Finite Group Databasebeta

← #8971

Subgroups — #8971

Order 256.

239 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 65 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#2990C211yesmaximal, characteristic
3#1994C211yesmaximal, characteristic, complemented
4C4×#104C211yesmaximal, characteristic
5#1485C211yesmaximal, characteristic, complemented
6#1299C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#485—21no—
11#475C411yescharacteristic, complemented
12#474C411yescharacteristic, complemented
13#462C411yescharacteristic, complemented
14#461C411yescharacteristic, complemented
15#452—21no—
16#446—21no—
17C4×Q16—21no—
18C4×SD16—21no—
19#435—22no—
20C8×D4—21no—
21#432—21no—
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25C2×C4⋊C8—21no—
26C2×#104—21no—
27C4×M16C2211yescharacteristic
28C43C2211yescharacteristic
29#339—22no—
30#339—21no—
31Q8⋊C8—21no—
32Q8⋊C8—21no—
33C2×#41D411yescharacteristic
34C2×Q16—21no—
35C2×SD16—21no—
36C2×M16—21no—
37C2×M16D412yes—
38C22×C8—22no—
39C4⋊Q8C2×C411yescharacteristic
40C4⋊Q8C2×C411yescharacteristic
41#125—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8C2×C411yescharacteristic
47C4×Q8D411yescharacteristic
48C4×D4—21no—
49C4×D4—21no—
50C4×D4C2×C411yescharacteristic
51C4×D4D411yescharacteristic
52#117—21no—
53#117—21no—
54#117D411yescharacteristic
55C2×C42—21no—
56C2×C42—21no—
57C2×C42—21no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#107—21no—
61#106—21no—
62C4⋊C8—22no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8C2×C411yescharacteristic
66C4⋊C8C2×C411yescharacteristic
67C4⋊C8D411yescharacteristic
68C4⋊C8D411yescharacteristic
69#104—24no—
70#103—21no—
71#103—21no—
72#103—21no—
73#102—21no—
74C22⋊C8—22no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77#97—22no—
78#97—21no—
79C4×C8—21no—
80C4×C8—21no—
81C4×C8—21no—
82#41—22no—
83#41—41no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8C22⋊C411yescharacteristic
87C2×D4—21no—
88C2×D4C22⋊C411yescharacteristic
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4#4111yescharacteristic
96Q16—42no—
97SD16—42no—
98M16—22no—
99M16—41no—
100M16—42no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—21no—
105C2×C8—22no—
106C2×C8—41no—
107C2×C8—82no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4C22⋊C411yescharacteristic
117C4⋊C4C22⋊C411yescharacteristic
118C22⋊C4—21no—
119C22⋊C4—21no—
120C22⋊C4—21no—
121C22⋊C4—41no—
122C42—21no—
123C42—21no—
124C42—21no—
125C42—22no—
126C42—22no—
127C42—22no—
128C42—41no—
129C42—41no—
130C42—41no—
131C42—41no—
132C42C22×C411yescharacteristic
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42#4111yescharacteristic
136C23—21no—
137C23#12111yescharacteristic
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—41no—
143Q8—41no—
144D4—21no—
145D4—42no—
146D4#10412yes—
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—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4C2×C22⋊C411yescharacteristic
167C2×C4C4×D411yescharacteristic
168C2×C4C4×D411yescharacteristic
169C2×C4#12011yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12311yescharacteristic
172C8—42no—
173C8—41no—
174C8—41no—
175C8—41no—
176C8—42no—
177C8—41no—
178C22—21no—
179C22—22no—
180C22—21no—
181C22—21no—
182C22—41no—
183C22—41no—
184C22#38611yescharacteristic
185C4—21no—
186C4—21no—
187C4—21no—
188C4—44no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—81no—
193C4—81no—
194C4C2×#10411yescharacteristic
195C4#42111yescharacteristic
196C4#44311yescharacteristic
197C4#44411yescharacteristic
198C4#45511yescharacteristic
199C4#45611yescharacteristic
200C2—41no—
201C2—42no—
202C2#187111yescharacteristic
203C2#187611yescharacteristic
204C2#188211yescharacteristic
subgroupG/H
centerC2×C4#120
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. #1994C2semidirect
5. #1485C2semidirect
6. #1299C2semidirect
7. #1276C2semidirect
11. #475C4semidirect
12. #474C4semidirect
13. #462C4semidirect
14. #461C4semidirect

Maximal subgroups

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

subgroupindex
#31072
#29902
#19942
C4×#1042
#14852
#12992
#12762