Finite Group Databasebeta

← #9346

Subgroups — #9346

Order 256.

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

normal subgroups only

This group has 222 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#3095C211yesmaximal, characteristic
2#2924C211yesmaximal, characteristic
3#1825C211yesmaximal, characteristic
4#1520C211yesmaximal, characteristic
5#1472C211yesmaximal, characteristic
6#1305C211yesmaximal, characteristic
7#1298C211yesmaximal, characteristic
8C2×C22⋊Q8C2211yescharacteristic
9C2×#117C2211yescharacteristic
10#485—21no—
11#485—21no—
12#484—21no—
13#483—21no—
14#433—22no—
15#432—21no—
16#431—21no—
17#428C411yescharacteristic, complemented
18#428C411yescharacteristic, complemented
19#424—21no—
20#419C411yescharacteristic, complemented
21#419C411yescharacteristic, complemented
22#407C2211yescharacteristic
23#407C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26#405—21no—
27#380C2211yescharacteristic
28#340—21no—
29#340—21no—
30#328—21no—
31#328—21no—
32#324—22no—
33C22×Q8C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C2×M16—21no—
36C2×M16C2×C411yescharacteristic
37C2×M16C2×C411yescharacteristic
38C22×C8—21no—
39C22×C8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44#117—21no—
45#117—21no—
46#117C2×C411yescharacteristic
47#117C2×C411yescharacteristic
48#117D411yescharacteristic
49#117D411yescharacteristic
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4C2×C411yescharacteristic
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C42—21no—
59#107—21no—
60#107—21no—
61#106—21no—
62#106—21no—
63C4⋊C8—22no—
64C4⋊C8—22no—
65#103—21no—
66#103—21no—
67#103—21no—
68#103—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76C22⋊C8D411yescharacteristic
77C22⋊C8D411yescharacteristic
78#97—22no—
79C4×C8—22no—
80#95—21no—
81C24#4111yescharacteristic
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—41no—
85C2×Q8—41no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C22×C411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4#4111yescharacteristic
99M16—41no—
100M16—41no—
101M16—41no—
102M16—41no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—22no—
107C2×C8—21no—
108C2×C8—41no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—41no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4C22⋊C411yescharacteristic
122C4⋊C4C22⋊C411yescharacteristic
123C4⋊C4C22⋊C411yescharacteristic
124C4⋊C4C22⋊C411yescharacteristic
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—81no—
132C42—21no—
133C42—21no—
134C42—21no—
135C42—21no—
136C23—21no—
137C23—21no—
138C23—41no—
139C23—41no—
140C23C4×D411yescharacteristic
141C23#12111yescharacteristic
142C23#12311yescharacteristic
143Q8—41no—
144Q8—41no—
145Q8—81no—
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—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4C2×C22⋊C411yescharacteristic
170C2×C4C4×D411yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C8—41no—
174C8—41no—
175C8—42no—
176C8—41no—
177C8—41no—
178C8—81no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—81no—
188C22—81no—
189C22#38611yescharacteristic
190C22#47111yescharacteristic
191C22#47311yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—41no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—81no—
201C4—161no—
202C4#41011yescharacteristic
203C4#41011yescharacteristic
204C2—21no—
205C2—21no—
206C2—41no—
207C2—41no—
208C2#188111yescharacteristic
209C2#189911yescharacteristic
210C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24#41
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
17. #428C4semidirect
18. #428C4semidirect
20. #419C4semidirect
21. #419C4semidirect

Maximal subgroups

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

subgroupindex
#30952
#29242
#18252
#15202
#14722
#13052
#12982