Finite Group Databasebeta

← #7599

Subgroups — #7599

Order 256.

294 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C4×#121C211yesmaximal, characteristic
2#1817C211yesmaximal, characteristic, complemented
3#1797C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C2×#121C411yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic
9C22×C42C2211yescharacteristic
10C2×C4⋊C8—21no—
11C2×C22⋊C8—21no—
12C2×C22⋊C8—21no—
13#390C2211yescharacteristic
14#389C2211yescharacteristic
15#388—21no—
16#386—21no—
17#380C411yescharacteristic
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C22×C8—22no—
30C22×C8—22no—
31#121—22no—
32#121C814yescomplemented
33C4×D4—42no—
34C4×D4—44no—
35C2×C4⋊C4—21no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C4⋊C4C2×C411yescharacteristic
39C2×C22⋊C4—21no—
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C42—21no—
45C2×C42—42no—
46C2×C42C2×C411yescharacteristic
47C2×C42D411yescharacteristic
48C2×C42D411yescharacteristic
49C2×C42C2311yescharacteristic
50C4⋊C8—42no—
51C22⋊C8—42no—
52C22⋊C8—42no—
53#95—21no—
54#95—21no—
55#95—21no—
56C24—21no—
57C24C22⋊C411yescharacteristic
58C2×D4—24no—
59C2×D4—21no—
60C2×D4—41no—
61C2×D4—41no—
62C2×D4—42no—
63C2×D4—42no—
64C2×D4C2×C812yes—
65C22×C4—21no—
66C22×C4—22no—
67C22×C4—21no—
68C22×C4—22no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—42no—
78C22×C4—42no—
79C22×C4—42no—
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22⋊C411yescharacteristic
83C22×C4C2×C812yes—
84C22×C4M1612yes—
85C22×C4C22×C411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—42no—
89C2×C8—42no—
90C2×C8—42no—
91C2×C8—42no—
92C2×C8—42no—
93C2×C8—42no—
94C4⋊C4—21no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—42no—
98C4⋊C4C2×C812yes—
99C22⋊C4—24no—
100C22⋊C4—42no—
101C22⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—42no—
104C22⋊C4—41no—
105C42—22no—
106C42—22no—
107C42—41no—
108C42—41no—
109C42—42no—
110C23—21no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—22no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23C22⋊C812yes—
124C23C2×C22⋊C411yescharacteristic
125D4—42no—
126D4—42no—
127D4—48no—
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—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—44no—
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—84no—
156C2×C4C22⋊C812yes—
157C2×C4#9912yes—
158C2×C4C22×C811yescharacteristic
159C2×C4C2×M1611yescharacteristic
160C2×C4AGL(1,ℤ/8)11yescharacteristic
161C2×C4AGL(1,ℤ/8)12yes—
162C2×C4#13711yescharacteristic
163C8—82no—
164C8—82no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22C2×C22⋊C811yescharacteristic
180C22C2×#9911yescharacteristic
181C22#41311yescharacteristic
182C22#41711yescharacteristic
183C22#41811yescharacteristic
184C22#42111yescharacteristic
185C22#42111yescharacteristic
186C4—22no—
187C4—21no—
188C4—21no—
189C4—22no—
190C4—22no—
191C4—21no—
192C4—21no—
193C4—42no—
194C4—41no—
195C4—81no—
196C4—81no—
197C2—42no—
198C2—82no—
199C2#145911yescharacteristic
200C2#147711yescharacteristic
201C2#147811yescharacteristic
202C2#151011yescharacteristic
203C2#151311yescharacteristic
204C2#153511yescharacteristic
205C2#155711yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC2×#121C4
LCSG ▷ C2×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. #1797C2semidirect
4. #1271C2semidirect
5. #1270C2semidirect
32. #121C8semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#18172
#17972
#12712
#12702