Finite Group Databasebeta

← #27213884

Subgroups — #27213884

Order 13504.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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#20269973C211yesmaximal, characteristic, complemented
2#20269972C211yesmaximal, characteristic, complemented
3#20269970C211yesmaximal, characteristic, complemented
4#20269962C211yesmaximal, characteristic, complemented
5#20269961C211yesmaximal, characteristic, complemented
6#20269960C211yesmaximal, characteristic, complemented
7#20269951C211yesmaximal, characteristic, complemented
8#20269950C211yesmaximal, characteristic, complemented
9#20269949C211yesmaximal, characteristic, complemented
10#20269949C211yesmaximal, characteristic, complemented
11#20269948C211yesmaximal, characteristic, complemented
12#20269947C211yesmaximal, characteristic, complemented
13#20269945C211yesmaximal, characteristic, complemented
14#20269945C211yesmaximal, characteristic, complemented
15#121×C211C211yesmaximal, characteristic, complemented
16C23×D211C2211yescharacteristic
17C2×D844C2211yescharacteristic
18C2×D844C2211yescharacteristic
19C2×C22⋊D211C2211yescharacteristic
20C2×C22⋊D211C2211yescharacteristic
21C2×C22⋊D211C2211yescharacteristic
22C2×C22⋊D211C2211yescharacteristic
23C2×C22⋊D211C2211yescharacteristic
24C2×C22⋊D211C2211yescharacteristic, complemented
25C2×C22⋊D211C2211yescharacteristic, complemented
26C22×Dic211C2211yescharacteristic, complemented
27C2×C4×D211C2211yescharacteristic, complemented
28C4⋊Dic211C2211yescharacteristic, complemented
29C4⋊Dic211C2211yescharacteristic, complemented
30Dic211⋊C4C2211yescharacteristic, complemented
31Dic211⋊C4C2211yescharacteristic, complemented
32Dic211⋊C4C2211yescharacteristic, complemented
33C22⋊Dic211C2211yescharacteristic
34C22⋊Dic211C2211yescharacteristic
35C22⋊Dic211C2211yescharacteristic, complemented
36(C2×D211)⋊C4C2211yescharacteristic
37(C2×D211)⋊C4C2211yescharacteristic
38(C2×D211)⋊C4C2211yescharacteristic
39(C2×D211)⋊C4C2211yescharacteristic
40(C2×D211)⋊C4C2211yescharacteristic
41(C2×D211)⋊C4C2211yescharacteristic
42(C2×D211)⋊C4C2211yescharacteristic, complemented
43C4×Dic211C2211yescharacteristic, complemented
44C2×D4×C211C2211yescharacteristic
45C2×D4×C211C2211yescharacteristic
46C2×D4×C211C2211yescharacteristic, complemented
47C22×C4×C211C2211yescharacteristic, complemented
48C4⋊C4×C211C2211yescharacteristic, complemented
49C22⋊C4×C211C2211yescharacteristic
50C22⋊C4×C211C2211yescharacteristic, complemented
51C22×D211—41no—
52C22×D211—41no—
53C22×D211—41no—
54C22×D211C2311yescharacteristic
55C22×D211C2311yescharacteristic
56C22×D211C2311yescharacteristic
57D844—41no—
58D844—41no—
59C22⋊D211—41no—
60C22⋊D211—41no—
61C22⋊D211—41no—
62C22⋊D211—41no—
63C22⋊D211—41no—
64C22⋊D211—41no—
65C22⋊D211—41no—
66C2×Dic211—41no—
67C2×Dic211C2311yescharacteristic
68C2×Dic211C2311yescharacteristic
69C2×Dic211C2311yescharacteristic
70C2×Dic211C2311yescharacteristic
71C2×Dic211C2311yescharacteristic
72C4×D211—41no—
73C23×C211C2311yescharacteristic
74C23×C211C2311yescharacteristic
75C23×C211C2311yescharacteristic
76D4×C211—41no—
77D4×C211—41no—
78D4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C2311yescharacteristic
81C2×C4×C211C2311yescharacteristic
82C2×C4×C211C2311yescharacteristic
83C2×C4×C211C2311yescharacteristic
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211—21no—
93C22×C211C2411yescharacteristic
94C2×D211—21no—
95C2×D211—21no—
96C2×D211—21no—
97C2×D211—21no—
98C2×D211—21no—
99C2×D211—21no—
100C2×D211—21no—
101C2×D211—21no—
102C2×D211—21no—
103C2×D211—44no—
104C4×C211—21no—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—21no—
108Dic211—21no—
109Dic211—21no—
110Dic211—21no—
111Dic211—21no—
112Dic211—21no—
113C2×C211—41no—
114C2×C211—41no—
115C2×C211—41no—
116C2×C211H(5,𝔽2)11yescharacteristic
117C2×C211H(5,𝔽2)11yescharacteristic
118C2×C211H(5,𝔽2)11yescharacteristic
119D211—41no—
120D211—41no—
121D211—41no—
122C211#56011yescharacteristic, complemented, Sylow
123#560—2111nomaximal, Sylow
124#127—2111no—
125#126—2111no—
126#126—2111no—
127#123—2111no—
128#123—2111no—
129#123—2111no—
130#121—2111no—
131#121—2111no—
132#121—2111no—
133#121—2111no—
134#121—2111no—
135#121D21111yescharacteristic, complemented
136#120—2111no—
137#120—2111no—
138#120—2111no—
139C24—2111no—
140C2×D4—2111no—
141C2×D4—2111no—
142C2×D4—2111no—
143C2×D4—2111no—
144C2×D4—2111no—
145C2×D4—2111no—
146C2×D4—2111no—
147C2×D4—2111no—
148C2×D4—2111no—
149C2×D4C2×D21111yescharacteristic
150C2×D4C2×D21111yescharacteristic
151C2×D4C2×D21111yescharacteristic, complemented
152C22×C4—2111no—
153C22×C4—2111no—
154C22×C4C2×D21111yescharacteristic, complemented
155C4⋊C4—2111no—
156C4⋊C4—2111no—
157C4⋊C4—2111no—
158C4⋊C4—2111no—
159C4⋊C4—2111no—
160C4⋊C4C2×D21111yescharacteristic, complemented
161C22⋊C4—2111no—
162C22⋊C4—2111no—
163C22⋊C4—2111no—
164C22⋊C4—2111no—
165C22⋊C4—2111no—
166C22⋊C4—2111no—
167C22⋊C4—2111no—
168C22⋊C4—2111no—
169C22⋊C4—2111no—
170C22⋊C4—2111no—
171C22⋊C4C2×D21111yescharacteristic
172C22⋊C4C2×D21111yescharacteristic, complemented
173C42—2111no—
174C23—8441no—
175C23—8441no—
176C23—8441no—
177C23—2111no—
178C23—2111no—
179C23—2111no—
180C23C22×D21111yescharacteristic
181C23C22×D21111yescharacteristic
182C23C22×D21111yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—8441no—
187D4—8441no—
188D4—8441no—
189D4—8441no—
190D4—8441no—
191D4—8441no—
192D4—8441no—
193D4—8441no—
194D4—8441no—
195C2×C4—41no—
196C2×C4—8441no—
197C2×C4—8441no—
198C2×C4—2111no—
199C2×C4—2111no—
200C2×C4—2111no—
201C2×C4—2111no—
202C2×C4—2111no—
203C2×C4C22×D21111yescharacteristic
204C2×C4C22×D21111yescharacteristic
205C2×C4C22×D21111yescharacteristic
206C2×C4C22×D21111yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—8444no—
217C22—4221no—
218C22—4221no—
219C22—4221no—
220C22—4221no—
221C22—4221no—
222C22—4221no—
223C22—4221no—
224C22—4221no—
225C22—4221no—
226C22C23×D21111yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—4221no—
232C4—4221no—
233C4—4221no—
234C4—4221no—
235C4—4221no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—8441no—
240C2—8441no—
241C2—8441no—
242C2#2027002211yescharacteristic
243C2#2027002211yescharacteristic
244C2#2027002311yescharacteristic
subgroupG/H
center, FrattiniC22C23×D211
commutator, socle, cosocleC22×C211C24
Fitting#121×C211C2
hypercenter#121D211
hypocenterC211#560
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20269973C2semidirect
2. #20269972C2semidirect
3. #20269970C2semidirect
4. #20269962C2semidirect
5. #20269961C2semidirect
6. #20269960C2semidirect
7. #20269951C2semidirect
8. #20269950C2semidirect
9. #20269949C2semidirect
10. #20269949C2semidirect
11. #20269948C2semidirect
12. #20269947C2semidirect
13. #20269945C2semidirect
14. #20269945C2semidirect
15. #121×C211C2semidirect
24. C2×C22⋊D211C22semidirect
25. C2×C22⋊D211C22semidirect
26. C22×Dic211C22semidirect
27. C2×C4×D211C22semidirect
28. C4⋊Dic211C22semidirect
29. C4⋊Dic211C22semidirect
30. Dic211⋊C4C22semidirect
31. Dic211⋊C4C22semidirect
32. Dic211⋊C4C22semidirect
35. C22⋊Dic211C22semidirect
42. (C2×D211)⋊C4C22semidirect
43. C4×Dic211C22semidirect
46. C2×D4×C211C22semidirect
47. C22×C4×C211C22semidirect
48. C4⋊C4×C211C22semidirect
50. C22⋊C4×C211C22semidirect
122. C211#560semidirect
135. #121D211semidirect
151. C2×D4C2×D211semidirect
154. C22×C4C2×D211semidirect
160. C4⋊C4C2×D211semidirect
172. C22⋊C4C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202699732
#202699722
#202699702
#202699622
#202699612
#202699602
#202699512
#202699502
#202699492
#202699482
#202699472
#202699452
#121×C2112
#560211