Finite Group Databasebeta

← #11875

Subgroups — #11875

Order 256.

268 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 57 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#2535C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1344C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#121—21no—
7C2×C4×D4C411yescharacteristic, complemented
8C2×C4×D4C2211yescharacteristic
9C2×#99—21no—
10C2×C22⋊C8—21no—
11#397C411yescharacteristic, complemented
12#397C2211yescharacteristic
13#392C411yescharacteristic, complemented
14#392C2211yescharacteristic
15#386—21no—
16#386—21no—
17#382C411yescharacteristic, complemented
18#380—21no—
19#380—21no—
20#342C2212yes—
21#324—21no—
22#323C2212yes—
23C22×D4—21no—
24C22×D4—21no—
25C22×D4C2×C411yescharacteristic
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C23×C4D411yescharacteristic
29C23×C4D411yescharacteristic
30C22×C8—41no—
31#121—42no—
32#121—42no—
33C4×D4—41no—
34C4×D4—41no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4C2×C411yescharacteristic
40C2×C4⋊C4D411yescharacteristic
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—21no—
44C2×C22⋊C4—22no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4C2311yescharacteristic
49C2×C42C2×C411yescharacteristic
50#99—42no—
51C22⋊C8—22no—
52C22⋊C8—41no—
53#95—21no—
54#95—21no—
55#95—21no—
56#95—41no—
57#95C2×C411yescharacteristic
58#95C2×C411yescharacteristic
59C24—21no—
60C24C22⋊C411yescharacteristic
61C24C2×D411yescharacteristic
62C2×D4—41no—
63C2×D4—41no—
64C2×D4—41no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4—42no—
68C2×D4—42no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4—81no—
72C2×D4—81no—
73C2×D4D812yes—
74C2×D4SD1612yes—
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—42no—
86C22×C4—81no—
87C22×C4—81no—
88C22×C4C22⋊C411yescharacteristic
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22×C411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C2×C8—42no—
94C2×C8—82no—
95C4⋊C4—41no—
96C4⋊C4—41no—
97C4⋊C4—42no—
98C4⋊C4—82no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—42no—
102C22⋊C4—41no—
103C22⋊C4—41no—
104C22⋊C4—41no—
105C22⋊C4—41no—
106C22⋊C4—42no—
107C22⋊C4—42no—
108C22⋊C4—81no—
109C22⋊C4—81no—
110C42—21no—
111C42—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—22no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23#10212yes—
128C23C2×C22⋊C411yescharacteristic
129C23C2×D811yescharacteristic
130C23C2×SD1611yescharacteristic
131D4—42no—
132D4—42no—
133D4—42no—
134D4—42no—
135D4—84no—
136D4—82no—
137D4—82no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—84no—
161C2×C4—82no—
162C2×C4—82no—
163C2×C4—82no—
164C2×C4#9911yescharacteristic
165C2×C4#9911yescharacteristic
166C2×C4#10212yes—
167C8—82no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—22no—
172C22—21no—
173C22—42no—
174C22—41no—
175C22—41no—
176C22—42no—
177C22—42no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—42no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22C2×#9911yescharacteristic
190C22C2×#10211yescharacteristic
191C22#42111yescharacteristic
192C4—21no—
193C4—21no—
194C4—22no—
195C4—41no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4—81no—
200C4—81no—
201C4—162no—
202C2—21no—
203C2—21no—
204C2—41no—
205C2—42no—
206C2—41no—
207C2—82no—
208C2#151911yescharacteristic
209C2#212011yescharacteristic
210C2#212711yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2535C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1799C2semidirect
3. #1459C2semidirect
4. #1344C2semidirect
5. #1343C2semidirect
7. C2×C4×D4C4semidirect
11. #397C4semidirect
13. #392C4semidirect
17. #382C4semidirect

Maximal subgroups

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

subgroupindex
#25352
#17992
#14592
#13442
#13432