Finite Group Databasebeta

← #27664

Subgroups — #27664

Order 256.

398 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 127 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×#435C211yesmaximal, characteristic
2C2×#427C211yesmaximal, characteristic, complemented
3#2485C212yesmaximal
4#2087C211yesmaximal, characteristic, complemented
5#2086C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2075C211yesmaximal, characteristic, complemented
8#1943C212yesmaximal, complemented
9#1825C212yesmaximal, complemented
10#1809C212yesmaximal, complemented
11#1808C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16#435—24no—
17#427—24no—
18C2×#107C2212yes—
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#106C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#397—22no—
27#394—22no—
28#389C2212yes—
29#389C2212yes—
30#388—22no—
31#387—22no—
32#380C2212yes—
33#380C2212yes—
34#380C2212yes—
35#340C2211yescharacteristic
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41C22×D4D411yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—22no—
45C2×M16—24no—
46C22×C8—24no—
47C22×C8D414yes—
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C4×D4—22no—
53C4×D4—22no—
54#117—24no—
55#117—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—22no—
70C2×C42C2311yescharacteristic
71#107—22no—
72#107—22no—
73#107—24no—
74#106—24no—
75C4⋊C8—22no—
76C22⋊C8—22no—
77C22⋊C8—22no—
78#97—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84C24C2×D411yescharacteristic
85C24C2×D411yescharacteristic
86C2×D4—22no—
87C2×D4—42no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D412yes—
103C22×C4#4112yes—
104C22×C4C2411yescharacteristic
105M16—44no—
106C2×C8—22no—
107C2×C8—28no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8C2×D412yes—
113C2×C8C2×D412yes—
114C2×C8#4114yes—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—44no—
125C4⋊C4—82no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—44no—
129C22⋊C4—44no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C42—21no—
133C42—21no—
134C42—44no—
135C23—22no—
136C23—22no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—22no—
141C23—22no—
142C23—21no—
143C23—21no—
144C23#12112yes—
145C23C22×D411yescharacteristic
146D4—44no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
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—42no—
172C2×C4—42no—
173C2×C4#12112yes—
174C2×C4C22×D411yescharacteristic
175C2×C4C2×#4112yes—
176C2×C4C2×#4111yescharacteristic
177C2×C42−1+412yes—
178C8—22no—
179C8—41no—
180C8—41no—
181C8#12314yes—
182C22—22no—
183C22—22no—
184C22—22no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—41no—
190C22—41no—
191C22—41no—
192C22C2×#12111yescharacteristic
193C22#54712yes—
194C22#57512yes—
195C22#57611yescharacteristic
196C22#57811yescharacteristic
197C4—22no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—82no—
202C4—82no—
203C4—82no—
204C4—82no—
205C4C2×#12311yescharacteristic
206C4#53611yescharacteristic
207C4#55212yes—
208C2—42no—
209C2—81no—
210C2#248611yescharacteristic
211C2#305411yescharacteristic
212C2#306111yescharacteristic
213C2#334712yes—
214C2#334912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#427C2semidirect
4. #2087C2semidirect
5. #2086C2semidirect
6. #2076C2semidirect
7. #2075C2semidirect
8. #1943C2semidirect
9. #1825C2semidirect
10. #1809C2semidirect
11. #1808C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4352
C2×#4272
#24852
#20872
#20862
#20762
#20752
#19432
#18252
#18092
#18082