Finite Group Databasebeta

← #8560

Subgroups — #8560

Order 256.

341 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 107 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×C4×C4⋊C4C211yesmaximal, characteristic
2#1797C212yesmaximal
3C2×#324C212yesmaximal
4#1280C212yesmaximal
5C22×C4⋊C4C411yescharacteristic
6C22×C4⋊C4C2211yescharacteristic
7C22×C42C411yescharacteristic
8C22×C42C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—22no—
11C2×C22⋊C8C2214yes—
12C4×C4⋊C4—22no—
13C4×C4⋊C4—22no—
14C2×#95C411yescharacteristic
15C2×#95C2211yescharacteristic
16#336—22no—
17#324—24no—
18C23×C4C2×C411yescharacteristic
19C23×C4C2×C411yescharacteristic
20C23×C4C2×C411yescharacteristic
21C23×C4C2×C411yescharacteristic
22C23×C4C2×C411yescharacteristic
23C23×C4C2×C411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—24no—
26C2×C4⋊C4—22no—
27C2×C4⋊C4—22no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4C814yescomplemented
31C2×C4⋊C4D412yes—
32C2×C4⋊C4Q812yes—
33C2×C42—21no—
34C2×C42—21no—
35C2×C42—22no—
36C2×C42—22no—
37C2×C42—21no—
38C2×C42—21no—
39C2×C42—42no—
40C2×C42—42no—
41C2×C42D412yes—
42C2×C42D412yes—
43C4⋊C8—44no—
44C22⋊C8—24no—
45C22⋊C8—24no—
46#95—22no—
47#95—22no—
48C24C22×C411yescharacteristic
49C22×C4—21no—
50C22×C4—21no—
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—22no—
56C22×C4—22no—
57C22×C4—21no—
58C22×C4—21no—
59C22×C4—21no—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—21no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4C22⋊C412yes—
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4C4⋊C412yes—
77C22×C4C4⋊C412yes—
78C22×C4C2×C812yes—
79C22×C4C2×C812yes—
80C22×C4C2×C812yes—
81C22×C4M1612yes—
82C22×C4M1612yes—
83C22×C4M1612yes—
84C22×C4C2×D412yes—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×Q811yescharacteristic
87C22×C4#4112yes—
88C2×C8—44no—
89C2×C8—44no—
90C2×C8—44no—
91C4⋊C4—28no—
92C4⋊C4—24no—
93C4⋊C4—24no—
94C42—24no—
95C42—24no—
96C42—24no—
97C42—42no—
98C42—42no—
99C42—42no—
100C23—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23C2×C22⋊C411yescharacteristic
105C23C2×C4⋊C411yescharacteristic
106C23#11711yescharacteristic
107C23C22×C811yescharacteristic
108C23C2×M1611yescharacteristic
109C23C2×M1611yescharacteristic
110C23C2×M1611yescharacteristic
111C2×C4—24no—
112C2×C4—24no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—21no—
121C2×C4—22no—
122C2×C4—21no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4C22⋊C814yes—
148C2×C4#9912yes—
149C2×C4#10112yes—
150C2×C4C4⋊C814yescomplemented
151C2×C4#12112yes—
152C2×C4C22⋊Q812yes—
153C8—84no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22#38011yescharacteristic
167C22C2×C22⋊C811yescharacteristic
168C22#40711yescharacteristic
169C22C2×#9911yescharacteristic
170C22#41011yescharacteristic
171C22C2×#10111yescharacteristic
172C22#41311yescharacteristic
173C22C2×C4⋊C811yescharacteristic
174C22#42311yescharacteristic
175C22#43111yescharacteristic
176C22#43211yescharacteristic
177C4—24no—
178C4—22no—
179C4—22no—
180C4—24no—
181C4—22no—
182C4—22no—
183C4—42no—
184C4—42no—
185C4—42no—
186C2—21no—
187C2—21no—
188C2—21no—
189C2—21no—
190C2#145911yescharacteristic
191C2#146111yescharacteristic
192C2#146311yescharacteristic
193C2#146411yescharacteristic
194C2#179711yescharacteristic
195C2#179911yescharacteristic
196C2#180111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC24C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
30. C2×C4⋊C4C8semidirect
150. C2×C4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C42
#17972
C2×#3242
#12802