Finite Group Databasebeta

← #10001

Subgroups — #10001

Order 256.

284 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 59 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#3101C211yesmaximal, characteristic
2#3078C211yesmaximal, characteristic, complemented
3#2536C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic
6#1479C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C22⋊Q8—21no—
10C2×C4×Q8C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#483—21no—
13#479—22no—
14#477—21no—
15#476—21no—
16#475—21no—
17#461—21no—
18#450—21no—
19C4×SD16—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#428—21no—
23#423C2211yescharacteristic
24#419—21no—
25#398—21no—
26#394—21no—
27#388—22no—
28#382C2211yescharacteristic
29#380—21no—
30#328—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×SD16—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#125—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—42no—
44C22⋊Q8—42no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55#117—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—21no—
64C2×C42—21no—
65C2×C42C2311yescharacteristic
66#107—22no—
67#106—22no—
68#106—22no—
69C4⋊C8—21no—
70C4⋊C8—21no—
71C4⋊C8—21no—
72C4⋊C8D412yes—
73#103—22no—
74#103—22no—
75#103—22no—
76#102—22no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C4×C8—21no—
80C4×C8D412yes—
81#95—22no—
82#95—21no—
83#95D412yes—
84C24—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8SD1612yescomplemented
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—42no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109SD16—44no—
110M16—41no—
111M16—41no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—41no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—44no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—41no—
135C42—41no—
136C42—41no—
137C42—41no—
138C42C2×D411yescharacteristic
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C42#4111yescharacteristic
142C23—22no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23#12011yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—42no—
153Q8—41no—
154Q8—41no—
155Q8—81no—
156D4—22no—
157D4—22no—
158D4—41no—
159D4—41no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4#12011yescharacteristic
180C2×C4#12011yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#12111yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4#12711yescharacteristic
185C2×C4C2×SD1611yescharacteristic
186C2×C4#13511yescharacteristic
187C2×C4AGL(1,ℤ/8)11yescharacteristic
188C2×C4#13711yescharacteristic
189C8—42no—
190C8—42no—
191C8—81no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22#39211yescharacteristic
202C22#45011yescharacteristic
203C22#45211yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—82no—
210C4—81no—
211C4—81no—
212C4—81no—
213C4—81no—
214C4#44811yescharacteristic
215C4#44911yescharacteristic
216C4#46511yescharacteristic
217C4#46811yescharacteristic
218C2—21no—
219C2—21no—
220C2—82no—
221C2#200011yescharacteristic
222C2#200911yescharacteristic
223C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3078C2semidirect
4. #1518C2semidirect
6. #1479C2semidirect
7. #1286C2semidirect
92. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#31012
#30782
#25362
#15182
#14862
#14792
#12862