Finite Group Databasebeta

← #27915

Subgroups — #27915

Order 256.

454 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 135 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×#485C212yesmaximal, complemented
2C2×#482C212yesmaximal
3#2850C211yesmaximal, characteristic
4#2285C211yesmaximal, characteristic, complemented
5#2097C212yesmaximal, complemented
6#2092C212yesmaximal, complemented
7#1959C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic, complemented
9#1847C211yesmaximal, characteristic, complemented
10#1847C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C2×#117C2212yes—
16C22×C42C2211yescharacteristic
17#485—28no—
18#482—28no—
19C2×#107C2212yes—
20C2×#107C2212yes—
21C2×#106C2212yes—
22C2×#106C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#388—22no—
33#386—22no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#382C2211yescharacteristic
37#382C2212yes—
38#380—22no—
39C4×C4⋊C4—22no—
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4D412yes—
45C23×C4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48#124—22no—
49#124—22no—
50C22⋊Q8—24no—
51#121—24no—
52#117—24no—
53#117—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—22no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C42—22no—
67C2×C42—22no—
68C2×C42—22no—
69C2×C42D412yes—
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—24no—
73#107—24no—
74#106—24no—
75#106—24no—
76#103—24no—
77#103—24no—
78#102—24no—
79#102—24no—
80C22⋊C8D418yes—
81#97—22no—
82#97—22no—
83#95—24no—
84#95—22no—
85#95—22no—
86C24—21no—
87C24C2×D411yescharacteristic
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×D4—22no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—44no—
97C2×D4—42no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—42no—
108C22×C4—42no—
109C22×C4C2×D412yes—
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D414yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8C2×D414yes—
119C2×C8C2×D414yes—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—44no—
127C4⋊C4—44no—
128C4⋊C4—42no—
129C22⋊C4—44no—
130C22⋊C4—44no—
131C22⋊C4—42no—
132C22⋊C4—44no—
133C22⋊C4—42no—
134C42—24no—
135C42—21no—
136C42—21no—
137C42—44no—
138C42#4112yes—
139C42#4112yes—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—42no—
149C23#12712yes—
150C23C22×D411yescharacteristic
151Q8—42no—
152Q8—42no—
153D4—42no—
154D4—42no—
155D4—84no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—48no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4#12314yes—
178C2×C4#12712yes—
179C2×C4C22×D412yes—
180C2×C4C2×#4111yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C2×C42−1+412yes—
183C8—24no—
184C8—24no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—81no—
192C22—81no—
193C22—82no—
194C22—81no—
195C22C2×#12311yescharacteristic
196C22C2×#12711yescharacteristic
197C22#53611yescharacteristic
198C22#57512yes—
199C22#57512yes—
200C4—24no—
201C4—22no—
202C4—42no—
203C4—82no—
204C4—81no—
205C4—82no—
206C4—81no—
207C4—81no—
208C4#54712yes—
209C4#54712yes—
210C2—42no—
211C2—161no—
212C2#250111yescharacteristic
213C2#309312yes—
214C2#315011yescharacteristic
215C2#315311yescharacteristic
216C2#324012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#485C2semidirect
4. #2285C2semidirect
5. #2097C2semidirect
6. #2092C2semidirect
7. #1959C2semidirect
8. #1959C2semidirect
9. #1847C2semidirect
10. #1847C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4822
#28502
#22852
#20972
#20922
#19592
#18472
#17492