Finite Group Databasebeta

← #10247

Subgroups — #10247

Order 256.

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

normal subgroups only

This group has 278 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#3229C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic
4#1546C211yesmaximal, characteristic, complemented
5#1503C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic
7#1303C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#124—21no—
11C2×C4×D4C2211yescharacteristic
12#498—21no—
13#487—21no—
14#483—22no—
15#478—22no—
16#476—21no—
17#475—22no—
18#474—21no—
19#431C2211yescharacteristic
20#431C2211yescharacteristic
21#428—21no—
22#423C2211yescharacteristic
23#418—21no—
24#397—22no—
25#391—21no—
26#380—22no—
27C4×C4⋊C4C2211yescharacteristic
28#332—22no—
29#328—22no—
30D4⋊C8—22no—
31C22×Q8—21no—
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C2×M16—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#125—21no—
42#124—42no—
43#124—42no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
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—21no—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D412yes—
57C2×C22⋊C4—24no—
58C2×C22⋊C4—21no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61#107—22no—
62#106—22no—
63#106—24no—
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#95—24no—
76#95D412yes—
77C24—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—41no—
81C2×Q8—41no—
82C2×Q8—41no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4SD1612yes—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—44no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101M16—41no—
102M16—41no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—22no—
113C4⋊C4—21no—
114C4⋊C4—42no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—48no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C42—41no—
124C42—41no—
125C42—41no—
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C23—22no—
131C23—42no—
132C23—42no—
133C23—41no—
134C23—41no—
135C23#12011yescharacteristic
136Q8—42no—
137Q8—81no—
138Q8—81no—
139Q8—81no—
140Q8—81no—
141Q8—81no—
142D4—22no—
143D4—22no—
144D4—41no—
145D4—41no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—82no—
159C2×C4—82no—
160C2×C4#12011yescharacteristic
161C2×C4#12011yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12711yescharacteristic
166C2×C4C2×SD1611yescharacteristic
167C2×C4#13511yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C8—42no—
171C8—42no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—44no—
176C22—42no—
177C22—82no—
178C22—81no—
179C22—81no—
180C22#39211yescharacteristic
181C22#45011yescharacteristic
182C22#45211yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—82no—
187C4—82no—
188C4—81no—
189C4—81no—
190C4—81no—
191C4#44811yescharacteristic
192C4#44911yescharacteristic
193C4#45412yes—
194C4#46511yescharacteristic
195C4#46811yescharacteristic
196C2—21no—
197C2—21no—
198C2—82no—
199C2#200011yescharacteristic
200C2#200311yescharacteristic
201C2#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. #3229C2semidirect
4. #1546C2semidirect
5. #1503C2semidirect
7. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#32292
#30772
#25472
#15462
#15032
#14782
#13032