Finite Group Databasebeta

← #28384165

Subgroups — #28384165

Order 14272.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#20297297C211yesmaximal, characteristic, complemented
2#20297297C211yesmaximal, characteristic, complemented
3#20297291C211yesmaximal, characteristic, complemented
4#20297290C211yesmaximal, characteristic, complemented
5#20297286C211yesmaximal, characteristic, complemented
6#20297285C211yesmaximal, characteristic, complemented
7#20297281C211yesmaximal, characteristic, complemented
8#20297276C211yesmaximal, characteristic, complemented
9#20297276C211yesmaximal, characteristic, complemented
10#20297275C211yesmaximal, characteristic, complemented
11#20297273C211yesmaximal, characteristic, complemented
12D4×Dic223C211yesmaximal, characteristic, complemented
13(C22⋊D223)⋊C4C211yesmaximal, characteristic, complemented
14C2×C22⋊Dic223C211yesmaximal, characteristic, complemented
15#120×C223C211yesmaximal, characteristic, complemented
16C2×Dic446C2211yescharacteristic, complemented
17C2×C22⋊D223C2211yescharacteristic
18C2×C22⋊D223C2211yescharacteristic
19C2×C22⋊D223C2211yescharacteristic
20C2×C22⋊D223C2211yescharacteristic, complemented
21C22×Dic223C2211yescharacteristic
22C22×Dic223C2211yescharacteristic
23C22×Dic223C2211yescharacteristic, complemented
24C2×C4×D223C2211yescharacteristic, complemented
25C4⋊Dic223C2211yescharacteristic, complemented
26C4⋊Dic223C2211yescharacteristic, complemented
27Dic223⋊C4C2211yescharacteristic, complemented
28Dic223⋊C4C2211yescharacteristic, complemented
29Dic223⋊C4C2211yescharacteristic, complemented
30Dic223⋊C4C2211yescharacteristic, complemented
31C22⋊Dic223C2211yescharacteristic
32C22⋊Dic223C2211yescharacteristic
33C22⋊Dic223C2211yescharacteristic
34C22⋊Dic223C2211yescharacteristic
35C22⋊Dic223C2211yescharacteristic
36C22⋊Dic223C2211yescharacteristic
37C22⋊Dic223C2211yescharacteristic
38C22⋊Dic223C2211yescharacteristic, complemented
39C22⋊Dic223C2211yescharacteristic, complemented
40(C2×D223)⋊C4C2211yescharacteristic, complemented
41(C2×D223)⋊C4C2211yescharacteristic, complemented
42C4×Dic223C2211yescharacteristic, complemented
43C4×Dic223C2211yescharacteristic, complemented
44C24×C223C2211yescharacteristic
45C2×D4×C223C2211yescharacteristic
46C2×D4×C223C2211yescharacteristic
47C2×D4×C223C2211yescharacteristic
48C22⋊C4×C223C2211yescharacteristic
49C22⋊C4×C223C2211yescharacteristic
50C22⋊C4×C223C2211yescharacteristic
51C22×D223C2311yescharacteristic
52Dic446—41no—
53C22⋊D223—22no—
54C22⋊D223—41no—
55C22⋊D223—41no—
56C22⋊D223—41no—
57C2×Dic223—41no—
58C2×Dic223—41no—
59C2×Dic223—41no—
60C2×Dic223C2311yescharacteristic
61C2×Dic223C2311yescharacteristic
62C2×Dic223C2311yescharacteristic
63C2×Dic223C2311yescharacteristic
64C2×Dic223C2311yescharacteristic
65C2×Dic223C2311yescharacteristic
66C2×Dic223C2311yescharacteristic
67C4×D223—41no—
68C23×C223—22no—
69C23×C223—41no—
70C23×C223—41no—
71C23×C223C2311yescharacteristic
72C23×C223C2311yescharacteristic
73C23×C223C2311yescharacteristic
74C23×C223C2311yescharacteristic
75D4×C223—22no—
76D4×C223—41no—
77D4×C223—41no—
78C2×C4×C223C2311yescharacteristic
79C2×C4×C223C2311yescharacteristic
80C2×C4×C223C2311yescharacteristic
81C22×C223—21no—
82C22×C223—21no—
83C22×C223—21no—
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—44no—
93C22×C223#4112yes—
94C22×C223C2411yescharacteristic
95C2×D223—21no—
96C2×D223—21no—
97C2×D223—21no—
98C4×C223—21no—
99C4×C223—21no—
100C4×C223—21no—
101Dic223—21no—
102Dic223—21no—
103Dic223—21no—
104Dic223—21no—
105Dic223—21no—
106Dic223—21no—
107Dic223—21no—
108C2×C223—22no—
109C2×C223—41no—
110C2×C223—41no—
111C2×C223—41no—
112C2×C223C2×#4111yescharacteristic
113C2×C223H(5,𝔽2)11yescharacteristic
114C2×C223H(5,𝔽2)11yescharacteristic
115D223—41no—
116C223#53811yescharacteristic, complemented, Sylow
117#538—2231nomaximal, Sylow
118#126—2231no—
119#126—2231no—
120#124—2231no—
121#124—2231no—
122#123—2231no—
123#123—2231no—
124C22⋊Q8—2231no—
125#121—2231no—
126#121—2231no—
127#121—2231no—
128#120—2231no—
129#120D22311yescharacteristic, complemented
130C4×D4—2231no—
131C4×D4—2231no—
132C2×C22⋊C4—2231no—
133C24C2×D22311yescharacteristic
134C2×Q8—2231no—
135C2×D4—2231no—
136C2×D4—2231no—
137C2×D4—2231no—
138C2×D4—2231no—
139C2×D4C2×D22311yescharacteristic
140C2×D4C2×D22311yescharacteristic
141C2×D4C2×D22311yescharacteristic
142C22×C4—2231no—
143C22×C4—2231no—
144C22×C4—2231no—
145C22×C4—2231no—
146C4⋊C4—2231no—
147C4⋊C4—2231no—
148C4⋊C4—2231no—
149C4⋊C4—2231no—
150C4⋊C4—2231no—
151C4⋊C4—2231no—
152C22⋊C4—2231no—
153C22⋊C4—2231no—
154C22⋊C4—2231no—
155C22⋊C4—2231no—
156C22⋊C4—2231no—
157C22⋊C4—2231no—
158C22⋊C4—2231no—
159C22⋊C4—2231no—
160C22⋊C4—2231no—
161C22⋊C4—2231no—
162C22⋊C4—2231no—
163C22⋊C4C2×D22311yescharacteristic
164C22⋊C4C2×D22311yescharacteristic
165C22⋊C4C2×D22311yescharacteristic
166C42—2231no—
167C42—2231no—
168C23—22no—
169C23—41no—
170C23—41no—
171C23—2231no—
172C23C22×D22311yescharacteristic
173C23C22×D22311yescharacteristic
174C23C22×D22311yescharacteristic
175C23C22×D22311yescharacteristic
176Q8—8921no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—8921no—
181D4—8921no—
182D4—8921no—
183D4—4462no—
184C2×C4—8921no—
185C2×C4—8921no—
186C2×C4—8921no—
187C2×C4—8921no—
188C2×C4—2231no—
189C2×C4—2231no—
190C2×C4—2231no—
191C2×C4—2231no—
192C2×C4—2231no—
193C2×C4—2231no—
194C2×C4—2231no—
195C2×C4C22×D22311yescharacteristic
196C2×C4C22×D22311yescharacteristic
197C2×C4C22×D22311yescharacteristic
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—44no—
210C22—4461no—
211C22—4461no—
212C22—4461no—
213C22#1681316912yes—
214C22C23×D22311yescharacteristic
215C4—21no—
216C4—21no—
217C4—21no—
218C4—4461no—
219C4—4461no—
220C4—4461no—
221C4—4461no—
222C4—4461no—
223C4—4461no—
224C4—4461no—
225C2—22no—
226C2—41no—
227C2—41no—
228C2—41no—
229C2—8921no—
230C2C2×#1681316911yescharacteristic
231C2#2029734911yescharacteristic
232C2#2029734911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#120×C223C2
hypercenter#120D223
hypocenterC223#538
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #20297297C2semidirect
2. #20297297C2semidirect
3. #20297291C2semidirect
4. #20297290C2semidirect
5. #20297286C2semidirect
6. #20297285C2semidirect
7. #20297281C2semidirect
8. #20297276C2semidirect
9. #20297276C2semidirect
10. #20297275C2semidirect
11. #20297273C2semidirect
12. D4×Dic223C2semidirect
13. (C22⋊D223)⋊C4C2semidirect
14. C2×C22⋊Dic223C2semidirect
15. #120×C223C2semidirect
16. C2×Dic446C22semidirect
20. C2×C22⋊D223C22semidirect
23. C22×Dic223C22semidirect
24. C2×C4×D223C22semidirect
25. C4⋊Dic223C22semidirect
26. C4⋊Dic223C22semidirect
27. Dic223⋊C4C22semidirect
28. Dic223⋊C4C22semidirect
29. Dic223⋊C4C22semidirect
30. Dic223⋊C4C22semidirect
38. C22⋊Dic223C22semidirect
39. C22⋊Dic223C22semidirect
40. (C2×D223)⋊C4C22semidirect
41. (C2×D223)⋊C4C22semidirect
42. C4×Dic223C22semidirect
43. C4×Dic223C22semidirect
116. C223#538semidirect
129. #120D223semidirect

Maximal subgroups

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

subgroupindex
#202972972
#202972912
#202972902
#202972862
#202972852
#202972812
#202972762
#202972752
#202972732
D4×Dic2232
(C22⋊D223)⋊C42
C2×C22⋊Dic2232
#120×C2232
#538223