Finite Group Databasebeta

← #28417111

Subgroups — #28417111

Order 14912.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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#21418528C211yesmaximal, characteristic, complemented
2#21418528C211yesmaximal, characteristic, complemented
3#21418521C211yesmaximal, characteristic, complemented
4#21418519C211yesmaximal, characteristic, complemented
5#21418518C211yesmaximal, characteristic, complemented
6#21418517C211yesmaximal, characteristic, complemented
7#21418514C211yesmaximal, characteristic, complemented
8#21418512C211yesmaximal, characteristic, complemented
9#21418504C211yesmaximal, characteristic
10C4×C22⋊D233C211yesmaximal, characteristic, complemented
11#21418495C211yesmaximal, characteristic, complemented
12#21418494C211yesmaximal, characteristic, complemented
13#21418491C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic233C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×C233C211yesmaximal, characteristic, complemented
16C2×Dic466C2211yescharacteristic, complemented
17C2×C22⋊D233C2211yescharacteristic
18C2×C22⋊D233C2211yescharacteristic
19C2×C22⋊D233C2211yescharacteristic
20C22×Dic233C2211yescharacteristic
21C22×Dic233C2211yescharacteristic
22C2×C4×D233C2211yescharacteristic, complemented
23C4⋊Dic233C2211yescharacteristic, complemented
24C4⋊Dic233C2211yescharacteristic, complemented
25C4⋊Dic233C2211yescharacteristic, complemented
26#16836573C2211yescharacteristic, complemented
27#16836573C2211yescharacteristic, complemented
28#16836573C2211yescharacteristic, complemented
29#16836573C2211yescharacteristic, complemented
30#16836573C2211yescharacteristic, complemented
31C22⋊Dic233C2211yescharacteristic
32C22⋊Dic233C2211yescharacteristic
33C22⋊Dic233C2211yescharacteristic
34C22⋊Dic233C2211yescharacteristic
35C22⋊Dic233C2211yescharacteristic
36C22⋊Dic233C2211yescharacteristic
37C22⋊Dic233C2211yescharacteristic
38#16836571C2211yescharacteristic, complemented
39#16836571C2211yescharacteristic, complemented
40#16836571C2211yescharacteristic, complemented
41C4×Dic233C2211yescharacteristic, complemented
42C4×Dic233C2211yescharacteristic, complemented
43C4×Dic233C2211yescharacteristic, complemented
44C24×C233C2211yescharacteristic
45C22×C4×C233C2211yescharacteristic
46C22×C4×C233C2211yescharacteristic
47C22⋊C4×C233C2211yescharacteristic
48C22⋊C4×C233C2211yescharacteristic
49C22⋊C4×C233C2211yescharacteristic
50C22⋊C4×C233C2211yescharacteristic
51C22×D233C2311yescharacteristic
52Dic466—41no—
53C22⋊D233—22no—
54C22⋊D233—22no—
55C22⋊D233—41no—
56C2×Dic233—22no—
57C2×Dic233—41no—
58C2×Dic233C2311yescharacteristic
59C2×Dic233C2311yescharacteristic
60C2×Dic233C2311yescharacteristic
61C2×Dic233C2311yescharacteristic
62C2×Dic233C2311yescharacteristic
63C2×Dic233C2311yescharacteristic
64C2×Dic233C2311yescharacteristic
65C4×D233—41no—
66C23×C233—22no—
67C23×C233—22no—
68C23×C233—41no—
69C23×C233C2311yescharacteristic
70C23×C233C2311yescharacteristic
71C23×C233C2311yescharacteristic
72C2×C4×C233—22no—
73C2×C4×C233—41no—
74C2×C4×C233C2311yescharacteristic
75C2×C4×C233C2311yescharacteristic
76C2×C4×C233C2311yescharacteristic
77C2×C4×C233C2311yescharacteristic
78C22×C233—21no—
79C22×C233—21no—
80C22×C233—21no—
81C22×C233—21no—
82C22×C233—21no—
83C22×C233—21no—
84C22×C233—21no—
85C22×C233—44no—
86C22×C233#4112yes—
87C22×C233#4112yes—
88C22×C233C2411yescharacteristic
89C2×D233—21no—
90C2×D233—21no—
91C2×D233—21no—
92C4×C233—21no—
93C4×C233—21no—
94C4×C233—21no—
95C4×C233—21no—
96Dic233—21no—
97Dic233—21no—
98Dic233—21no—
99Dic233—21no—
100Dic233—21no—
101Dic233—21no—
102Dic233—21no—
103C2×C233—22no—
104C2×C233—22no—
105C2×C233—41no—
106C2×C233C2×#4111yescharacteristic
107C2×C233C2×#4111yescharacteristic
108C2×C233H(5,𝔽2)11yescharacteristic
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⋊Q8—2331no—
120#120—2331no—
121C4×D4—2331no—
122C4×D4—2331no—
123#117—2331no—
124#117—2331no—
125C2×C22⋊C4—2331no—
126C2×C22⋊C4D23311yescharacteristic, complemented
127C24C2×D23311yescharacteristic
128C2×Q8—2331no—
129C2×D4—2331no—
130C2×D4—2331no—
131C2×D4—2331no—
132C22×C4—2331no—
133C22×C4—2331no—
134C22×C4—2331no—
135C22×C4C2×D23311yescharacteristic
136C22×C4C2×D23311yescharacteristic
137C4⋊C4—2331no—
138C4⋊C4—2331no—
139C4⋊C4—2331no—
140C4⋊C4—2331no—
141C4⋊C4—2331no—
142C4⋊C4—2331no—
143C4⋊C4—2331no—
144C4⋊C4—2331no—
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⋊C4C2×D23311yescharacteristic
156C22⋊C4C2×D23311yescharacteristic
157C22⋊C4C2×D23311yescharacteristic
158C22⋊C4C2×D23311yescharacteristic
159C42—2331no—
160C42—2331no—
161C42—2331no—
162C23—22no—
163C23—22no—
164C23—41no—
165C23—2331no—
166C23C22×D23311yescharacteristic
167C23C22×D23311yescharacteristic
168C23C22×D23311yescharacteristic
169Q8—9321no—
170D4—9321no—
171D4—4662no—
172D4—4662no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—9321no—
176C2×C4—9321no—
177C2×C4—4662no—
178C2×C4—2331no—
179C2×C4—2331no—
180C2×C4—2331no—
181C2×C4—2331no—
182C2×C4—2331no—
183C2×C4—2331no—
184C2×C4—2331no—
185C2×C4C22×D23311yescharacteristic
186C2×C4C22×D23311yescharacteristic
187C2×C4C22×D23311yescharacteristic
188C2×C4C22×D23311yescharacteristic
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—44no—
197C22—4661no—
198C22—4661no—
199C22—4661no—
200C22#1683659312yes—
201C22#1683659412yes—
202C22C23×D23311yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—4661no—
208C4—4661no—
209C4—4661no—
210C4—4661no—
211C4—4661no—
212C4—4661no—
213C4—4661no—
214C2—22no—
215C2—22no—
216C2—41no—
217C2—9321no—
218C2C2×#1683659311yescharacteristic
219C2C2×#1683659411yescharacteristic
220C2#2141858011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D233
commutator, socle, cosocleC22×C233C24
FittingC2×C22⋊C4×C233C2
prime radical#21418504C2
hypercenterC2×C22⋊C4D233
hypocenterC233#551
LCSG ▷ C22×C233 ▷ C233
UCSC1 ◁ C22 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #21418528C2semidirect
2. #21418528C2semidirect
3. #21418521C2semidirect
4. #21418519C2semidirect
5. #21418518C2semidirect
6. #21418517C2semidirect
7. #21418514C2semidirect
8. #21418512C2semidirect
10. C4×C22⋊D233C2semidirect
11. #21418495C2semidirect
12. #21418494C2semidirect
13. #21418491C2semidirect
14. C2×C22⋊Dic233C2semidirect
15. C2×C22⋊C4×C233C2semidirect
16. C2×Dic466C22semidirect
22. C2×C4×D233C22semidirect
23. C4⋊Dic233C22semidirect
24. C4⋊Dic233C22semidirect
25. C4⋊Dic233C22semidirect
26. #16836573C22semidirect
27. #16836573C22semidirect
28. #16836573C22semidirect
29. #16836573C22semidirect
30. #16836573C22semidirect
38. #16836571C22semidirect
39. #16836571C22semidirect
40. #16836571C22semidirect
41. C4×Dic233C22semidirect
42. C4×Dic233C22semidirect
43. C4×Dic233C22semidirect
110. C233#551semidirect
126. C2×C22⋊C4D233semidirect

Maximal subgroups

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

subgroupindex
#214185282
#214185212
#214185192
#214185182
#214185172
#214185142
#214185122
#214185042
C4×C22⋊D2332
#214184952
#214184942
#214184912
C2×C22⋊Dic2332
C2×C22⋊C4×C2332
#551233