Finite Group Databasebeta

← #30054

Subgroups — #30054

Order 256.

379 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 379 conjugacy classes of subgroups — too many to draw. Only its 107 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#2642C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2072C212yesmaximal, complemented
4#2026C211yesmaximal, characteristic, complemented
5#2025C211yesmaximal, characteristic
6#2016C211yesmaximal, characteristic, complemented
7#2015C212yesmaximal
8#2013C211yesmaximal, characteristic
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal
11#1840C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23#400—22no—
24#400—22no—
25#391C2211yescharacteristic
26#391C2212yes—
27#391C2212yes—
28#390C2211yescharacteristic
29#388—22no—
30#385—22no—
31#384C2211yescharacteristic
32#376C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#336C2212yes—
36#336C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C22×Q8C2312yes—
40C22×Q8C2311yescharacteristic
41C22×D4C2311yescharacteristic
42C23×C4—21no—
43C2×Q16—24no—
44C2×Q16—42no—
45C2×Q16—42no—
46C2×SD16—24no—
47C2×SD16—42no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—42no—
52#125—42no—
53#124—22no—
54#124—22no—
55C4×Q8—42no—
56C4×D4—42no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C42—21no—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#103—42no—
72#103—42no—
73#103—42no—
74#102—42no—
75#95—22no—
76#95—22no—
77#95—24no—
78#95—22no—
79#95D412yes—
80#95D412yes—
81C24—21no—
82C2×Q8—21no—
83C2×Q8—22no—
84C2×Q8—22no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8#4112yes—
92C2×D4—21no—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4#4112yes—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—42no—
102C22×C4—82no—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108Q16—48no—
109SD16—48no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—44no—
122C4⋊C4—44no—
123C4⋊C4—82no—
124C22⋊C4—28no—
125C22⋊C4—42no—
126C22⋊C4—82no—
127C42—21no—
128C42—21no—
129C42—22no—
130C42—22no—
131C42—42no—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—42no—
137C23C22×D411yescharacteristic
138Q8—24no—
139Q8—42no—
140Q8—42no—
141Q8—42no—
142Q8—42no—
143Q8—42no—
144D4—24no—
145D4—42no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4C2×#4111yescharacteristic
171C2×C4C2×#4111yescharacteristic
172C2×C4H(5,𝔽2)12yes—
173C2×C4H(5,𝔽2)12yes—
174C8—42no—
175C8—42no—
176C22—44no—
177C22—42no—
178C22—81no—
179C22—81no—
180C22—82no—
181C22#52511yescharacteristic
182C22#53511yescharacteristic
183C22#53511yescharacteristic
184C22#57712yes—
185C22#57812yes—
186C4—42no—
187C4—42no—
188C4—41no—
189C4—41no—
190C4—42no—
191C4—82no—
192C4—81no—
193C4—81no—
194C4—82no—
195C4#53812yes—
196C4#53812yes—
197C2—82no—
198C2#263111yescharacteristic
199C2#312012yes—
200C2#322211yescharacteristic
201C2#322311yescharacteristic
202C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2642C2semidirect
3. #2072C2semidirect
4. #2026C2semidirect
6. #2016C2semidirect
9. #1878C2semidirect
11. #1840C2semidirect

Maximal subgroups

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

subgroupindex
#26422
#23262
#20722
#20262
#20252
#20162
#20152
#20132
#18782
#18772
#18402