Finite Group Databasebeta

← #7694

Subgroups — #7694

Order 256.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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
3#1728C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C22×M16—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
28C2×M16—24no—
29C2×M16—42no—
30C2×M16—42no—
31C22×C8—22no—
32C22×C8—22no—
33#123—44no—
34#121—42no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C42—42no—
47C2×C42C2×C411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42C2311yescharacteristic
51C22⋊C8—42no—
52#95—21no—
53#95—21no—
54#95—21no—
55C24—21no—
56C24C22⋊C411yescharacteristic
57C2×D4—22no—
58C2×D4—41no—
59C2×D4—41no—
60C2×D4—42no—
61C2×D4—42no—
62C2×D4—42no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—22no—
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—22no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4C22⋊C411yescharacteristic
77C22×C4C22⋊C411yescharacteristic
78C22×C4C22⋊C411yescharacteristic
79C22×C4C22×C411yescharacteristic
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D411yescharacteristic
82M16—48no—
83C2×C8—24no—
84C2×C8—42no—
85C2×C8—42no—
86C2×C8—42no—
87C2×C8—42no—
88C2×C8—42no—
89C4⋊C4—22no—
90C4⋊C4—41no—
91C4⋊C4—41no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—41no—
99C22⋊C4—42no—
100C22⋊C4—41no—
101C42—41no—
102C42—41no—
103C42—41no—
104C42—41no—
105C23—21no—
106C23—21no—
107C23—21no—
108C23—21no—
109C23—21no—
110C23—21no—
111C23—41no—
112C23—41no—
113C23—41no—
114C23—41no—
115C23—41no—
116C23—41no—
117C23—41no—
118C23#9912yes—
119C23C2×C22⋊C411yescharacteristic
120D4—82no—
121D4—84no—
122C2×C4—24no—
123C2×C4—24no—
124C2×C4—21no—
125C2×C4—21no—
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—44no—
139C2×C4—44no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4#10412yes—
150C2×C4#13112yes—
151C2×C4AGL(1,ℤ/8)11yescharacteristic
152C2×C4AGL(1,ℤ/8)12yes—
153C2×C4#13711yescharacteristic
154C8—44no—
155C8—82no—
156C22—22no—
157C22—22no—
158C22—22no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22#40811yescharacteristic
171C22C2×#9911yescharacteristic
172C22#41311yescharacteristic
173C22#41711yescharacteristic
174C22#41811yescharacteristic
175C22C2×#10411yescharacteristic
176C22#42111yescharacteristic
177C4—22no—
178C4—21no—
179C4—22no—
180C4—21no—
181C4—22no—
182C4—41no—
183C4—41no—
184C4—81no—
185C4—81no—
186C4—81no—
187C2—42no—
188C2—161no—
189C2#146811yescharacteristic
190C2#148411yescharacteristic
191C2#149511yescharacteristic
192C2#150411yescharacteristic
193C2#151011yescharacteristic
194C2#152911yescharacteristic
195C2#155711yescharacteristic
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. #1728C2semidirect
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
#17282
#12712
#12702