Finite Group Databasebeta

← #29108

Subgroups — #29108

Order 256.

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

normal subgroups only

This group has 439 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#2297C211yesmaximal, characteristic
3#2074C211yesmaximal, characteristic, complemented
4#2073C211yesmaximal, characteristic, complemented
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#1934C212yesmaximal, complemented
8#1933C212yesmaximal, complemented
9#1825C212yesmaximal, complemented
10#1788C212yesmaximal
11C4×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C42C2211yescharacteristic
16C2×#103C2212yes—
17C2×#103C2212yes—
18C2×#102C2212yes—
19C2×#102C2212yes—
20C2×C22⋊C8C2212yes—
21C2×C4×C8C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#389C2212yes—
27#387—22no—
28#386—22no—
29#385—22no—
30#382C2211yescharacteristic
31#380C2212yes—
32C4×C22⋊C4—22no—
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—24no—
42C2×#41—42no—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4D412yes—
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51#125—42no—
52#117—24no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C42—21no—
64C2×C42—42no—
65C2×C42D412yes—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#103—24no—
69#103—42no—
70#102—24no—
71#102—42no—
72C22⋊C8—24no—
73C4×C8—42no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79C24—21no—
80C24C2×D411yescharacteristic
81#41—48no—
82#41—48no—
83C2×Q8—21no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8#4112yes—
87C2×D4—24no—
88C2×D4—22no—
89C2×D4—21no—
90C2×D4—22no—
91C2×D4—42no—
92C2×D4—42no—
93C2×D4#4112yes—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—24no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—42no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D412yes—
111C22×C4#4114yes—
112C22×C4C2411yescharacteristic
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—44no—
126C4⋊C4—44no—
127C22⋊C4—44no—
128C22⋊C4—44no—
129C22⋊C4—42no—
130C22⋊C4—44no—
131C42—22no—
132C42—22no—
133C42—42no—
134C42—42no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—22no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23#12412yes—
145C23C22×D411yescharacteristic
146Q8—24no—
147Q8—42no—
148D4—24no—
149D4—48no—
150D4—42no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—48no—
162C2×C4—48no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#12412yes—
176C2×C4#13514yes—
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4C2×#4112yes—
180C2×C4H(5,𝔽2)12yes—
181C8—42no—
182C8—42no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—44no—
190C22—81no—
191C22—81no—
192C22—82no—
193C22#52511yescharacteristic
194C22C2×#12411yescharacteristic
195C22#53411yescharacteristic
196C22C2×#13512yes—
197C22#57512yes—
198C4—22no—
199C4—24no—
200C4—42no—
201C4—42no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4—82no—
206C4#55112yes—
207C4#55112yes—
208C2—42no—
209C2—82no—
210C2#256111yescharacteristic
211C2#310212yes—
212C2#313111yescharacteristic
213C2#314211yescharacteristic
214C2#319112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2074C2semidirect
4. #2073C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #1934C2semidirect
8. #1933C2semidirect
9. #1825C2semidirect
11. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#28512
#22972
#20742
#20732
#20262
#20252
#19342
#19332
#18252
#17882
C4×C22⋊C82