Finite Group Databasebeta

← #10002

Subgroups — #10002

Order 256.

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

normal subgroups only

This group has 274 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#3104C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic
3#2580C211yesmaximal, characteristic
4#1518C211yesmaximal, characteristic, complemented
5#1487C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic
7#1286C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#484—21no—
12#478—22no—
13#477—21no—
14#476—21no—
15#474—21no—
16#462—21no—
17#451—21no—
18C4×Q16—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#428—21no—
22#423C2211yescharacteristic
23#418—21no—
24#399—21no—
25#393—21no—
26#390—21no—
27#389—21no—
28#385—22no—
29#382C2211yescharacteristic
30#328—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C22×Q8D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×Q16—22no—
37C2×M16—21no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40#125—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C4×Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×Q8—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53#117—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C42—21no—
63C2×C42C2311yescharacteristic
64#107—22no—
65#107—22no—
66#106—22no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70C4⋊C8D412yes—
71#103—22no—
72#103—22no—
73#102—22no—
74#102—22no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C4×C8—21no—
78C4×C8D412yes—
79#95—22no—
80#95—21no—
81#95D412yes—
82C24—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—42no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4Q1612yescomplemented
94C22×C4—21no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—82no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104Q16—44no—
105M16—41no—
106M16—41no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—41no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C4⋊C4—82no—
125C22⋊C4—44no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—82no—
130C42—41no—
131C42—41no—
132C42—41no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42#4111yescharacteristic
137C23—22no—
138C23—42no—
139C23—42no—
140C23—41no—
141C23—41no—
142C23#12011yescharacteristic
143Q8—22no—
144Q8—22no—
145Q8—41no—
146Q8—41no—
147Q8—42no—
148Q8—81no—
149D4—22no—
150D4—22no—
151D4—41no—
152D4—41no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4—82no—
170C2×C4#12011yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12711yescharacteristic
176C2×C4C2×Q1611yescharacteristic
177C2×C4#13511yescharacteristic
178C2×C4#13711yescharacteristic
179C2×C4#13711yescharacteristic
180C8—42no—
181C8—42no—
182C8—81no—
183C22—21no—
184C22—21no—
185C22—44no—
186C22—42no—
187C22—82no—
188C22—81no—
189C22—81no—
190C22#39211yescharacteristic
191C22#45111yescharacteristic
192C22#45211yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4#45111yescharacteristic
204C4#45211yescharacteristic
205C4#46711yescharacteristic
206C4#47011yescharacteristic
207C2—21no—
208C2—21no—
209C2—82no—
210C2#200111yescharacteristic
211C2#200811yescharacteristic
212C2#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
1. #3104C2semidirect
4. #1518C2semidirect
5. #1487C2semidirect
7. #1286C2semidirect
93. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#31042
#30772
#25802
#15182
#14872
#14782
#12862