Finite Group Databasebeta

← #3297×C227 #44397253

Subgroups — #3297×C227 #44397253

Order 29056.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#553×C227C211yesmaximal, characteristic, complemented
2#546×C227C211yesmaximal, characteristic, complemented
3#488×C227C211yesmaximal, characteristic, complemented
4#484×C227C211yesmaximal, characteristic, complemented
5#482×C227C211yesmaximal, characteristic, complemented
6#478×C227C211yesmaximal, characteristic, complemented
7#466×C227C211yesmaximal, characteristic, complemented
8#465×C227C211yesmaximal, characteristic, complemented
9#463×C227C211yesmaximal, characteristic, complemented
10#449×C227C211yesmaximal, characteristic, complemented
11#447×C227C211yesmaximal, characteristic, complemented
12C4×D8×C227C211yesmaximal, characteristic, complemented
13C8×D4×C227C211yesmaximal, characteristic, complemented
14#416×C227C211yesmaximal, characteristic, complemented
15C2×#102×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227C2211yescharacteristic
17C22×D4×C227C2211yescharacteristic
18C2×SD16×C227C2211yescharacteristic, complemented
19C2×D8×C227C2211yescharacteristic
20C2×D8×C227C2211yescharacteristic
21C22×C8×C227C2211yescharacteristic
22C22×C8×C227C2211yescharacteristic, complemented
23#126×C227—21no—
24#125×C227C2211yescharacteristic, complemented
25#124×C227C2211yescharacteristic
26#123×C227—21no—
27#123×C227—21no—
28C22⋊Q8×C227C2211yescharacteristic
29#121×C227—21no—
30#121×C227—21no—
31#121×C227C2211yescharacteristic
32#121×C227C2211yescharacteristic
33#121×C227C2211yescharacteristic, complemented
34#120×C227—21no—
35C4×D4×C227—21no—
36C4×D4×C227C2211yescharacteristic
37C4×D4×C227C2211yescharacteristic
38C4×D4×C227C2211yescharacteristic, complemented
39#117×C227C2211yescharacteristic, complemented
40C2×C4⋊C4×C227C2211yescharacteristic
41C2×C22⋊C4×C227—21no—
42#107×C227C2211yescharacteristic, complemented
43#107×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
58C4×C8×C227C2211yescharacteristic, complemented
59C24×C227—21no—
60#41×C227—22no—
61#41×C227—41no—
62C2×Q8×C227C2311yescharacteristic
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—41no—
70C2×D4×C227D411yescharacteristic
71C2×D4×C227C2311yescharacteristic
72C2×D4×C227C2311yescharacteristic
73C2×D4×C227C2311yescharacteristic
74C22×C4×C227—21no—
75C22×C4×C227—21no—
76C22×C4×C227—21no—
77C22×C4×C227—21no—
78C22×C4×C227C2311yescharacteristic
79C22×C4×C227C2311yescharacteristic
80SD16×C227—41no—
81D8×C227—22no—
82D8×C227—41no—
83C2×C8×C227—41no—
84C2×C8×C227—41no—
85C2×C8×C227C2311yescharacteristic
86C2×C8×C227C2311yescharacteristic
87C2×C8×C227C2311yescharacteristic
88C2×C8×C227C2311yescharacteristic
89C4⋊C4×C227—21no—
90C4⋊C4×C227—21no—
91C4⋊C4×C227—21no—
92C4⋊C4×C227—21no—
93C4⋊C4×C227D411yescharacteristic
94C4⋊C4×C227C2311yescharacteristic
95C4⋊C4×C227C2311yescharacteristic
96C4⋊C4×C227C2311yescharacteristic
97C4⋊C4×C227C2311yescharacteristic
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×C227—21no—
106C22⋊C4×C227—21no—
107C22⋊C4×C227D411yescharacteristic
108C22⋊C4×C227D411yescharacteristic
109C42×C227—21no—
110C42×C227C2311yescharacteristic
111C23×C227—21no—
112C23×C227—21no—
113C23×C227—21no—
114C23×C227—42no—
115C23×C227—42no—
116C23×C227—81no—
117C23×C227C2×D411yescharacteristic
118C23×C227C2×D411yescharacteristic
119Q8×C227—21no—
120Q8×C227—21no—
121D4×C227—22no—
122D4×C227—21no—
123D4×C227—21no—
124D4×C227—21no—
125D4×C227—21no—
126D4×C227—21no—
127D4×C227—42no—
128D4×C227—42no—
129D4×C227—42no—
130D4×C227—81no—
131D4×C227—81no—
132D4×C227#4112yes—
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—21no—
142C2×C4×C227—42no—
143C2×C4×C227—81no—
144C2×C4×C227—81no—
145C2×C4×C227—81no—
146C2×C4×C227C2×D411yescharacteristic
147C2×C4×C227C2×D411yescharacteristic
148C2×C4×C227C2×D411yescharacteristic
149C2×C4×C227C2×D411yescharacteristic
150C2×C4×C227C2411yescharacteristic
151C8×C227—21no—
152C8×C227—21no—
153C8×C227—21no—
154C8×C227—21no—
155C22×C227—21no—
156C22×C227—21no—
157C22×C227—21no—
158C22×C227—22no—
159C22×C227—21no—
160C22×C227—21no—
161C22×C227—41no—
162C22×C227—41no—
163C22×C227—41no—
164C22×C227—41no—
165C22×C227—41no—
166C22×C227—41no—
167C22×C227—41no—
168C22×C227—41no—
169C22×C227—84no—
170C22×C227#13512yes—
171C22×C227C22×D411yescharacteristic
172C4×C227—21no—
173C4×C227—21no—
174C4×C227—21no—
175C4×C227—21no—
176C4×C227—41no—
177C4×C227—41no—
178C4×C227—41no—
179C4×C227—41no—
180C4×C227—41no—
181C4×C227C2×#4111yescharacteristic
182C4×C227H(5,𝔽2)11yescharacteristic
183C2×C227—22no—
184C2×C227—41no—
185C2×C227—42no—
186C2×C227—81no—
187C2×C227—81no—
188C2×C227#54611yescharacteristic
189C2×C227C2×#13511yescharacteristic
190C2×C227#57611yescharacteristic
191C227#329711yescharacteristic, direct product factor, Sylow
192#3297C22711yesmaximal, characteristic, direct product factor, Sylow
193#553C2×C22711yescharacteristic, complemented
194#546C2×C22711yescharacteristic, complemented
195#488C2×C22711yescharacteristic, complemented
196#484C2×C22711yescharacteristic, complemented
197#482C2×C22711yescharacteristic, complemented
198#478C2×C22711yescharacteristic, complemented
199#466C2×C22711yescharacteristic, complemented
200#465C2×C22711yescharacteristic, complemented
201#463C2×C22711yescharacteristic, complemented
202#449C2×C22711yescharacteristic, complemented
203#447C2×C22711yescharacteristic, complemented
204C4×D8C2×C22711yescharacteristic, complemented
205C8×D4C2×C22711yescharacteristic, complemented
206#416C2×C22711yescharacteristic, complemented
207C2×#102C2×C22711yescharacteristic, complemented
208C2×#41C22×C22711yescharacteristic
209C22×D4C22×C22711yescharacteristic
210C2×SD16C22×C22711yescharacteristic, complemented
211C2×D8C22×C22711yescharacteristic
212C2×D8C22×C22711yescharacteristic
213C22×C8C22×C22711yescharacteristic
214C22×C8C22×C22711yescharacteristic, complemented
215#126—21no—
216#125C22×C22711yescharacteristic, complemented
217#124C22×C22711yescharacteristic
218#123—21no—
219#123—21no—
220C22⋊Q8C22×C22711yescharacteristic
221#121—21no—
222#121—21no—
223#121C22×C22711yescharacteristic
224#121C22×C22711yescharacteristic
225#121C22×C22711yescharacteristic, complemented
226#120—21no—
227C4×D4—21no—
228C4×D4C22×C22711yescharacteristic
229C4×D4C22×C22711yescharacteristic
230C4×D4C22×C22711yescharacteristic, complemented
231#117C22×C22711yescharacteristic, complemented
232C2×C4⋊C4C22×C22711yescharacteristic
233C2×C22⋊C4—21no—
234#107C22×C22711yescharacteristic, complemented
235#107C22×C22711yescharacteristic, complemented
236#106C22×C22711yescharacteristic, complemented
237C4⋊C8C22×C22711yescharacteristic, complemented
238#103C22×C22711yescharacteristic, complemented
239#103C22×C22711yescharacteristic, complemented
240#103C22×C22711yescharacteristic, complemented
241#102C22×C22711yescharacteristic
242#102C22×C22711yescharacteristic
243#102C22×C22711yescharacteristic
244#102C22×C22711yescharacteristic
245#102C22×C22711yescharacteristic
246#102C22×C22711yescharacteristic
247#102C22×C22711yescharacteristic, complemented
248C22⋊C8C22×C22711yescharacteristic
249C22⋊C8C22×C22711yescharacteristic, complemented
250C4×C8C22×C22711yescharacteristic, complemented
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C23×C22711yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—41no—
262C2×D4D4×C22711yescharacteristic
263C2×D4C23×C22711yescharacteristic
264C2×D4C23×C22711yescharacteristic
265C2×D4C23×C22711yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4C23×C22711yescharacteristic
271C22×C4C23×C22711yescharacteristic
272SD16—41no—
273D8—22no—
274D8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8C23×C22711yescharacteristic
278C2×C8C23×C22711yescharacteristic
279C2×C8C23×C22711yescharacteristic
280C2×C8C23×C22711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4D4×C22711yescharacteristic
286C4⋊C4C23×C22711yescharacteristic
287C4⋊C4C23×C22711yescharacteristic
288C4⋊C4C23×C22711yescharacteristic
289C4⋊C4C23×C22711yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4D4×C22711yescharacteristic
300C22⋊C4D4×C22711yescharacteristic
301C42—21no—
302C42C23×C22711yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23C2×D4×C22711yescharacteristic
310C23C2×D4×C22711yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323D4—81no—
324D4#41×C22712yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4C2×D4×C22711yescharacteristic
339C2×C4C2×D4×C22711yescharacteristic
340C2×C4C2×D4×C22711yescharacteristic
341C2×C4C2×D4×C22711yescharacteristic
342C2×C4C24×C22711yescharacteristic
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—84no—
362C22#135×C22712yes—
363C22C22×D4×C22711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4C2×#41×C22711yescharacteristic
374C4H(5,𝔽2)×C22711yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—81no—
380C2#546×C22711yescharacteristic
381C2C2×#135×C22711yescharacteristic
382C2#576×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. #553×C227C2semidirect
2. #546×C227C2semidirect
3. #488×C227C2semidirect
4. #484×C227C2semidirect
5. #482×C227C2semidirect
6. #478×C227C2semidirect
7. #466×C227C2semidirect
8. #465×C227C2semidirect
9. #463×C227C2semidirect
10. #449×C227C2semidirect
11. #447×C227C2semidirect
12. C4×D8×C227C2semidirect
13. C8×D4×C227C2semidirect
14. #416×C227C2semidirect
15. C2×#102×C227C2semidirect
18. C2×SD16×C227C22semidirect
22. C22×C8×C227C22semidirect
24. #125×C227C22semidirect
33. #121×C227C22semidirect
38. C4×D4×C227C22semidirect
39. #117×C227C22semidirect
42. #107×C227C22semidirect
43. #107×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. C4×C8×C227C22semidirect
191. C227#3297direct (N × H)
192. #3297C227direct (N × H)
193. #553C2×C227semidirect
194. #546C2×C227semidirect
195. #488C2×C227semidirect
196. #484C2×C227semidirect
197. #482C2×C227semidirect
198. #478C2×C227semidirect
199. #466C2×C227semidirect
200. #465C2×C227semidirect
201. #463C2×C227semidirect
202. #449C2×C227semidirect
203. #447C2×C227semidirect
204. C4×D8C2×C227semidirect
205. C8×D4C2×C227semidirect
206. #416C2×C227semidirect
207. C2×#102C2×C227semidirect
210. C2×SD16C22×C227semidirect
214. C22×C8C22×C227semidirect
216. #125C22×C227semidirect
225. #121C22×C227semidirect
230. C4×D4C22×C227semidirect
231. #117C22×C227semidirect
234. #107C22×C227semidirect
235. #107C22×C227semidirect
236. #106C22×C227semidirect
237. C4⋊C8C22×C227semidirect
238. #103C22×C227semidirect
239. #103C22×C227semidirect
240. #103C22×C227semidirect
247. #102C22×C227semidirect
249. C22⋊C8C22×C227semidirect
250. C4×C8C22×C227semidirect

Maximal subgroups

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

subgroupindex
#553×C2272
#546×C2272
#488×C2272
#484×C2272
#482×C2272
#478×C2272
#466×C2272
#465×C2272
#463×C2272
#449×C2272
#447×C2272
C4×D8×C2272
C8×D4×C2272
#416×C2272
C2×#102×C2272
#3297227