Finite Group Databasebeta

← #27665

Subgroups — #27665

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#2088C212yesmaximal, complemented
5#2077C212yesmaximal, complemented
6#1944C211yesmaximal, characteristic, complemented
7#1944C211yesmaximal, characteristic, complemented
8#1825C212yesmaximal, complemented
9#1809C211yesmaximal, characteristic, complemented
10#1809C211yesmaximal, characteristic, complemented
11#1807C211yesmaximal, characteristic, complemented
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C2×#117C2212yes—
16#435—24no—
17#427—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#106C2212yes—
23C2×C4⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#397—22no—
28#394—22no—
29#389C2212yes—
30#389C2212yes—
31#388—22no—
32#387—22no—
33#380C2212yes—
34#380C2212yes—
35#380C2212yes—
36#340C2212yes—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41C22×D4D411yescharacteristic
42C23×C4C2311yescharacteristic
43C23×C4C2311yescharacteristic
44C2×M16—22no—
45C2×M16—22no—
46C2×M16—22no—
47C22×C8—22no—
48C22×C8—22no—
49C22×C8D414yes—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C4×D4—22no—
55C4×D4—22no—
56#117—24no—
57#117—24no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C22⋊C4D411yescharacteristic
70C2×C22⋊C4D411yescharacteristic
71C2×C42—22no—
72C2×C42C2311yescharacteristic
73#107—22no—
74#107—22no—
75#106—24no—
76#106—22no—
77#106—22no—
78C4⋊C8—22no—
79C22⋊C8—22no—
80C22⋊C8—22no—
81#97—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—22no—
87C24C2×D411yescharacteristic
88C24C2×D411yescharacteristic
89C2×D4—22no—
90C2×D4—42no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4#4112yes—
111C22×C4C2411yescharacteristic
112M16—44no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—28no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8C2×D412yes—
123C2×C8C2×D412yes—
124C2×C8#4112yes—
125C2×C8#4112yes—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—44no—
136C4⋊C4—82no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—44no—
140C22⋊C4—44no—
141C22⋊C4—82no—
142C22⋊C4—82no—
143C42—21no—
144C42—21no—
145C42—44no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—22no—
150C23—21no—
151C23—21no—
152C23—21no—
153C23—22no—
154C23—22no—
155C23#12112yes—
156C23C22×D411yescharacteristic
157D4—44no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4#12112yes—
188C2×C4C22×D411yescharacteristic
189C2×C4C2×#4111yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4C2×#4111yescharacteristic
192C2×C42−1+411yescharacteristic
193C2×C42−1+411yescharacteristic
194C8—22no—
195C8—41no—
196C8—41no—
197C8#12314yes—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22C2×#12111yescharacteristic
209C22#54711yescharacteristic
210C22#54711yescharacteristic
211C22#57512yes—
212C22#57712yes—
213C4—22no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4—82no—
218C4—82no—
219C4—82no—
220C4—82no—
221C4C2×#12311yescharacteristic
222C4#53611yescharacteristic
223C4#55211yescharacteristic
224C4#55211yescharacteristic
225C2—42no—
226C2—81no—
227C2#248611yescharacteristic
228C2#305411yescharacteristic
229C2#306011yescharacteristic
230C2#334812yes—
231C2#335012yes—
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. #2088C2semidirect
5. #2077C2semidirect
6. #1944C2semidirect
7. #1944C2semidirect
8. #1825C2semidirect
9. #1809C2semidirect
10. #1809C2semidirect
11. #1807C2semidirect

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
#20882
#20772
#19442
#18252
#18092
#18072