Finite Group Databasebeta

← #10249

Subgroups — #10249

Order 256.

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

normal subgroups only

This group has 268 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#3230C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#1549C211yesmaximal, characteristic, complemented
5#1503C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic
7#1303C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#500—21no—
12#487—21no—
13#484—22no—
14#478—22no—
15#477—22no—
16#476—21no—
17#474—21no—
18#431C2211yescharacteristic
19#431C2211yescharacteristic
20#428—21no—
21#423C2211yescharacteristic
22#418—21no—
23#393—22no—
24#390—21no—
25#382—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—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C4×Q8—21no—
45C4×Q8—21no—
46C4×D4—21no—
47C4×D4—21no—
48C4×D4—21no—
49C4×D4—21no—
50#117—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D412yes—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—24no—
57C2×C42—21no—
58C2×C42C2311yescharacteristic
59#107—22no—
60#107—24no—
61#106—22no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8D412yes—
65C4⋊C8D412yes—
66#103—24no—
67#102—22no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C4×C8—21no—
72C4×C8—21no—
73#95—24no—
74#95D412yes—
75C24—21no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×Q8—82no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4Q1612yes—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—44no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96M16—41no—
97M16—41no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—22no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—22no—
108C4⋊C4—21no—
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—41no—
117C22⋊C4—84no—
118C42—41no—
119C42—41no—
120C42—41no—
121C42C2×D411yescharacteristic
122C42C2×D411yescharacteristic
123C42C2×D411yescharacteristic
124C42#4111yescharacteristic
125C23—22no—
126C23—42no—
127C23—42no—
128C23—41no—
129C23—41no—
130C23#12011yescharacteristic
131Q8—42no—
132Q8—81no—
133Q8—82no—
134Q8—82no—
135D4—22no—
136D4—22no—
137D4—41no—
138D4—41no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4#12011yescharacteristic
153C2×C4#12011yescharacteristic
154C2×C4#12111yescharacteristic
155C2×C4#12111yescharacteristic
156C2×C4#12111yescharacteristic
157C2×C4#12711yescharacteristic
158C2×C4C2×Q1611yescharacteristic
159C2×C4#13511yescharacteristic
160C2×C4#13711yescharacteristic
161C2×C4#13711yescharacteristic
162C8—42no—
163C8—42no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—44no—
168C22—42no—
169C22—82no—
170C22—81no—
171C22—81no—
172C22#39211yescharacteristic
173C22#45111yescharacteristic
174C22#45211yescharacteristic
175C4—21no—
176C4—21no—
177C4—21no—
178C4—82no—
179C4—81no—
180C4—81no—
181C4—82no—
182C4—81no—
183C4#45111yescharacteristic
184C4#45211yescharacteristic
185C4#45412yes—
186C4#46711yescharacteristic
187C4#47011yescharacteristic
188C2—21no—
189C2—21no—
190C2—82no—
191C2#200111yescharacteristic
192C2#200311yescharacteristic
193C2#200811yescharacteristic
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. #3230C2semidirect
4. #1549C2semidirect
5. #1503C2semidirect
7. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#32302
#30772
#25632
#15492
#15032
#14782
#13032