Finite Group Databasebeta

← #7586

Subgroups — #7586

Order 256.

331 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 91 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×#121C211yesmaximal, characteristic
2#1797C211yesmaximal, characteristic, complemented
3C2×#336C211yesmaximal, characteristic, complemented
4#1271C212yesmaximal, complemented
5#1270C212yesmaximal, complemented
6C23×C8—21no—
7C2×#121C411yescharacteristic
8C2×C4×D4—21no—
9C2×C4×D4C411yescharacteristic
10C22×C42C2211yescharacteristic
11C2×C4⋊C8—21no—
12C2×C22⋊C8—21no—
13#390C2211yescharacteristic
14#389C2211yescharacteristic
15#388—21no—
16#386—21no—
17#380C411yescharacteristic
18C4×C4⋊C4C411yescharacteristic
19C4×C22⋊C4—21no—
20#336—21no—
21#336C2212yes—
22#336C2212yes—
23C22×D4—21no—
24C22×D4C2×C411yescharacteristic
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C23×C4D411yescharacteristic
29C22×C8—24no—
30C22×C8—22no—
31C22×C8—22no—
32C22×C8—42no—
33C22×C8—42no—
34#121—22no—
35#121C814yescomplemented
36C4×D4—42no—
37C4×D4—42no—
38C4×D4—42no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4C2×C411yescharacteristic
41C2×C4⋊C4C2×C411yescharacteristic
42C2×C4⋊C4C2×C411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C42—21no—
49C2×C42—42no—
50C2×C42C2×C411yescharacteristic
51C2×C42D411yescharacteristic
52C2×C42D411yescharacteristic
53C2×C42C2311yescharacteristic
54C4⋊C8—42no—
55C22⋊C8—42no—
56#95—21no—
57#95—21no—
58#95—21no—
59C24—21no—
60C24C22⋊C411yescharacteristic
61C2×D4—24no—
62C2×D4—21no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4C2×C812yescomplemented
68C22×C4—22no—
69C22×C4—22no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—21no—
74C22×C4—21no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—42no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4C22⋊C411yescharacteristic
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C2×C812yes—
86C22×C4M1612yes—
87C22×C4D812yes—
88C22×C4SD1612yes—
89C22×C4C22×C411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92C2×C8—24no—
93C2×C8—48no—
94C2×C8—42no—
95C2×C8—42no—
96C2×C8—42no—
97C2×C8—42no—
98C2×C8—42no—
99C4⋊C4—21no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4C2×C812yescomplemented
105C22⋊C4—24no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C42—22no—
113C42—22no—
114C42—41no—
115C42—41no—
116C42—42no—
117C23—21no—
118C23—21no—
119C23—21no—
120C23—21no—
121C23—22no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23—41no—
128C23C22⋊C812yes—
129C23#10212yes—
130C23#10412yes—
131C23C2×C22⋊C411yescharacteristic
132D4—42no—
133D4—42no—
134D4—48no—
135C2×C4—24no—
136C2×C4—24no—
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—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—84no—
163C2×C4C22⋊C812yes—
164C2×C4#9912yes—
165C2×C4#10212yes—
166C2×C4C22×C811yescharacteristic
167C2×C4C2×M1611yescharacteristic
168C2×C4C2×D811yescharacteristic
169C2×C4C2×SD1611yescharacteristic
170C2×C4AGL(1,ℤ/8)11yescharacteristic
171C2×C4#13711yescharacteristic
172C8—44no—
173C8—82no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—22no—
179C22—22no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22D4⋊C814yes—
188C22C2×C22⋊C811yescharacteristic
189C22C2×#9911yescharacteristic
190C22#41311yescharacteristic
191C22C2×#10211yescharacteristic
192C22#41711yescharacteristic
193C22C2×#10411yescharacteristic
194C22#42111yescharacteristic
195C4—22no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—22no—
200C4—21no—
201C4—21no—
202C4—22no—
203C4—21no—
204C4—42no—
205C4—81no—
206C4—81no—
207C2—24no—
208C2—82no—
209C2#145911yescharacteristic
210C2C2×D4⋊C811yescharacteristic
211C2#147711yescharacteristic
212C2#150411yescharacteristic
213C2#151911yescharacteristic
214C2#153211yescharacteristic
215C2#154711yescharacteristic
subgroupG/H
center, socleC23C2×C22⋊C4
commutatorC2×C4C22×C8
Frattini, cosocleC2×C42C23
prime radicalC2×#121C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1797C2semidirect
3. C2×#336C2semidirect
4. #1271C2semidirect
5. #1270C2semidirect
35. #121C8semidirect
67. C2×D4C2×C8semidirect
104. C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C4×#1212
#17972
C2×#3362
#12712
#12702