Finite Group Databasebeta

← #25657

Subgroups — #25657

Order 256.

490 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 135 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×#557C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic
3C2×#479C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic
5#2043C212yesmaximal
6#2043C212yesmaximal
7#1985C211yesmaximal, characteristic
8#1981C211yesmaximal, characteristic
9#1865C212yesmaximal
10#1863C212yesmaximal
11#1763C211yesmaximal, characteristic
12#557—24no—
13#557—24no—
14C2×C4⋊Q8C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—22no—
17C2×#125—21no—
18C2×#125C2211yescharacteristic
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#491—24no—
24#479—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#97C2211yescharacteristic
36#398—21no—
37#395—21no—
38#389—22no—
39#389C2212yes—
40#389C2212yes—
41C4×C4⋊C4C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C4⋊Q8—24no—
52C4⋊Q8—24no—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55#125—216no—
56#125—28no—
57#125—22no—
58#125—22no—
59C4×Q8—28no—
60C4×Q8—24no—
61C4×Q8—24no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8D414yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D412yes—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2312yes—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#106—22no—
89C4⋊C8—22no—
90#103—22no—
91#103—22no—
92#103—24no—
93#103—24no—
94#103—24no—
95#103—24no—
96#103—24no—
97#97—22no—
98#95—21no—
99#95—22no—
100#95—22no—
101C2×Q8—22no—
102C2×Q8—24no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C2×D414yes—
112C2×Q8C2×D412yes—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—28no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—42no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4—82no—
149C4⋊C4C2×D412yes—
150C4⋊C4#4114yes—
151C42—28no—
152C42—24no—
153C42—22no—
154C42—22no—
155C42—81no—
156C42C2×D412yes—
157C42C2×D412yes—
158C23C22×D411yescharacteristic
159Q8—28no—
160Q8—22no—
161Q8—22no—
162Q8—24no—
163Q8—24no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4#12012yes—
192C2×C4#12012yes—
193C2×C4#13714yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C2×#4112yes—
197C2×C42−1+412yes—
198C8—41no—
199C8—41no—
200C8—42no—
201C22C2×#12011yescharacteristic
202C22#54812yes—
203C22C2×#13712yes—
204C22#57711yescharacteristic
205C22#57811yescharacteristic
206C4—22no—
207C4—24no—
208C4—22no—
209C4—44no—
210C4—42no—
211C4—44no—
212C4—42no—
213C4—42no—
214C4—81no—
215C4—81no—
216C4#53211yescharacteristic
217C4#53611yescharacteristic
218C4#54712yes—
219C2#243211yescharacteristic
220C2#300411yescharacteristic
221C2#301311yescharacteristic
222C2#337212yes—
223C2#337412yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4912
C2×#4792
#25842
#20432
#19852
#19812
#18652
#18632
#17632