Finite Group Databasebeta

← #27803

Subgroups — #27803

Order 256.

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

normal subgroups only

This group has 396 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×#498C211yesmaximal, characteristic, complemented
2C2×#470C211yesmaximal, characteristic
3#2492C212yesmaximal
4#2035C212yesmaximal
5#2024C212yesmaximal, complemented
6#1971C211yesmaximal, characteristic, complemented
7#1969C211yesmaximal, characteristic, complemented
8#1927C212yesmaximal, complemented
9#1809C211yesmaximal, characteristic
10#1776C211yesmaximal, characteristic, complemented
11#1753C211yesmaximal, characteristic
12C22×Q16C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C2×C4⋊Q8C2212yes—
15C2×C22⋊Q8C2212yes—
16#498—24no—
17#470—24no—
18C2×#107C2211yescharacteristic
19C2×#106C2211yescharacteristic
20C2×#106C2212yes—
21C2×#106C2211yescharacteristic
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#393—22no—
30#391—22no—
31#391C2212yes—
32#391C2212yes—
33#388—22no—
34#380C2212yes—
35C4×C4⋊C4C2212yes—
36C4×C22⋊C4C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#336C2211yescharacteristic
40#336C2211yescharacteristic
41C22×Q8—22no—
42C22×Q8C2312yes—
43C23×C4C2311yescharacteristic
44C2×Q16—22no—
45C2×Q16—22no—
46C2×Q16—22no—
47C2×M16—22no—
48C2×M16—22no—
49C2×M16—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C22×C8C2311yescharacteristic
54C4⋊Q8—24no—
55C4⋊Q8—42no—
56C22⋊Q8—24no—
57C22⋊Q8—42no—
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#106—22no—
71#106—24no—
72#106D414yes—
73#103—24no—
74#103—24no—
75#103—42no—
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×Q8—22no—
93C2×Q8—44no—
94C2×Q8—44no—
95C2×Q8—82no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D412yes—
109C22×C4C2411yescharacteristic
110Q16—44no—
111M16—44no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8C2×D412yes—
121C2×C8#4112yes—
122C2×C8#4112yes—
123C2×C8#4112yes—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—44no—
130C4⋊C4—44no—
131C4⋊C4C2×D414yes—
132C22⋊C4—24no—
133C22⋊C4—44no—
134C22⋊C4—82no—
135C42—21no—
136C42—21no—
137C42—21no—
138C42—21no—
139C42—82no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—22no—
145C23C22×D411yescharacteristic
146Q8—42no—
147Q8—42no—
148Q8—42no—
149Q8—42no—
150Q8—88no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—24no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4#13714yes—
176C2×C4C22×D411yescharacteristic
177C2×C4C2×#4111yescharacteristic
178C2×C4C2×#4111yescharacteristic
179C2×C4C2×#4111yescharacteristic
180C2×C4H(5,𝔽2)11yescharacteristic
181C2×C42−1+411yescharacteristic
182C8—22no—
183C8—41no—
184C8—41no—
185C8#12414yes—
186C22—42no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—42no—
191C22#53211yescharacteristic
192C22D4211yescharacteristic
193C22#54711yescharacteristic
194C22C2×#13712yes—
195C22#57712yes—
196C4—24no—
197C4—42no—
198C4—41no—
199C4—44no—
200C4—82no—
201C4—82no—
202C4—82no—
203C4C2×#12411yescharacteristic
204C4#54211yescharacteristic
205C4#54811yescharacteristic
206C4#55511yescharacteristic
207C2—81no—
208C2#249111yescharacteristic
209C2#317311yescharacteristic
210C2#318511yescharacteristic
211C2#326612yes—
212C2#335612yes—
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
1. C2×#498C2semidirect
5. #2024C2semidirect
6. #1971C2semidirect
7. #1969C2semidirect
8. #1927C2semidirect
10. #1776C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4982
C2×#4702
#24922
#20352
#20242
#19712
#19692
#19272
#18092
#17762
#17532