Finite Group Databasebeta

← #9314

Subgroups — #9314

Order 256.

203 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 59 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#3116C211yesmaximal, characteristic
2#2923C211yesmaximal, characteristic
3#1531C211yesmaximal, characteristic, complemented
4#1488C211yesmaximal, characteristic
5#1485C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1291C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#485—21no—
11#477C411yescharacteristic
12#476C411yescharacteristic
13#462C411yescharacteristic
14#461C411yescharacteristic
15#452—21no—
16#441—21no—
17#440—21no—
18#433—22no—
19#432—22no—
20#432C2211yescharacteristic
21#432C2211yescharacteristic
22#431C2211yescharacteristic
23#424—21no—
24#408—21no—
25C4×M16C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#330—22no—
28Q8⋊C8—21no—
29Q8⋊C8—22no—
30D4⋊C8—21no—
31C2×#41D411yescharacteristic
32C2×Q16—21no—
33C2×SD16—21no—
34C2×M16—21no—
35C22×C8—21no—
36C4⋊Q8C2×C411yescharacteristic
37C4⋊Q8C2×C411yescharacteristic
38#125—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C4×Q8—21no—
42C4×Q8—21no—
43C4×Q8C2×C411yescharacteristic
44C4×Q8D411yescharacteristic
45C4×D4—21no—
46C4×D4—21no—
47C4×D4C2×C411yescharacteristic
48C4×D4D411yescharacteristic
49#117—21no—
50#117—21no—
51#117D411yescharacteristic
52C2×C42—21no—
53C2×C42C2311yescharacteristic
54#107—21no—
55#106—21no—
56C4⋊C8—24no—
57C4⋊C8—21no—
58C4⋊C8—21no—
59C4⋊C8C2×C411yescharacteristic
60C4⋊C8C2×C411yescharacteristic
61#103—21no—
62#103—21no—
63#103—21no—
64#102—21no—
65C22⋊C8—24no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69#97—21no—
70#97—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75C4×C8D412yes—
76#41—41no—
77#41—41no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8C22⋊C411yescharacteristic
81C2×D4—21no—
82C2×D4C22⋊C411yescharacteristic
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4C2×D411yescharacteristic
87C22×C4#4111yescharacteristic
88C22×C4#4111yescharacteristic
89Q16—42no—
90SD16—42no—
91M16—41no—
92M16—41no—
93C2×C8—21no—
94C2×C8—21no—
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4C22⋊C411yescharacteristic
110C4⋊C4C22⋊C411yescharacteristic
111C22⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—21no—
114C22⋊C4—41no—
115C42—21no—
116C42—21no—
117C42—21no—
118C42—41no—
119C42C22×C411yescharacteristic
120C42C2×D411yescharacteristic
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C23—21no—
124C23#12311yescharacteristic
125Q8—21no—
126Q8—21no—
127Q8—42no—
128Q8—41no—
129Q8—41no—
130D4—21no—
131D4—21no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4C2×C22⋊C411yescharacteristic
146C2×C4C4×D411yescharacteristic
147C2×C4C4×D411yescharacteristic
148C2×C4#12011yescharacteristic
149C2×C4#12111yescharacteristic
150C2×C4#12111yescharacteristic
151C8—42no—
152C8—42no—
153C8—41no—
154C8—41no—
155C8—81no—
156C8—81no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22#38611yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—81no—
173C4—81no—
174C4#42111yescharacteristic
175C4#42111yescharacteristic
176C2—41no—
177C2—81no—
178C2#188111yescharacteristic
179C2#189711yescharacteristic
180C2#190211yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC2×#41D4
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
3. #1531C2semidirect
5. #1485C2semidirect
6. #1302C2semidirect
7. #1291C2semidirect

Maximal subgroups

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

subgroupindex
#31162
#29232
#15312
#14882
#14852
#13022
#12912