Finite Group Databasebeta

← #29844

Subgroups — #29844

Order 256.

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

normal subgroups only

This group has 425 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×#487C211yesmaximal, characteristic, complemented
2C2×#431C211yesmaximal, characteristic
3#2857C211yesmaximal, characteristic
4#2857C211yesmaximal, characteristic
5#2068C214yesmaximal, complemented
6#2036C214yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1845C211yesmaximal, characteristic, complemented
10C2×C4⋊Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C22⋊Q8—21no—
13C2×C22⋊Q8C2212yes—
14C22×C42C2211yescharacteristic
15#487—24no—
16#431—24no—
17C2×#107C2214yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2214yes—
20C2×#103C2214yes—
21C2×C22⋊C8C2212yes—
22C2×C4×C8C2211yescharacteristic
23C2×C4×C8C2211yescharacteristic
24#391—21no—
25#391C2212yes—
26#389—21no—
27#389—21no—
28#389—21no—
29#389C2212yes—
30#384C2211yescharacteristic
31#384C2211yescharacteristic
32#380—21no—
33#380—21no—
34#380—21no—
35#380C2212yes—
36#340C2214yes—
37C22×Q8—21no—
38C22×Q8C2312yes—
39C23×C4D411yescharacteristic
40C23×C4D411yescharacteristic
41C23×C4C2311yescharacteristic
42C22×C8C2312yes—
43C22×C8C2312yes—
44C4⋊Q8—22no—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C22⋊Q8—44no—
49C22⋊Q8—44no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—22no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C42—22no—
65C2×C42—22no—
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—44no—
71C4⋊C8—24no—
72#103—28no—
73#103—44no—
74C22⋊C8—24no—
75C4×C8—22no—
76C4×C8D414yes—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24C2×D411yescharacteristic
82C2×Q8—22no—
83C2×Q8—22no—
84C2×Q8—24no—
85C2×Q8—24no—
86C2×Q8—42no—
87C2×Q8—42no—
88C2×Q8—42no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4Q1614yes—
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8C2×D414yes—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—24no—
118C4⋊C4—24no—
119C4⋊C4—44no—
120C4⋊C4—44no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C42—22no—
131C42—22no—
132C42—21no—
133C42—21no—
134C42C2×D412yes—
135C42#4112yes—
136C23—21no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—21no—
142C23C2×Q1612yes—
143C23C22×D411yescharacteristic
144Q8—44no—
145Q8—44no—
146Q8—84no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—28no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—44no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4#12112yes—
172C2×C4#12112yes—
173C2×C4C2×Q1612yes—
174C2×C4C2×Q1612yes—
175C2×C4#13512yes—
176C2×C4AGL(1,ℤ/8)12yes—
177C2×C4#13712yes—
178C2×C4C22×D411yescharacteristic
179C2×C4C2×#4111yescharacteristic
180C2×C4H(5,𝔽2)12yes—
181C2×C42−1+412yes—
182C8—24no—
183C8—42no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22C2×#12111yescharacteristic
192C22#53511yescharacteristic
193C22#53611yescharacteristic
194C22C22×Q1611yescharacteristic
195C22C2×#13511yescharacteristic
196C22C2×AGL(1,ℤ/8)11yescharacteristic
197C22C2×#13711yescharacteristic
198C4—22no—
199C4—41no—
200C4—41no—
201C4—82no—
202C4—82no—
203C4—82no—
204C4—82no—
205C4#46714yes—
206C4#46814yes—
207C4#53712yes—
208C4#54212yes—
209C2—42no—
210C2#262011yescharacteristic
211C2C2×#46711yescharacteristic
212C2C2×#46811yescharacteristic
213C2#320711yescharacteristic
214C2#320911yescharacteristic
215C2#323011yescharacteristic
216C2#323111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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×#487C2semidirect
5. #2068C2semidirect
7. #1990C2semidirect
8. #1956C2semidirect
9. #1845C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4312
#28572
#20682
#20362
#19902
#19562
#18452