Finite Group Databasebeta

← #9984

Subgroups — #9984

Order 256.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#3095C211yesmaximal, characteristic, complemented
2#3094C211yesmaximal, characteristic
3#2816C211yesmaximal, characteristic
4#1820C211yesmaximal, characteristic, complemented
5#1511C211yesmaximal, characteristic, complemented
6#1510C211yesmaximal, characteristic
7#1306C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C2×#117C2211yescharacteristic
11#485—22no—
12#484—21no—
13#483—21no—
14#482—22no—
15#481—21no—
16#480—21no—
17#428—21no—
18#428—21no—
19#419—21no—
20#418—21no—
21#407C2211yescharacteristic
22#407C2211yescharacteristic
23#399—21no—
24#398—21no—
25#394—21no—
26#394—22no—
27#393—21no—
28#381C2211yescharacteristic
29#381C2211yescharacteristic
30#342—22no—
31#328—22no—
32#327—22no—
33C22×Q8D411yescharacteristic
34C22×D4—21no—
35C22×D4D411yescharacteristic
36C23×C4C2311yescharacteristic
37C2×M16—21no—
38C2×M16—21no—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#117—21no—
48#117—21no—
49#117—21no—
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4D412yes—
64C2×C42—21no—
65#107—22no—
66#107—22no—
67#106—22no—
68#106—22no—
69#103—22no—
70#103—22no—
71#102—22no—
72#102—22no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76C22⋊C8D412yes—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82C24—21no—
83C24C2×D411yescharacteristic
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—42no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—82no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—41no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4#4111yescharacteristic
107M16—42no—
108M16—42no—
109C2×C8—22no—
110C2×C8—22no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C4⋊C4—82no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C42—41no—
133C42—41no—
134C23—22no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23—41no—
140C23—42no—
141C23#12011yescharacteristic
142C23#12111yescharacteristic
143C23#12711yescharacteristic
144Q8—42no—
145Q8—81no—
146D4—42no—
147D4—81no—
148D4—82no—
149D4—82no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—81no—
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4#12111yescharacteristic
177C8—42no—
178C8—42no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—82no—
188C22—84no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22#39211yescharacteristic
193C22#45812yes—
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4—81no—
202C4—81no—
203C4—81no—
204C4—161no—
205C2—22no—
206C2—41no—
207C2—41no—
208C2—161no—
209C2#200811yescharacteristic
210C2#200911yescharacteristic
211C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
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. #3095C2semidirect
4. #1820C2semidirect
5. #1511C2semidirect
7. #1306C2semidirect

Maximal subgroups

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

subgroupindex
#30952
#30942
#28162
#18202
#15112
#15102
#13062