Finite Group Databasebeta

← #9166

Subgroups — #9166

Order 256.

218 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 63 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#3113C211yesmaximal, characteristic
2#2984C211yesmaximal, characteristic
3#1763C211yesmaximal, characteristic
4#1531C211yesmaximal, characteristic
5#1503C211yesmaximal, characteristic
6#1302C211yesmaximal, characteristic
7#1285C211yesmaximal, characteristic
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#483—21no—
11#475C411yescharacteristic, complemented
12#475C411yescharacteristic, complemented
13#462C411yescharacteristic, complemented
14#462C411yescharacteristic, complemented
15#451—21no—
16#446—21no—
17#446—22no—
18C8×Q8—21no—
19#441—21no—
20#441—21no—
21#432—21no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431—21no—
25#431C2211yescharacteristic
26C2×#103—21no—
27C2×#97C2211yescharacteristic
28C4×C4⋊C4C2211yescharacteristic
29#340—21no—
30#330—22no—
31#328—21no—
32#328—21no—
33C22×Q8D411yescharacteristic
34C2×Q16—21no—
35C2×Q16—21no—
36C22×C8—21no—
37C4⋊Q8C2×C411yescharacteristic
38C4⋊Q8C2×C411yescharacteristic
39#125—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C4×Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8C2×C411yescharacteristic
47C4×Q8C2×C411yescharacteristic
48C4×Q8D411yescharacteristic
49C4×Q8D411yescharacteristic
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D411yescharacteristic
54C2×C42—21no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#106—21no—
58#106—21no—
59C4⋊C8—22no—
60C4⋊C8—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C411yescharacteristic
64C4⋊C8C2×C411yescharacteristic
65#103—24no—
66#103—21no—
67#103—21no—
68#103—21no—
69#103—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8D411yescharacteristic
73C22⋊C8D411yescharacteristic
74#97—22no—
75#97—21no—
76#97—21no—
77#97—21no—
78#97D412yes—
79C4×C8—22no—
80C4×C8—21no—
81#95—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—41no—
87C2×Q8C22⋊C411yescharacteristic
88C2×Q8C22⋊C411yescharacteristic
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4#4111yescharacteristic
95Q16—42no—
96Q16—42no—
97C2×C8—22no—
98C2×C8—21no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—22no—
103C2×C8—41no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4C22⋊C411yescharacteristic
119C4⋊C4C22⋊C411yescharacteristic
120C22⋊C4—41no—
121C42—21no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—41no—
126C42—81no—
127C42C22×C411yescharacteristic
128C42C2×D411yescharacteristic
129C42C2×D411yescharacteristic
130C42#4111yescharacteristic
131C23#12111yescharacteristic
132Q8—21no—
133Q8—21no—
134Q8—21no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—81no—
139Q8#10412yes—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—81no—
156C2×C4—81no—
157C2×C4C2×C22⋊C411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4C4×D411yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12311yescharacteristic
163C8—41no—
164C8—41no—
165C8—42no—
166C8—42no—
167C8—41no—
168C8—41no—
169C22—21no—
170C22—21no—
171C22#38611yescharacteristic
172C22#47111yescharacteristic
173C22#47311yescharacteristic
174C4—21no—
175C4—22no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—82no—
184C4—81no—
185C4—81no—
186C4C2×#10411yescharacteristic
187C4#42111yescharacteristic
188C2—21no—
189C2—21no—
190C2#187111yescharacteristic
191C2#190511yescharacteristic
192C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC23#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #475C4semidirect
12. #475C4semidirect
13. #462C4semidirect
14. #462C4semidirect

Maximal subgroups

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

subgroupindex
#31132
#29842
#17632
#15312
#15032
#13022
#12852