Finite Group Databasebeta

← #29862

Subgroups — #29862

Order 256.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#491C211yesmaximal, characteristic, complemented
2#3080C211yesmaximal, characteristic
3#3039C211yesmaximal, characteristic
4#2923C211yesmaximal, characteristic
5#2064C214yesmaximal, complemented
6#2052C214yesmaximal
7#1998C211yesmaximal, characteristic, complemented
8#1966C211yesmaximal, characteristic, complemented
9#1849C211yesmaximal, characteristic, complemented
10C2×#137C2212yes—
11C2×C4⋊Q8C2211yescharacteristic
12C2×#125C2211yescharacteristic
13C4×#41C2211yescharacteristic
14#491—22no—
15#491—22no—
16#479—22no—
17#478—22no—
18#462—22no—
19#461—22no—
20#433—22no—
21#432—22no—
22C2×#108C2214yes—
23#428C2212yes—
24#424C2212yes—
25#423C2211yescharacteristic
26#423C2211yescharacteristic
27#419C2212yes—
28#417C2212yes—
29C2×#103C2214yes—
30C2×#101C2214yes—
31#408C2212yes—
32C4×M16C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#343C2214yes—
35C2×#41C2311yescharacteristic
36C22×Q8C2312yes—
37#137—44no—
38C2×Q16—22no—
39C2×SD16—22no—
40C2×M16C2312yes—
41C2×M16C2312yes—
42C2×M16C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47#125—22no—
48#125—21no—
49#125—21no—
50C4×Q8D412yes—
51C4×D4—22no—
52C4×D4D412yes—
53#117—21no—
54#117C2311yescharacteristic
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2312yes—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#108—44no—
61#107—22no—
62#106—22no—
63C4⋊C8—24no—
64C4⋊C8—22no—
65C4⋊C8—22no—
66#103—22no—
67#103—28no—
68#103—24no—
69#102—22no—
70#101—44no—
71C22⋊C8—24no—
72#97—21no—
73#97D412yes—
74#97D412yes—
75C4×C8—21no—
76#41—22no—
77#41—22no—
78C2×Q8—22no—
79C2×Q8—22no—
80C2×Q8—24no—
81C2×Q8—24no—
82C2×Q8C2×D411yescharacteristic
83C2×D4—21no—
84C2×D4C2×D411yescharacteristic
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4#4112yes—
90C22×C4C2411yescharacteristic
91Q16—44no—
92SD16—44no—
93M16—22no—
94M16—24no—
95M16—24no—
96M16—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8C2×D414yes—
103C4⋊C4—22no—
104C4⋊C4—22no—
105C4⋊C4—22no—
106C4⋊C4—42no—
107C4⋊C4—42no—
108C4⋊C4—42no—
109C4⋊C4—42no—
110C4⋊C4C2×D412yes—
111C22⋊C4—21no—
112C22⋊C4—22no—
113C42—22no—
114C42—21no—
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C42C2×D412yes—
118C23—21no—
119C23C2×#4111yescharacteristic
120Q8—21no—
121Q8—21no—
122Q8—42no—
123Q8—42no—
124Q8—44no—
125D4—21no—
126D4—21no—
127D4—42no—
128C2×C4—22no—
129C2×C4—21no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—44no—
135C2×C4—82no—
136C2×C4—82no—
137C2×C4#12112yes—
138C2×C4#12112yes—
139C2×C4C22×D411yescharacteristic
140C2×C4C22×D411yescharacteristic
141C2×C4H(5,𝔽2)12yes—
142C2×C42−1+412yes—
143C8—24no—
144C8—42no—
145C8—42no—
146C8—42no—
147C22—22no—
148C22—41no—
149C22—42no—
150C22C2×#12111yescharacteristic
151C22#54212yes—
152C4—22no—
153C4—22no—
154C4—41no—
155C4—42no—
156C4—82no—
157C4—82no—
158C4#53511yescharacteristic
159C4#53611yescharacteristic
160C4#53712yes—
161C2—22no—
162C2—81no—
163C2#262011yescharacteristic
164C2#306912yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#491C2semidirect
5. #2064C2semidirect
7. #1998C2semidirect
8. #1966C2semidirect
9. #1849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
#30802
#30392
#29232
#20642
#20522
#19982
#19662
#18492