Finite Group Databasebeta

← #3343×C227 #44397299

Subgroups — #3343×C227 #44397299

Order 29056.

376 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#552×C227C211yesmaximal, characteristic, complemented
2#547×C227C211yesmaximal, characteristic
3#501×C227C211yesmaximal, characteristic, complemented
4#484×C227C211yesmaximal, characteristic, complemented
5#483×C227C211yesmaximal, characteristic, complemented
6#482×C227C211yesmaximal, characteristic, complemented
7#480×C227C211yesmaximal, characteristic, complemented
8#478×C227C211yesmaximal, characteristic, complemented
9#477×C227C211yesmaximal, characteristic, complemented
10#468×C227C211yesmaximal, characteristic, complemented
11#465×C227C211yesmaximal, characteristic, complemented
12#440×C227C211yesmaximal, characteristic, complemented
13#435×C227C211yesmaximal, characteristic, complemented
14#428×C227C211yesmaximal, characteristic, complemented
15C2×#107×C227C211yesmaximal, characteristic, complemented
16C2×#41×C227—21no—
17C2×SD16×C227C2211yescharacteristic
18C2×M16×C227C2211yescharacteristic
19C22×C8×C227C2211yescharacteristic
20C4⋊Q8×C227C2211yescharacteristic
21#126×C227—21no—
22#125×C227—21no—
23#125×C227C2211yescharacteristic
24#123×C227—21no—
25#123×C227—21no—
26#123×C227—21no—
27C22⋊Q8×C227—21no—
28C22⋊Q8×C227C2211yescharacteristic
29C22⋊Q8×C227C2211yescharacteristic
30#121×C227C2211yescharacteristic
31#121×C227C2211yescharacteristic
32C4×Q8×C227C2211yescharacteristic
33C4×D4×C227C2211yescharacteristic
34C4×D4×C227C2211yescharacteristic
35#117×C227—21no—
36#117×C227C2211yescharacteristic
37C2×C4⋊C4×C227C2211yescharacteristic
38C2×C4⋊C4×C227C2211yescharacteristic
39C2×C4⋊C4×C227C2211yescharacteristic
40#107×C227C2211yescharacteristic
41#107×C227C2211yescharacteristic
42#107×C227C2211yescharacteristic
43#107×C227C2211yescharacteristic
44#107×C227C2211yescharacteristic
45#107×C227C2211yescharacteristic
46#106×C227C2211yescharacteristic
47#106×C227C2211yescharacteristic
48#106×C227C2211yescharacteristic
49C4⋊C8×C227C2211yescharacteristic
50#103×C227C2211yescharacteristic
51#103×C227C2211yescharacteristic
52#103×C227C2211yescharacteristic
53#102×C227C2211yescharacteristic
54#102×C227C2211yescharacteristic
55#102×C227C2211yescharacteristic
56C22⋊C8×C227C2211yescharacteristic
57C22⋊C8×C227C2211yescharacteristic
58#97×C227C2211yescharacteristic
59#41×C227—44no—
60C2×Q8×C227—21no—
61C2×Q8×C227C2311yescharacteristic
62C2×D4×C227—21no—
63C2×D4×C227—21no—
64C2×D4×C227D411yescharacteristic
65C2×D4×C227C2311yescharacteristic
66C22×C4×C227—21no—
67C22×C4×C227—21no—
68C22×C4×C227—21no—
69C22×C4×C227—21no—
70C22×C4×C227C2311yescharacteristic
71C22×C4×C227C2311yescharacteristic
72SD16×C227—22no—
73M16×C227—22no—
74C2×C8×C227—22no—
75C2×C8×C227C2311yescharacteristic
76C2×C8×C227C2311yescharacteristic
77C2×C8×C227C2311yescharacteristic
78C2×C8×C227C2311yescharacteristic
79C4⋊C4×C227—21no—
80C4⋊C4×C227—21no—
81C4⋊C4×C227—21no—
82C4⋊C4×C227—21no—
83C4⋊C4×C227—21no—
84C4⋊C4×C227—21no—
85C4⋊C4×C227—21no—
86C4⋊C4×C227—21no—
87C4⋊C4×C227—21no—
88C4⋊C4×C227D411yescharacteristic
89C4⋊C4×C227C2311yescharacteristic
90C4⋊C4×C227C2311yescharacteristic
91C4⋊C4×C227C2311yescharacteristic
92C4⋊C4×C227C2311yescharacteristic
93C4⋊C4×C227C2311yescharacteristic
94C4⋊C4×C227C2311yescharacteristic
95C22⋊C4×C227—21no—
96C22⋊C4×C227—21no—
97C22⋊C4×C227—21no—
98C22⋊C4×C227—21no—
99C22⋊C4×C227—21no—
100C22⋊C4×C227—21no—
101C22⋊C4×C227D411yescharacteristic
102C22⋊C4×C227D411yescharacteristic
103C42×C227—21no—
104C42×C227—21no—
105C42×C227C2311yescharacteristic
106C23×C227—21no—
107C23×C227C2×D411yescharacteristic
108C23×C227C2×D411yescharacteristic
109Q8×C227—21no—
110Q8×C227—21no—
111Q8×C227—42no—
112D4×C227—22no—
113D4×C227—21no—
114D4×C227—21no—
115D4×C227—42no—
116D4×C227—42no—
117C2×C4×C227—21no—
118C2×C4×C227—21no—
119C2×C4×C227—21no—
120C2×C4×C227—21no—
121C2×C4×C227—21no—
122C2×C4×C227—21no—
123C2×C4×C227—21no—
124C2×C4×C227—21no—
125C2×C4×C227—21no—
126C2×C4×C227—21no—
127C2×C4×C227—21no—
128C2×C4×C227—42no—
129C2×C4×C227—42no—
130C2×C4×C227—42no—
131C2×C4×C227—42no—
132C2×C4×C227C2×D411yescharacteristic
133C2×C4×C227C2×D411yescharacteristic
134C2×C4×C227C2×D411yescharacteristic
135C2×C4×C227C2×D411yescharacteristic
136C2×C4×C227C2411yescharacteristic
137C8×C227—21no—
138C8×C227—21no—
139C8×C227—21no—
140C8×C227#4112yescomplemented
141C22×C227—21no—
142C22×C227—21no—
143C22×C227—21no—
144C22×C227—21no—
145C22×C227—21no—
146C22×C227—41no—
147C22×C227—41no—
148C22×C227—41no—
149C22×C227AGL(1,ℤ/8)12yes—
150C22×C227C22×D411yescharacteristic
151C4×C227—21no—
152C4×C227—21no—
153C4×C227—21no—
154C4×C227—21no—
155C4×C227—41no—
156C4×C227—41no—
157C4×C227—41no—
158C4×C227—41no—
159C4×C227—41no—
160C4×C227—41no—
161C4×C227—41no—
162C4×C227C2×#4111yescharacteristic
163C4×C2272−1+411yescharacteristic
164C2×C227—22no—
165C2×C227—41no—
166C2×C227—81no—
167C2×C227#54711yescharacteristic
168C2×C227C2×AGL(1,ℤ/8)11yescharacteristic
169C2×C227#57811yescharacteristic
170C227#334311yescharacteristic, direct product factor, Sylow
171#3343C22711yesmaximal, characteristic, direct product factor, Sylow
172#552C2×C22711yescharacteristic, complemented
173#547C2×C22711yescharacteristic
174#501C2×C22711yescharacteristic, complemented
175#484C2×C22711yescharacteristic, complemented
176#483C2×C22711yescharacteristic, complemented
177#482C2×C22711yescharacteristic, complemented
178#480C2×C22711yescharacteristic, complemented
179#478C2×C22711yescharacteristic, complemented
180#477C2×C22711yescharacteristic, complemented
181#468C2×C22711yescharacteristic, complemented
182#465C2×C22711yescharacteristic, complemented
183#440C2×C22711yescharacteristic, complemented
184#435C2×C22711yescharacteristic, complemented
185#428C2×C22711yescharacteristic, complemented
186C2×#107C2×C22711yescharacteristic, complemented
187C2×#41—21no—
188C2×SD16C22×C22711yescharacteristic
189C2×M16C22×C22711yescharacteristic
190C22×C8C22×C22711yescharacteristic
191C4⋊Q8C22×C22711yescharacteristic
192#126—21no—
193#125—21no—
194#125C22×C22711yescharacteristic
195#123—21no—
196#123—21no—
197#123—21no—
198C22⋊Q8—21no—
199C22⋊Q8C22×C22711yescharacteristic
200C22⋊Q8C22×C22711yescharacteristic
201#121C22×C22711yescharacteristic
202#121C22×C22711yescharacteristic
203C4×Q8C22×C22711yescharacteristic
204C4×D4C22×C22711yescharacteristic
205C4×D4C22×C22711yescharacteristic
206#117—21no—
207#117C22×C22711yescharacteristic
208C2×C4⋊C4C22×C22711yescharacteristic
209C2×C4⋊C4C22×C22711yescharacteristic
210C2×C4⋊C4C22×C22711yescharacteristic
211#107C22×C22711yescharacteristic
212#107C22×C22711yescharacteristic
213#107C22×C22711yescharacteristic
214#107C22×C22711yescharacteristic
215#107C22×C22711yescharacteristic
216#107C22×C22711yescharacteristic
217#106C22×C22711yescharacteristic
218#106C22×C22711yescharacteristic
219#106C22×C22711yescharacteristic
220C4⋊C8C22×C22711yescharacteristic
221#103C22×C22711yescharacteristic
222#103C22×C22711yescharacteristic
223#103C22×C22711yescharacteristic
224#102C22×C22711yescharacteristic
225#102C22×C22711yescharacteristic
226#102C22×C22711yescharacteristic
227C22⋊C8C22×C22711yescharacteristic
228C22⋊C8C22×C22711yescharacteristic
229#97C22×C22711yescharacteristic
230#41—44no—
231C2×Q8—21no—
232C2×Q8C23×C22711yescharacteristic
233C2×D4—21no—
234C2×D4—21no—
235C2×D4D4×C22711yescharacteristic
236C2×D4C23×C22711yescharacteristic
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4C23×C22711yescharacteristic
242C22×C4C23×C22711yescharacteristic
243SD16—22no—
244M16—22no—
245C2×C8—22no—
246C2×C8C23×C22711yescharacteristic
247C2×C8C23×C22711yescharacteristic
248C2×C8C23×C22711yescharacteristic
249C2×C8C23×C22711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4D4×C22711yescharacteristic
260C4⋊C4C23×C22711yescharacteristic
261C4⋊C4C23×C22711yescharacteristic
262C4⋊C4C23×C22711yescharacteristic
263C4⋊C4C23×C22711yescharacteristic
264C4⋊C4C23×C22711yescharacteristic
265C4⋊C4C23×C22711yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4D4×C22711yescharacteristic
273C22⋊C4D4×C22711yescharacteristic
274C42—21no—
275C42—21no—
276C42C23×C22711yescharacteristic
277C23—21no—
278C23C2×D4×C22711yescharacteristic
279C23C2×D4×C22711yescharacteristic
280Q8—21no—
281Q8—21no—
282Q8—42no—
283D4—22no—
284D4—21no—
285D4—21no—
286D4—42no—
287D4—42no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4C2×D4×C22711yescharacteristic
304C2×C4C2×D4×C22711yescharacteristic
305C2×C4C2×D4×C22711yescharacteristic
306C2×C4C2×D4×C22711yescharacteristic
307C2×C4C24×C22711yescharacteristic
308C8—21no—
309C8—21no—
310C8—21no—
311C8#41×C22712yescomplemented
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22AGL(1,ℤ/8)×C22712yes—
321C22C22×D4×C22711yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4C2×#41×C22711yescharacteristic
334C42−1+4×C22711yescharacteristic
335C2—22no—
336C2—41no—
337C2—81no—
338C2#547×C22711yescharacteristic
339C2C2×AGL(1,ℤ/8)×C22711yescharacteristic
340C2#578×C22711yescharacteristic
subgroupG/H
center, socleC22×C227C22×D4
commutator, Frattini, cosocleC2×C4C24×C227
prime radical#547×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. #501×C227C2semidirect
4. #484×C227C2semidirect
5. #483×C227C2semidirect
6. #482×C227C2semidirect
7. #480×C227C2semidirect
8. #478×C227C2semidirect
9. #477×C227C2semidirect
10. #468×C227C2semidirect
11. #465×C227C2semidirect
12. #440×C227C2semidirect
13. #435×C227C2semidirect
14. #428×C227C2semidirect
15. C2×#107×C227C2semidirect
140. C8×C227#41semidirect
170. C227#3343direct (N × H)
171. #3343C227direct (N × H)
172. #552C2×C227semidirect
174. #501C2×C227semidirect
175. #484C2×C227semidirect
176. #483C2×C227semidirect
177. #482C2×C227semidirect
178. #480C2×C227semidirect
179. #478C2×C227semidirect
180. #477C2×C227semidirect
181. #468C2×C227semidirect
182. #465C2×C227semidirect
183. #440C2×C227semidirect
184. #435C2×C227semidirect
185. #428C2×C227semidirect
186. C2×#107C2×C227semidirect
311. C8#41×C227semidirect

Maximal subgroups

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

subgroupindex
#552×C2272
#547×C2272
#501×C2272
#484×C2272
#483×C2272
#482×C2272
#480×C2272
#478×C2272
#477×C2272
#468×C2272
#465×C2272
#440×C2272
#435×C2272
#428×C2272
C2×#107×C2272
#3343227