Finite Group Databasebeta

← #9214

Subgroups — #9214

Order 256.

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

normal subgroups only

This group has 214 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#3107C211yesmaximal, characteristic
2#2922C211yesmaximal, characteristic
3#1526C211yesmaximal, characteristic, complemented
4#1493C211yesmaximal, characteristic, complemented
5#1489C211yesmaximal, characteristic
6#1299C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#485—21no—
11#475C411yescharacteristic
12#474C411yescharacteristic
13#462C411yescharacteristic
14#461C411yescharacteristic
15#452—21no—
16C4×Q16—21no—
17C4×SD16—21no—
18#433—22no—
19#431—22no—
20#431C2211yescharacteristic
21#424—21no—
22#423C2211yescharacteristic
23#408—21no—
24C4×M16C2211yescharacteristic
25C4×M16C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#330—22no—
28Q8⋊C8—21no—
29Q8⋊C8—22no—
30D4⋊C8—21no—
31C2×#41D411yescharacteristic
32C2×Q16—21no—
33C2×SD16—21no—
34C2×M16—21no—
35C2×M16—21no—
36C2×M16—21no—
37C22×C8—21no—
38C4⋊Q8C2×C411yescharacteristic
39C4⋊Q8C2×C411yescharacteristic
40#125—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4C2×C411yescharacteristic
50C4×D4D411yescharacteristic
51#117—21no—
52#117—21no—
53#117D411yescharacteristic
54C2×C42—21no—
55C2×C42C2311yescharacteristic
56#107—21no—
57#106—21no—
58C4⋊C8—24no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8C2×C411yescharacteristic
62C4⋊C8C2×C411yescharacteristic
63#103—21no—
64#103—21no—
65#103—21no—
66#102—21no—
67C22⋊C8—24no—
68C22⋊C8—21no—
69#97—21no—
70#97D412yes—
71C4×C8—21no—
72C4×C8—21no—
73C4×C8—21no—
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—22no—
92M16—41no—
93M16—41no—
94M16—41no—
95M16—81no—
96C2×C8—21no—
97C2×C8—21no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4C22⋊C411yescharacteristic
113C4⋊C4C22⋊C411yescharacteristic
114C22⋊C4—21no—
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—41no—
118C42—21no—
119C42—21no—
120C42—21no—
121C42—41no—
122C42C22×C411yescharacteristic
123C42C2×D411yescharacteristic
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C23—21no—
127C23#12311yescharacteristic
128Q8—21no—
129Q8—21no—
130Q8—41no—
131Q8—41no—
132Q8—41no—
133Q8—41no—
134D4—21no—
135D4—21no—
136D4—42no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4C2×C22⋊C411yescharacteristic
154C2×C4C4×D411yescharacteristic
155C2×C4C4×D411yescharacteristic
156C2×C4#12011yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#12111yescharacteristic
159C8—42no—
160C8—42no—
161C8—42no—
162C8—41no—
163C8—41no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22#38611yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—41no—
176C4—41no—
177C4—41no—
178C4—81no—
179C4—81no—
180C4#42111yescharacteristic
181C4#42111yescharacteristic
182C4#44311yescharacteristic
183C4#44411yescharacteristic
184C4#47111yescharacteristic
185C4#47311yescharacteristic
186C2—41no—
187C2—81no—
188C2#187611yescharacteristic
189C2#189711yescharacteristic
190C2#190811yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radicalC2×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1526C2semidirect
4. #1493C2semidirect
6. #1299C2semidirect
7. #1278C2semidirect

Maximal subgroups

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

subgroupindex
#31072
#29222
#15262
#14932
#14892
#12992
#12782