Finite Group Databasebeta

← #23698

Subgroups — #23698

Order 256.

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

normal subgroups only

This group has 447 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×C4×Q16C211yesmaximal, characteristic, complemented
2C2×#431C211yesmaximal, characteristic
3C4×C22⋊Q8C212yesmaximal
4#2087C211yesmaximal, characteristic, complemented
5#2076C211yesmaximal, characteristic
6#2068C212yesmaximal, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2036C212yesmaximal
9#2001C211yesmaximal, characteristic
10C4×#107C211yesmaximal, characteristic, complemented
11C4×#103C212yesmaximal, complemented
12C22×Q16C2211yescharacteristic
13C2×C22⋊Q8C2212yes—
14C2×C4×Q8C2212yes—
15C22×C42C2211yescharacteristic
16C4×Q16—24no—
17#431—24no—
18C2×#107C2212yes—
19C2×#107C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28C2×C4×C8C2211yescharacteristic
29#391C2212yes—
30#389C2212yes—
31#387—22no—
32#385—22no—
33#380C2212yes—
34C4×C4⋊C4—22no—
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4—22no—
37#340C2211yescharacteristic
38#340C2212yes—
39#340C2211yescharacteristic
40#340C2212yes—
41C22×Q8C2312yes—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×Q16—24no—
46C2×Q16—22no—
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22⋊Q8—24no—
52C22⋊Q8—24no—
53C4×Q8—24no—
54C4×Q8—24no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C42—22no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42—22no—
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71#107—22no—
72#107—24no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75#103—24no—
76#103—24no—
77#103—24no—
78C22⋊C8—22no—
79C22⋊C8—22no—
80C4×C8—22no—
81C4×C8D414yes—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87C24C2×D411yescharacteristic
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8#4114yes—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4Q1614yescomplemented
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2411yescharacteristic
113Q16—28no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8C2×D412yes—
121C2×C8C2×D412yes—
122C4⋊C4—28no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4#4114yes—
131C22⋊C4—28no—
132C22⋊C4—44no—
133C42—22no—
134C42—22no—
135C42—21no—
136C42—21no—
137C42—44no—
138C42—44no—
139C42C2×D412yes—
140C42#4112yes—
141C23—21no—
142C23—22no—
143C23—21no—
144C23—22no—
145C23C2×Q1612yes—
146C23C22×D411yescharacteristic
147Q8—24no—
148Q8—24no—
149Q8—42no—
150Q8—42no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—28no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—44no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#12112yes—
176C2×C4#12112yes—
177C2×C4C2×Q1612yes—
178C2×C4C2×Q1612yes—
179C2×C4#13512yes—
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4112yes—
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4112yes—
184C8—22no—
185C8—22no—
186C8—41no—
187C8—41no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—41no—
192C22—41no—
193C22C2×#12111yescharacteristic
194C22#52512yes—
195C22C22×Q1611yescharacteristic
196C22C2×#13511yescharacteristic
197C22#57512yes—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—41no—
202C4—44no—
203C4—41no—
204C4—82no—
205C4—82no—
206C4#46714yes—
207C4#52511yescharacteristic
208C4#52912yes—
209C4#53211yescharacteristic
210C2—42no—
211C2#239211yescharacteristic
212C2C2×#46711yescharacteristic
213C2#305411yescharacteristic
214C2#310212yes—
215C2#310412yes—
subgroupG/H
centerC22×C4C2×D4
commutator, Frattini, cosocleC22×C4C24
socleC23C22×D4
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×Q16C2semidirect
4. #2087C2semidirect
6. #2068C2semidirect
7. #2056C2semidirect
10. C4×#107C2semidirect
11. C4×#103C2semidirect
106. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q162
C2×#4312
C4×C22⋊Q82
#20872
#20762
#20682
#20562
#20362
#20012
C4×#1072
C4×#1032