Finite Group Databasebeta

← #19395

Subgroups — #19395

Order 256.

314 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 109 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#3231C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic
3#3101C211yesmaximal, characteristic
4#2972C211yesmaximal, characteristic
5#1683C212yesmaximal, complemented
6#1679C212yesmaximal, complemented
7#1673C212yesmaximal, complemented
8#1671C212yesmaximal, complemented
9#1572C211yesmaximal, characteristic, complemented
10#1554C211yesmaximal, characteristic, complemented
11#1544C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#533C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15#500C2211yescharacteristic, complemented
16#498C2211yescharacteristic, complemented
17#487C2211yescharacteristic, complemented
18#487C2211yescharacteristic, complemented
19#485—22no—
20#484—21no—
21#483—21no—
22#479—22no—
23#476C2212yes—
24#474C2212yes—
25#461C2212yes—
26#461C2212yes—
27#450—21no—
28#450—21no—
29#440—22no—
30C4×SD16—22no—
31C8×D4—22no—
32C8×D4C2212yes—
33C8×D4C2212yes—
34#432C2211yescharacteristic
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38C2×C22⋊C8—22no—
39#335C2212yescomplemented
40#334C2212yescomplemented
41#333C2212yescomplemented
42#332C2212yescomplemented
43#332C2212yescomplemented
44#332C2212yescomplemented
45C22×D4D411yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C22×C8—22no—
50C22×C8—22no—
51C4⋊Q8C2311yescharacteristic
52C4⋊Q8C2311yescharacteristic
53C4⋊Q8C2311yescharacteristic
54C4⋊Q8C2311yescharacteristic
55#125—21no—
56#125—21no—
57C22⋊Q8—21no—
58C22⋊Q8—21no—
59C22⋊Q8—21no—
60C22⋊Q8—21no—
61C4×Q8—21no—
62C4×Q8—21no—
63C4×Q8—21no—
64C4×Q8—21no—
65C4×D4—21no—
66C4×D4—21no—
67C4×D4D412yes—
68C4×D4C2312yes—
69#117—21no—
70#117—21no—
71C2×C4⋊C4D411yescharacteristic
72C2×C22⋊C4—21no—
73C2×C42C2311yescharacteristic
74#107—22no—
75#107—22no—
76#106—22no—
77#106—22no—
78C4⋊C8D412yes—
79C4⋊C8C2312yes—
80C4⋊C8C2312yes—
81C4⋊C8C2312yes—
82#103—22no—
83#103—22no—
84#103—22no—
85#103—22no—
86#102—22no—
87#102—22no—
88C22⋊C8—21no—
89C22⋊C8—21no—
90C22⋊C8—21no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93C22⋊C8D412yes—
94#97—21no—
95C4×C8—21no—
96C4×C8—21no—
97C4×C8C2311yescharacteristic
98C4×C8C2311yescharacteristic
99C24—21no—
100C2×Q8—21no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×D4—24no—
105C2×D4—21no—
106C2×D4C2×D412yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—82no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4#4111yescharacteristic
114SD16—44no—
115SD16—44no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—82no—
120C2×C8—82no—
121C2×C8C2×D412yes—
122C2×C8C2×D412yes—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—41no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4C2×D412yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C42—41no—
142C42—41no—
143C42C2×D411yescharacteristic
144C42C2×D411yescharacteristic
145C42#4111yescharacteristic
146C42C2411yescharacteristic
147C23—24no—
148C23—22no—
149C23—81no—
150C23—81no—
151C23#12111yescharacteristic
152Q8—42no—
153Q8—42no—
154Q8—42no—
155Q8—81no—
156D4—24no—
157D4—22no—
158D4#13514yes—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—82no—
170C2×C4—81no—
171C2×C4#12111yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4C22×D411yescharacteristic
175C2×C4C22×D411yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C8—22no—
178C8—22no—
179C8—42no—
180C8—42no—
181C8—42no—
182C22—21no—
183C22—21no—
184C22—24no—
185C22—21no—
186C22—42no—
187C22—42no—
188C22—84no—
189C22C2×#12111yescharacteristic
190C4—21no—
191C4—21no—
192C4—21no—
193C4—21no—
194C4—42no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4C2×#13512yes—
200C4#57712yes—
201C2—41no—
202C2—44no—
203C2#304211yescharacteristic
204C2#305011yescharacteristic
205C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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. #3231C2semidirect
5. #1683C2semidirect
6. #1679C2semidirect
7. #1673C2semidirect
8. #1671C2semidirect
9. #1572C2semidirect
10. #1554C2semidirect
11. #1544C2semidirect
15. #500C22semidirect
16. #498C22semidirect
17. #487C22semidirect
18. #487C22semidirect
39. #335C22semidirect
40. #334C22semidirect
41. #333C22semidirect
42. #332C22semidirect
43. #332C22semidirect
44. #332C22semidirect

Maximal subgroups

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

subgroupindex
#32312
#31102
#31012
#29722
#16832
#16792
#16732
#16712
#15722
#15542
#15442