Finite Group Databasebeta

← #9521

Subgroups — #9521

Order 256.

236 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 61 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#3092C211yesmaximal, characteristic
2#2976C211yesmaximal, characteristic
3#1824C211yesmaximal, characteristic, complemented
4#1821C211yesmaximal, characteristic, complemented
5#1513C211yesmaximal, characteristic
6#1305C211yesmaximal, characteristic, complemented
7#1293C211yesmaximal, characteristic, complemented
8#525C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10#484—21no—
11#483C412yescomplemented
12#481—21no—
13#480C412yescomplemented
14#436—21no—
15#435—21no—
16#435—21no—
17C8×D4—21no—
18#433—21no—
19#428—21no—
20#417—21no—
21#408—21no—
22#407C2211yescharacteristic
23#407C2211yescharacteristic
24#407C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26#380C2211yescharacteristic
27#343—21no—
28#343—21no—
29#340—21no—
30#340—21no—
31#328—21no—
32#327—21no—
33C2×#41D411yescharacteristic
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C2×M16—21no—
37C2×M16—21no—
38C2×M16—21no—
39C2×M16D411yescharacteristic
40C2×M16D411yescharacteristic
41C22×C8—21no—
42C22×C8D411yescharacteristic
43C22×C8D411yescharacteristic
44#123—21no—
45C22⋊Q8C2×C411yescharacteristic
46C22⋊Q8D411yescharacteristic
47#121C2×C411yescharacteristic
48#121D411yescharacteristic
49#120—21no—
50C4×D4—21no—
51C4×D4—21no—
52#117D411yescharacteristic
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2×C412yes—
60C2×C22⋊C4—21no—
61#107—22no—
62#106—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65#103—22no—
66#102—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8C2×C412yes—
77#97—21no—
78C4×C8—21no—
79#95—21no—
80C24#4111yescharacteristic
81#41—41no—
82#41—41no—
83C2×Q8C22⋊C411yescharacteristic
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4C22⋊C411yescharacteristic
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—82no—
96C22×C4C22×C411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4#4111yescharacteristic
102M16—41no—
103M16—41no—
104M16—41no—
105M16—41no—
106M16—41no—
107M16—41no—
108M16—81no—
109C2×C8—22no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—81no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4C22⋊C411yescharacteristic
127C4⋊C4C22⋊C411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—81no—
134C42—21no—
135C23—21no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23C4×D411yescharacteristic
141C23#12111yescharacteristic
142C23#12311yescharacteristic
143Q8—41no—
144D4—41no—
145D4—41no—
146D4—41no—
147D4—81no—
148D4—81no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—82no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C4×D411yescharacteristic
172C2×C4#12011yescharacteristic
173C2×C4#12111yescharacteristic
174C8—42no—
175C8—41no—
176C8—41no—
177C8—41no—
178C8—41no—
179C8—81no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—42no—
189C22—42no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22#38611yescharacteristic
194C22#47111yescharacteristic
195C22#47311yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—41no—
201C4—41no—
202C4—82no—
203C4—161no—
204C4#45811yescharacteristic
205C4#45811yescharacteristic
206C2—21no—
207C2—21no—
208C2—41no—
209C2—41no—
210C2—81no—
211C2#189011yescharacteristic
212C2#190311yescharacteristic
213C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#525C22
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
3. #1824C2semidirect
4. #1821C2semidirect
6. #1305C2semidirect
7. #1293C2semidirect
11. #483C4semidirect
13. #480C4semidirect

Maximal subgroups

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

subgroupindex
#30922
#29762
#18242
#18212
#15132
#13052
#12932