Finite Group Databasebeta

← #25049

Subgroups — #25049

Order 256.

388 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 123 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×#470C211yesmaximal, characteristic
2C2×#441C211yesmaximal, characteristic, complemented
3#2421C212yesmaximal
4#2074C212yesmaximal, complemented
5#2036C212yesmaximal
6#1969C211yesmaximal, characteristic, complemented
7#1943C211yesmaximal, characteristic, complemented
8#1863C212yesmaximal, complemented
9#1846C211yesmaximal, characteristic, complemented
10#1809C211yesmaximal, characteristic
11#1753C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C2×C4×Q8C2212yes—
16#470—24no—
17#441—24no—
18C2×#107C2211yescharacteristic
19C2×#107C2212yes—
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#397—22no—
30#391C2212yes—
31#389C2212yes—
32#388—22no—
33#387—22no—
34#382C2212yes—
35#380C2212yes—
36C4×C22⋊C4C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8C2312yes—
42C23×C4C2311yescharacteristic
43C2×Q16—22no—
44C2×Q16—22no—
45C2×Q16D414yes—
46C2×M16—22no—
47C2×M16—22no—
48C2×M16—22no—
49C22×C8C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22⋊Q8—24no—
54C22⋊Q8—42no—
55C4×Q8—24no—
56C4×Q8—42no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4D412yes—
66C2×C42—22no—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—22no—
70#107—42no—
71#106—22no—
72#106—22no—
73#103—24no—
74#103—24no—
75#103—24no—
76C22⋊C8—22no—
77#97—22no—
78C4×C8—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95D412yes—
86C24C2×D411yescharacteristic
87C2×Q8—22no—
88C2×Q8—22no—
89C2×Q8—22no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8C2×D414yes—
93C22×C4—22no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2411yescharacteristic
107Q16—28no—
108M16—44no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8C2×D412yes—
118C2×C8#4112yes—
119C2×C8#4112yes—
120C2×C8#4112yes—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—44no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C22⋊C4—42no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C42—21no—
134C42—21no—
135C42—21no—
136C42—21no—
137C42—44no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23—22no—
142C23—22no—
143C23C22×D411yescharacteristic
144Q8—24no—
145Q8—24no—
146Q8—42no—
147Q8—42no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
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×C4C22×D411yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)11yescharacteristic
178C2×C42−1+411yescharacteristic
179C8—22no—
180C8—41no—
181C8—41no—
182C8#12314yes—
183C22—42no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—42no—
188C22#53211yescharacteristic
189C22D4211yescharacteristic
190C22#54711yescharacteristic
191C22#57512yes—
192C22#57812yes—
193C4—42no—
194C4—41no—
195C4—44no—
196C4—42no—
197C4—82no—
198C4—82no—
199C4—82no—
200C4C2×#12311yescharacteristic
201C4#54211yescharacteristic
202C4#54711yescharacteristic
203C4#55311yescharacteristic
204C2—81no—
205C2#242211yescharacteristic
206C2#317711yescharacteristic
207C2#318311yescharacteristic
208C2#327712yes—
209C2#334912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#441C2semidirect
4. #2074C2semidirect
6. #1969C2semidirect
7. #1943C2semidirect
8. #1863C2semidirect
9. #1846C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4412
#24212
#20742
#20362
#19692
#19432
#18632
#18462
#18092
#17532