Finite Group Databasebeta

← #28417112

Subgroups — #28417112

Order 14912.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21418530C211yesmaximal, characteristic, complemented
2#21418530C211yesmaximal, characteristic, complemented
3#21418524C211yesmaximal, characteristic, complemented
4#21418520C211yesmaximal, characteristic, complemented
5#21418520C211yesmaximal, characteristic, complemented
6#21418519C211yesmaximal, characteristic, complemented
7#21418511C211yesmaximal, characteristic, complemented
8#21418505C211yesmaximal, characteristic
9#21418497C211yesmaximal, characteristic, complemented
10C4×C22⋊D233C211yesmaximal, characteristic, complemented
11#21418493C211yesmaximal, characteristic, complemented
12#21418491C211yesmaximal, characteristic, complemented
13C2×#16836571C211yesmaximal, characteristic, complemented
14C22⋊C4×D233C211yesmaximal, characteristic, complemented
15C22⋊Q8×C233C211yesmaximal, characteristic, complemented
16C23×D233C2211yescharacteristic
17C2×D932C2211yescharacteristic
18C2×D932C2211yescharacteristic
19C2×C22⋊D233C2211yescharacteristic
20C22×Dic233C2211yescharacteristic
21C2×C4×D233C2211yescharacteristic
22C2×C4×D233C2211yescharacteristic
23C2×C4×D233C2211yescharacteristic, complemented
24C4⋊Dic233C2211yescharacteristic, complemented
25C4⋊Dic233C2211yescharacteristic, complemented
26#16836573C2211yescharacteristic, complemented
27#16836573C2211yescharacteristic, complemented
28#16836573C2211yescharacteristic, complemented
29C22⋊Dic233C2211yescharacteristic
30#16836571C2211yescharacteristic
31#16836571C2211yescharacteristic
32#16836571C2211yescharacteristic
33#16836571C2211yescharacteristic
34#16836571C2211yescharacteristic
35#16836571C2211yescharacteristic
36#16836571C2211yescharacteristic
37#16836571C2211yescharacteristic
38#16836571C2211yescharacteristic, complemented
39#16836571C2211yescharacteristic, complemented
40#16836571C2211yescharacteristic, complemented
41C4×Dic233C2211yescharacteristic, complemented
42C4×Dic233C2211yescharacteristic, complemented
43C4×Dic233C2211yescharacteristic, complemented
44C2×Q8×C233C2211yescharacteristic, complemented
45C22×C4×C233C2211yescharacteristic
46C4⋊C4×C233C2211yescharacteristic, complemented
47C4⋊C4×C233C2211yescharacteristic, complemented
48C4⋊C4×C233C2211yescharacteristic, complemented
49C22⋊C4×C233C2211yescharacteristic
50C22⋊C4×C233C2211yescharacteristic
51C22×D233—22no—
52C22×D233—22no—
53C22×D233—41no—
54C22×D233C2311yescharacteristic
55C22×D233C2311yescharacteristic
56C22×D233C2311yescharacteristic
57D932—22no—
58D932—41no—
59C22⋊D233—22no—
60C2×Dic233—22no—
61C2×Dic233C2311yescharacteristic
62C2×Dic233C2311yescharacteristic
63C2×Dic233C2311yescharacteristic
64C2×Dic233C2311yescharacteristic
65C2×Dic233C2311yescharacteristic
66C4×D233—22no—
67C4×D233—41no—
68C4×D233—41no—
69C23×C233C2311yescharacteristic
70Q8×C233—41no—
71C2×C4×C233—41no—
72C2×C4×C233C2311yescharacteristic
73C2×C4×C233C2311yescharacteristic
74C2×C4×C233C2311yescharacteristic
75C2×C4×C233C2311yescharacteristic
76C2×C4×C233C2311yescharacteristic
77C2×C4×C233C2311yescharacteristic
78C22×C233—21no—
79C22×C233—21no—
80C22×C233#4112yes—
81C22×C233C2411yescharacteristic
82C2×D233—21no—
83C2×D233—21no—
84C2×D233—21no—
85C2×D233—21no—
86C2×D233—21no—
87C2×D233—21no—
88C2×D233—21no—
89C2×D233—21no—
90C2×D233—44no—
91C2×D233#4112yes—
92C4×C233—21no—
93C4×C233—21no—
94C4×C233—21no—
95C4×C233—21no—
96C4×C233—21no—
97C4×C233—21no—
98Dic233—21no—
99Dic233—21no—
100Dic233—21no—
101Dic233—21no—
102Dic233—21no—
103C2×C233—22no—
104C2×C233C2×#4111yescharacteristic
105C2×C233C2×#4111yescharacteristic
106C2×C233H(5,𝔽2)11yescharacteristic
107D233—22no—
108D233—41no—
109D233—41no—
110C233#55111yescharacteristic, complemented, Sylow
111#551—2331nomaximal, Sylow
112#126—2331no—
113#126—2331no—
114#124—2331no—
115#123—2331no—
116#123—2331no—
117#123—2331no—
118C22⋊Q8—2331no—
119C22⋊Q8D23311yescharacteristic, complemented
120#120—2331no—
121C4×D4—2331no—
122C4×D4—2331no—
123#117—2331no—
124#117—2331no—
125C2×C22⋊C4—2331no—
126C2×C22⋊C4—2331no—
127C24—2331no—
128C2×Q8C2×D23311yescharacteristic, complemented
129C2×D4—2331no—
130C2×D4—2331no—
131C2×D4—2331no—
132C22×C4—2331no—
133C22×C4—2331no—
134C22×C4—2331no—
135C22×C4—2331no—
136C22×C4C2×D23311yescharacteristic
137C4⋊C4—2331no—
138C4⋊C4—2331no—
139C4⋊C4—2331no—
140C4⋊C4—2331no—
141C4⋊C4—2331no—
142C4⋊C4C2×D23311yescharacteristic, complemented
143C4⋊C4C2×D23311yescharacteristic, complemented
144C4⋊C4C2×D23311yescharacteristic, complemented
145C22⋊C4—2331no—
146C22⋊C4—2331no—
147C22⋊C4—2331no—
148C22⋊C4—2331no—
149C22⋊C4—2331no—
150C22⋊C4—2331no—
151C22⋊C4—2331no—
152C22⋊C4—2331no—
153C22⋊C4—2331no—
154C22⋊C4—2331no—
155C22⋊C4—2331no—
156C22⋊C4—2331no—
157C22⋊C4C2×D23311yescharacteristic
158C22⋊C4C2×D23311yescharacteristic
159C42—2331no—
160C42—2331no—
161C42—2331no—
162C23—9321no—
163C23—4662no—
164C23—4662no—
165C23—2331no—
166C23—2331no—
167C23—2331no—
168C23C22×D23311yescharacteristic
169Q8—41no—
170D4—9321no—
171D4—4662no—
172D4—4662no—
173C2×C4—41no—
174C2×C4—9321no—
175C2×C4—9321no—
176C2×C4—4662no—
177C2×C4—4662no—
178C2×C4—2331no—
179C2×C4—2331no—
180C2×C4—2331no—
181C2×C4—2331no—
182C2×C4—2331no—
183C2×C4C22×D23311yescharacteristic
184C2×C4C22×D23311yescharacteristic
185C2×C4C22×D23311yescharacteristic
186C2×C4C22×D23311yescharacteristic
187C2×C4C22×D23311yescharacteristic
188C2×C4C22×D23311yescharacteristic
189C22—21no—
190C22—21no—
191C22—9324no—
192C22—4661no—
193C22—4661no—
194C22—4661no—
195C22—4661no—
196C22—4661no—
197C22—4661no—
198C22—4661no—
199C22—4661no—
200C22—2332no—
201C22#1683659512yes—
202C22C23×D23311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—4661no—
210C4—4661no—
211C4—4661no—
212C4—4661no—
213C4—4661no—
214C2—22no—
215C2—9321no—
216C2—9321no—
217C2—4662no—
218C2#41×D23311yescharacteristic
219C2C2×#1683659511yescharacteristic
220C2#2141858011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
FittingC22⋊Q8×C233C2
prime radical#21418505C2
hypercenterC22⋊Q8D233
hypocenterC233#551
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21418530C2semidirect
2. #21418530C2semidirect
3. #21418524C2semidirect
4. #21418520C2semidirect
5. #21418520C2semidirect
6. #21418519C2semidirect
7. #21418511C2semidirect
9. #21418497C2semidirect
10. C4×C22⋊D233C2semidirect
11. #21418493C2semidirect
12. #21418491C2semidirect
13. C2×#16836571C2semidirect
14. C22⋊C4×D233C2semidirect
15. C22⋊Q8×C233C2semidirect
23. C2×C4×D233C22semidirect
24. C4⋊Dic233C22semidirect
25. C4⋊Dic233C22semidirect
26. #16836573C22semidirect
27. #16836573C22semidirect
28. #16836573C22semidirect
38. #16836571C22semidirect
39. #16836571C22semidirect
40. #16836571C22semidirect
41. C4×Dic233C22semidirect
42. C4×Dic233C22semidirect
43. C4×Dic233C22semidirect
44. C2×Q8×C233C22semidirect
46. C4⋊C4×C233C22semidirect
47. C4⋊C4×C233C22semidirect
48. C4⋊C4×C233C22semidirect
110. C233#551semidirect
119. C22⋊Q8D233semidirect
128. C2×Q8C2×D233semidirect
142. C4⋊C4C2×D233semidirect
143. C4⋊C4C2×D233semidirect
144. C4⋊C4C2×D233semidirect

Maximal subgroups

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

subgroupindex
#214185302
#214185242
#214185202
#214185192
#214185112
#214185052
#214184972
C4×C22⋊D2332
#214184932
#214184912
C2×#168365712
C22⋊C4×D2332
C22⋊Q8×C2332
#551233