Finite Group Databasebeta

← #11205

Subgroups — #11205

Order 256.

459 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 119 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
1#2315C211yesmaximal, characteristic
2#2315C211yesmaximal, characteristic
3C2×#342C212yesmaximal
4C2×#342C212yesmaximal
5C2×#342C211yesmaximal, characteristic
6C22×C22⋊C4C2211yescharacteristic
7C22×C22⋊C4C2212yes—
8C22×C22⋊C4C2212yes—
9C22×C22⋊C4C2211yescharacteristic
10C22×C22⋊C4C2211yescharacteristic
11#385—22no—
12#385—24no—
13#385C414yescomplemented
14#380—22no—
15#380—22no—
16#342—22no—
17#342—24no—
18#342—24no—
19C25C2311yescharacteristic
20C23×C4—21no—
21C23×C4—22no—
22C23×C4—22no—
23C23×C4C2×C412yes—
24C23×C4D412yes—
25C2×C4⋊C4—24no—
26C2×C4⋊C4—24no—
27C2×C22⋊C4—22no—
28C2×C22⋊C4—21no—
29C2×C22⋊C4—21no—
30C2×C22⋊C4—21no—
31C2×C22⋊C4—22no—
32C2×C22⋊C4—22no—
33C2×C22⋊C4—22no—
34C2×C22⋊C4—22no—
35C2×C22⋊C4—22no—
36C2×C22⋊C4—22no—
37C2×C22⋊C4—22no—
38C2×C22⋊C4—22no—
39C2×C22⋊C4—22no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—22no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—22no—
46C2×C22⋊C4C2×C412yes—
47C2×C22⋊C4C2×C412yes—
48C2×C22⋊C4D414yes—
49C2×C22⋊C4D412yes—
50C2×C22⋊C4D412yes—
51C2×C22⋊C4Q814yes—
52C2×C22⋊C4Q812yes—
53#95—24no—
54#95—24no—
55C24—82no—
56C24C22×C411yescharacteristic
57C24C2×D411yescharacteristic
58C24C2×D411yescharacteristic
59C24C2×D412yes—
60C24C2×D411yescharacteristic
61C24C2×Q811yescharacteristic
62C24C2×Q812yes—
63C24#4112yes—
64C24#4111yescharacteristic
65C24#4111yescharacteristic
66C24#4112yes—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—22no—
72C22×C4—22no—
73C22×C4—24no—
74C22×C4—24no—
75C22×C4—24no—
76C22×C4—42no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4—82no—
81C22×C4—82no—
82C22×C4—84no—
83C22×C4—82no—
84C22×C4—84no—
85C22×C4C22⋊C414yes—
86C4⋊C4—84no—
87C4⋊C4—84no—
88C22⋊C4—22no—
89C22⋊C4—24no—
90C22⋊C4—22no—
91C22⋊C4—24no—
92C22⋊C4—22no—
93C22⋊C4—24no—
94C22⋊C4—24no—
95C22⋊C4—24no—
96C22⋊C4—24no—
97C22⋊C4—41no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4C4⋊C414yescomplemented
105C23—42no—
106C23—42no—
107C23—42no—
108C23—44no—
109C23—44no—
110C23—81no—
111C23—81no—
112C23—82no—
113C23—82no—
114C23—81no—
115C23—81no—
116C23C2×C22⋊C411yescharacteristic
117C23C2×C4⋊C411yescharacteristic
118C23#11711yescharacteristic
119C23C4×D412yes—
120C23C4×Q812yes—
121C23#12011yescharacteristic
122C23#12112yes—
123C23#12111yescharacteristic
124C23#12111yescharacteristic
125C23#12112yes—
126C23C22⋊Q811yescharacteristic
127C23C22⋊Q811yescharacteristic
128C23C22⋊Q812yes—
129C23C22⋊Q812yes—
130C23C22⋊Q812yes—
131C23#12311yescharacteristic
132C23#12312yes—
133C23#12411yescharacteristic
134C23#12512yes—
135C23#12512yes—
136C23#12612yes—
137C23C4⋊Q812yes—
138C23C4⋊Q811yescharacteristic
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—44no—
142C2×C4—44no—
143C2×C4—44no—
144C2×C4—48no—
145C2×C4—44no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—44no—
149C2×C4—82no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—84no—
153C2×C4—84no—
154C2×C4—84no—
155C2×C4—84no—
156C2×C4—84no—
157C2×C4—84no—
158C22—22no—
159C22—22no—
160C22—24no—
161C22—24no—
162C22—22no—
163C22—41no—
164C22—42no—
165C22—41no—
166C22—42no—
167C22—42no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—41no—
173C22—41no—
174C22—42no—
175C22—42no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—41no—
181C22#38011yescharacteristic
182C22#38611yescharacteristic
183C22#38712yes—
184C22#38912yes—
185C22#39111yescharacteristic
186C22#39411yescharacteristic
187C22#39511yescharacteristic
188C22#39612yes—
189C22#39812yes—
190C22#39811yescharacteristic
191C22#40011yescharacteristic
192C4—44no—
193C4—44no—
194C4—84no—
195C4—84no—
196C4—82no—
197C2—21no—
198C2—21no—
199C2—22no—
200C2—22no—
201C2—22no—
202C2—21no—
203C2—21no—
204C2—22no—
205C2—21no—
206C2—21no—
207C2#144111yescharacteristic
208C2#191312yes—
subgroupG/H
center, socleC22#386
commutatorC24C22×C4
Frattini, prime radical, cosocleC25C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #385C4semidirect
104. C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#23152
C2×#3422