Finite Group Databasebeta

← #29643

Subgroups — #29643

Order 256.

507 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×#125C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic
5#2059C214yesmaximal, complemented
6#2026C214yesmaximal, complemented
7#1824C211yesmaximal, characteristic
8#1823C211yesmaximal, characteristic
9#1817C211yesmaximal, characteristic
10C2×#125—24no—
11C2×#125—24no—
12C2×#125—22no—
13C2×#125C2214yes—
14C2×C22⋊Q8C2211yescharacteristic
15C22×C4⋊C4—21no—
16C22×C4⋊C4—21no—
17C22×C4⋊C4C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#484—24no—
21#483—24no—
22C2×#107C2212yes—
23C2×#106C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#391C2212yes—
29#389C2212yes—
30#387—22no—
31#385—21no—
32#381—21no—
33#380C2211yescharacteristic
34#340C2214yes—
35#340C2214yes—
36#336C2214yes—
37C22×Q8C2311yescharacteristic
38C23×C4—21no—
39C23×C4—21no—
40C23×C4D411yescharacteristic
41C23×C4D411yescharacteristic
42C23×C4C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2312yes—
45#125—216no—
46#125—28no—
47#125—28no—
48C22⋊Q8—22no—
49C22⋊Q8—22no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—28no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D414yes—
61C2×C4⋊C4D414yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C42—22no—
69C2×C42—22no—
70C2×C42D412yes—
71C2×C42C2312yes—
72#107—24no—
73#106—24no—
74#103—24no—
75#103—24no—
76C22⋊C8—22no—
77C22⋊C8—22no—
78#95—21no—
79#95—22no—
80#95—22no—
81#95—22no—
82C24C2×D411yescharacteristic
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—24no—
86C22×C4—24no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—24no—
90C22×C4—24no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—42no—
99C22×C4—44no—
100C22×C4—42no—
101C22×C4—42no—
102C22×C4C2×D412yes—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—24no—
111C2×C8—24no—
112C4⋊C4—216no—
113C4⋊C4—216no—
114C4⋊C4—24no—
115C4⋊C4—24no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—24no—
121C4⋊C4—44no—
122C4⋊C4—42no—
123C4⋊C4C2×D414yes—
124C4⋊C4C2×D414yes—
125C22⋊C4—42no—
126C22⋊C4—42no—
127C42—24no—
128C42—22no—
129C42—22no—
130C23—22no—
131C23—22no—
132C23#12012yes—
133C23#13512yes—
134C23#13712yes—
135C23C22×D411yescharacteristic
136Q8—42no—
137Q8—42no—
138C2×C4—22no—
139C2×C4—24no—
140C2×C4—24no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—44no—
150C2×C4—48no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—44no—
154C2×C4—44no—
155C2×C4—48no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4#12012yes—
164C2×C4C22×D411yescharacteristic
165C2×C4C22×D411yescharacteristic
166C2×C42−1+414yes—
167C8—42no—
168C8—42no—
169C22—22no—
170C22—24no—
171C22—24no—
172C22—22no—
173C22#45214yes—
174C22C2×#12011yescharacteristic
175C22#53612yes—
176C22C2×#13511yescharacteristic
177C22C2×#13711yescharacteristic
178C22#57512yes—
179C4—22no—
180C4—42no—
181C4—44no—
182C4—42no—
183C4—44no—
184C4—81no—
185C4—82no—
186C4—81no—
187C4#53612yes—
188C4#53612yes—
189C2—24no—
190C2#261411yescharacteristic
191C2C2×#45211yescharacteristic
192C2#300511yescharacteristic
193C2#323212yes—
194C2#323612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2059C2semidirect
6. #2026C2semidirect

Maximal subgroups

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

subgroupindex
C22×#1252
C2×#4842
C2×#4832
#28552
#20592
#20262
#18242
#18232
#18172