Finite Group Databasebeta

← #10376

Subgroups — #10376

Order 256.

232 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 53 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#2569C211yesmaximal, characteristic, complemented
2#1964C211yesmaximal, characteristic
3#1519C211yesmaximal, characteristic
4C2×#327C211yesmaximal, characteristic
5#1294C211yesmaximal, characteristic, complemented
6#1289C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#99—21no—
10C2×#99—21no—
11C2×C22⋊C8C2211yescharacteristic
12C2×C22⋊C8C2211yescharacteristic
13#400—21no—
14#398—21no—
15#387—21no—
16#387—21no—
17#385—21no—
18#380—21no—
19#380C2211yescharacteristic
20#380C2211yescharacteristic
21C4×C22⋊C4C2211yescharacteristic
22C2×#95C2211yescharacteristic
23#327—21no—
24#327—21no—
25#327—21no—
26#327—21no—
27C22×D4—21no—
28C22×D4Q811yescharacteristic
29C23×C4—21no—
30C23×C4C2311yescharacteristic
31C22×C8—21no—
32C22×C8—21no—
33#121—21no—
34#121—21no—
35#121—21no—
36#121—21no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4Q811yescharacteristic
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4D411yescharacteristic
48C2×C22⋊C4D411yescharacteristic
49C2×C42—21no—
50#99—42no—
51#99—42no—
52C22⋊C8—21no—
53C22⋊C8—21no—
54C22⋊C8—21no—
55C22⋊C8—21no—
56#95—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—21no—
62#95—21no—
63#95D411yescharacteristic
64#95D411yescharacteristic
65#95D411yescharacteristic
66#95D411yescharacteristic
67C24—21no—
68C24C2×D411yescharacteristic
69C2×D4—21no—
70C2×D4—21no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4—41no—
76C2×D4—41no—
77C2×D4—81no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—42no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—81no—
90C22×C4—81no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×Q811yescharacteristic
94C22×C4#4111yescharacteristic
95C22×C4#4111yescharacteristic
96C22×C4#4111yescharacteristic
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—81no—
100C2×C8—81no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—42no—
104C4⋊C4—42no—
105C4⋊C4—81no—
106C4⋊C4—81no—
107C4⋊C4—81no—
108C4⋊C4—81no—
109C22⋊C4—21no—
110C22⋊C4—21no—
111C22⋊C4—21no—
112C22⋊C4—21no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C42—81no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—81no—
126C23—81no—
127C23#12011yescharacteristic
128C23C22⋊Q811yescharacteristic
129C23#12411yescharacteristic
130C23#13511yescharacteristic
131C23#13511yescharacteristic
132C23AGL(1,ℤ/8)11yescharacteristic
133C23AGL(1,ℤ/8)11yescharacteristic
134D4—81no—
135D4—81no—
136D4—81no—
137D4—81no—
138D4—81no—
139D4—81no—
140C2×C4—22no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
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—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—82no—
168C2×C4—82no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4C22⋊Q811yescharacteristic
173C2×C4C22⋊Q811yescharacteristic
174C2×C4#12411yescharacteristic
175C2×C4#12411yescharacteristic
176C8—81no—
177C8—81no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—41no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—161no—
192C22—161no—
193C22#39311yescharacteristic
194C22#44911yescharacteristic
195C22#44911yescharacteristic
196C22#47811yescharacteristic
197C22#47811yescharacteristic
198C22#48811yescharacteristic
199C22#49011yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—42no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4—81no—
209C4—81no—
210C4—81no—
211C4—161no—
212C2—21no—
213C2—21no—
214C2—21no—
215C2—21no—
216C2—21no—
217C2—21no—
218C2—161no—
219C2#202511yescharacteristic
220C2#203111yescharacteristic
221C2#203211yescharacteristic
subgroupG/H
center, socleC22#393
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2569C2semidirect
5. #1294C2semidirect
6. #1289C2semidirect
7. #1282C2semidirect

Maximal subgroups

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

subgroupindex
#25692
#19642
#15192
C2×#3272
#12942
#12892
#12822