Finite Group Databasebeta

← #28384318

Subgroups — #28384318

Order 14272.

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#20297300C211yesmaximal, characteristic, complemented
2#20297299C211yesmaximal, characteristic, complemented
3#20297297C211yesmaximal, characteristic, complemented
4#20297289C211yesmaximal, characteristic, complemented
5#20297288C211yesmaximal, characteristic, complemented
6#20297287C211yesmaximal, characteristic, complemented
7#20297278C211yesmaximal, characteristic, complemented
8#20297277C211yesmaximal, characteristic, complemented
9#20297276C211yesmaximal, characteristic, complemented
10#20297276C211yesmaximal, characteristic, complemented
11#20297275C211yesmaximal, characteristic, complemented
12#20297274C211yesmaximal, characteristic, complemented
13#20297272C211yesmaximal, characteristic, complemented
14#20297272C211yesmaximal, characteristic, complemented
15#121×C223C211yesmaximal, characteristic, complemented
16C23×D223C2211yescharacteristic
17C2×D892C2211yescharacteristic
18C2×D892C2211yescharacteristic
19C2×C22⋊D223C2211yescharacteristic
20C2×C22⋊D223C2211yescharacteristic
21C2×C22⋊D223C2211yescharacteristic
22C2×C22⋊D223C2211yescharacteristic
23C2×C22⋊D223C2211yescharacteristic
24C2×C22⋊D223C2211yescharacteristic, complemented
25C2×C22⋊D223C2211yescharacteristic, complemented
26C22×Dic223C2211yescharacteristic, complemented
27C2×C4×D223C2211yescharacteristic, complemented
28C4⋊Dic223C2211yescharacteristic, complemented
29C4⋊Dic223C2211yescharacteristic, complemented
30Dic223⋊C4C2211yescharacteristic, complemented
31Dic223⋊C4C2211yescharacteristic, complemented
32Dic223⋊C4C2211yescharacteristic, complemented
33C22⋊Dic223C2211yescharacteristic
34C22⋊Dic223C2211yescharacteristic
35C22⋊Dic223C2211yescharacteristic, complemented
36(C2×D223)⋊C4C2211yescharacteristic
37(C2×D223)⋊C4C2211yescharacteristic
38(C2×D223)⋊C4C2211yescharacteristic
39(C2×D223)⋊C4C2211yescharacteristic
40(C2×D223)⋊C4C2211yescharacteristic
41(C2×D223)⋊C4C2211yescharacteristic
42(C2×D223)⋊C4C2211yescharacteristic, complemented
43C4×Dic223C2211yescharacteristic, complemented
44C2×D4×C223C2211yescharacteristic
45C2×D4×C223C2211yescharacteristic
46C2×D4×C223C2211yescharacteristic, complemented
47C22×C4×C223C2211yescharacteristic, complemented
48C4⋊C4×C223C2211yescharacteristic, complemented
49C22⋊C4×C223C2211yescharacteristic
50C22⋊C4×C223C2211yescharacteristic, complemented
51C22×D223—41no—
52C22×D223—41no—
53C22×D223—41no—
54C22×D223C2311yescharacteristic
55C22×D223C2311yescharacteristic
56C22×D223C2311yescharacteristic
57D892—41no—
58D892—41no—
59C22⋊D223—41no—
60C22⋊D223—41no—
61C22⋊D223—41no—
62C22⋊D223—41no—
63C22⋊D223—41no—
64C22⋊D223—41no—
65C22⋊D223—41no—
66C2×Dic223—41no—
67C2×Dic223C2311yescharacteristic
68C2×Dic223C2311yescharacteristic
69C2×Dic223C2311yescharacteristic
70C2×Dic223C2311yescharacteristic
71C2×Dic223C2311yescharacteristic
72C4×D223—41no—
73C23×C223C2311yescharacteristic
74C23×C223C2311yescharacteristic
75C23×C223C2311yescharacteristic
76D4×C223—41no—
77D4×C223—41no—
78D4×C223—41no—
79C2×C4×C223—41no—
80C2×C4×C223C2311yescharacteristic
81C2×C4×C223C2311yescharacteristic
82C2×C4×C223C2311yescharacteristic
83C2×C4×C223C2311yescharacteristic
84C22×C223—21no—
85C22×C223—21no—
86C22×C223—21no—
87C22×C223—21no—
88C22×C223—21no—
89C22×C223—21no—
90C22×C223—21no—
91C22×C223—21no—
92C22×C223—21no—
93C22×C223C2411yescharacteristic
94C2×D223—21no—
95C2×D223—21no—
96C2×D223—21no—
97C2×D223—21no—
98C2×D223—21no—
99C2×D223—21no—
100C2×D223—21no—
101C2×D223—21no—
102C2×D223—21no—
103C2×D223—44no—
104C4×C223—21no—
105C4×C223—21no—
106C4×C223—21no—
107C4×C223—21no—
108Dic223—21no—
109Dic223—21no—
110Dic223—21no—
111Dic223—21no—
112Dic223—21no—
113C2×C223—41no—
114C2×C223—41no—
115C2×C223—41no—
116C2×C223H(5,𝔽2)11yescharacteristic
117C2×C223H(5,𝔽2)11yescharacteristic
118C2×C223H(5,𝔽2)11yescharacteristic
119D223—41no—
120D223—41no—
121D223—41no—
122C223#56011yescharacteristic, complemented, Sylow
123#560—2231nomaximal, Sylow
124#127—2231no—
125#126—2231no—
126#126—2231no—
127#123—2231no—
128#123—2231no—
129#123—2231no—
130#121—2231no—
131#121—2231no—
132#121—2231no—
133#121—2231no—
134#121—2231no—
135#121D22311yescharacteristic, complemented
136#120—2231no—
137#120—2231no—
138#120—2231no—
139C24—2231no—
140C2×D4—2231no—
141C2×D4—2231no—
142C2×D4—2231no—
143C2×D4—2231no—
144C2×D4—2231no—
145C2×D4—2231no—
146C2×D4—2231no—
147C2×D4—2231no—
148C2×D4—2231no—
149C2×D4C2×D22311yescharacteristic
150C2×D4C2×D22311yescharacteristic
151C2×D4C2×D22311yescharacteristic, complemented
152C22×C4—2231no—
153C22×C4—2231no—
154C22×C4C2×D22311yescharacteristic, complemented
155C4⋊C4—2231no—
156C4⋊C4—2231no—
157C4⋊C4—2231no—
158C4⋊C4—2231no—
159C4⋊C4—2231no—
160C4⋊C4C2×D22311yescharacteristic, complemented
161C22⋊C4—2231no—
162C22⋊C4—2231no—
163C22⋊C4—2231no—
164C22⋊C4—2231no—
165C22⋊C4—2231no—
166C22⋊C4—2231no—
167C22⋊C4—2231no—
168C22⋊C4—2231no—
169C22⋊C4—2231no—
170C22⋊C4—2231no—
171C22⋊C4C2×D22311yescharacteristic
172C22⋊C4C2×D22311yescharacteristic, complemented
173C42—2231no—
174C23—8921no—
175C23—8921no—
176C23—8921no—
177C23—2231no—
178C23—2231no—
179C23—2231no—
180C23C22×D22311yescharacteristic
181C23C22×D22311yescharacteristic
182C23C22×D22311yescharacteristic
183D4—41no—
184D4—41no—
185D4—41no—
186D4—8921no—
187D4—8921no—
188D4—8921no—
189D4—8921no—
190D4—8921no—
191D4—8921no—
192D4—8921no—
193D4—8921no—
194D4—8921no—
195C2×C4—41no—
196C2×C4—8921no—
197C2×C4—8921no—
198C2×C4—2231no—
199C2×C4—2231no—
200C2×C4—2231no—
201C2×C4—2231no—
202C2×C4—2231no—
203C2×C4C22×D22311yescharacteristic
204C2×C4C22×D22311yescharacteristic
205C2×C4C22×D22311yescharacteristic
206C2×C4C22×D22311yescharacteristic
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—21no—
216C22—8924no—
217C22—4461no—
218C22—4461no—
219C22—4461no—
220C22—4461no—
221C22—4461no—
222C22—4461no—
223C22—4461no—
224C22—4461no—
225C22—4461no—
226C22C23×D22311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—4461no—
232C4—4461no—
233C4—4461no—
234C4—4461no—
235C4—4461no—
236C2—41no—
237C2—41no—
238C2—41no—
239C2—8921no—
240C2—8921no—
241C2—8921no—
242C2#2029734911yescharacteristic
243C2#2029734911yescharacteristic
244C2#2029735011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#121×C223C2
hypercenter#121D223
hypocenterC223#560
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20297300C2semidirect
2. #20297299C2semidirect
3. #20297297C2semidirect
4. #20297289C2semidirect
5. #20297288C2semidirect
6. #20297287C2semidirect
7. #20297278C2semidirect
8. #20297277C2semidirect
9. #20297276C2semidirect
10. #20297276C2semidirect
11. #20297275C2semidirect
12. #20297274C2semidirect
13. #20297272C2semidirect
14. #20297272C2semidirect
15. #121×C223C2semidirect
24. C2×C22⋊D223C22semidirect
25. C2×C22⋊D223C22semidirect
26. C22×Dic223C22semidirect
27. C2×C4×D223C22semidirect
28. C4⋊Dic223C22semidirect
29. C4⋊Dic223C22semidirect
30. Dic223⋊C4C22semidirect
31. Dic223⋊C4C22semidirect
32. Dic223⋊C4C22semidirect
35. C22⋊Dic223C22semidirect
42. (C2×D223)⋊C4C22semidirect
43. C4×Dic223C22semidirect
46. C2×D4×C223C22semidirect
47. C22×C4×C223C22semidirect
48. C4⋊C4×C223C22semidirect
50. C22⋊C4×C223C22semidirect
122. C223#560semidirect
135. #121D223semidirect
151. C2×D4C2×D223semidirect
154. C22×C4C2×D223semidirect
160. C4⋊C4C2×D223semidirect
172. C22⋊C4C2×D223semidirect

Maximal subgroups

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

subgroupindex
#202973002
#202972992
#202972972
#202972892
#202972882
#202972872
#202972782
#202972772
#202972762
#202972752
#202972742
#202972722
#121×C2232
#560223