Finite Group Databasebeta

← #19524

Subgroups — #19524

Order 256.

287 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 105 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#3255C211yesmaximal, characteristic, complemented
2#3246C211yesmaximal, characteristic
3#3210C211yesmaximal, characteristic
4#2985C211yesmaximal, characteristic, complemented
5#1702C212yesmaximal, complemented
6#1690C212yesmaximal, complemented
7#1680C212yesmaximal, complemented
8#1666C212yesmaximal, complemented
9#1570C211yesmaximal, characteristic, complemented
10#1557C211yesmaximal, characteristic
11#1552C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#533C2211yescharacteristic
14#532C2211yescharacteristic
15#494C2211yescharacteristic
16#492C2211yescharacteristic
17#487C2211yescharacteristic
18#486C2211yescharacteristic
19#476—21no—
20#475—21no—
21#475C2212yes—
22#475C2212yes—
23#470—22no—
24#465—22no—
25#464—22no—
26#462—21no—
27#462C2212yes—
28#461—21no—
29#460C2212yes—
30#452—22no—
31#446—22no—
32#446C2212yes—
33C8×Q8C2212yes—
34#433—22no—
35#432C2211yescharacteristic
36#431C2211yescharacteristic
37#423C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39#334C2212yes—
40#334C2212yes—
41#332C2212yes—
42#332C2212yes—
43#332C2212yes—
44#331C2212yes—
45C2×#41—21no—
46C2×Q16—22no—
47C2×Q16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C2×M16—21no—
51C22×C8—21no—
52C4⋊Q8D411yescharacteristic
53C4⋊Q8D411yescharacteristic
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127C2311yescharacteristic
58#125—21no—
59#125—21no—
60#124—21no—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65#121—21no—
66#121—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×Q8C2312yes—
71C4×D4—21no—
72C4×D4—21no—
73#117—21no—
74#117—21no—
75C2×C42C2311yescharacteristic
76#106—22no—
77#106—22no—
78C4⋊C8—22no—
79C4⋊C8—22no—
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8D412yes—
83C4⋊C8C2312yes—
84C4⋊C8C2312yes—
85C4⋊C8C2312yes—
86#103—22no—
87#103—22no—
88#103—22no—
89#103—22no—
90#102—22no—
91#102—22no—
92C22⋊C8—21no—
93C22⋊C8D412yes—
94#97—22no—
95#97—21no—
96C4×C8—22no—
97C4×C8C2311yescharacteristic
98C4×C8C2311yescharacteristic
99#41—82no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8C2×D412yes—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—41no—
109C22×C4—41no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4#4111yescharacteristic
113Q16—82no—
114Q16—82no—
115SD16—82no—
116SD16—82no—
117M16—81no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—81no—
122C2×C8C2×D412yes—
123C2×C8C2×D412yes—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4C2×D412yes—
138C22⋊C4—22no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C42—22no—
143C42—41no—
144C42C2×D411yescharacteristic
145C42C2×D411yescharacteristic
146C42#4111yescharacteristic
147C42C2411yescharacteristic
148C23—41no—
149C23#12111yescharacteristic
150Q8—22no—
151Q8—22no—
152Q8—42no—
153Q8—41no—
154Q8—42no—
155Q8—41no—
156Q8—81no—
157D4—41no—
158D4—42no—
159D4—41no—
160D4—81no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—82no—
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4C22×D411yescharacteristic
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C8—22no—
179C8—22no—
180C8—42no—
181C8—42no—
182C8—42no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22C2×#12111yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—42no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4#57712yes—
200C4#57711yescharacteristic
201C4#57711yescharacteristic
202C2—41no—
203C2—161no—
204C2#304311yescharacteristic
205C2#306011yescharacteristic
206C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3255C2semidirect
4. #2985C2semidirect
5. #1702C2semidirect
6. #1690C2semidirect
7. #1680C2semidirect
8. #1666C2semidirect
9. #1570C2semidirect
11. #1552C2semidirect

Maximal subgroups

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

subgroupindex
#32552
#32462
#32102
#29852
#17022
#16902
#16802
#16662
#15702
#15572
#15522