Finite Group Databasebeta

← #29339

Subgroups — #29339

Order 256.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#2850C211yesmaximal, characteristic
2#2296C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2042C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1935C212yesmaximal, complemented
7#1875C211yesmaximal, characteristic
8#1875C211yesmaximal, characteristic
9#1873C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24C2×#97C2212yes—
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#389C2212yes—
28#389C2212yes—
29#388—22no—
30#386—22no—
31#385—22no—
32#382C2211yescharacteristic
33C4×C4⋊C4—22no—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4C2311yescharacteristic
42C23×C4D412yes—
43C23×C4C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#124—42no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49#121—22no—
50#121—22no—
51#117—24no—
52#117—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42—42no—
69C2×C42D412yes—
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#103—24no—
73#103—42no—
74#102—24no—
75#102—42no—
76C22⋊C8—22no—
77C22⋊C8—22no—
78#97—42no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83C24—21no—
84C24C2×D411yescharacteristic
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×D4—22no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—22no—
93C2×D4—44no—
94C2×D4—42no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4#4112yes—
109C22×C4#4112yes—
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—44no—
125C4⋊C4—44no—
126C4⋊C4—42no—
127C4⋊C4#4114yes—
128C22⋊C4—24no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C22⋊C4—42no—
132C22⋊C4—42no—
133C22⋊C4—42no—
134C42—24no—
135C42—22no—
136C42—22no—
137C42—42no—
138C42—42no—
139C23—21no—
140C23—21no—
141C23—22no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—42no—
148C23#12412yes—
149C23C22×D411yescharacteristic
150Q8—42no—
151Q8—42no—
152D4—42no—
153D4—42no—
154D4—84no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—48no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4#12412yes—
180C2×C4C2×#4112yes—
181C2×C4C2×#4111yescharacteristic
182C2×C4C2×#4111yescharacteristic
183C2×C4H(5,𝔽2)12yes—
184C8—42no—
185C8—42no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—42no—
190C22—42no—
191C22—81no—
192C22—82no—
193C22—81no—
194C22—81no—
195C22—82no—
196C22#52511yescharacteristic
197C22C2×#12411yescharacteristic
198C22#53411yescharacteristic
199C22#57512yes—
200C22#57512yes—
201C4—22no—
202C4—44no—
203C4—42no—
204C4—42no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4—82no—
209C4#55312yes—
210C4#55312yes—
211C2—42no—
212C2—161no—
213C2#259111yescharacteristic
214C2#311112yes—
215C2#313511yescharacteristic
216C2#314011yescharacteristic
217C2#319712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2296C2semidirect
3. #2101C2semidirect
4. #2042C2semidirect
5. #1936C2semidirect
6. #1935C2semidirect
9. #1873C2semidirect
10. #1873C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
#28502
#22962
#21012
#20422
#19362
#19352
#18752
#18732
#17492