Finite Group Databasebeta

← #11902

Subgroups — #11902

Order 256.

226 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 57 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#2564C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic
3#1460C211yesmaximal, characteristic
4#1352C212yesmaximal, complemented
5#1351C212yesmaximal, complemented
6C2×C22⋊Q8C411yescharacteristic, complemented
7C2×C4×D4C2211yescharacteristic
8#431—21no—
9C2×#99—21no—
10#400C411yescharacteristic, complemented
11#399C2211yescharacteristic
12#395C411yescharacteristic, complemented
13#393C2211yescharacteristic
14#387—21no—
15#386—21no—
16#385—21no—
17#380—21no—
18#380—21no—
19C2×#95C411yescharacteristic, complemented
20#342C2212yes—
21#323—21no—
22#323C2212yes—
23C22×Q8C2×C411yescharacteristic
24C22×D4D411yescharacteristic
25C23×C4—21no—
26C23×C4C2×C411yescharacteristic
27C23×C4D411yescharacteristic
28C22⋊Q8—41no—
29C22⋊Q8—41no—
30C4×D4—41no—
31C4×D4—41no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—41no—
36C2×C4⋊C4C2×C411yescharacteristic
37C2×C4⋊C4C2×C411yescharacteristic
38C2×C22⋊C4—21no—
39C2×C22⋊C4—21no—
40C2×C22⋊C4—22no—
41C2×C22⋊C4—21no—
42C2×C22⋊C4D411yescharacteristic
43C2×C22⋊C4C2311yescharacteristic
44C2×C42D411yescharacteristic
45C4⋊C8—41no—
46#99—42no—
47C22⋊C8—22no—
48C4×C8—41no—
49#95—21no—
50#95—21no—
51#95—21no—
52#95—21no—
53#95—21no—
54#95—41no—
55#95C2×C411yescharacteristic
56#95C2×C411yescharacteristic
57C24C22⋊C411yescharacteristic
58C24C2×D411yescharacteristic
59C2×Q8—41no—
60C2×Q8—81no—
61C2×D4—41no—
62C2×D4—41no—
63C2×D4SD1612yes—
64C2×D4Q1612yes—
65C22×C4—21no—
66C22×C4—21no—
67C22×C4—21no—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—42no—
76C22×C4—81no—
77C22×C4—81no—
78C22×C4—81no—
79C22×C4C22⋊C411yescharacteristic
80C22×C4C22⋊C411yescharacteristic
81C22×C4C22⋊C411yescharacteristic
82C22×C4C22×C411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C2×C8—42no—
85C4⋊C4—41no—
86C4⋊C4—41no—
87C4⋊C4—42no—
88C4⋊C4—41no—
89C4⋊C4—41no—
90C4⋊C4—81no—
91C4⋊C4—82no—
92C22⋊C4—41no—
93C22⋊C4—41no—
94C22⋊C4—42no—
95C22⋊C4—42no—
96C22⋊C4—42no—
97C22⋊C4—42no—
98C22⋊C4—81no—
99C42—21no—
100C42—21no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—41no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23#10312yes—
110C23C2×C22⋊C411yescharacteristic
111C23C2×SD1611yescharacteristic
112C23C2×Q1611yescharacteristic
113Q8—82no—
114D4—42no—
115D4—42no—
116C2×C4—21no—
117C2×C4—21no—
118C2×C4—22no—
119C2×C4—21no—
120C2×C4—21no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—81no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—84no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4—82no—
145C2×C4#9911yescharacteristic
146C2×C4#9911yescharacteristic
147C2×C4#10312yes—
148C8—82no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—22no—
153C22—21no—
154C22—42no—
155C22—41no—
156C22—41no—
157C22—42no—
158C22—41no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—81no—
163C22C2×#9911yescharacteristic
164C22C2×#10311yescharacteristic
165C22#42111yescharacteristic
166C4—21no—
167C4—21no—
168C4—22no—
169C4—42no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—81no—
174C4—81no—
175C4—81no—
176C4—162no—
177C2—21no—
178C2—21no—
179C2—41no—
180C2—42no—
181C2—41no—
182C2#152011yescharacteristic
183C2#212111yescharacteristic
184C2#212611yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1352C2semidirect
5. #1351C2semidirect
6. C2×C22⋊Q8C4semidirect
10. #400C4semidirect
12. #395C4semidirect
19. C2×#95C4semidirect

Maximal subgroups

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

subgroupindex
#25642
#17992
#14602
#13522
#13512