Finite Group Databasebeta

← #7654

Subgroups — #7654

Order 256.

290 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 75 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#2854C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic, complemented
3C2×#336C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C23×C8—21no—
7C2×#123—21no—
8C2×#121C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C22⋊C8—21no—
11#390C411yescharacteristic
12#390C2211yescharacteristic
13#389C411yescharacteristic
14#389C2211yescharacteristic
15#388—21no—
16#388—21no—
17#386—21no—
18#380C411yescharacteristic
19#336—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C22×C8—24no—
29C22×C8—22no—
30C22×C8—22no—
31C22×C8—42no—
32C22×C8—42no—
33#123—42no—
34#123—42no—
35#121—42no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C42—42no—
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C2×C42C2311yescharacteristic
52C22⋊C8—42no—
53#95—21no—
54#95—21no—
55#95—21no—
56C24—21no—
57C24C22⋊C411yescharacteristic
58C2×D4—22no—
59C2×D4—41no—
60C2×D4—41no—
61C2×D4—42no—
62C2×D4—42no—
63C2×D4—42no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4C22⋊C411yescharacteristic
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4D812yes—
79C22×C4SD1612yes—
80C22×C4C22×C411yescharacteristic
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C2×C8—24no—
84C2×C8—48no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—42no—
89C2×C8—42no—
90C4⋊C4—22no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—42no—
94C4⋊C4—41no—
95C4⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C22⋊C4—41no—
101C22⋊C4—41no—
102C22⋊C4—42no—
103C42—41no—
104C42—41no—
105C42—41no—
106C42—41no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23#9912yes—
119C23#10212yes—
120C23#10412yes—
121C23C2×C22⋊C411yescharacteristic
122D4—82no—
123D4—84no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—44no—
141C2×C4—44no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4#10212yes—
152C2×C4#13112yes—
153C2×C4C2×D811yescharacteristic
154C2×C4C2×SD1611yescharacteristic
155C2×C4#13512yes—
156C8—44no—
157C8—82no—
158C22—22no—
159C22—22no—
160C22—22no—
161C22—22no—
162C22—22no—
163C22—22no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22#32714yes—
172C22#40811yescharacteristic
173C22C2×#9911yescharacteristic
174C22#41311yescharacteristic
175C22C2×#10211yescharacteristic
176C22#41611yescharacteristic
177C22C2×#10411yescharacteristic
178C22#42111yescharacteristic
179C4—22no—
180C4—21no—
181C4—22no—
182C4—21no—
183C4—22no—
184C4—41no—
185C4—41no—
186C4—81no—
187C4—81no—
188C4—81no—
189C2—24no—
190C2—161no—
191C2#146811yescharacteristic
192C2#148011yescharacteristic
193C2#149411yescharacteristic
194C2C2×#32711yescharacteristic
195C2#151611yescharacteristic
196C2#153211yescharacteristic
197C2#154011yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1817C2semidirect
3. C2×#336C2semidirect
4. #1271C2semidirect
5. #1270C2semidirect

Maximal subgroups

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

subgroupindex
#28542
#18172
C2×#3362
#12712
#12702