Finite Group Databasebeta

← #10027

Subgroups — #10027

Order 256.

271 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 51 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#3246C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic, complemented
3#3109C211yesmaximal, characteristic
4#1955C211yesmaximal, characteristic, complemented
5#1531C211yesmaximal, characteristic, complemented
6#1530C211yesmaximal, characteristic, complemented
7#1290C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#484—21no—
12#481—21no—
13#476—21no—
14#476—21no—
15#475—21no—
16#474—21no—
17#465—22no—
18#461—21no—
19#461—21no—
20#460—21no—
21#459—21no—
22#450—21no—
23#447—21no—
24#442—22no—
25#440—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C22⋊C8—22no—
30C2×#97C2211yescharacteristic
31#330—22no—
32#329—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×SD16—22no—
37C2×SD16—22no—
38C2×D8—22no—
39C22×C8—21no—
40C22×C8D412yes—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#125—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49#121—21no—
50#121—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59#117—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—21no—
62C2×C42C2311yescharacteristic
63#107—22no—
64#106—22no—
65#106—22no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#103—22no—
71#103—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75#102—22no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—44no—
79#97—21no—
80#97—21no—
81#97D412yes—
82#97D412yes—
83C24—21no—
84#41—82no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—42no—
94C2×D4—41no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4#4111yescharacteristic
104SD16—44no—
105SD16—82no—
106D8—44no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—41no—
112C2×C8—41no—
113C2×C8—42no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C42—41no—
130C42C2×D411yescharacteristic
131C42C2×D411yescharacteristic
132C42C2×D411yescharacteristic
133C42C2×D411yescharacteristic
134C23—22no—
135C23—42no—
136C23—41no—
137C23—41no—
138C23—42no—
139C23—41no—
140C23#12111yescharacteristic
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—81no—
145D4—22no—
146D4—22no—
147D4—41no—
148D4—41no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—81no—
153D4—82no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—82no—
164C2×C4—82no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4#12011yescharacteristic
168C2×C4#12011yescharacteristic
169C2×C4#12011yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12711yescharacteristic
173C8—42no—
174C8—42no—
175C8—42no—
176C8—42no—
177C22—21no—
178C22—21no—
179C22—44no—
180C22—42no—
181C22—82no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22#39211yescharacteristic
188C22#47112yes—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—81no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4#45311yescharacteristic
199C4#45411yescharacteristic
200C2—22no—
201C2—82no—
202C2—161no—
203C2#200611yescharacteristic
204C2#201011yescharacteristic
205C2#201911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3109C2
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. #3246C2semidirect
2. #3110C2semidirect
4. #1955C2semidirect
5. #1531C2semidirect
6. #1530C2semidirect
7. #1290C2semidirect

Maximal subgroups

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

subgroupindex
#32462
#31102
#31092
#19552
#15312
#15302
#12902