Finite Group Databasebeta

← #23555

Subgroups — #23555

Order 256.

555 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 191 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×#544C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic
3C22×#107C211yesmaximal, characteristic
4#2317C211yesmaximal, characteristic
5#2059C214yesmaximal, complemented
6#2057C214yesmaximal, complemented
7#1736C211yesmaximal, characteristic
8#1735C211yesmaximal, characteristic
9#1729C211yesmaximal, characteristic
10C22×M16C2211yescharacteristic
11C23×C8C2211yescharacteristic
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2212yes—
16C2×#125—21no—
17C2×#125C2212yes—
18C2×C22⋊Q8—22no—
19C2×#117C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#428—24no—
25C2×#107—24no—
26C2×#107C2212yes—
27C2×#107C2214yes—
28C2×#106C2212yes—
29#389C2214yes—
30#385—22no—
31#384—21no—
32#382—21no—
33#340C2212yes—
34#340C2214yes—
35#340C2212yes—
36#336C2214yes—
37C22×Q8—22no—
38C23×C4—21no—
39C23×C4—21no—
40C23×C4C2311yescharacteristic
41C2×M16—22no—
42C2×M16—22no—
43C2×M16Q814yescomplemented
44C22×C8—22no—
45C22×C8—22no—
46C22×C8Q814yescomplemented
47C22×C8C2312yes—
48C22×C8C2312yes—
49C4⋊Q8—28no—
50C4⋊Q8—24no—
51C4⋊Q8—24no—
52#125—28no—
53#125—24no—
54#125—24no—
55C22⋊Q8—216no—
56#117—22no—
57#117—22no—
58#117—22no—
59#117—22no—
60#117D414yes—
61#117Q814yescomplemented
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4—22no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—24no—
71C2×C4⋊C4D411yescharacteristic
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4D414yes—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4D412yes—
82C2×C42—21no—
83C2×C42—21no—
84C2×C42C2312yes—
85#107—28no—
86#107—24no—
87#106—24no—
88#95—24no—
89C24C2×D411yescharacteristic
90C2×Q8—28no—
91C2×Q8—84no—
92C22×C4—22no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—24no—
97C22×C4—22no—
98C22×C4—42no—
99C22×C4—82no—
100C22×C4—82no—
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×Q812yes—
107C22×C4C2×Q812yes—
108C22×C4C2×Q812yes—
109C22×C4#4112yes—
110C22×C4C2411yescharacteristic
111M16—28no—
112C2×C8—28no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8C2×Q814yes—
118C2×C8C2×Q814yes—
119C4⋊C4—28no—
120C4⋊C4—28no—
121C4⋊C4—28no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—84no—
133C4⋊C4C2×D412yes—
134C4⋊C4C2×D414yes—
135C4⋊C4C2×D412yes—
136C4⋊C4C2×Q814yes—
137C22⋊C4—24no—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C42—24no—
141C42—24no—
142C42—22no—
143C42—22no—
144C23—22no—
145C23—22no—
146C23#12012yes—
147C23C22⋊Q812yes—
148C23C4⋊Q812yes—
149C23C22×D411yescharacteristic
150Q8—88no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—84no—
167C2×C4—84no—
168C2×C4#12012yes—
169C2×C4C22⋊Q814yes—
170C2×C4C22⋊Q814yes—
171C2×C4C22⋊Q812yes—
172C2×C4C4⋊Q814yes—
173C2×C4C4⋊Q814yes—
174C2×C4C4⋊Q812yes—
175C2×C4C22×D411yescharacteristic
176C2×C4C22×D411yescharacteristic
177C2×C4C22×Q811yescharacteristic
178C2×C4C22×Q811yescharacteristic
179C2×C4C22×Q811yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C8—24no—
182C8—24no—
183C22—24no—
184C22—24no—
185C22—22no—
186C22—22no—
187C22#39514yes—
188C22C2×#12011yescharacteristic
189C22C2×C22⋊Q811yescharacteristic
190C22C2×C4⋊Q811yescharacteristic
191C22#57612yes—
192C22#57812yes—
193C4—44no—
194C4—42no—
195C4—44no—
196C4—42no—
197C4—42no—
198C4—42no—
199C4—82no—
200C4#39514yes—
201C4C2×C22⋊Q811yescharacteristic
202C4C2×C22⋊Q811yescharacteristic
203C4C2×C4⋊Q811yescharacteristic
204C4C2×C4⋊Q811yescharacteristic
205C2—24no—
206C2C2×#39511yescharacteristic
207C2#300811yescharacteristic
208C2#301111yescharacteristic
209C2#308312yes—
210C2#316412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2059C2semidirect
6. #2057C2semidirect
43. C2×M16Q8semidirect
46. C22×C8Q8semidirect
61. #117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4282
C22×#1072
#23172
#20592
#20572
#17362
#17352
#17292