Finite Group Databasebeta

← #8454

Subgroups — #8454

Order 256.

419 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 75 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#3492C211yesmaximal, characteristic
2#1821C211yesmaximal, characteristic, complemented
3#1504C212yesmaximal, complemented
4#1504C212yesmaximal, complemented
5#1445C211yesmaximal, characteristic, complemented
6#554—22no—
7#553—22no—
8#552—22no—
9#547—22no—
10#544C411yescharacteristic, complemented
11#539—22no—
12#537C411yescharacteristic, complemented
13#525—22no—
14#525C2212yes—
15#520C412yescomplemented
16C22×C4⋊C4C2211yescharacteristic
17#407C2212yes—
18C2×#95—22no—
19C2×#95C2211yescharacteristic
20C2×#95C2211yescharacteristic
21#343—22no—
22#328—22no—
23#328—22no—
24#327—22no—
25#327—22no—
26C2×#41—22no—
27C2×#41C2×C412yes—
28C23×C4—21no—
29C23×C4—21no—
30C23×C4D412yes—
31C23×C4C2311yescharacteristic
32C2×M16—22no—
33C4⋊Q8—22no—
34#126—24no—
35#125—24no—
36#125—22no—
37#123—24no—
38#123—24no—
39#123—22no—
40C22⋊Q8—22no—
41C22⋊Q8—24no—
42C22⋊Q8—22no—
43C22⋊Q8—22no—
44C22⋊Q8D412yes—
45C22⋊Q8D412yes—
46#121—22no—
47#121—22no—
48#121—22no—
49#121D412yes—
50#121D412yes—
51#120—22no—
52C4×Q8—22no—
53C4×D4—24no—
54C4×D4—22no—
55C4×D4—24no—
56C4×D4—22no—
57C4×D4—22no—
58#117—22no—
59#117—22no—
60#117C2×C412yes—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4C2×C411yescharacteristic
65C2×C4⋊C4C2×C411yescharacteristic
66C2×C4⋊C4D412yes—
67C22⋊C8—22no—
68C22⋊C8—22no—
69#95—22no—
70#95—22no—
71#95—44no—
72C24C2×D411yescharacteristic
73#41—44no—
74#41—82no—
75#41—82no—
76C2×Q8—22no—
77C2×Q8C22⋊C412yes—
78C2×D4—22no—
79C2×D4—22no—
80C2×D4—22no—
81C2×D4—22no—
82C2×D4C22⋊C412yes—
83C22×C4—22no—
84C22×C4—24no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—42no—
90C22×C4—42no—
91C22×C4—42no—
92C22×C4—42no—
93C22×C4—42no—
94C22×C4—42no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4C22⋊C412yes—
101C22×C4C22⋊C412yes—
102C22×C4C22×C411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106M16—44no—
107C2×C8—42no—
108C4⋊C4—24no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—24no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4C22⋊C412yes—
118C4⋊C4C22⋊C412yes—
119C22⋊C4—24no—
120C22⋊C4—24no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C42—24no—
127C42—22no—
128C23—22no—
129C23—21no—
130C23—21no—
131C23—21no—
132C23—21no—
133C23—22no—
134C23C2×C22⋊C411yescharacteristic
135C23#12011yescharacteristic
136C23#12011yescharacteristic
137Q8—42no—
138Q8—82no—
139D4—42no—
140D4—44no—
141D4—42no—
142D4—42no—
143D4—42no—
144D4—42no—
145D4—82no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—44no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—44no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—82no—
165C2×C4—84no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4C2×C22⋊C412yes—
173C2×C4#12012yes—
174C8—82no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22#37911yescharacteristic
189C22#41012yes—
190C22#45311yescharacteristic
191C22#45411yescharacteristic
192C22#45511yescharacteristic
193C22#45611yescharacteristic
194C4—22no—
195C4—22no—
196C4—42no—
197C4—42no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—82no—
203C2—21no—
204C2—21no—
205C2—22no—
206C2—21no—
207C2—21no—
208C2—82no—
209C2#178211yescharacteristic
210C2#179211yescharacteristic
211C2#179311yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3492C2
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. #1821C2semidirect
3. #1504C2semidirect
4. #1504C2semidirect
5. #1445C2semidirect
10. #544C4semidirect
12. #537C4semidirect
15. #520C4semidirect

Maximal subgroups

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

subgroupindex
#34922
#18212
#15042
#14452