Finite Group Databasebeta

← #29269

Subgroups — #29269

Order 256.

363 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 123 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#2330C211yesmaximal, characteristic, complemented
2#2326C211yesmaximal, characteristic
3#2096C211yesmaximal, characteristic, complemented
4#2094C211yesmaximal, characteristic
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic
7#1983C212yesmaximal
8#1948C212yesmaximal, complemented
9#1865C212yesmaximal, complemented
10#1864C212yesmaximal
11#1832C211yesmaximal, characteristic, complemented
12C2×#125C2211yescharacteristic
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16C2×#106C2212yes—
17C2×#106C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23#391C2211yescharacteristic
24#390C2211yescharacteristic
25#389—22no—
26#389C2212yes—
27#388—22no—
28#387—22no—
29#385—22no—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32C4×C4⋊C4C2212yes—
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#336C2211yescharacteristic
37#336C2211yescharacteristic
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×D4C2311yescharacteristic
41C23×C4—21no—
42C22×C8C2312yes—
43C22×C8C2312yes—
44#125—24no—
45#124—22no—
46#124Q814yes—
47C4×Q8—42no—
48C4×D4—42no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4C2311yescharacteristic
55C2×C4⋊C4C2311yescharacteristic
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C22⋊C4—22no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—22no—
61C2×C42—21no—
62C2×C42—22no—
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65C2×C42C2311yescharacteristic
66#106—42no—
67#106—42no—
68C4⋊C8—24no—
69#103—24no—
70#103—42no—
71#102—24no—
72#102—42no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95D412yes—
77#95D412yes—
78C24—21no—
79C2×Q8—21no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8C2×Q812yes—
83C2×D4—22no—
84C2×D4—21no—
85C2×D4—22no—
86C2×D4C2×Q812yes—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—42no—
93C22×C4—82no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D412yes—
98C22×C4C2411yescharacteristic
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C2×C8—22no—
105C4⋊C4—28no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—42no—
112C4⋊C4—82no—
113C4⋊C4—82no—
114C4⋊C4—82no—
115C4⋊C4#4114yes—
116C22⋊C4—28no—
117C22⋊C4—42no—
118C22⋊C4—82no—
119C42—22no—
120C42—22no—
121C42—42no—
122C42—82no—
123C42C2×Q812yes—
124C42#4112yes—
125C23—22no—
126C23—41no—
127C23—41no—
128C23—42no—
129C23—42no—
130C23C22×D411yescharacteristic
131Q8—24no—
132Q8—42no—
133D4—24no—
134D4—42no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—84no—
158C2×C4C22⋊Q814yes—
159C2×C4C22×Q811yescharacteristic
160C2×C4C2×#4112yes—
161C2×C4C2×#4111yescharacteristic
162C2×C4H(5,𝔽2)12yes—
163C8—42no—
164C8—42no—
165C22—44no—
166C22—42no—
167C22—82no—
168C22—81no—
169C22—81no—
170C22C2×C22⋊Q811yescharacteristic
171C22#52511yescharacteristic
172C22#53511yescharacteristic
173C22#57712yes—
174C22#57712yes—
175C4—24no—
176C4—42no—
177C4—42no—
178C4—44no—
179C4—41no—
180C4—41no—
181C4—82no—
182C4—81no—
183C4—81no—
184C4#55412yes—
185C4#55912yes—
186C2—82no—
187C2#258011yescharacteristic
188C2#308412yes—
189C2#312512yes—
190C2#322111yescharacteristic
191C2#322211yescharacteristic
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. #2330C2semidirect
3. #2096C2semidirect
5. #2026C2semidirect
8. #1948C2semidirect
9. #1865C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
#23302
#23262
#20962
#20942
#20262
#20252
#19832
#19482
#18652
#18642
#18322