Finite Group Databasebeta

← #19425

Subgroups — #19425

Order 256.

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

normal subgroups only

This group has 330 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#3207C211yesmaximal, characteristic, complemented
2#3109C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic, complemented
4#2975C211yesmaximal, characteristic, complemented
5#1703C212yesmaximal, complemented
6#1697C212yesmaximal, complemented
7#1687C212yesmaximal, complemented
8#1685C212yesmaximal, complemented
9#1572C211yesmaximal, characteristic, complemented
10#1553C211yesmaximal, characteristic, complemented
11#1543C211yesmaximal, characteristic, complemented
12#532C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15#493C2211yescharacteristic, complemented
16#492C2211yescharacteristic, complemented
17#487C2211yescharacteristic, complemented
18#486C2211yescharacteristic, complemented
19#481—21no—
20#480—21no—
21#476C2212yes—
22#474C2212yes—
23#463—22no—
24#459C2212yes—
25#459C2212yes—
26#449—22no—
27#447—21no—
28#447—21no—
29#442—22no—
30C4×D8—22no—
31#436—22no—
32#436C2212yes—
33#436C2212yes—
34#432C2211yescharacteristic
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#407—22no—
39#335C2212yescomplemented
40#334C2212yescomplemented
41#333C2212yescomplemented
42#332C2212yescomplemented
43#331C2212yescomplemented
44#331C2212yescomplemented
45C2×#41—21no—
46C2×#41—21no—
47C22×D4D411yescharacteristic
48C23×C4—21no—
49C2×SD16—22no—
50C2×D8—22no—
51C2×D8—22no—
52C2×D8—22no—
53C2×M16—22no—
54C2×M16—22no—
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127C2311yescharacteristic
58#127C2311yescharacteristic
59#124—21no—
60#124—21no—
61#121—21no—
62#121—21no—
63#121—21no—
64#121—21no—
65C4×D4—21no—
66C4×D4—21no—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4—21no—
71C4×D4D412yes—
72C4×D4C2312yes—
73C2×C4⋊C4D411yescharacteristic
74C2×C22⋊C4—21no—
75C2×C42C2311yescharacteristic
76#107—22no—
77#106—22no—
78C4⋊C8D412yes—
79C4⋊C8C2312yes—
80C4⋊C8C2312yes—
81C4⋊C8C2312yes—
82#103—22no—
83#102—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—
101#41—82no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—42no—
112C2×D4—41no—
113C2×D4—41no—
114C2×D4C2×D412yes—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—82no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4#4111yescharacteristic
124SD16—82no—
125D8—44no—
126D8—44no—
127D8—82no—
128M16—82no—
129M16—82no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8C2×D412yes—
134C2×C8C2×D412yes—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4C2×D412yes—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C22⋊C4—41no—
146C22⋊C4—41no—
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C42#4111yescharacteristic
150C42C2411yescharacteristic
151C23—22no—
152C23—42no—
153C23—41no—
154C23—41no—
155C23—81no—
156C23—81no—
157C23#12111yescharacteristic
158Q8—42no—
159Q8—81no—
160D4—22no—
161D4—22no—
162D4—42no—
163D4—42no—
164D4—42no—
165D4—42no—
166D4—42no—
167D4—81no—
168D4—81no—
169D4—82no—
170D4—82no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—81no—
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4C22×D411yescharacteristic
187C2×C4C22×D411yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C8—22no—
190C8—22no—
191C8—42no—
192C8—42no—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—84no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22C2×#12111yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—42no—
213C4—81no—
214C4—81no—
215C4#57512yes—
216C4#57612yes—
217C2—41no—
218C2—82no—
219C2—161no—
220C2—161no—
221C2#304111yescharacteristic
222C2#305411yescharacteristic
223C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
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. #3207C2semidirect
2. #3109C2semidirect
3. #3100C2semidirect
4. #2975C2semidirect
5. #1703C2semidirect
6. #1697C2semidirect
7. #1687C2semidirect
8. #1685C2semidirect
9. #1572C2semidirect
10. #1553C2semidirect
11. #1543C2semidirect
15. #493C22semidirect
16. #492C22semidirect
17. #487C22semidirect
18. #486C22semidirect
39. #335C22semidirect
40. #334C22semidirect
41. #333C22semidirect
42. #332C22semidirect
43. #331C22semidirect
44. #331C22semidirect

Maximal subgroups

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

subgroupindex
#32072
#31092
#31002
#29752
#17032
#16972
#16872
#16852
#15722
#15532
#15432