Finite Group Databasebeta

← #9261

Subgroups — #9261

Order 256.

283 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#481C211yesmaximal, characteristic
2#2345C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1824C211yesmaximal, characteristic, complemented
5#1519C211yesmaximal, characteristic
6#1289C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10#481—22no—
11#481—21no—
12#481—21no—
13C2×#106C412yescomplemented
14C2×#102C412yescomplemented
15C2×#99—21no—
16C2×#99—21no—
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#389—21no—
20#387—22no—
21#380—21no—
22#380C2211yescharacteristic
23#380C2211yescharacteristic
24C4×C4⋊C4—21no—
25C4×C22⋊C4—21no—
26C4×C22⋊C4C2211yescharacteristic
27#340—22no—
28#327—22no—
29C22×D4—21no—
30C22×D4C2×C411yescharacteristic
31C23×C4—21no—
32C23×C4C2311yescharacteristic
33C22×C8—21no—
34C22×C8C2×C412yes—
35#121—21no—
36#121—21no—
37#121—21no—
38#121—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—22no—
45C2×C4⋊C4C2×C411yescharacteristic
46C2×C4⋊C4C2×C412yes—
47C2×C4⋊C4D412yes—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4D411yescharacteristic
54C2×C22⋊C4D411yescharacteristic
55C2×C42—21no—
56C2×C42—22no—
57#106—22no—
58#106—22no—
59#102—22no—
60#102—22no—
61#99—42no—
62#99—42no—
63C22⋊C8—21no—
64C22⋊C8—21no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67#95—22no—
68#95—22no—
69#95—21no—
70#95—21no—
71#95D412yes—
72C24—21no—
73C24C2×D411yescharacteristic
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—42no—
77C2×D4—41no—
78C2×D4—41no—
79C2×D4—41no—
80C2×D4—41no—
81C2×D4—41no—
82C2×D4—41no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—42no—
92C22×C4—41no—
93C22×C4—82no—
94C22×C4C22×C411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4#4111yescharacteristic
99C22×C4#4111yescharacteristic
100C2×C8—22no—
101C2×C8—42no—
102C2×C8—42no—
103C2×C8—41no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—42no—
111C4⋊C4—42no—
112C4⋊C4—82no—
113C4⋊C4C22⋊C412yes—
114C4⋊C4C22⋊C412yes—
115C22⋊C4—21no—
116C22⋊C4—21no—
117C22⋊C4—21no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4SD1612yes—
125C42—42no—
126C42—82no—
127C23—41no—
128C23—41no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—81no—
133C23—81no—
134C23C4×D411yescharacteristic
135C23#12011yescharacteristic
136C23#12111yescharacteristic
137C23C2×SD1611yescharacteristic
138C23#13511yescharacteristic
139C23AGL(1,ℤ/8)11yescharacteristic
140C23#13711yescharacteristic
141D4—42no—
142D4—81no—
143D4—81no—
144D4—81no—
145D4—81no—
146D4—81no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—82no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4—81no—
174C2×C4C2×C22⋊C411yescharacteristic
175C2×C4C4×D411yescharacteristic
176C2×C4#12111yescharacteristic
177C2×C4#12311yescharacteristic
178C8—42no—
179C8—81no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—41no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22—161no—
194C22—161no—
195C22#38611yescharacteristic
196C22C4×SD1611yescharacteristic
197C22#44011yescharacteristic
198C22#44811yescharacteristic
199C22#44911yescharacteristic
200C22#46011yescharacteristic
201C22#46411yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—42no—
206C4—42no—
207C4—42no—
208C4—81no—
209C4—82no—
210C4—81no—
211C4—161no—
212C4#41012yes—
213C2—21no—
214C2—21no—
215C2—21no—
216C2—21no—
217C2—21no—
218C2—21no—
219C2—161no—
220C2#188011yescharacteristic
221C2#189911yescharacteristic
222C2#190411yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×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
2. #2345C2semidirect
4. #1824C2semidirect
6. #1289C2semidirect
7. #1282C2semidirect
13. C2×#106C4semidirect
14. C2×#102C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
#23452
#19642
#18242
#15192
#12892
#12822