Finite Group Databasebeta

← #8216

Subgroups — #8216

Order 256.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#1749C211yesmaximal, characteristic, complemented
3C2×#327C212yesmaximal
4#1434C211yesmaximal, characteristic, complemented
5#1294C212yesmaximal, complemented
6C2×#121C2211yescharacteristic
7C2×C4×D4—21no—
8C2×C4×D4C411yescharacteristic, complemented
9C22×C42C2211yescharacteristic
10C2×C22⋊C8C2212yes—
11C2×#97—21no—
12#390C411yescharacteristic, complemented
13#389C411yescharacteristic, complemented
14#388—21no—
15#386—21no—
16#380C2211yescharacteristic
17C4×C4⋊C4C411yescharacteristic, complemented
18C4×C22⋊C4—21no—
19C2×#95—21no—
20C2×#95—21no—
21C2×#95C2212yes—
22#336—21no—
23#327C418yescomplemented
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C23×C4—22no—
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32#121—22no—
33#121C2×C414yes—
34C4×D4—42no—
35C4×D4—44no—
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×C42—21no—
46C2×C42—42no—
47C2×C42C2×C411yescharacteristic
48C2×C42C2×C411yescharacteristic
49C2×C42D411yescharacteristic
50C2×C42D411yescharacteristic
51C22⋊C8—22no—
52C22⋊C8C2×C414yescomplemented
53#97—42no—
54#95—21no—
55#95—21no—
56#95—21no—
57#95—22no—
58#95—42no—
59#95—42no—
60#95C2×C414yescomplemented
61C24—21no—
62C24C2×D411yescharacteristic
63C2×D4—24no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4—42no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4C4212yes—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—24no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4—41no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—42no—
85C22×C4—42no—
86C22×C4—42no—
87C22×C4—42no—
88C22×C4—42no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4C22⋊C411yescharacteristic
94C22×C4C22⋊C411yescharacteristic
95C22×C4C22⋊C411yescharacteristic
96C22×C4C22⋊C411yescharacteristic
97C22×C4C22×C411yescharacteristic
98C22×C4C22×C412yes—
99C22×C4C2×D411yescharacteristic
100C22×C4#4112yes—
101C2×C8—24no—
102C2×C8—42no—
103C2×C8—42no—
104C4⋊C4—21no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—42no—
108C4⋊C4C4212yes—
109C22⋊C4—24no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4—42no—
113C22⋊C4—42no—
114C42—22no—
115C42—22no—
116C42—41no—
117C42—41no—
118C42—42no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—22no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23C2×C22⋊C411yescharacteristic
131C23C4×D412yes—
132C23AGL(1,ℤ/8)11yescharacteristic
133C23AGL(1,ℤ/8)12yes—
134C23#13711yescharacteristic
135D4—42no—
136D4—42no—
137D4—44no—
138D4—44no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—44no—
163C2×C4—44no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—44no—
171C2×C4—84no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#9912yes—
177C2×C4C2×C4211yescharacteristic
178C2×C4#11711yescharacteristic
179C2×C4C4×D412yes—
180C8—44no—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—22no—
189C22—42no—
190C22—42no—
191C22—42no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22C4×C22⋊C411yescharacteristic
197C22C2×#9911yescharacteristic
198C22#41011yescharacteristic
199C22#41711yescharacteristic
200C22#41811yescharacteristic
201C22#42111yescharacteristic
202C22#42111yescharacteristic
203C22#44012yes—
204C22#44212yes—
205C4—22no—
206C4—21no—
207C4—21no—
208C4—22no—
209C4—42no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—84no—
214C4—81no—
215C4—81no—
216C2—22no—
217C2—22no—
218C2—82no—
219C2#151011yescharacteristic
220C2#151211yescharacteristic
221C2#151311yescharacteristic
222C2#151411yescharacteristic
223C2C4×#9911yescharacteristic
224C2#175911yescharacteristic
225C2#176211yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C2×C42
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1749C2semidirect
4. #1434C2semidirect
5. #1294C2semidirect
8. C2×C4×D4C4semidirect
12. #390C4semidirect
13. #389C4semidirect
17. C4×C4⋊C4C4semidirect
23. #327C4semidirect
52. C22⋊C8C2×C4semidirect
60. #95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#17492
C2×#3272
#14342
#12942