Finite Group Databasebeta

← #8562

Subgroups — #8562

Order 256.

285 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 83 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×#387C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic
3#1797C211yesmaximal, characteristic
4C2×#324C212yesmaximal
5#1280C212yesmaximal
6C22×C4⋊C4C411yescharacteristic
7C22×C4⋊C4C2211yescharacteristic
8C22×C42C2211yescharacteristic
9C2×C4⋊C8—21no—
10C2×C4⋊C8—21no—
11C2×C22⋊C8C2212yes—
12C2×C22⋊C8C2212yes—
13#387—21no—
14#387—21no—
15#387—21no—
16#387—21no—
17C2×#95C411yescharacteristic
18C2×#95C411yescharacteristic
19C2×#95C411yescharacteristic
20C2×#95C2211yescharacteristic
21#336—21no—
22#336—21no—
23#324—24no—
24C23×C4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4C2×C411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—42no—
40C2×C4⋊C4D412yes—
41C2×C4⋊C4Q812yes—
42C2×C42—42no—
43C2×C42D411yescharacteristic
44C2×C42D411yescharacteristic
45C2×C42D411yescharacteristic
46C2×C42D411yescharacteristic
47C4⋊C8—42no—
48C4⋊C8—42no—
49C22⋊C8—22no—
50C22⋊C8—22no—
51C22⋊C8—22no—
52C22⋊C8—22no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24C22×C411yescharacteristic
62C22×C4—21no—
63C22×C4—21no—
64C22×C4—21no—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—22no—
75C22×C4—22no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4C22⋊C411yescharacteristic
87C22×C4C22⋊C411yescharacteristic
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C4⋊C412yes—
91C22×C4C4⋊C412yes—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×Q811yescharacteristic
96C22×C4#4111yescharacteristic
97C22×C4#4111yescharacteristic
98C2×C8—42no—
99C2×C8—42no—
100C2×C8—42no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—42no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C42—22no—
111C42—22no—
112C42—41no—
113C42—41no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23C2×C22⋊C411yescharacteristic
119C23C2×C4⋊C411yescharacteristic
120C23#11711yescharacteristic
121C23#13112yes—
122C23#13112yes—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—22no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—24no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—44no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4#9912yes—
162C2×C4#10112yes—
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4C22⋊Q811yescharacteristic
166C2×C4C22⋊Q811yescharacteristic
167C8—82no—
168C8—82no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22#38011yescharacteristic
178C22#40811yescharacteristic
179C22#40811yescharacteristic
180C22C2×#9911yescharacteristic
181C22#41011yescharacteristic
182C22C2×#10111yescharacteristic
183C22#41311yescharacteristic
184C22#42412yes—
185C22#43312yes—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—21no—
190C4—22no—
191C4—22no—
192C4—42no—
193C4—42no—
194C4—41no—
195C4—41no—
196C4—81no—
197C4—81no—
198C2—22no—
199C2—22no—
200C2#146811yescharacteristic
201C2#147011yescharacteristic
202C2#147211yescharacteristic
203C2#147311yescharacteristic
204C2#179811yescharacteristic
205C2#179911yescharacteristic
206C2#180111yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutator, prime radicalC24C22×C4
Frattini, cosocleC23×C4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#3872
#17972
C2×#3242
#12802