Finite Group Databasebeta

← #9447

Subgroups — #9447

Order 256.

284 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 61 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#2973C211yesmaximal, characteristic, complemented
2#2544C211yesmaximal, characteristic
3#1533C212yesmaximal, complemented
4#1466C211yesmaximal, characteristic, complemented
5#1285C212yesmaximal, complemented
6C2×#127C411yescharacteristic
7C2×#124C411yescharacteristic
8C2×#124C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#435—24no—
11#432C2212yes—
12#432C2212yes—
13#407—21no—
14C2×C22⋊C8—21no—
15#394—21no—
16#394—21no—
17#391C411yescharacteristic
18#390C411yescharacteristic
19#386—21no—
20#386—21no—
21C4×C4⋊C4C2211yescharacteristic
22#329—24no—
23#323—21no—
24#323—21no—
25C22×Q8C2×C411yescharacteristic
26C22×D4—21no—
27C22×D4—21no—
28C22×D4C2×C411yescharacteristic
29C22×D4C2×C411yescharacteristic
30C23×C4D411yescharacteristic
31C23×C4D411yescharacteristic
32C2×M16—22no—
33C22×C8—22no—
34#127—42no—
35#124—22no—
36#124—22no—
37#124—42no—
38C4×D4—22no—
39C4×D4—22no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4D411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C42C2×C411yescharacteristic
51C2×C42C2×C411yescharacteristic
52C2×C42C2311yescharacteristic
53C4⋊C8—22no—
54C4⋊C8—22no—
55C22⋊C8—22no—
56C22⋊C8—22no—
57C22⋊C8—22no—
58C22⋊C8—22no—
59#97—22no—
60#97D414yescomplemented
61#95—21no—
62#95—21no—
63#95—21no—
64C24—21no—
65C24C22⋊C411yescharacteristic
66C24C22⋊C411yescharacteristic
67C2×Q8—22no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×D4—22no—
71C2×D4—22no—
72C2×D4—22no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—42no—
77C2×D4—42no—
78C2×D4—42no—
79C2×D4—42no—
80C2×D4—81no—
81C2×D4—81no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—81no—
90C22×C4—81no—
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22⋊C411yescharacteristic
93C22×C4C22×C411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96M16—82no—
97C2×C8—24no—
98C2×C8—42no—
99C2×C8—82no—
100C4⋊C4—22no—
101C4⋊C4—42no—
102C22⋊C4—22no—
103C22⋊C4—22no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—41no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—81no—
110C22⋊C4—81no—
111C42—41no—
112C42—41no—
113C42C2×D412yes—
114C42#4112yes—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23#10012yes—
132C23C2×C22⋊C411yescharacteristic
133Q8—82no—
134D4—44no—
135D4—82no—
136D4—84no—
137D4—84no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—81no—
158C2×C4—81no—
159C2×C4C4×D412yes—
160C2×C4#12011yescharacteristic
161C2×C4#12112yes—
162C2×C4#12311yescharacteristic
163C8—44no—
164C8—82no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—41no—
174C22—41no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22#38611yescharacteristic
185C22C2×#10011yescharacteristic
186C22#41311yescharacteristic
187C4—22no—
188C4—22no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—41no—
193C4—41no—
194C4—81no—
195C4—81no—
196C2—21no—
197C2—21no—
198C2—42no—
199C2—81no—
200C2—161no—
201C2#188411yescharacteristic
202C2#190011yescharacteristic
203C2#191111yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2973C2semidirect
3. #1533C2semidirect
4. #1466C2semidirect
5. #1285C2semidirect
60. #97D4semidirect

Maximal subgroups

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

subgroupindex
#29732
#25442
#15332
#14662
#12852