Finite Group Databasebeta

← #13069752

Subgroups — #13069752

Order 1408.

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

normal subgroups only

This group has 326 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
1C2×#66922C211yesmaximal, characteristic, complemented
2C11×#390C211yesmaximal, characteristic
3C2×#66896C212yesmaximal
4#10622167C212yesmaximal, complemented
5#10622162C211yesmaximal, characteristic, complemented
6C22×D4×C11C2211yescharacteristic
7C22×Dic22—21no—
8C2×C11×C4⋊C4C2211yescharacteristic
9C2×C11×C22⋊C4—21no—
10C2×C11×C22⋊C4—21no—
11C2×C42×C11C2211yescharacteristic
12C2×C4⋊Dic11—21no—
13C2×C4⋊Dic11C2212yes—
14C22×C11⋊C8C2212yes—
15#66922—24no—
16#66896—24no—
17C24×C11—21no—
18C2×D4×C11—22no—
19C2×D4×C11—22no—
20C2×D4×C11—21no—
21C2×D4×C11Q812yescomplemented
22C22×C4×C11—21no—
23C22×C4×C11D411yescharacteristic
24C22×C4×C11D411yescharacteristic
25C22×C4×C11C2311yescharacteristic
26C22×Dic11—22no—
27C2×Dic22—24no—
28C2×Dic22—42no—
29C2×Dic22—42no—
30C11×C4⋊C4—22no—
31C11×C22⋊C4—42no—
32C11×C22⋊C4—42no—
33C42×C11—22no—
34C4⋊Dic11—24no—
35C4⋊Dic11—22no—
36C4⋊Dic11D414yes—
37C2×C11⋊C8—22no—
38C2×C11⋊C8—22no—
39C2×C11⋊C8—22no—
40#2023—111nomaximal, Sylow
41C23×C11—22no—
42C23×C11—41no—
43C23×C11—41no—
44C23×C11—41no—
45C23×C11—41no—
46C23×C11—41no—
47C23×C11—41no—
48C23×C11C2×D411yescharacteristic
49D4×C11—24no—
50D4×C11—42no—
51C2×C4×C11—21no—
52C2×C4×C11—21no—
53C2×C4×C11—21no—
54C2×C4×C11—21no—
55C2×C4×C11—41no—
56C2×C4×C11—41no—
57C2×C4×C11—41no—
58C2×C4×C11D812yes—
59C2×C4×C11SD1612yes—
60C2×C4×C11C2×D412yes—
61C2×C4×C11C2×Q811yescharacteristic
62C2×C4×C11#4111yescharacteristic
63C2×C4×C11#4112yes—
64C2×Dic11—24no—
65C2×Dic11—42no—
66C2×Dic11—42no—
67Dic22—48no—
68C11⋊C8—42no—
69C2×C4⋊Q8—111no—
70C2×#102—112no—
71#390D1111yescharacteristic
72#340—112no—
73#336—111no—
74C22×C11—42no—
75C22×C11—42no—
76C22×C11—42no—
77C22×C11—81no—
78C22×C11—81no—
79C22×C11—81no—
80C22×C11—81no—
81C22×C11#12011yescharacteristic
82C22×C11C22⋊Q811yescharacteristic
83C22×C11#12411yescharacteristic
84C22×C11C2×D811yescharacteristic
85C22×C11C2×SD1611yescharacteristic
86C22×C11AGL(1,ℤ/8)11yescharacteristic
87C22×C11#13711yescharacteristic
88C4×C11—22no—
89C4×C11—22no—
90C4×C11—81no—
91C4×C11C22⋊Q812yes—
92C4×C11#12412yes—
93Dic11—44no—
94C22×Q8—221no—
95C22×D4C2×D1111yescharacteristic
96C22×C8—112no—
97C4⋊Q8—224no—
98C2×C4⋊C4—112no—
99C2×C4⋊C4—221no—
100C2×C4⋊C4C2×D1111yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C42C2×D1111yescharacteristic
104#102—224no—
105C2×C11—82no—
106C2×C11#39311yescharacteristic
107C2×C11#44711yescharacteristic
108C2×C11#44811yescharacteristic
109C2×C11#47411yescharacteristic
110C2×C11#47611yescharacteristic
111C2×C11#48611yescharacteristic
112C2×C11#48911yescharacteristic
113C24—21no—
114C2×Q8—224no—
115C2×Q8—442no—
116C2×Q8—442no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4Dic2212yescomplemented
121C22×C4—21no—
122C22×C4—222no—
123C22×C4C22⋊D1111yescharacteristic
124C22×C4C22⋊D1111yescharacteristic
125C22×C4C22×D1111yescharacteristic
126C2×C8—222no—
127C2×C8—222no—
128C2×C8—222no—
129C4⋊C4—22no—
130C4⋊C4—114no—
131C4⋊C4—224no—
132C4⋊C4—222no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C42—22no—
136C11#202311yescharacteristic, complemented, Sylow
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23C2×C22⋊D1111yescharacteristic
145Q8—448no—
146D4—24no—
147D4—42no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—224no—
156C2×C4—442no—
157C2×C4—442no—
158C2×C4#432312yes—
159C2×C4#432412yes—
160C2×C4C2×Dic2211yescharacteristic
161C2×C4#433911yescharacteristic
162C2×C4D4×D1112yes—
163C2×C4#434112yes—
164C8—442no—
165C22—42no—
166C22—42no—
167C22—42no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22#6697711yescharacteristic
173C22C2×#432311yescharacteristic
174C22C2×#432411yescharacteristic
175C22#6698911yescharacteristic
176C22#6699011yescharacteristic
177C22#6700211yescharacteristic
178C22#6700411yescharacteristic
179C4—22no—
180C4—22no—
181C4—81no—
182C4—444no—
183C4#6693112yes—
184C4#6693812yes—
185C2—82no—
186C2#1062256011yescharacteristic
187C2#1062262011yescharacteristic
188C2#1062262111yescharacteristic
189C2#1062264911yescharacteristic
190C2#1062265311yescharacteristic
191C2#1062266811yescharacteristic
192C2#1062268311yescharacteristic
subgroupG/H
centerC23C2×C22⋊D11
commutator, cosocleC22×C4×C11C23
FrattiniC22×C4C22×D11
FittingC11×#390C2
socleC23×C11C2×D4
prime radicalC22×D4×C11C22
hypercenter#390D11
hypocenterC11#2023
LCSG ▷ C22×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#66922C2semidirect
4. #10622167C2semidirect
5. #10622162C2semidirect
21. C2×D4×C11Q8semidirect
120. C2×D4Dic22semidirect
136. C11#2023semidirect

Maximal subgroups

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

subgroupindex
C2×#669222
C11×#3902
C2×#668962
#106221672
#106221622
#202311