Finite Group Databasebeta

← #9165

Subgroups — #9165

Order 256.

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

normal subgroups only

This group has 243 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#3110C211yesmaximal, characteristic
2#2974C211yesmaximal, characteristic
3#1762C211yesmaximal, characteristic
4#1531C211yesmaximal, characteristic, complemented
5#1503C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10#484—21no—
11#474C411yescharacteristic, complemented
12#474C411yescharacteristic, complemented
13#461C411yescharacteristic, complemented
14#461C411yescharacteristic, complemented
15#450—21no—
16#440—21no—
17#440—21no—
18#436—21no—
19#435—22no—
20C8×D4—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#431C2211yescharacteristic
24C2×#102—21no—
25#407—21no—
26C2×C22⋊C8—21no—
27C2×#97C2211yescharacteristic
28C4×C4⋊C4C2211yescharacteristic
29#340—21no—
30#330—22no—
31#328—21no—
32#328—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35C2×SD16—21no—
36C2×SD16—21no—
37C2×M16—22no—
38C22×C8—21no—
39C22×C8—22no—
40C4⋊Q8C2×C411yescharacteristic
41C4⋊Q8C2×C411yescharacteristic
42#125—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4C2×C411yescharacteristic
50C4×D4C2×C411yescharacteristic
51C4×D4D411yescharacteristic
52C4×D4D411yescharacteristic
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4D411yescharacteristic
56C2×C22⋊C4—21no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59#107—21no—
60#107—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C411yescharacteristic
64C4⋊C8C2×C411yescharacteristic
65#103—21no—
66#103—21no—
67#102—24no—
68#102—21no—
69#102—21no—
70C22⋊C8—22no—
71C22⋊C8—22no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8D411yescharacteristic
75C22⋊C8D411yescharacteristic
76#97—21no—
77#97—21no—
78#97—21no—
79#97D412yes—
80C4×C8—21no—
81#95—21no—
82C24—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4C22⋊C411yescharacteristic
89C2×D4C22⋊C411yescharacteristic
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—82no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4#4111yescharacteristic
100SD16—42no—
101SD16—42no—
102M16—82no—
103C2×C8—22no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—22no—
109C2×C8—41no—
110C2×C8—41no—
111C2×C8—82no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4C22⋊C411yescharacteristic
122C4⋊C4C22⋊C411yescharacteristic
123C22⋊C4—21no—
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—41no—
128C42—41no—
129C42—81no—
130C42C22×C411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C42#4111yescharacteristic
134C23—21no—
135C23—21no—
136C23—42no—
137C23—41no—
138C23—41no—
139C23—81no—
140C23#12111yescharacteristic
141Q8—41no—
142Q8—41no—
143Q8—81no—
144D4—21no—
145D4—21no—
146D4—21no—
147D4—42no—
148D4#10412yes—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4C2×C22⋊C411yescharacteristic
167C2×C4C4×D411yescharacteristic
168C2×C4C4×D411yescharacteristic
169C2×C4#12011yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12311yescharacteristic
172C8—42no—
173C8—41no—
174C8—42no—
175C8—41no—
176C8—41no—
177C8—41no—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—82no—
187C22—82no—
188C22#38611yescharacteristic
189C22#47111yescharacteristic
190C22#47311yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4C2×#10411yescharacteristic
202C4#42111yescharacteristic
203C2—21no—
204C2—21no—
205C2—42no—
206C2—81no—
207C2#187111yescharacteristic
208C2#190411yescharacteristic
209C2#190811yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
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
4. #1531C2semidirect
5. #1503C2semidirect
6. #1302C2semidirect
7. #1285C2semidirect
11. #474C4semidirect
12. #474C4semidirect
13. #461C4semidirect
14. #461C4semidirect

Maximal subgroups

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

subgroupindex
#31102
#29742
#17622
#15312
#15032
#13022
#12852