Finite Group Databasebeta

← #19463

Subgroups — #19463

Order 256.

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

normal subgroups only

This group has 313 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#3229C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic
3C2×#476C211yesmaximal, characteristic
4#2974C211yesmaximal, characteristic
5#1703C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1683C212yesmaximal, complemented
8#1671C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic, complemented
10#1558C211yesmaximal, characteristic, complemented
11#1552C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15#498C2212yescomplemented
16#487C2212yescomplemented
17#484—21no—
18#483—21no—
19#483—21no—
20#476—21no—
21#476—21no—
22#476C2212yes—
23#476C2212yes—
24#475—22no—
25#474C2211yescharacteristic
26#474C2211yescharacteristic
27#461C2211yescharacteristic
28#461C2211yescharacteristic
29#450—21no—
30#440—21no—
31#440—21no—
32#436C2212yes—
33#435—22no—
34C8×D4C2212yes—
35#432C2211yescharacteristic
36#432C2211yescharacteristic
37#431C2211yescharacteristic
38C2×#106—21no—
39C2×C4⋊C8C2211yescharacteristic
40C2×#102—21no—
41#407—21no—
42C2×C22⋊C8—21no—
43#335C2212yescomplemented
44#334C2212yescomplemented
45#333C2212yescomplemented
46#333C2212yescomplemented
47#332C2212yescomplemented
48#332C2212yescomplemented
49C22×Q8—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C2×SD16—21no—
53C2×SD16—21no—
54C2×M16—22no—
55C22×C8—21no—
56C22×C8—22no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8C2312yes—
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63#125—21no—
64C22⋊Q8—21no—
65C22⋊Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4D411yescharacteristic
71C4×D4D411yescharacteristic
72C4×D4C2311yescharacteristic
73C4×D4C2311yescharacteristic
74#117—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C22⋊C4—21no—
79C2×C42C2311yescharacteristic
80#107—21no—
81#107—21no—
82#106—22no—
83#106—22no—
84#106—22no—
85#106—22no—
86C4⋊C8—21no—
87C4⋊C8D412yes—
88C4⋊C8C2312yes—
89C4⋊C8C2312yes—
90C4⋊C8C2311yescharacteristic
91C4⋊C8C2311yescharacteristic
92#103—22no—
93#103—22no—
94#103—21no—
95#103—21no—
96#102—22no—
97#102—22no—
98#102—21no—
99#102—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8—22no—
103C22⋊C8—22no—
104C22⋊C8D411yescharacteristic
105C22⋊C8D411yescharacteristic
106#97—21no—
107#97—21no—
108C4×C8C2312yes—
109C24—21no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—41no—
114C2×Q8—82no—
115C2×D4—22no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4C2×D411yescharacteristic
119C2×D4C2×D411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—82no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4#4111yescharacteristic
130SD16—42no—
131SD16—42no—
132M16—82no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—42no—
138C2×C8—82no—
139C2×C8C2×D412yes—
140C2×C8C2×D412yes—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4C2×D411yescharacteristic
153C4⋊C4C2×D411yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—41no—
159C42—41no—
160C42C2×D411yescharacteristic
161C42C2×D411yescharacteristic
162C42#4111yescharacteristic
163C42C2411yescharacteristic
164C23—21no—
165C23—21no—
166C23—42no—
167C23—42no—
168C23—81no—
169C23#12111yescharacteristic
170Q8—41no—
171Q8—41no—
172Q8—42no—
173Q8—82no—
174Q8—82no—
175D4—21no—
176D4—21no—
177D4—21no—
178D4—42no—
179D4#13712yes—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—82no—
195C2×C4#12111yescharacteristic
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C8—22no—
202C8—22no—
203C8—41no—
204C8—42no—
205C8—41no—
206C8—42no—
207C22—21no—
208C22—21no—
209C22—22no—
210C22—41no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—84no—
216C22#47112yes—
217C22C2×#12111yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—81no—
225C4—81no—
226C4—82no—
227C4C2×#13711yescharacteristic
228C4#57511yescharacteristic
229C4#57712yes—
230C2—22no—
231C2—42no—
232C2—81no—
233C2#304211yescharacteristic
234C2#305611yescharacteristic
235C2C2×#47111yescharacteristic
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. #3229C2semidirect
5. #1703C2semidirect
6. #1691C2semidirect
7. #1683C2semidirect
8. #1671C2semidirect
9. #1573C2semidirect
10. #1558C2semidirect
11. #1552C2semidirect
15. #498C22semidirect
16. #487C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #333C22semidirect
46. #333C22semidirect
47. #332C22semidirect
48. #332C22semidirect

Maximal subgroups

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

subgroupindex
#32292
#31102
C2×#4762
#29742
#17032
#16912
#16832
#16712
#15732
#15582
#15522