Finite Group Databasebeta

← #7648

Subgroups — #7648

Order 256.

250 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C4×C22⋊Q8C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic
4#1272C212yesmaximal, complemented
5#1271C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic
7C2×C4×Q8C411yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C22⋊C8—21no—
11C2×C22⋊C8—21no—
12#391C2211yescharacteristic
13#389C2211yescharacteristic
14#387—21no—
15#385—21no—
16#380C411yescharacteristic
17C4×C4⋊C4—21no—
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×Q8C2×C411yescharacteristic
24C23×C4C2×C411yescharacteristic
25C23×C4D411yescharacteristic
26C23×C4D411yescharacteristic
27C22×C8—22no—
28C22×C8—22no—
29C22⋊Q8—22no—
30C22⋊Q8—22no—
31C4×Q8—42no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C42—21no—
41C2×C42—21no—
42C2×C42—42no—
43C2×C42C2×C411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42C2311yescharacteristic
47C4⋊C8—42no—
48C22⋊C8—42no—
49C22⋊C8—42no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—21no—
55C24C22⋊C411yescharacteristic
56C2×Q8—21no—
57C2×Q8—41no—
58C2×Q8—41no—
59C2×Q8M1612yes—
60C22×C4—21no—
61C22×C4—21no—
62C22×C4—22no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—22no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4C22⋊C411yescharacteristic
74C22×C4C22⋊C411yescharacteristic
75C22×C4C22⋊C411yescharacteristic
76C22×C4SD1612yes—
77C22×C4Q1612yes—
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C2×C8—42no—
82C2×C8—42no—
83C2×C8—42no—
84C2×C8—42no—
85C2×C8—42no—
86C2×C8—42no—
87C4⋊C4—24no—
88C4⋊C4—21no—
89C4⋊C4—41no—
90C4⋊C4—41no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4M1612yes—
94C22⋊C4—24no—
95C22⋊C4—42no—
96C42—22no—
97C42—22no—
98C42—41no—
99C42—41no—
100C42—42no—
101C42—42no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23#10312yes—
109C23C2×C22⋊C411yescharacteristic
110Q8—42no—
111Q8—42no—
112C2×C4—21no—
113C2×C4—21no—
114C2×C4—21no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—21no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—44no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4#9912yes—
143C2×C4#10312yes—
144C2×C4C2×M1611yescharacteristic
145C2×C4C2×M1611yescharacteristic
146C2×C4C2×SD1611yescharacteristic
147C2×C4C2×Q1611yescharacteristic
148C2×C4#13512yes—
149C8—82no—
150C8—82no—
151C22—22no—
152C22—22no—
153C22—22no—
154C22—41no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#40711yescharacteristic
159C22C2×#9911yescharacteristic
160C22#41311yescharacteristic
161C22C2×#10311yescharacteristic
162C22#41611yescharacteristic
163C22#42111yescharacteristic
164C22#42111yescharacteristic
165C4—21no—
166C4—21no—
167C4—22no—
168C4—22no—
169C4—21no—
170C4—22no—
171C4—21no—
172C4—41no—
173C4—42no—
174C4—42no—
175C4—81no—
176C4—81no—
177C2—42no—
178C2#146511yescharacteristic
179C2#148711yescharacteristic
180C2#148811yescharacteristic
181C2#151611yescharacteristic
182C2#152011yescharacteristic
183C2#153511yescharacteristic
184C2#154211yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC24C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1272C2semidirect
5. #1271C2semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊Q82
#18172
#17972
#12722
#12712