Finite Group Databasebeta

← #46822930

Subgroups — #46822930

Order 34432.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×Dic538C211yesmaximal, characteristic
2#31770975C211yesmaximal, characteristic, complemented
3#31770946C211yesmaximal, characteristic, complemented
4#31770935C211yesmaximal, characteristic, complemented
5#31770909C211yesmaximal, characteristic, complemented
6#31770861C211yesmaximal, characteristic, complemented
7#31770844C211yesmaximal, characteristic, complemented
8#31770815C211yesmaximal, characteristic, complemented
9#31770760C211yesmaximal, characteristic, complemented
10#31770760C211yesmaximal, characteristic, complemented
11#31770710C211yesmaximal, characteristic, complemented
12#31770677C211yesmaximal, characteristic
13#31770613C211yesmaximal, characteristic, complemented
14C2×#22575376C211yesmaximal, characteristic, complemented
15#552×C269C211yesmaximal, characteristic
16C22×Dic538C2211yescharacteristic
17C22×Dic538C2211yescharacteristic
18C2×#16864392C2211yescharacteristic
19C2×#16864392C2211yescharacteristic
20C2×#16864392C2211yescharacteristic
21C2×#16864386C2211yescharacteristic
22C22×#14524130C2211yescharacteristic
23#22575453C2211yescharacteristic
24#22575452—21no—
25#22575432C2211yescharacteristic
26#22575432C2211yescharacteristic
27#22575430—21no—
28#22575430—21no—
29C4×Dic538C2211yescharacteristic
30D4×Dic269—21no—
31#22575386C2211yescharacteristic
32#22575384C2211yescharacteristic
33#22575384C2211yescharacteristic
34C4⋊#14524130C2211yescharacteristic
35#22575377C2211yescharacteristic
36#22575376C2211yescharacteristic
37#22575376C2211yescharacteristic
38#22575376C2211yescharacteristic
39#22575376C2211yescharacteristic
40#22575376C2211yescharacteristic
41#22575376C2211yescharacteristic
42#22575376C2211yescharacteristic
43#22575376C2211yescharacteristic
44#22575376C2211yescharacteristic
45C22⋊#14524130C2211yescharacteristic
46C22⋊#14524130C2211yescharacteristic
47#22575364C2211yescharacteristic
48#126×C269—21no—
49#125×C269—21no—
50#125×C269C2211yescharacteristic
51#123×C269—21no—
52C22⋊Q8×C269C2211yescharacteristic
53C22⋊Q8×C269C2211yescharacteristic
54C4×Q8×C269C2211yescharacteristic
55C4×D4×C269C2211yescharacteristic
56#117×C269—21no—
57#117×C269C2211yescharacteristic
58C2×C4⋊C4×C269C2211yescharacteristic
59C2×Dic538—22no—
60C2×Dic538—22no—
61C2×Dic538—21no—
62C2×Dic538—21no—
63C2×Dic538—41no—
64C2×Dic538—41no—
65C2×Dic538C2311yescharacteristic
66C2×Dic538C2311yescharacteristic
67C2×Dic538C2311yescharacteristic
68C22×Dic269—21no—
69C22×Dic269—21no—
70#16864392—22no—
71#16864392—41no—
72#16864392—41no—
73#16864386—41no—
74C2×#14524130—41no—
75C2×#14524130C2311yescharacteristic
76C2×#14524130C2311yescharacteristic
77C2×#14524130C2311yescharacteristic
78C2×#14524130C2311yescharacteristic
79C4⋊Dic269—21no—
80C4⋊Dic269—21no—
81C4⋊Dic269C2311yescharacteristic
82#16864381—21no—
83#16864381—21no—
84#16864381—21no—
85C22⋊Dic269—21no—
86C22⋊Dic269—21no—
87C4×Dic269—21no—
88C2×Q8×C269C2311yescharacteristic
89C2×D4×C269D411yescharacteristic
90C22×C4×C269—21no—
91C22×C4×C269C2311yescharacteristic
92C22×C4×C269C2311yescharacteristic
93C4⋊C4×C269—21no—
94C4⋊C4×C269—21no—
95C4⋊C4×C269—21no—
96C4⋊C4×C269—21no—
97C4⋊C4×C269—21no—
98C4⋊C4×C269—21no—
99C4⋊C4×C269D411yescharacteristic
100C4⋊C4×C269C2311yescharacteristic
101C4⋊C4×C269C2311yescharacteristic
102C4⋊C4×C269C2311yescharacteristic
103C22⋊C4×C269—21no—
104C22⋊C4×C269—21no—
105C22⋊C4×C269—21no—
106C22⋊C4×C269D411yescharacteristic
107C22⋊C4×C269D411yescharacteristic
108C42×C269—21no—
109C42×C269—21no—
110C42×C269C2311yescharacteristic
111Dic538—21no—
112Dic538—21no—
113Dic538—21no—
114Dic538—21no—
115Dic538—21no—
116Dic538—42no—
117Dic538—42no—
118Dic538#4112yes—
119C2×Dic269—21no—
120C2×Dic269—21no—
121C2×Dic269—21no—
122C2×Dic269—21no—
123C2×Dic269—42no—
124C2×Dic269—42no—
125#14524130—21no—
126#14524130—21no—
127#14524130—21no—
128#14524130—21no—
129C23×C269C2×D411yescharacteristic
130C23×C269C2×D411yescharacteristic
131Q8×C269—21no—
132Q8×C269—21no—
133D4×C269—22no—
134C2×C4×C269—21no—
135C2×C4×C269—21no—
136C2×C4×C269—21no—
137C2×C4×C269—21no—
138C2×C4×C269—21no—
139C2×C4×C269—21no—
140C2×C4×C269—21no—
141C2×C4×C269—21no—
142C2×C4×C269—42no—
143C2×C4×C269C2×D411yescharacteristic
144C2×C4×C269C2×D411yescharacteristic
145C2×C4×C269C2×D411yescharacteristic
146C2×C4×C269C2×D411yescharacteristic
147C2×C4×C269C2411yescharacteristic
148C22×C269—21no—
149C22×C269—21no—
150C22×C269—21no—
151C22×C269—21no—
152C22×C269—21no—
153C22×C269#13712yes—
154C22×C269C22×D411yescharacteristic
155C4×C269—21no—
156C4×C269—21no—
157C4×C269—21no—
158C4×C269—21no—
159C4×C269—41no—
160C4×C269—41no—
161C4×C269—41no—
162C4×C269—41no—
163C4×C269C2×#4111yescharacteristic
164C4×C269H(5,𝔽2)11yescharacteristic
165Dic269—22no—
166Dic269—41no—
167Dic269—41no—
168Dic269—41no—
169C2×C269—22no—
170C2×C269—41no—
171C2×C269#54611yescharacteristic
172C2×C269C2×#13711yescharacteristic
173C2×C269#57811yescharacteristic
174C269#331211yescharacteristic, complemented, Sylow
175#3312—2691nomaximal, Sylow
176#552D26911yescharacteristic
177D4×Q8—2691no—
178#491—2691no—
179#485—2691no—
180#483—2691no—
181#479—2691no—
182#470—2691no—
183#467—2691no—
184#462—2691no—
185#451—2691no—
186#451—2691no—
187#441—2691no—
188#435—2691no—
189#419—2691no—
190C2×#103—2691no—
191C22×Q8—2691no—
192C22×Q8—2691no—
193C2×Q16—2691no—
194C2×Q16—2691no—
195C2×Q16—2691no—
196C2×M16—2691no—
197C22×C8—2691no—
198C4⋊Q8—2691no—
199C4⋊Q8—5381no—
200#126—21no—
201#125—21no—
202#125C2×D26911yescharacteristic
203#123—21no—
204C22⋊Q8—2691no—
205C22⋊Q8—2691no—
206C22⋊Q8—5381no—
207C22⋊Q8—5381no—
208C22⋊Q8C2×D26911yescharacteristic
209C22⋊Q8C2×D26911yescharacteristic
210C4×Q8—2691no—
211C4×Q8C2×D26911yescharacteristic
212C4×D4—5381no—
213C4×D4C2×D26911yescharacteristic
214#117—21no—
215#117C2×D26911yescharacteristic
216C2×C4⋊C4C2×D26911yescharacteristic
217#107—2691no—
218#106—2691no—
219#106—2691no—
220C4⋊C8—2691no—
221#103—2691no—
222#103—2691no—
223#103—2691no—
224#103—2691no—
225#103—2691no—
226#103—2691no—
227#103—2691no—
228#103—2691no—
229#103—2691no—
230#103—2691no—
231C22⋊C8—2691no—
232C22⋊C8—2691no—
233#97—2691no—
234C2×Q8—2691no—
235C2×Q8—2691no—
236C2×Q8—2691no—
237C2×Q8—5382no—
238C2×Q8—5382no—
239C2×Q8—5381no—
240C2×Q8—5381no—
241C2×Q8—10761no—
242C2×Q8—10761no—
243C2×Q8C22×D26911yescharacteristic
244C2×D4C22⋊D26911yescharacteristic
245C22×C4—21no—
246C22×C4—5381no—
247C22×C4—5381no—
248C22×C4C22×D26911yescharacteristic
249C22×C4C22×D26911yescharacteristic
250Q16—5382no—
251Q16—10761no—
252Q16—10761no—
253M16—10761no—
254C2×C8—2691no—
255C2×C8—2691no—
256C2×C8—2691no—
257C2×C8—2691no—
258C2×C8—10761no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—2691no—
266C4⋊C4—5381no—
267C4⋊C4—5381no—
268C4⋊C4—5381no—
269C4⋊C4—5381no—
270C4⋊C4—5381no—
271C4⋊C4C22⋊D26911yescharacteristic
272C4⋊C4C22×D26911yescharacteristic
273C4⋊C4C22×D26911yescharacteristic
274C4⋊C4C22×D26911yescharacteristic
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—5381no—
279C22⋊C4—5381no—
280C22⋊C4C22⋊D26911yescharacteristic
281C22⋊C4C22⋊D26911yescharacteristic
282C42—21no—
283C42—21no—
284C42—5381no—
285C42C22×D26911yescharacteristic
286C23C2×C22⋊D26911yescharacteristic
287C23C2×C22⋊D26911yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—2692no—
291Q8—5381no—
292Q8—5381no—
293Q8—5381no—
294Q8—5381no—
295Q8—5381no—
296Q8—10762no—
297Q8—10762no—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—5381no—
309C2×C4—5381no—
310C2×C4—5381no—
311C2×C4—5381no—
312C2×C4—10762no—
313C2×C4—10762no—
314C2×C4C2×C22⋊D26911yescharacteristic
315C2×C4C2×C22⋊D26911yescharacteristic
316C2×C4C2×C22⋊D26911yescharacteristic
317C2×C4C2×C22⋊D26911yescharacteristic
318C2×C4C23×D26911yescharacteristic
319C8—5381no—
320C8—5381no—
321C8—5381no—
322C8—5381no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#2257548312yes—
329C22C22×C22⋊D26911yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—5382no—
339C4—10761no—
340C4—10761no—
341C4—10761no—
342C4#41×D26911yescharacteristic
343C4#2257549911yescharacteristic
344C2—22no—
345C2—41no—
346C2#3177124311yescharacteristic
347C2C2×#2257548311yescharacteristic
348C2#3177142011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#552×C269C2
socleC22×C269C22×D4
prime radicalC2×D4×C269D4
hypercenter#552D269
hypocenterC269#3312
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #31770975C2semidirect
3. #31770946C2semidirect
4. #31770935C2semidirect
5. #31770909C2semidirect
6. #31770861C2semidirect
7. #31770844C2semidirect
8. #31770815C2semidirect
9. #31770760C2semidirect
10. #31770760C2semidirect
11. #31770710C2semidirect
13. #31770613C2semidirect
14. C2×#22575376C2semidirect
174. C269#3312semidirect

Maximal subgroups

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

subgroupindex
D4×Dic5382
#317709752
#317709462
#317709352
#317709092
#317708612
#317708442
#317708152
#317707602
#317707102
#317706772
#317706132
C2×#225753762
#552×C2692
#3312269