Finite Group Databasebeta

← #10248

Subgroups — #10248

Order 256.

302 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 61 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#3209C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic
3#2542C211yesmaximal, characteristic
4#1548C211yesmaximal, characteristic, complemented
5#1503C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic
7#1303C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C22⋊Q8—22no—
11C2×C4×D4C2211yescharacteristic
12#494—21no—
13#487—21no—
14#478—22no—
15#476—21no—
16#474—21no—
17#462—22no—
18#451—22no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#428—21no—
22#423C2211yescharacteristic
23#418—21no—
24#399—22no—
25#390—21no—
26#384—21no—
27C4×C4⋊C4C2211yescharacteristic
28#333—22no—
29#328—22no—
30D4⋊C8—22no—
31C22×Q8D412yes—
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C2×Q16—24no—
35C2×M16—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42#125—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C22⋊Q8—48no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—24no—
59C2×C22⋊C4—21no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—22no—
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8D412yes—
67C4⋊C8D412yes—
68#103—24no—
69#102—22no—
70#102—22no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75#95D412yes—
76C24—21no—
77C2×Q8—22no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—22no—
81C2×Q8—44no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4Q1612yescomplemented
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—44no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97Q16—84no—
98M16—41no—
99M16—41no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—41no—
103C4⋊C4—24no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—48no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C22⋊C4—48no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C42—41no—
122C42—41no—
123C42—41no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C23—22no—
129C23—42no—
130C23—42no—
131C23—41no—
132C23—41no—
133C23#12011yescharacteristic
134Q8—42no—
135Q8—44no—
136Q8—81no—
137Q8—82no—
138D4—22no—
139D4—22no—
140D4—41no—
141D4—41no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—44no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4#12011yescharacteristic
156C2×C4#12011yescharacteristic
157C2×C4#12111yescharacteristic
158C2×C4#12111yescharacteristic
159C2×C4#12111yescharacteristic
160C2×C4#12711yescharacteristic
161C2×C4C2×Q1611yescharacteristic
162C2×C4#13511yescharacteristic
163C2×C4#13711yescharacteristic
164C2×C4#13711yescharacteristic
165C8—42no—
166C8—42no—
167C8—81no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22—42no—
172C22—82no—
173C22—81no—
174C22—81no—
175C22#39211yescharacteristic
176C22#45111yescharacteristic
177C22#45211yescharacteristic
178C4—21no—
179C4—21no—
180C4—21no—
181C4—44no—
182C4—81no—
183C4—82no—
184C4—81no—
185C4—81no—
186C4#45111yescharacteristic
187C4#45211yescharacteristic
188C4#45312yes—
189C4#46711yescharacteristic
190C4#47011yescharacteristic
191C2—21no—
192C2—21no—
193C2—82no—
194C2#200111yescharacteristic
195C2#200211yescharacteristic
196C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3209C2semidirect
4. #1548C2semidirect
5. #1503C2semidirect
7. #1303C2semidirect
87. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#32092
#30772
#25422
#15482
#15032
#14782
#13032