Finite Group Databasebeta

← #27213735

Subgroups — #27213735

Order 13504.

250 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 93 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#20269972C211yesmaximal, characteristic, complemented
2#20269965C211yesmaximal, characteristic, complemented
3#20269963C211yesmaximal, characteristic, complemented
4#20269962C211yesmaximal, characteristic, complemented
5#20269959C211yesmaximal, characteristic, complemented
6#20269955C211yesmaximal, characteristic, complemented
7#20269952C211yesmaximal, characteristic, complemented
8#20269952C211yesmaximal, characteristic, complemented
9#20269950C211yesmaximal, characteristic, complemented
10#20269947C211yesmaximal, characteristic, complemented
11#20269947C211yesmaximal, characteristic, complemented
12C4×D844C211yesmaximal, characteristic, complemented
13#20269939C211yesmaximal, characteristic, complemented
14C22⋊C4×D211C211yesmaximal, characteristic, complemented
15#126×C211C211yesmaximal, characteristic, complemented
16C23×D211C2211yescharacteristic
17C2×Dic422C2211yescharacteristic, complemented
18C2×D844C2211yescharacteristic
19C2×D844C2211yescharacteristic
20C2×D844C2211yescharacteristic
21C2×D844C2211yescharacteristic
22C2×D844C2211yescharacteristic, complemented
23C2×C22⋊D211C2211yescharacteristic
24C2×C22⋊D211C2211yescharacteristic
25C2×C4×D211C2211yescharacteristic
26C2×C4×D211C2211yescharacteristic
27C2×C4×D211C2211yescharacteristic, complemented
28C2×C4×D211C2211yescharacteristic, complemented
29C4⋊Dic211C2211yescharacteristic, complemented
30Dic211⋊C4C2211yescharacteristic, complemented
31Dic211⋊C4C2211yescharacteristic, complemented
32C22⋊Dic211C2211yescharacteristic
33(C2×D211)⋊C4C2211yescharacteristic
34(C2×D211)⋊C4C2211yescharacteristic
35(C2×D211)⋊C4C2211yescharacteristic
36(C2×D211)⋊C4C2211yescharacteristic
37(C2×D211)⋊C4C2211yescharacteristic
38(C2×D211)⋊C4C2211yescharacteristic
39(C2×D211)⋊C4C2211yescharacteristic, complemented
40(C2×D211)⋊C4C2211yescharacteristic, complemented
41(C2×D211)⋊C4C2211yescharacteristic, complemented
42(C2×D211)⋊C4C2211yescharacteristic, complemented
43C4×Dic211C2211yescharacteristic, complemented
44C4⋊C4×C211C2211yescharacteristic, complemented
45C4⋊C4×C211C2211yescharacteristic, complemented
46C4⋊C4×C211C2211yescharacteristic, complemented
47C22⋊C4×C211C2211yescharacteristic
48C22⋊C4×C211C2211yescharacteristic
49C22⋊C4×C211C2211yescharacteristic
50C42×C211C2211yescharacteristic, complemented
51C22×D211—22no—
52C22×D211—41no—
53C22×D211—41no—
54C22×D211C2311yescharacteristic
55C22×D211C2311yescharacteristic
56C22×D211C2311yescharacteristic
57C22×D211C2311yescharacteristic
58Dic422—41no—
59D844—22no—
60D844—22no—
61D844—41no—
62D844—41no—
63D844—41no—
64C22⋊D211—41no—
65C22⋊D211—41no—
66C2×Dic211C2311yescharacteristic
67C2×Dic211C2311yescharacteristic
68C2×Dic211C2311yescharacteristic
69C2×Dic211C2311yescharacteristic
70C4×D211—41no—
71C4×D211—41no—
72C4×D211—41no—
73C4×D211—41no—
74C23×C211C2311yescharacteristic
75C2×C4×C211C2311yescharacteristic
76C2×C4×C211C2311yescharacteristic
77C2×C4×C211C2311yescharacteristic
78C2×C4×C211C2311yescharacteristic
79C2×C4×C211C2311yescharacteristic
80C2×C4×C211C2311yescharacteristic
81C22×C211—21no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211C2411yescharacteristic
85C2×D211—21no—
86C2×D211—21no—
87C2×D211—21no—
88C2×D211—21no—
89C2×D211—21no—
90C2×D211—21no—
91C2×D211—21no—
92C2×D211—21no—
93C2×D211—21no—
94C2×D211—21no—
95C2×D211—21no—
96C2×D211—44no—
97C2×D211#4112yes—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—21no—
101C4×C211—21no—
102C4×C211—21no—
103C4×C211—21no—
104Dic211—21no—
105Dic211—21no—
106Dic211—21no—
107Dic211—21no—
108C2×C211—41no—
109C2×C211C2×#4111yescharacteristic
110C2×C211H(5,𝔽2)11yescharacteristic
111C2×C211H(5,𝔽2)11yescharacteristic
112D211—22no—
113D211—41no—
114D211—41no—
115D211—41no—
116C211#53811yescharacteristic, complemented, Sylow
117#538—2111nomaximal, Sylow
118#126—2111no—
119#126D21111yescharacteristic, complemented
120#124—2111no—
121#124—2111no—
122#123—2111no—
123#123—2111no—
124C22⋊Q8—2111no—
125#121—2111no—
126#121—2111no—
127#121—2111no—
128#120—2111no—
129#120—2111no—
130C4×D4—2111no—
131C4×D4—2111no—
132C2×C22⋊C4—2111no—
133C24—2111no—
134C2×Q8—2111no—
135C2×D4—2111no—
136C2×D4—2111no—
137C2×D4—2111no—
138C2×D4—2111no—
139C2×D4—2111no—
140C2×D4—2111no—
141C2×D4—2111no—
142C22×C4—2111no—
143C22×C4—2111no—
144C22×C4—2111no—
145C22×C4—2111no—
146C4⋊C4—2111no—
147C4⋊C4—2111no—
148C4⋊C4—2111no—
149C4⋊C4C2×D21111yescharacteristic, complemented
150C4⋊C4C2×D21111yescharacteristic, complemented
151C4⋊C4C2×D21111yescharacteristic, complemented
152C22⋊C4—2111no—
153C22⋊C4—2111no—
154C22⋊C4—2111no—
155C22⋊C4—2111no—
156C22⋊C4—2111no—
157C22⋊C4—2111no—
158C22⋊C4—2111no—
159C22⋊C4—2111no—
160C22⋊C4—2111no—
161C22⋊C4—2111no—
162C22⋊C4—2111no—
163C22⋊C4C2×D21111yescharacteristic
164C22⋊C4C2×D21111yescharacteristic
165C22⋊C4C2×D21111yescharacteristic
166C42—2111no—
167C42C2×D21111yescharacteristic, complemented
168C23—8441no—
169C23—8441no—
170C23—4222no—
171C23—2111no—
172C23—2111no—
173C23—2111no—
174C23—2111no—
175C23C22×D21111yescharacteristic
176Q8—8441no—
177D4—8441no—
178D4—8441no—
179D4—8441no—
180D4—8441no—
181D4—8441no—
182D4—4222no—
183D4—4222no—
184C2×C4—8441no—
185C2×C4—8441no—
186C2×C4—8441no—
187C2×C4—8441no—
188C2×C4—2111no—
189C2×C4—2111no—
190C2×C4—2111no—
191C2×C4—2111no—
192C2×C4C22×D21111yescharacteristic
193C2×C4C22×D21111yescharacteristic
194C2×C4C22×D21111yescharacteristic
195C2×C4C22×D21111yescharacteristic
196C2×C4C22×D21111yescharacteristic
197C2×C4C22×D21111yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—8444no—
202C22—4221no—
203C22—4221no—
204C22—4221no—
205C22—4221no—
206C22—4221no—
207C22—4221no—
208C22—4221no—
209C22—4221no—
210C22—4221no—
211C22—4221no—
212C22—4221no—
213C22—2112no—
214C22C23×D21111yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—4221no—
222C4—4221no—
223C4—4221no—
224C4—4221no—
225C2—41no—
226C2—8441no—
227C2—8441no—
228C2—8441no—
229C2—4222no—
230C2#41×D21111yescharacteristic
231C2#2027002311yescharacteristic
232C2#2027002311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
Fitting#126×C211C2
hypercenter#126D211
hypocenterC211#538
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #20269972C2semidirect
2. #20269965C2semidirect
3. #20269963C2semidirect
4. #20269962C2semidirect
5. #20269959C2semidirect
6. #20269955C2semidirect
7. #20269952C2semidirect
8. #20269952C2semidirect
9. #20269950C2semidirect
10. #20269947C2semidirect
11. #20269947C2semidirect
12. C4×D844C2semidirect
13. #20269939C2semidirect
14. C22⋊C4×D211C2semidirect
15. #126×C211C2semidirect
17. C2×Dic422C22semidirect
22. C2×D844C22semidirect
27. C2×C4×D211C22semidirect
28. C2×C4×D211C22semidirect
29. C4⋊Dic211C22semidirect
30. Dic211⋊C4C22semidirect
31. Dic211⋊C4C22semidirect
39. (C2×D211)⋊C4C22semidirect
40. (C2×D211)⋊C4C22semidirect
41. (C2×D211)⋊C4C22semidirect
42. (C2×D211)⋊C4C22semidirect
43. C4×Dic211C22semidirect
44. C4⋊C4×C211C22semidirect
45. C4⋊C4×C211C22semidirect
46. C4⋊C4×C211C22semidirect
50. C42×C211C22semidirect
116. C211#538semidirect
119. #126D211semidirect
149. C4⋊C4C2×D211semidirect
150. C4⋊C4C2×D211semidirect
151. C4⋊C4C2×D211semidirect
167. C42C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202699722
#202699652
#202699632
#202699622
#202699592
#202699552
#202699522
#202699502
#202699472
C4×D8442
#202699392
C22⋊C4×D2112
#126×C2112
#538211