Finite Group Databasebeta

← #27911

Subgroups — #27911

Order 256.

460 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 139 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#2096C211yesmaximal, characteristic, complemented
6#2095C211yesmaximal, characteristic, complemented
7#2094C211yesmaximal, characteristic, complemented
8#2093C211yesmaximal, characteristic, complemented
9#1958C212yesmaximal, complemented
10#1846C212yesmaximal, complemented
11C4×C22⋊C8C211yesmaximal, 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×C4×C8C2212yes—
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#388—22no—
32#386—22no—
33#384C2211yescharacteristic
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#382C2212yes—
37#380—22no—
38C4×C4⋊C4—22no—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2311yescharacteristic
42C22×D4—21no—
43C22×D4C2311yescharacteristic
44C23×C4D412yes—
45C23×C4C2311yescharacteristic
46C22×C8C2312yes—
47C22×C8C2312yes—
48#124—24no—
49C22⋊Q8—24no—
50#121—24no—
51#117—24no—
52#117—24no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2312yes—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C22⋊C4—22no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C42—24no—
66C2×C42—22no—
67C2×C42D412yes—
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—24no—
71#107—24no—
72#106—24no—
73#106—24no—
74#103—24no—
75#103—24no—
76#102—24no—
77#102—24no—
78C22⋊C8D418yes—
79C4×C8—24no—
80#95—24no—
81#95—22no—
82#95—22no—
83C24—21no—
84C24C2×D411yescharacteristic
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—44no—
94C2×D4—42no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—42no—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D414yes—
109C22×C4C2×D412yes—
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8C2×D414yes—
116C2×C8C2×D414yes—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—44no—
124C4⋊C4—42no—
125C4⋊C4—44no—
126C22⋊C4—44no—
127C22⋊C4—42no—
128C22⋊C4—44no—
129C22⋊C4—42no—
130C22⋊C4—44no—
131C42—24no—
132C42—22no—
133C42—44no—
134C42#4114yes—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—42no—
144C23#12712yes—
145C23C22×D411yescharacteristic
146Q8—42no—
147Q8—42no—
148D4—42no—
149D4—42no—
150D4—84no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—48no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—44no—
170C2×C4—42no—
171C2×C4#12314yes—
172C2×C4#12712yes—
173C2×C4#13514yes—
174C2×C4C22×D412yes—
175C2×C4C2×#4112yes—
176C2×C42−1+411yescharacteristic
177C2×C42−1+411yescharacteristic
178C8—24no—
179C8—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22—81no—
187C22—81no—
188C22—82no—
189C22—81no—
190C22C2×#12311yescharacteristic
191C22C2×#12711yescharacteristic
192C22#53611yescharacteristic
193C22C2×#13512yes—
194C22#57512yes—
195C4—24no—
196C4—24no—
197C4—22no—
198C4—82no—
199C4—81no—
200C4—82no—
201C4—81no—
202C4—81no—
203C4#54712yes—
204C4#54712yes—
205C2—42no—
206C2—161no—
207C2#250111yescharacteristic
208C2#309112yes—
209C2#314711yescharacteristic
210C2#315311yescharacteristic
211C2#323312yes—
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. #2096C2semidirect
6. #2095C2semidirect
7. #2094C2semidirect
8. #2093C2semidirect
9. #1958C2semidirect
10. #1846C2semidirect
11. C4×C22⋊C8C2semidirect

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
#20962
#20952
#20942
#20932
#19582
#18462
C4×C22⋊C82