Finite Group Databasebeta

← #28628

Subgroups — #28628

Order 256.

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

normal subgroups only

This group has 416 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×#497C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3#2640C211yesmaximal, characteristic
4#2327C211yesmaximal, characteristic, complemented
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic
9#1880C212yesmaximal
10#1878C212yesmaximal, complemented
11#1768C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2212yes—
18#497—24no—
19#487—24no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×#97C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#397—22no—
28#393—22no—
29#391C2212yes—
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#387—22no—
35#387—22no—
36#382C2211yescharacteristic
37#376C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C2×Q16—24no—
46C2×Q16—24no—
47C2×Q16—42no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—42no—
51C22×C8C2312yes—
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8D414yes—
57#124—22no—
58#124—22no—
59C4×Q8—42no—
60C4×Q8—42no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—22no—
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C4⋊C8—24no—
77#103—24no—
78#103—24no—
79#103—42no—
80#102—42no—
81#97—22no—
82C4×C8—22no—
83#95—21no—
84#95—21no—
85#95—24no—
86#95—22no—
87#95—22no—
88#95D412yes—
89#95D412yes—
90C24—21no—
91C2×Q8—22no—
92C2×Q8—21no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—42no—
101C2×Q8—82no—
102C2×Q8C2×D414yes—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4C2×D412yes—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117Q16—48no—
118SD16—48no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8#4114yes—
125C4⋊C4—24no—
126C4⋊C4—24no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—44no—
134C4⋊C4—42no—
135C22⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—82no—
138C22⋊C4—82no—
139C42—22no—
140C42—22no—
141C42—44no—
142C42C2×D412yes—
143C42#4112yes—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—42no—
148C23—42no—
149C23C22×D411yescharacteristic
150Q8—24no—
151Q8—24no—
152Q8—42no—
153Q8—42no—
154Q8—42no—
155Q8—42no—
156Q8—84no—
157D4—42no—
158D4—42no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—24no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4#12112yes—
181C2×C4#12112yes—
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4112yes—
184C2×C4C2×#4111yescharacteristic
185C2×C4H(5,𝔽2)12yes—
186C8—24no—
187C8—42no—
188C22—82no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22C2×#12111yescharacteristic
194C22#53211yescharacteristic
195C22#53511yescharacteristic
196C22#57712yes—
197C22#57812yes—
198C4—22no—
199C4—22no—
200C4—42no—
201C4—42no—
202C4—44no—
203C4—82no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4#54612yes—
208C4#55912yes—
209C2—161no—
210C2#254411yescharacteristic
211C2#304311yescharacteristic
212C2#304411yescharacteristic
213C2#317812yes—
214C2#322512yes—
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
2. C2×#487C2semidirect
4. #2327C2semidirect
5. #2056C2semidirect
7. #1993C2semidirect
10. #1878C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4872
#26402
#23272
#20562
#20552
#19932
#19582
#18802
#18782
#17682