Finite Group Databasebeta

← #27926

Subgroups — #27926

Order 256.

460 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic
3C2×#465C211yesmaximal, characteristic
4C22×#106C211yesmaximal, characteristic, complemented
5#2504C211yesmaximal, characteristic
6#2503C211yesmaximal, characteristic, complemented
7#2097C212yesmaximal, complemented
8#2092C212yesmaximal, complemented
9#1968C212yesmaximal, complemented
10#1846C212yesmaximal, complemented
11C8×C22⋊C4C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C22⋊Q8C2211yescharacteristic
15C2×#121C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18#484—24no—
19#480—24no—
20#465—24no—
21C2×#107C2212yes—
22C2×#107C2212yes—
23C2×#106—24no—
24C2×#106C2212yes—
25C2×#106C2211yescharacteristic
26C2×#106C2212yes—
27C2×#103C2212yes—
28C2×#103C2211yescharacteristic
29C2×#102C2212yes—
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C4×C8C2212yes—
33#400—22no—
34#397—22no—
35#391C2212yes—
36#390C2212yes—
37#386—22no—
38#382C2212yes—
39#382C2212yes—
40#381—22no—
41C4×C22⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C2×SD16—22no—
50C2×SD16—24no—
51C22×C8—24no—
52C22×C8D414yes—
53C22×C8C2311yescharacteristic
54C22×C8C2312yes—
55C22×C8C2311yescharacteristic
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58#121—22no—
59#121—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4D412yes—
75C2×C42C2312yes—
76#107—24no—
77#107—24no—
78#106—24no—
79#106—24no—
80#106—22no—
81#103—22no—
82#103—24no—
83#102—22no—
84#102—24no—
85C22⋊C8D414yes—
86C4×C8—24no—
87#95—24no—
88#95—24no—
89#95—24no—
90#95D412yes—
91C24—21no—
92C24C2×D411yescharacteristic
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—44no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—44no—
113C22×C4—82no—
114C22×C4C2×D414yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D412yes—
117C22×C4C2×D412yes—
118C22×C4C2411yescharacteristic
119SD16—44no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—28no—
124C2×C8C2×D414yes—
125C2×C8C2×D412yes—
126C2×C8C2×D412yes—
127C2×C8#4114yes—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—42no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C22⋊C4—24no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—84no—
143C42—22no—
144C42—22no—
145C23—22no—
146C23—22no—
147C23—41no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23#12712yes—
152C23#13514yes—
153C23C22×D411yescharacteristic
154Q8—41no—
155Q8—41no—
156Q8—42no—
157D4—41no—
158D4—41no—
159D4—42no—
160D4—84no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—42no—
178C2×C4—84no—
179C2×C4—84no—
180C2×C4#12712yes—
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4112yes—
184C2×C42−1+412yes—
185C8—22no—
186C8—24no—
187C8#12314yes—
188C22—28no—
189C22—22no—
190C22—22no—
191C22—84no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22#49514yes—
196C22C2×#12711yescharacteristic
197C22#54712yes—
198C22C2×#13512yes—
199C22#57712yes—
200C4—22no—
201C4—44no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4—82no—
206C4—81no—
207C4C2×#12311yescharacteristic
208C4#53611yescharacteristic
209C4#54712yes—
210C2—24no—
211C2—161no—
212C2#250111yescharacteristic
213C2C2×#49511yescharacteristic
214C2#315411yescharacteristic
215C2#333714yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
4. C22×#106C2semidirect
6. #2503C2semidirect
7. #2097C2semidirect
8. #2092C2semidirect
9. #1968C2semidirect
10. #1846C2semidirect
11. C8×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4802
C2×#4652
C22×#1062
#25042
#25032
#20972
#20922
#19682
#18462
C8×C22⋊C42