Finite Group Databasebeta

← #30186

Subgroups — #30186

Order 256.

398 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 119 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#2851C211yesmaximal, characteristic, complemented
2#2850C211yesmaximal, characteristic
3#2101C212yesmaximal, complemented
4#2074C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic, complemented
6#2042C212yesmaximal, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1893C212yesmaximal
10#1892C212yesmaximal, complemented
11C2×#336C211yesmaximal, characteristic, complemented
12C23×C8C2212yes—
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C22⋊Q8C2211yescharacteristic
16C2×#121C2211yescharacteristic
17C22×C42C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#102C2212yes—
21C2×#102C2212yes—
22#391C2211yescharacteristic
23#390C2211yescharacteristic
24#389C2212yes—
25#388—22no—
26#387—22no—
27#386—22no—
28#385—22no—
29#382C2211yescharacteristic
30#382C2211yescharacteristic
31#380C2212yes—
32#340C2212yes—
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2211yescharacteristic
37#336C2211yescharacteristic
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4—21no—
41C22×D4C2311yescharacteristic
42C23×C4D412yes—
43C23×C4C2311yescharacteristic
44C22×C8—24no—
45C22×C8—42no—
46C22×C8—42no—
47C22×C8C2312yes—
48C22×C8C2312yes—
49#125—42no—
50#124—42no—
51C22⋊Q8—24no—
52#121—24no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C42—42no—
66C2×C42D412yes—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#103—42no—
70#103—42no—
71#102—42no—
72#102—42no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78C24—21no—
79C24C2×D411yescharacteristic
80C2×Q8—22no—
81C2×Q8—21no—
82C2×Q8—22no—
83C2×Q8—21no—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—22no—
87C2×D4—21no—
88C2×D4—42no—
89C2×D4—44no—
90C22×C4—22no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—42no—
100C22×C4—42no—
101C22×C4C2×D412yes—
102C22×C4C2×D412yes—
103C22×C4C2×D411yescharacteristic
104C22×C4#4114yes—
105C22×C4C2411yescharacteristic
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—48no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C22⋊C4—44no—
123C22⋊C4—44no—
124C22⋊C4—42no—
125C22⋊C4—44no—
126C22⋊C4—42no—
127C42—22no—
128C42—22no—
129C42—42no—
130C23—21no—
131C23—21no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—42no—
137C23—42no—
138C23#12412yes—
139C23#13514yes—
140C23C22×D411yescharacteristic
141Q8—42no—
142Q8—42no—
143D4—42no—
144D4—42no—
145D4—84no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—48no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4#12412yes—
171C2×C4C2×#4112yes—
172C2×C4H(5,𝔽2)11yescharacteristic
173C2×C4H(5,𝔽2)12yes—
174C2×C42−1+411yescharacteristic
175C8—42no—
176C8—42no—
177C22—22no—
178C22—22no—
179C22—24no—
180C22—24no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—82no—
185C22—82no—
186C22#48814yes—
187C22C2×#12411yescharacteristic
188C22#53411yescharacteristic
189C22#53711yescharacteristic
190C22C2×#13512yes—
191C22#57512yes—
192C4—22no—
193C4—42no—
194C4—42no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—82no—
199C4—82no—
200C4#53812yes—
201C4#53912yes—
202C2—24no—
203C2—161no—
204C2#263911yescharacteristic
205C2C2×#48811yescharacteristic
206C2#313511yescharacteristic
207C2#319112yes—
208C2#325012yes—
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
1. #2851C2semidirect
3. #2101C2semidirect
4. #2074C2semidirect
5. #2073C2semidirect
6. #2042C2semidirect
7. #2026C2semidirect
8. #2025C2semidirect
10. #1892C2semidirect
11. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#28502
#21012
#20742
#20732
#20422
#20262
#20252
#18932
#18922
C2×#3362