Finite Group Databasebeta

← #8970

Subgroups — #8970

Order 256.

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

normal subgroups only

This group has 257 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#3106C211yesmaximal, characteristic
2#2991C211yesmaximal, characteristic, complemented
3#1962C211yesmaximal, characteristic, complemented
4C4×#104C211yesmaximal, characteristic, complemented
5#1484C211yesmaximal, characteristic, complemented
6#1299C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C4×#41C2211yescharacteristic
10#482—21no—
11#477C411yescharacteristic, complemented
12#476C411yescharacteristic, complemented
13#460C411yescharacteristic, complemented
14#459C411yescharacteristic, complemented
15#449—21no—
16C8×Q8—21no—
17C4×SD16—21no—
18C4×D8—21no—
19#436—22no—
20#436—21no—
21#431—21no—
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#423—21no—
25#423C2211yescharacteristic
26C2×#104—21no—
27C2×#104—22no—
28C4×M16C2211yescharacteristic
29C43C2211yescharacteristic
30#339—21no—
31D4⋊C8—21no—
32D4⋊C8—21no—
33C2×#41—21no—
34C2×#41D411yescharacteristic
35C2×SD16—21no—
36C2×D8—21no—
37C2×M16—22no—
38C2×M16—21no—
39C2×M16D412yes—
40C4⋊Q8C2×C411yescharacteristic
41#127C2×C411yescharacteristic
42#124—21no—
43#121—21no—
44#121—21no—
45C4×Q8C2×C411yescharacteristic
46C4×Q8D411yescharacteristic
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4C2×C411yescharacteristic
52C4×D4D411yescharacteristic
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#104—44no—
71#103—21no—
72#102—21no—
73#102—21no—
74#102—21no—
75C22⋊C8—22no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78#97—21no—
79C4×C8—22no—
80C4×C8—21no—
81C4×C8—21no—
82C4×C8—21no—
83#41—22no—
84#41—41no—
85#41—42no—
86#41—42no—
87C2×Q8—21no—
88C2×Q8C22⋊C411yescharacteristic
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—41no—
94C2×D4C22⋊C411yescharacteristic
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103SD16—42no—
104D8—42no—
105M16—22no—
106M16—41no—
107M16—42no—
108M16—82no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—22no—
113C2×C8—21no—
114C2×C8—41no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4C22⋊C411yescharacteristic
120C4⋊C4C22⋊C411yescharacteristic
121C22⋊C4—21no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C42—21no—
127C42—21no—
128C42—21no—
129C42—22no—
130C42—22no—
131C42—22no—
132C42—41no—
133C42—41no—
134C42—41no—
135C42C22×C411yescharacteristic
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C42#4111yescharacteristic
139C23—21no—
140C23—41no—
141C23#12111yescharacteristic
142Q8—21no—
143Q8—42no—
144Q8#10412yescomplemented
145D4—21no—
146D4—21no—
147D4—42no—
148D4—42no—
149D4—41no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—82no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4C4×D411yescharacteristic
176C2×C4C4×D411yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12311yescharacteristic
180C8—42no—
181C8—41no—
182C8—41no—
183C8—42no—
184C8—41no—
185C8—41no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22#38611yescharacteristic
196C4—21no—
197C4—22no—
198C4—21no—
199C4—21no—
200C4—44no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4C2×#10411yescharacteristic
205C4#42111yescharacteristic
206C4#44311yescharacteristic
207C4#44411yescharacteristic
208C4#45311yescharacteristic
209C4#45411yescharacteristic
210C2—41no—
211C2—81no—
212C2—161no—
213C2#187111yescharacteristic
214C2#187611yescharacteristic
215C2#188311yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3106C2
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
2. #2991C2semidirect
3. #1962C2semidirect
4. C4×#104C2semidirect
5. #1484C2semidirect
6. #1299C2semidirect
7. #1276C2semidirect
11. #477C4semidirect
12. #476C4semidirect
13. #460C4semidirect
14. #459C4semidirect
144. Q8#104semidirect

Maximal subgroups

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

subgroupindex
#31062
#29912
#19622
C4×#1042
#14842
#12992
#12762