Finite Group Databasebeta

← #29153

Subgroups — #29153

Order 256.

343 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 127 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#2660C211yesmaximal, characteristic
2#2327C211yesmaximal, characteristic
3#2026C212yesmaximal
4#2024C212yesmaximal
5#1960C211yesmaximal, characteristic
6#1956C211yesmaximal, characteristic
7#1836C211yesmaximal, characteristic
8#1835C211yesmaximal, characteristic
9#1805C212yesmaximal
10#1803C212yesmaximal
11#1768C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×Q8C2212yes—
16C2×C4⋊C8C2212yes—
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#97C2211yescharacteristic
22C2×C4×C8C2211yescharacteristic
23#398—22no—
24#395—22no—
25#391C2212yes—
26#387—22no—
27#387—22no—
28#384C2211yescharacteristic
29#384C2211yescharacteristic
30#382C2211yescharacteristic
31#382C2211yescharacteristic
32#382C2211yescharacteristic
33#376C2211yescharacteristic
34#340C2212yes—
35#340C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8C2312yes—
41C22×C8C2312yes—
42C22×C8C2312yes—
43C4⋊Q8—22no—
44C4⋊Q8—42no—
45C4⋊Q8Q814yescomplemented
46#125—42no—
47C4×Q8—24no—
48C4×Q8—24no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C42—22no—
62C2×C42C2311yescharacteristic
63C2×C42C2311yescharacteristic
64C2×C42C2311yescharacteristic
65C4⋊C8—24no—
66#103—42no—
67#103—42no—
68#103—42no—
69#103—42no—
70#97—22no—
71C4×C8—22no—
72#95—22no—
73#95—22no—
74#95—22no—
75#95—22no—
76#95—22no—
77#95D412yes—
78#95D412yes—
79C2×Q8—22no—
80C2×Q8—22no—
81C2×Q8—22no—
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—42no—
85C2×Q8—82no—
86C2×Q8C2×Q814yes—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D412yes—
96C22×C4C2×D412yes—
97C22×C4C2411yescharacteristic
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—22no—
104C4⋊C4—28no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—22no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—42no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4—44no—
115C4⋊C4—82no—
116C4⋊C4—82no—
117C42—22no—
118C42—44no—
119C42C2×Q812yes—
120C42#4112yes—
121C42#4114yes—
122C23C22×D411yescharacteristic
123Q8—24no—
124Q8—24no—
125Q8—42no—
126Q8—42no—
127Q8—84no—
128C2×C4—22no—
129C2×C4—22no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—24no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4C22⋊Q814yes—
151C2×C4#12414yes—
152C2×C4C22×Q811yescharacteristic
153C2×C4C2×#4112yes—
154C2×C4C2×#4111yescharacteristic
155C2×C4H(5,𝔽2)12yes—
156C8—42no—
157C8—42no—
158C22C2×C22⋊Q811yescharacteristic
159C22C2×#12411yescharacteristic
160C22#53511yescharacteristic
161C22#57712yes—
162C22#57812yes—
163C4—24no—
164C4—24no—
165C4—44no—
166C4—41no—
167C4—41no—
168C4—82no—
169C4—81no—
170C4—81no—
171C4—81no—
172C4—81no—
173C4#55412yes—
174C4#55512yes—
175C2#256311yescharacteristic
176C2#308212yes—
177C2#313712yes—
178C2#322211yescharacteristic
179C2#322311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
45. C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#26602
#23272
#20262
#20242
#19602
#19562
#18362
#18352
#18052
#18032
#17682