Finite Group Databasebeta

← #9107

Subgroups — #9107

Order 256.

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

normal subgroups only

This group has 337 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
1#2972C211yesmaximal, characteristic, complemented
2#2321C211yesmaximal, characteristic
3#1459C211yesmaximal, characteristic, complemented
4C2×#323C212yesmaximal, complemented
5#1282C212yesmaximal, complemented
6C2×#120C411yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C22⋊C4C2211yescharacteristic
9C8×D4—24no—
10#431—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8C2212yes—
13C2×C22⋊C8C2212yes—
14#390—21no—
15#388—21no—
16#388—21no—
17#386—21no—
18#386C411yescharacteristic
19#385C411yescharacteristic
20#379C411yescharacteristic
21C4×C22⋊C4C2211yescharacteristic
22#324—21no—
23#323—21no—
24#323—24no—
25C25C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C22×C8—22no—
33C22×C8—22no—
34#120—22no—
35#120C814yescomplemented
36C4×D4—22no—
37C4×D4—22no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4C2×C411yescharacteristic
49C2×C22⋊C4C2×C411yescharacteristic
50C2×C22⋊C4C2×C411yescharacteristic
51C2×C22⋊C4D411yescharacteristic
52C2×C42—21no—
53C2×C42D411yescharacteristic
54C4⋊C8—22no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—22no—
58C22⋊C8—22no—
59C22⋊C8D414yescomplemented
60C4×C8—22no—
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24—22no—
66C24—21no—
67C24—22no—
68C24—81no—
69C24—81no—
70C24C22⋊C411yescharacteristic
71C24C2×C812yescomplemented
72C24C22×C411yescharacteristic
73C2×D4—22no—
74C2×D4—22no—
75C2×D4—24no—
76C2×D4—81no—
77C2×D4—81no—
78C2×D4C2×C812yes—
79C2×D4M1612yes—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—81no—
87C22×C4—81no—
88C22×C4—81no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C2×D412yes—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4112yes—
96C2×C8—24no—
97C2×C8—42no—
98C2×C8—82no—
99C2×C8—82no—
100C2×C8—82no—
101C4⋊C4—22no—
102C4⋊C4—81no—
103C22⋊C4—22no—
104C22⋊C4—21no—
105C22⋊C4—24no—
106C22⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—41no—
109C22⋊C4—42no—
110C22⋊C4—41no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C22⋊C4—81no—
114C22⋊C4C2×C812yescomplemented
115C42—21no—
116C42—21no—
117C42—81no—
118C23—21no—
119C23—21no—
120C23—22no—
121C23—24no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—42no—
133C23—82no—
134C23—82no—
135C23—82no—
136C23—82no—
137C23—84no—
138C23C22⋊C812yes—
139C23C2×C22⋊C411yescharacteristic
140C23C4×D412yes—
141C23C22×C811yescharacteristic
142C23C2×M1611yescharacteristic
143C23#13112yes—
144D4—24no—
145D4—24no—
146D4—84no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4—82no—
169C2×C4C22⋊C812yes—
170C2×C4#12011yescharacteristic
171C2×C4#12112yes—
172C2×C4#12311yescharacteristic
173C8—44no—
174C8—82no—
175C22—21no—
176C22—22no—
177C22—21no—
178C22—22no—
179C22—22no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—41no—
188C22—42no—
189C22—81no—
190C22—84no—
191C22—81no—
192C22—81no—
193C22—84no—
194C22—81no—
195C22—81no—
196C22#38611yescharacteristic
197C22C2×C22⋊C811yescharacteristic
198C22#40811yescharacteristic
199C22C8×D412yes—
200C22#43512yes—
201C4—22no—
202C4—21no—
203C4—22no—
204C4—21no—
205C4—42no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—81no—
210C2—21no—
211C2—21no—
212C2—22no—
213C2—22no—
214C2—42no—
215C2—82no—
216C2#186911yescharacteristic
217C2#191311yescharacteristic
218C2#191411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC23C22×C8
Frattini, cosocleC23×C4C23
prime radicalC2×#120C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2972C2semidirect
3. #1459C2semidirect
4. C2×#323C2semidirect
5. #1282C2semidirect
35. #120C8semidirect
59. C22⋊C8D4semidirect
71. C24C2×C8semidirect
114. C22⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
#29722
#23212
#14592
C2×#3232
#12822