Finite Group Databasebeta

← #8380

Subgroups — #8380

Order 256.

409 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 89 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×#554C211yesmaximal, characteristic
2#1824C212yesmaximal, complemented
3#1799C211yesmaximal, characteristic
4#1520C212yesmaximal, complemented
5#1459C211yesmaximal, characteristic
6#554—24no—
7#554—24no—
8C2×C4⋊Q8C411yescharacteristic, complemented
9C2×#125—21no—
10C2×C22⋊Q8—21no—
11C2×C22⋊Q8—21no—
12C2×C22⋊Q8C411yescharacteristic, complemented
13C2×C22⋊Q8C2211yescharacteristic
14C2×C4×Q8C411yescharacteristic, complemented
15C2×C4×D4—21no—
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4C411yescharacteristic, complemented
18C22×C4⋊C4C2211yescharacteristic
19C2×#99—21no—
20C2×C22⋊C8C2212yes—
21#380C2212yes—
22#342—21no—
23#340—24no—
24#328—24no—
25#324—21no—
26#323—21no—
27C22×Q8—21no—
28C22×Q8C2×C411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C23×C4—21no—
32C23×C4C2×C411yescharacteristic
33C23×C4C2311yescharacteristic
34C22×C8—22no—
35C4⋊Q8—22no—
36C4⋊Q8—22no—
37#125—28no—
38C22⋊Q8—28no—
39C22⋊Q8—28no—
40C22⋊Q8—22no—
41C22⋊Q8—22no—
42C22⋊Q8—22no—
43C22⋊Q8D414yes—
44C4×Q8—22no—
45C4×Q8—22no—
46C4×D4—28no—
47C4×D4—22no—
48C4×D4—22no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4C2×C411yescharacteristic
60C2×C4⋊C4C2×C411yescharacteristic
61C2×C4⋊C4C2×C411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C4⋊C4D414yes—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—21no—
70C2×C42C2×C411yescharacteristic
71#99—42no—
72C22⋊C8—22no—
73C22⋊C8—22no—
74#95—22no—
75C24C22⋊C411yescharacteristic
76C24C2×D411yescharacteristic
77C2×Q8—24no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×Q8—82no—
81C2×Q8C22⋊C412yes—
82C2×D4—22no—
83C2×D4—22no—
84C2×D4SD1612yes—
85C2×D4Q1612yes—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—24no—
91C22×C4—24no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—42no—
95C22×C4—42no—
96C22×C4—42no—
97C22×C4—82no—
98C22×C4—82no—
99C22×C4C22⋊C411yescharacteristic
100C22×C4C22⋊C411yescharacteristic
101C22×C4C22⋊C411yescharacteristic
102C22×C4C22×C411yescharacteristic
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C2×C8—44no—
107C2×C8—42no—
108C4⋊C4—24no—
109C4⋊C4—24no—
110C4⋊C4—24no—
111C4⋊C4—24no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—24no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—44no—
119C4⋊C4C22⋊C412yes—
120C4⋊C4C22⋊C412yes—
121C4⋊C4C22⋊C412yes—
122C22⋊C4—24no—
123C22⋊C4—24no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—42no—
127C22⋊C4—42no—
128C42—24no—
129C42—21no—
130C42—21no—
131C23—21no—
132C23—21no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23#10312yes—
138C23C2×C22⋊C411yescharacteristic
139C23#12012yes—
140C23C2×SD1611yescharacteristic
141C23C2×Q1611yescharacteristic
142C23#13712yes—
143Q8—42no—
144Q8—42no—
145Q8—84no—
146D4—24no—
147D4—24no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—24no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—84no—
169C2×C4—84no—
170C2×C4—82no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4#10312yes—
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4C2×C22⋊C411yescharacteristic
176C2×C4#12012yes—
177C8—82no—
178C22—21no—
179C22—22no—
180C22—21no—
181C22—22no—
182C22—22no—
183C22—41no—
184C22—41no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—41no—
189C22—41no—
190C22#37911yescharacteristic
191C22C2×#10311yescharacteristic
192C22#41911yescharacteristic
193C22#44812yes—
194C22#45112yes—
195C4—22no—
196C4—21no—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—42no—
204C4—162no—
205C4#41012yes—
206C2—22no—
207C2—21no—
208C2—21no—
209C2—22no—
210C2—42no—
211C2#178211yescharacteristic
212C2#178711yescharacteristic
213C2#179511yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1824C2semidirect
4. #1520C2semidirect
8. C2×C4⋊Q8C4semidirect
12. C2×C22⋊Q8C4semidirect
14. C2×C4×Q8C4semidirect
17. C22×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
#18242
#17992
#15202
#14592