Finite Group Databasebeta

← #8406

Subgroups — #8406

Order 256.

434 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#544C211yesmaximal, characteristic
2#1872C212yesmaximal
3C2×#328C212yesmaximal
4C2×#324C211yesmaximal, characteristic
5#1441C211yesmaximal, characteristic
6#544—24no—
7#544—22no—
8#544C414yescomplemented
9C2×C4⋊Q8—22no—
10C2×#125—22no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8C2212yes—
13C2×#117—21no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#103—22no—
17C2×C22⋊C8C2212yes—
18C2×#95—21no—
19C2×#95—21no—
20C2×#95C2211yescharacteristic
21#340—22no—
22#328—24no—
23#324—22no—
24C22×Q8—21no—
25C22×Q8D412yes—
26C23×C4—21no—
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C23×C4C2311yescharacteristic
31C22×C8—22no—
32C4⋊Q8—28no—
33C4⋊Q8—28no—
34#125—28no—
35#125—28no—
36C22⋊Q8—24no—
37C22⋊Q8—24no—
38C22⋊Q8—22no—
39C22⋊Q8—22no—
40C22⋊Q8C2×C414yes—
41C22⋊Q8D414yes—
42#117—24no—
43#117—24no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—22no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—42no—
53C2×C4⋊C4C2×C412yes—
54C2×C4⋊C4D412yes—
55C2×C4⋊C4D412yes—
56C2×C22⋊C4—22no—
57C2×C42—22no—
58#103—44no—
59C22⋊C8—22no—
60C22⋊C8—22no—
61#95—22no—
62#95—42no—
63#95—42no—
64C24C2×D411yescharacteristic
65C2×Q8—24no—
66C2×Q8—22no—
67C2×Q8—42no—
68C2×Q8—42no—
69C2×Q8—81no—
70C2×Q8—81no—
71C2×Q8C22⋊C414yes—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—22no—
77C22×C4—22no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—22no—
81C22×C4—22no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—41no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C22⋊C412yes—
99C22×C4C22⋊C412yes—
100C22×C4SD1612yes—
101C22×C4Q1612yes—
102C22×C4C22×C411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—42no—
109C4⋊C4—24no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—28no—
113C4⋊C4—24no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—24no—
119C4⋊C4—42no—
120C4⋊C4C22⋊C414yes—
121C22⋊C4—24no—
122C22⋊C4—24no—
123C42—28no—
124C23—21no—
125C23—21no—
126C23—21no—
127C23—21no—
128C23C2×C22⋊C411yescharacteristic
129C23#12011yescharacteristic
130C23#12011yescharacteristic
131C23C2×SD1611yescharacteristic
132C23C2×Q1611yescharacteristic
133C23#13711yescharacteristic
134C23#13711yescharacteristic
135Q8—44no—
136Q8—82no—
137Q8—84no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—24no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4#10314yescomplemented
175C2×C4C2×C22⋊C412yes—
176C2×C4#12012yes—
177C8—82no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22#37911yescharacteristic
191C22C2×#10311yescharacteristic
192C22#41911yescharacteristic
193C22#44811yescharacteristic
194C22#44811yescharacteristic
195C22#45111yescharacteristic
196C22#45111yescharacteristic
197C22#45411yescharacteristic
198C22#45611yescharacteristic
199C22B(4,𝔽2)11yescharacteristic
200C22#45811yescharacteristic
201C4—24no—
202C4—22no—
203C4—22no—
204C4—44no—
205C4—42no—
206C4—44no—
207C4—82no—
208C4—82no—
209C2—21no—
210C2—21no—
211C2—21no—
212C2—21no—
213C2#159911yescharacteristic
214C2#160511yescharacteristic
215C2#161111yescharacteristic
216C2#161711yescharacteristic
217C2#178711yescharacteristic
218C2#179011yescharacteristic
219C2#179311yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #544C4semidirect
174. C2×C4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
#18722
C2×#3282
C2×#3242
#14412