Finite Group Databasebeta

← #19388

Subgroups — #19388

Order 256.

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

normal subgroups only

This group has 331 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#3210C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#2972C211yesmaximal, characteristic, complemented
5#1681C212yesmaximal, complemented
6#1679C212yesmaximal, complemented
7#1669C212yesmaximal, complemented
8#1665C212yesmaximal, complemented
9#1572C211yesmaximal, characteristic, complemented
10#1553C211yesmaximal, characteristic, complemented
11#1544C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15#494C2211yescharacteristic, complemented
16#492C2211yescharacteristic, complemented
17#487C2211yescharacteristic, complemented
18#486C2211yescharacteristic, complemented
19#484—21no—
20#480—21no—
21#474C2212yes—
22#474C2212yes—
23#464—22no—
24#461C2212yes—
25#459C2212yes—
26#452—22no—
27#450—21no—
28#447—21no—
29#440—22no—
30C4×D8—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#332C2212yescomplemented
42#332C2212yescomplemented
43#332C2212yescomplemented
44#331C2212yescomplemented
45C2×#41—21no—
46C22×D4D411yescharacteristic
47C23×C4—21no—
48C2×Q16—22no—
49C2×SD16—22no—
50C2×SD16—22no—
51C2×D8—22no—
52C22×C8—22no—
53C22×C8—22no—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127C2311yescharacteristic
58#125—21no—
59#124—21no—
60C22⋊Q8—21no—
61C22⋊Q8—21no—
62#121—21no—
63#121—21no—
64C4×Q8—21no—
65C4×Q8—21no—
66C4×D4—21no—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4D412yes—
71C4×D4C2312yes—
72#117—21no—
73C2×C4⋊C4D411yescharacteristic
74C2×C22⋊C4—21no—
75C2×C42C2311yescharacteristic
76#107—22no—
77#107—22no—
78C4⋊C8D412yes—
79C4⋊C8C2312yes—
80C4⋊C8C2312yes—
81C4⋊C8C2312yes—
82#103—22no—
83#103—22no—
84#102—22no—
85#102—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—
100#41—82no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×D4—24no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—41no—
110C2×D4C2×D412yes—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4—82no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4#4111yescharacteristic
119Q16—82no—
120SD16—44no—
121SD16—82no—
122D8—44no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—82no—
127C2×C8—82no—
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4C2×D412yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C42—41no—
146C42C2×D411yescharacteristic
147C42C2×D411yescharacteristic
148C42#4111yescharacteristic
149C42C2411yescharacteristic
150C23—24no—
151C23—22no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#12111yescharacteristic
156Q8—42no—
157Q8—42no—
158Q8—42no—
159D4—24no—
160D4—22no—
161D4—42no—
162D4—42no—
163D4—81no—
164D4—82no—
165D4#13514yes—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—82no—
176C2×C4—82no—
177C2×C4—81no—
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C8—22no—
185C8—22no—
186C8—42no—
187C8—42no—
188C8—42no—
189C22—21no—
190C22—21no—
191C22—24no—
192C22—21no—
193C22—42no—
194C22—42no—
195C22—84no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22C2×#12111yescharacteristic
200C4—21no—
201C4—21no—
202C4—21no—
203C4—21no—
204C4—42no—
205C4—81no—
206C4—81no—
207C4—81no—
208C4C2×#13512yes—
209C4#57612yes—
210C2—41no—
211C2—44no—
212C2—161no—
213C2#304111yescharacteristic
214C2#305011yescharacteristic
215C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3100C2
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. #3210C2semidirect
2. #3110C2semidirect
4. #2972C2semidirect
5. #1681C2semidirect
6. #1679C2semidirect
7. #1669C2semidirect
8. #1665C2semidirect
9. #1572C2semidirect
10. #1553C2semidirect
11. #1544C2semidirect
15. #494C22semidirect
16. #492C22semidirect
17. #487C22semidirect
18. #486C22semidirect
39. #335C22semidirect
40. #334C22semidirect
41. #332C22semidirect
42. #332C22semidirect
43. #332C22semidirect
44. #331C22semidirect

Maximal subgroups

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

subgroupindex
#32102
#31102
#31002
#29722
#16812
#16792
#16692
#16652
#15722
#15532
#15442