Finite Group Databasebeta

← #11799

Subgroups — #11799

Order 256.

287 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 49 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#2617C211yesmaximal, characteristic
2#1819C211yesmaximal, characteristic, complemented
3#1462C211yesmaximal, characteristic, complemented
4#1344C212yesmaximal, complemented
5#1343C212yesmaximal, complemented
6C2×#121C411yescharacteristic, complemented
7C2×#120—21no—
8C22×C22⋊C4C2211yescharacteristic
9#407—21no—
10#397C411yescharacteristic, complemented
11#397C2211yescharacteristic
12#394—21no—
13#394—21no—
14#392C411yescharacteristic, complemented
15#392C2211yescharacteristic
16#386—21no—
17#380C411yescharacteristic, complemented
18#379—21no—
19#342—21no—
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C25D411yescharacteristic
24C22×D4—21no—
25C22×D4—21no—
26C22×D4—21no—
27C22×D4C2×C411yescharacteristic
28C23×C4C2×C411yescharacteristic
29C23×C4D411yescharacteristic
30C2×M16—41no—
31#121—41no—
32#121—41no—
33#120—44no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4C2×C411yescharacteristic
36C2×C22⋊C4—21no—
37C2×C22⋊C4—21no—
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4—21no—
43C2×C22⋊C4—41no—
44C2×C22⋊C4—41no—
45C2×C22⋊C4—41no—
46C2×C22⋊C4—41no—
47C2×C22⋊C4C2×C411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C22⋊C4D411yescharacteristic
50C2×C22⋊C4C2311yescharacteristic
51C22⋊C8—22no—
52C22⋊C8—41no—
53#95—21no—
54#95—21no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24—21no—
58C24—21no—
59C24—22no—
60C24—21no—
61C24—41no—
62C24—81no—
63C24—81no—
64C24C22⋊C411yescharacteristic
65C24C22⋊C411yescharacteristic
66C24C2×D411yescharacteristic
67C2×D4—41no—
68C2×D4—42no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—42no—
74C2×D4—81no—
75C2×D4—81no—
76C2×D4—81no—
77C2×D4—81no—
78C2×D4—81no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—42no—
87C22×C4—81no—
88C22×C4—81no—
89C22×C4C22⋊C411yescharacteristic
90C22×C4C22⋊C411yescharacteristic
91C22×C4C22×C411yescharacteristic
92C22×C4C2×D411yescharacteristic
93M16—82no—
94C2×C8—42no—
95C4⋊C4—41no—
96C4⋊C4—81no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C22⋊C4—41no—
100C22⋊C4—41no—
101C22⋊C4—42no—
102C22⋊C4—42no—
103C22⋊C4—42no—
104C22⋊C4—42no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C22⋊C4—81no—
108C22⋊C4—81no—
109C22⋊C4—82no—
110C22⋊C4—82no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23—21no—
116C23—21no—
117C23—21no—
118C23—41no—
119C23—41no—
120C23—42no—
121C23—42no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—81no—
136C23—81no—
137C23—84no—
138C23—84no—
139C23#9911yescharacteristic
140C23#9911yescharacteristic
141C23C2×C22⋊C411yescharacteristic
142C23AGL(1,ℤ/8)12yes—
143D4—82no—
144D4—82no—
145D4—82no—
146D4—84no—
147D4—84no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—42no—
153C2×C4—41no—
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—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—82no—
172C8—82no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—22no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—84no—
198C22—84no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22C2×#9911yescharacteristic
203C22#41811yescharacteristic
204C22#42111yescharacteristic
205C4—41no—
206C4—41no—
207C4—41no—
208C4—41no—
209C4—81no—
210C4—81no—
211C4—81no—
212C4—162no—
213C2—21no—
214C2—21no—
215C2—42no—
216C2—41no—
217C2—41no—
218C2—81no—
219C2—161no—
220C2#151011yescharacteristic
221C2#212111yescharacteristic
222C2#212711yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2617C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1819C2semidirect
3. #1462C2semidirect
4. #1344C2semidirect
5. #1343C2semidirect
6. C2×#121C4semidirect
10. #397C4semidirect
14. #392C4semidirect
17. #380C4semidirect

Maximal subgroups

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

subgroupindex
#26172
#18192
#14622
#13442
#13432