Finite Group Databasebeta

← #27595

Subgroups — #27595

Order 256.

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

normal subgroups only

This group has 421 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#2314C211yesmaximal, characteristic, complemented
2#2311C211yesmaximal, characteristic
3#2095C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2074C211yesmaximal, characteristic, complemented
6#2073C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1925C212yesmaximal, complemented
9#1852C212yesmaximal, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal
12C22×#41C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C2×#107C2212yes—
16C2×#107C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×C22⋊C8C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#390C2211yescharacteristic
26#388—22no—
27#387—22no—
28#386—22no—
29#384C2211yescharacteristic
30#382C2211yescharacteristic
31#380—22no—
32#380C2212yes—
33C4×C4⋊C4C2212yes—
34#340C2212yes—
35#340C2212yes—
36#336C2211yescharacteristic
37#336C2211yescharacteristic
38#336C2212yes—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—24no—
42C2×#41—42no—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51#117—24no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—42no—
55C2×C4⋊C4D412yes—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4D412yes—
64C2×C42—22no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67#107—42no—
68#107—42no—
69#103—24no—
70#103—42no—
71#102—24no—
72#102—42no—
73C22⋊C8—24no—
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—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—82no—
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4#4114yes—
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C4⋊C4—24no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—42no—
126C4⋊C4—82no—
127C22⋊C4—24no—
128C22⋊C4—82no—
129C22⋊C4—82no—
130C22⋊C4—82no—
131C42—22no—
132C42—22no—
133C42—82no—
134C23—21no—
135C23—21no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23#12312yes—
144C23C22×D411yescharacteristic
145Q8—24no—
146Q8—42no—
147D4—24no—
148D4—48no—
149D4—42no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—24no—
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—48no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—84no—
175C2×C4#12312yes—
176C2×C4C2×#4112yes—
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4H(5,𝔽2)12yes—
180C8—42no—
181C8—42no—
182C22—24no—
183C22—22no—
184C22—42no—
185C22—41no—
186C22—41no—
187C22—44no—
188C22—42no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22C2×#12311yescharacteristic
193C22#52511yescharacteristic
194C22#53511yescharacteristic
195C22#57612yes—
196C22#57812yes—
197C4—22no—
198C4—42no—
199C4—42no—
200C4—42no—
201C4—82no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4#55112yes—
206C4#55112yes—
207C2—42no—
208C2—82no—
209C2#248211yescharacteristic
210C2#309812yes—
211C2#312412yes—
212C2#322011yescharacteristic
213C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2314C2semidirect
3. #2095C2semidirect
4. #2093C2semidirect
5. #2074C2semidirect
6. #2073C2semidirect
7. #1928C2semidirect
8. #1925C2semidirect
9. #1852C2semidirect
10. #1818C2semidirect

Maximal subgroups

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

subgroupindex
#23142
#23112
#20952
#20932
#20742
#20732
#19282
#19252
#18522
#18182
#17882