Finite Group Databasebeta

← #28517

Subgroups — #28517

Order 256.

427 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 427 conjugacy classes of subgroups — too many to draw. Only its 143 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×#479C212yesmaximal, complemented
2C2×#478C212yesmaximal
3#2859C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic
5#2097C212yesmaximal, complemented
6#2092C212yesmaximal
7#1959C211yesmaximal, characteristic
8#1959C211yesmaximal, characteristic
9#1847C211yesmaximal, characteristic, complemented
10#1847C211yesmaximal, characteristic, complemented
11#1767C211yesmaximal, characteristic, complemented
12C2×#125C2212yes—
13C2×#124C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#479—28no—
17#478—28no—
18C2×#107C2212yes—
19C2×#107C2212yes—
20C2×#106C2212yes—
21C2×#106C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#102C2212yes—
26C2×#102C2212yes—
27C2×#97C2211yescharacteristic
28C2×#97C2211yescharacteristic
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#389—22no—
32#389—22no—
33#388—22no—
34#384C2211yescharacteristic
35#384C2211yescharacteristic
36#382C2212yes—
37#382C2211yescharacteristic
38C4×C22⋊C4—22no—
39C43C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×D4C2311yescharacteristic
43C23×C4—21no—
44C22×C8C2312yes—
45C22×C8C2312yes—
46#125—24no—
47#125—24no—
48#124—22no—
49#124Q814yes—
50C4×Q8—24no—
51C4×D4—24no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2311yescharacteristic
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2312yes—
60C2×C4⋊C4C2312yes—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C42—21no—
65C2×C42D412yes—
66C2×C42D412yes—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69C2×C42C2311yescharacteristic
70#107—24no—
71#107—24no—
72#106—24no—
73#106—24no—
74C4⋊C8D418yes—
75#103—24no—
76#103—24no—
77#102—24no—
78#102—24no—
79#97—22no—
80#97—22no—
81#95—22no—
82#95—22no—
83#95—24no—
84C24—21no—
85C2×Q8—21no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8C2×Q812yes—
89C2×D4—22no—
90C2×D4—21no—
91C2×D4—22no—
92C2×D4C2×Q812yes—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—44no—
99C22×C4—42no—
100C22×C4C2×D412yes—
101C22×C4C2×D412yes—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2411yescharacteristic
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C2×C8C2×D414yes—
110C2×C8C2×D414yes—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—22no—
116C4⋊C4—22no—
117C4⋊C4—44no—
118C4⋊C4—44no—
119C4⋊C4—42no—
120C4⋊C4—44no—
121C4⋊C4—44no—
122C22⋊C4—28no—
123C22⋊C4—44no—
124C22⋊C4—42no—
125C42—22no—
126C42—42no—
127C42—42no—
128C42—42no—
129C42C2×D414yes—
130C42C2×Q812yes—
131C42#4112yes—
132C23—22no—
133C23—41no—
134C23—41no—
135C23—42no—
136C23—42no—
137C23C22×D411yescharacteristic
138Q8—24no—
139Q8—42no—
140D4—24no—
141D4—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—44no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4C22⋊Q814yes—
162C2×C4#12712yes—
163C2×C4#12712yes—
164C2×C4C22×D412yes—
165C2×C4C22×Q811yescharacteristic
166C2×C4C2×#4111yescharacteristic
167C2×C42−1+412yes—
168C8—24no—
169C8—24no—
170C22—44no—
171C22—42no—
172C22—81no—
173C22—81no—
174C22—82no—
175C22C2×C22⋊Q811yescharacteristic
176C22C2×#12711yescharacteristic
177C22#53611yescharacteristic
178C22#57512yes—
179C22#57512yes—
180C4—24no—
181C4—22no—
182C4—22no—
183C4—42no—
184C4—42no—
185C4—81no—
186C4—82no—
187C4—82no—
188C4—81no—
189C4#54712yes—
190C4D4×Q812yes—
191C2—82no—
192C2#254211yescharacteristic
193C2#308012yes—
194C2#315011yescharacteristic
195C2#315311yescharacteristic
196C2#324012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#479C2semidirect
3. #2859C2semidirect
5. #2097C2semidirect
9. #1847C2semidirect
10. #1847C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4792
C2×#4782
#28592
C4×#1242
#20972
#20922
#19592
#18472
#17672