Finite Group Databasebeta

← #28662

Subgroups — #28662

Order 256.

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

normal subgroups only

This group has 370 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×#498C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic
3#2640C211yesmaximal, characteristic
4#2331C211yesmaximal, characteristic, complemented
5#2066C212yesmaximal, complemented
6#2065C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic
9#1927C212yesmaximal, complemented
10#1926C212yesmaximal
11#1768C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124C2211yescharacteristic
15#498—24no—
16#489—24no—
17C2×#106C2212yes—
18C2×#106C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#97C2211yescharacteristic
25C2×C4×C8C2211yescharacteristic
26#397—22no—
27#393—22no—
28#391—22no—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#389C2212yes—
34#387—22no—
35#382C2211yescharacteristic
36C4×C4⋊C4C2212yes—
37#376C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8—21no—
43C22×Q8C2311yescharacteristic
44C22×D4C2311yescharacteristic
45C22×C8C2312yes—
46C22×C8C2312yes—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8D414yes—
51#124—22no—
52#124—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4C2311yescharacteristic
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C42—22no—
65C2×C42C2311yescharacteristic
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68#106—24no—
69#106—24no—
70C4⋊C8—24no—
71#103—24no—
72#103—24no—
73#102—24no—
74#102—24no—
75#97—22no—
76C4×C8—22no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95—22no—
81#95—22no—
82#95—22no—
83#95D412yes—
84#95D412yes—
85C24—21no—
86C2×Q8—24no—
87C2×Q8—21no—
88C2×Q8—22no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—42no—
92C2×Q8—82no—
93C2×Q8—82no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—21no—
97C2×D4—22no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4C2×D412yes—
104C22×C4C2×D412yes—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8#4114yes—
114C4⋊C4—24no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—42no—
123C4⋊C4—82no—
124C4⋊C4C2×D414yes—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C22⋊C4—82no—
128C22⋊C4—82no—
129C42—22no—
130C42—22no—
131C42—82no—
132C42C2×D412yes—
133C42#4112yes—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23C22×D411yescharacteristic
140Q8—42no—
141Q8—42no—
142Q8—84no—
143Q8—84no—
144D4—42no—
145D4—42no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4#12112yes—
168C2×C4#12112yes—
169C2×C4C22×D411yescharacteristic
170C2×C4C2×#4112yes—
171C2×C4C2×#4111yescharacteristic
172C2×C42−1+412yes—
173C8—24no—
174C8—42no—
175C22—82no—
176C22—82no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22C2×#12111yescharacteristic
181C22#53211yescharacteristic
182C22#53611yescharacteristic
183C22#57712yes—
184C22#57712yes—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—44no—
190C4—82no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4#54812yes—
195C4#55612yes—
196C2—161no—
197C2#254511yescharacteristic
198C2#304211yescharacteristic
199C2#304311yescharacteristic
200C2#317312yes—
201C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#498C2semidirect
4. #2331C2semidirect
5. #2066C2semidirect
7. #1993C2semidirect
9. #1927C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4982
C2×#4892
#26402
#23312
#20662
#20652
#19932
#19582
#19272
#19262
#17682