056.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#547×C227C211yesmaximal, characteristic, complemented
2#546×C227C211yesmaximal, characteristic, complemented
3#489×C227C211yesmaximal, characteristic, complemented
4#483×C227C211yesmaximal, characteristic, complemented
5#481×C227C211yesmaximal, characteristic, complemented
6#476×C227C211yesmaximal, characteristic, complemented
7#469×C227C211yesmaximal, characteristic, complemented
8#465×C227C211yesmaximal, characteristic, complemented
9#463×C227C211yesmaximal, characteristic, complemented
10#449×C227C211yesmaximal, characteristic, complemented
11#447×C227C211yesmaximal, characteristic, complemented
12#442×C227C211yesmaximal, characteristic, complemented
13#435×C227C211yesmaximal, characteristic, complemented
14#417×C227C211yesmaximal, characteristic, complemented
15C2×#102×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227—21no—
17C2×#41×C227C2211yescharacteristic
18C22×D4×C227C2211yescharacteristic
19C2×SD16×C227C2211yescharacteristic, complemented
20C2×D8×C227C2211yescharacteristic
21C2×D8×C227C2211yescharacteristic
22C2×M16×C227C2211yescharacteristic, complemented
23C22×C8×C227C2211yescharacteristic
24C4⋊Q8×C227C2211yescharacteristic, complemented
25#124×C227C2211yescharacteristic
26#123×C227—21no—
27#123×C227—21no—
28#123×C227—21no—
29C22⋊Q8×C227—21no—
30C22⋊Q8×C227C2211yescharacteristic
31#121×C227—21no—
32#121×C227C2211yescharacteristic
33#121×C227C2211yescharacteristic
34#121×C227C2211yescharacteristic, complemented
35#120×C227—21no—
36C4×D4×C227C2211yescharacteristic
37C4×D4×C227C2211yescharacteristic
38C4×D4×C227C2211yescharacteristic, complemented
39C2×C4⋊C4×C227C2211yescharacteristic
40C2×C4⋊C4×C227C2211yescharacteristic, complemented
41C2×C22⋊C4×C227—21no—
42#106×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#102×C227C2211yescharacteristic
50#102×C227C2211yescharacteristic
51#102×C227C2211yescharacteristic
52#102×C227C2211yescharacteristic
53#102×C227C2211yescharacteristic
54#102×C227C2211yescharacteristic
55#102×C227C2211yescharacteristic, complemented
56C22⋊C8×C227C2211yescharacteristic
57C22⋊C8×C227C2211yescharacteristic, complemented
58#97×C227C2211yescharacteristic, complemented
59C24×C227—21no—
60#41×C227—22no—
61#41×C227—41no—
62#41×C227—44no—
63C2×Q8×C227—21no—
64C2×Q8×C227C2311yescharacteristic
65C2×D4×C227—22no—
66C2×D4×C227—21no—
67C2×D4×C227—21no—
68C2×D4×C227—21no—
69C2×D4×C227—21no—
70C2×D4×C227—21no—
71C2×D4×C227—41no—
72C2×D4×C227D411yescharacteristic
73C2×D4×C227C2311yescharacteristic
74C2×D4×C227C2311yescharacteristic
75C2×D4×C227C2311yescharacteristic
76C22×C4×C227—21no—
77C22×C4×C227—21no—
78C22×C4×C227—21no—
79C22×C4×C227—21no—
80C22×C4×C227—21no—
81C22×C4×C227C2311yescharacteristic
82C22×C4×C227C2311yescharacteristic
83SD16×C227—41no—
84D8×C227—22no—
85D8×C227—41no—
86M16×C227—41no—
87C2×C8×C227—41no—
88C2×C8×C227C2311yescharacteristic
89C2×C8×C227C2311yescharacteristic
90C2×C8×C227C2311yescharacteristic
91C2×C8×C227C2311yescharacteristic
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×C227—21no—
108C22⋊C4×C227—21no—
109C22⋊C4×C227D411yescharacteristic
110C22⋊C4×C227D411yescharacteristic
111C42×C227C2311yescharacteristic
112C23×C227—21no—
113C23×C227—21no—
114C23×C227—21no—
115C23×C227—42no—
116C23×C227—42no—
117C23×C227—81no—
118C23×C227C2×D411yescharacteristic
119C23×C227C2×D411yescharacteristic
120Q8×C227—21no—
121Q8×C227—21no—
122Q8×C227—42no—
123D4×C227—22no—
124D4×C227—21no—
125D4×C227—21no—
126D4×C227—21no—
127D4×C227—21no—
128D4×C227—21no—
129D4×C227—42no—
130D4×C227—42no—
131D4×C227—42no—
132D4×C227—42no—
133D4×C227—81no—
134D4×C227#4112yescomplemented
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—21no—
142C2×C4×C227—21no—
143C2×C4×C227—21no—
144C2×C4×C227—42no—
145C2×C4×C227—42no—
146C2×C4×C227—42no—
147C2×C4×C227—42no—
148C2×C4×C227—81no—
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×C227AGL(1,ℤ/8)12yes—
174C22×C227C22×D411yescharacteristic
175C4×C227—21no—
176C4×C227—21no—
177C4×C227—21no—
178C4×C227—21no—
179C4×C227—41no—
180C4×C227—41no—
181C4×C227—41no—
182C4×C227—41no—
183C4×C227—41no—
184C4×C227C2×#4111yescharacteristic
185C4×C227H(5,𝔽2)11yescharacteristic
186C2×C227—22no—
187C2×C227—41no—
188C2×C227—42no—
189C2×C227—81no—
190C2×C227—81no—
191C2×C227#54611yescharacteristic
192C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
193C2×C227#57711yescharacteristic
194C227#330611yescharacteristic, direct product factor, Sylow
195#3306C22711yesmaximal, characteristic, direct product factor, Sylow
196#547C2×C22711yescharacteristic, complemented
197#546C2×C22711yescharacteristic, complemented
198#489C2×C22711yescharacteristic, complemented
199#483C2×C22711yescharacteristic, complemented
200#481C2×C22711yescharacteristic, complemented
201#476C2×C22711yescharacteristic, complemented
202#469C2×C22711yescharacteristic, complemented
203#465C2×C22711yescharacteristic, complemented
204#463C2×C22711yescharacteristic, complemented
205#449C2×C22711yescharacteristic, complemented
206#447C2×C22711yescharacteristic, complemented
207#442C2×C22711yescharacteristic, complemented
208#435C2×C22711yescharacteristic, complemented
209#417C2×C22711yescharacteristic, complemented
210C2×#102C2×C22711yescharacteristic, complemented
211C2×#41—21no—
212C2×#41C22×C22711yescharacteristic
213C22×D4C22×C22711yescharacteristic
214C2×SD16C22×C22711yescharacteristic, complemented
215C2×D8C22×C22711yescharacteristic
216C2×D8C22×C22711yescharacteristic
217C2×M16C22×C22711yescharacteristic, complemented
218C22×C8C22×C22711yescharacteristic
219C4⋊Q8C22×C22711yescharacteristic, complemented
220#124C22×C22711yescharacteristic
221#123—21no—
222#123—21no—
223#123—21no—
224C22⋊Q8—21no—
225C22⋊Q8C22×C22711yescharacteristic
226#121—21no—
227#121C22×C22711yescharacteristic
228#121C22×C22711yescharacteristic
229#121C22×C22711yescharacteristic, complemented
230#120—21no—
231C4×D4C22×C22711yescharacteristic
232C4×D4C22×C22711yescharacteristic
233C4×D4C22×C22711yescharacteristic, complemented
234C2×C4⋊C4C22×C22711yescharacteristic
235C2×C4⋊C4C22×C22711yescharacteristic, complemented
236C2×C22⋊C4—21no—
237#106C22×C22711yescharacteristic, complemented
238#106C22×C22711yescharacteristic, complemented
239#106C22×C22711yescharacteristic, complemented
240C4⋊C8C22×C22711yescharacteristic, complemented
241#103C22×C22711yescharacteristic, complemented
242#103C22×C22711yescharacteristic, complemented
243#103C22×C22711yescharacteristic, complemented
244#102C22×C22711yescharacteristic
245#102C22×C22711yescharacteristic
246#102C22×C22711yescharacteristic
247#102C22×C22711yescharacteristic
248#102C22×C22711yescharacteristic
249#102C22×C22711yescharacteristic
250#102C22×C22711yescharacteristic, complemented
251C22⋊C8C22×C22711yescharacteristic
252C22⋊C8C22×C22711yescharacteristic, complemented
253#97C22×C22711yescharacteristic, complemented
254C24—21no—
255#41—22no—
256#41—41no—
257#41—44no—
258C2×Q8—21no—
259C2×Q8C23×C22711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4D4×C22711yescharacteristic
268C2×D4C23×C22711yescharacteristic
269C2×D4C23×C22711yescharacteristic
270C2×D4C23×C22711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4C23×C22711yescharacteristic
277C22×C4C23×C22711yescharacteristic
278SD16—41no—
279D8—22no—
280D8—41no—
281M16—41no—
282C2×C8—41no—
283C2×C8C23×C22711yescharacteristic
284C2×C8C23×C22711yescharacteristic
285C2×C8C23×C22711yescharacteristic
286C2×C8C23×C22711yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4D4×C22711yescharacteristic
292C4⋊C4C23×C22711yescharacteristic
293C4⋊C4C23×C22711yescharacteristic
294C4⋊C4C23×C22711yescharacteristic
295C4⋊C4C23×C22711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4D4×C22711yescharacteristic
305C22⋊C4D4×C22711yescharacteristic
306C42C23×C22711yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23C2×D4×C22711yescharacteristic
314C23C2×D4×C22711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—42no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—81no—
329D4#41×C22712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4C2×D4×C22711yescharacteristic
345C2×C4C2×D4×C22711yescharacteristic
346C2×C4C2×D4×C22711yescharacteristic
347C2×C4C2×D4×C22711yescharacteristic
348C2×C4C24×C22711yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—84no—
368C22AGL(1,ℤ/8)×C22712yes—
369C22C22×D4×C22711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4C2×#41×C22711yescharacteristic
380C4H(5,𝔽2)×C22711yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—81no—
386C2#546×C22711yescharacteristic
387C2C2×AGL(1,ℤ/8)×C22711yescharacteristic
388C2#577×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
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. #547×C227C2semidirect
2. #546×C227C2semidirect
3. #489×C227C2semidirect
4. #483×C227C2semidirect
5. #481×C227C2semidirect
6. #476×C227C2semidirect
7. #469×C227C2semidirect
8. #465×C227C2semidirect
9. #463×C227C2semidirect
10. #449×C227C2semidirect
11. #447×C227C2semidirect
12. #442×C227C2semidirect
13. #435×C227C2semidirect
14. #417×C227C2semidirect
15. C2×#102×C227C2semidirect
19. C2×SD16×C227C22semidirect
22. C2×M16×C227C22semidirect
24. C4⋊Q8×C227C22semidirect
34. #121×C227C22semidirect
38. C4×D4×C227C22semidirect
40. C2×C4⋊C4×C227C22semidirect
42. #106×C227C22semidirect
43. #106×C227C22semidirect
44. #106×C227C22semidirect
45. C4⋊C8×C227C22semidirect
46. #103×C227C22semidirect
47. #103×C227C22semidirect
48. #103×C227C22semidirect
55. #102×C227C22semidirect
57. C22⋊C8×C227C22semidirect
58. #97×C227C22semidirect
134. D4×C227#41semidirect
194. C227#3306direct (N × H)
195. #3306C227direct (N × H)
196. #547C2×C227semidirect
197. #546C2×C227semidirect
198. #489C2×C227semidirect
199. #483C2×C227semidirect
200. #481C2×C227semidirect
201. #476C2×C227semidirect
202. #469C2×C227semidirect
203. #465C2×C227semidirect
204. #463C2×C227semidirect
205. #449C2×C227semidirect
206. #447C2×C227semidirect
207. #442C2×C227semidirect
208. #435C2×C227semidirect
209. #417C2×C227semidirect
210. C2×#102C2×C227semidirect
214. C2×SD16C22×C227semidirect
217. C2×M16C22×C227semidirect
219. C4⋊Q8C22×C227semidirect
229. #121C22×C227semidirect
233. C4×D4C22×C227semidirect
235. C2×C4⋊C4C22×C227semidirect
237. #106C22×C227semidirect
238. #106C22×C227semidirect
239. #106C22×C227semidirect
240. C4⋊C8C22×C227semidirect
241. #103C22×C227semidirect
242. #103C22×C227semidirect
243. #103C22×C227semidirect
250. #102C22×C227semidirect
252. C22⋊C8C22×C227semidirect
253. #97C22×C227semidirect
329. D4#41×C227semidirect

Maximal subgroups

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

subgroupindex
#547×C2272
#546×C2272
#489×C2272
#483×C2272
#481×C2272
#476×C2272
#469×C2272
#465×C2272
#463×C2272
#449×C2272
#447×C2272
#442×C2272
#435×C2272
#417×C2272
C2×#102×C2272
#3306227