Finite Group Databasebeta

← #12794

Subgroups — #12794

Order 256.

270 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C4×#123C211yesmaximal, characteristic
2C2×#352C212yesmaximal
3#1800C211yesmaximal, characteristic
4#1799C211yesmaximal, characteristic
5#1390C212yesmaximal
6C2×#123C2211yescharacteristic
7C2×C4×D4C2211yescharacteristic
8C22×C42C411yescharacteristic, complemented
9#423—21no—
10C2×#100C2212yes—
11C2×#99C2212yes—
12#388—21no—
13#387—21no—
14#386C2211yescharacteristic
15#385C411yescharacteristic, complemented
16#381C411yescharacteristic, complemented
17#380—21no—
18C4×C4⋊C4—21no—
19C4×C22⋊C4—21no—
20C4×C22⋊C4C411yescharacteristic, complemented
21#352—24no—
22#343—21no—
23#342—21no—
24C22×D4C2311yescharacteristic
25C23×C4C2×C411yescharacteristic
26C23×C4C2×C411yescharacteristic
27C23×C4C2×C411yescharacteristic
28C23×C4D411yescharacteristic
29C2×M16—22no—
30#123—21no—
31#123—21no—
32#123D412yes—
33#123Q812yes—
34C4×D4—41no—
35C4×D4—41no—
36C2×C4⋊C4—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—21no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4C2×C411yescharacteristic
45C2×C22⋊C4C2×C411yescharacteristic
46C2×C22⋊C4D411yescharacteristic
47C2×C42—21no—
48C2×C42—21no—
49C2×C42—21no—
50C2×C42—41no—
51C2×C42—41no—
52C2×C42D411yescharacteristic
53C4⋊C8—41no—
54C4⋊C8—41no—
55#100—22no—
56#100—22no—
57#99—22no—
58#99—22no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95—21no—
64#95—41no—
65C24C22×C411yescharacteristic
66C24C2×D411yescharacteristic
67C2×D4—41no—
68C2×D4—41no—
69C2×D4C2×D411yescharacteristic
70C2×D4C2×Q811yescharacteristic
71C2×D4#4111yescharacteristic
72C2×D4#4111yescharacteristic
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—42no—
83C22×C4—42no—
84C22×C4—42no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—42no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22⋊C411yescharacteristic
92C22×C4C22⋊C411yescharacteristic
93C22×C4C4⋊C412yescomplemented
94C22×C4C2×D411yescharacteristic
95M16—42no—
96M16—42no—
97C2×C8—42no—
98C4⋊C4—22no—
99C4⋊C4—22no—
100C4⋊C4—41no—
101C4⋊C4—42no—
102C4⋊C4—82no—
103C22⋊C4—21no—
104C22⋊C4—22no—
105C22⋊C4—22no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—42no—
111C22⋊C4—42no—
112C22⋊C4C4⋊C412yescomplemented
113C42—21no—
114C42—21no—
115C42—42no—
116C42—42no—
117C42—41no—
118C42—41no—
119C23—21no—
120C23—21no—
121C23—21no—
122C23—21no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23C2×C22⋊C411yescharacteristic
128C23C2×C4⋊C411yescharacteristic
129C23#11711yescharacteristic
130C23#12111yescharacteristic
131C23C22⋊Q811yescharacteristic
132D4—41no—
133D4—41no—
134D4—41no—
135D4—41no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4—82no—
167C2×C4—82no—
168C2×C4#9911yescharacteristic
169C2×C4#9911yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4C22⋊Q811yescharacteristic
172C8—82no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—42no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22#38011yescharacteristic
190C22C2×#9911yescharacteristic
191C22#41011yescharacteristic
192C4—21no—
193C4—21no—
194C4—21no—
195C4—42no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—42no—
201C4—81no—
202C4—162no—
203C4#35212yescomplemented
204C2—21no—
205C2—21no—
206C2—41no—
207C2—41no—
208C2—41no—
209C2—41no—
210C2#179911yescharacteristic
211C2C2×#35211yescharacteristic
212C2#212011yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC24C22×C4
Frattini, prime radical, cosocleC22×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C22×C42C4semidirect
15. #385C4semidirect
16. #381C4semidirect
20. C4×C22⋊C4C4semidirect
93. C22×C4C4⋊C4semidirect
112. C22⋊C4C4⋊C4semidirect
203. C4#352semidirect

Maximal subgroups

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

subgroupindex
C4×#1232
C2×#3522
#18002
#17992
#13902