Finite Group Databasebeta

← #8889

Subgroups — #8889

Order 256.

273 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 85 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
1#2354C211yesmaximal, characteristic
2#1947C211yesmaximal, characteristic, complemented
3#1844C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×D4⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4×D4C2211yescharacteristic
9C2×#117—21no—
10C2×#107—21no—
11C2×#106—22no—
12C2×C4⋊C8C2211yescharacteristic
13C2×C4⋊C8C2211yescharacteristic
14C2×#102—21no—
15C2×C4×C8C2211yescharacteristic
16#390—21no—
17#388—22no—
18#384C2211yescharacteristic
19#384C2211yescharacteristic
20#380—21no—
21C4×C4⋊C4—21no—
22C4×C4⋊C4C2211yescharacteristic
23#340—21no—
24#340—21no—
25D4⋊C8C414yescomplemented
26C22×D4D411yescharacteristic
27C23×C4—21no—
28C22×C8—21no—
29C22×C8—21no—
30C22×C8—21no—
31C4×D4—21no—
32C4×D4—21no—
33C4×D4C2×C412yes—
34C4×D4Q812yes—
35#117—42no—
36#117—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4D411yescharacteristic
43C2×C4⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D412yes—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—22no—
48C2×C22⋊C4—22no—
49C2×C42—22no—
50C2×C42—21no—
51C2×C42C2311yescharacteristic
52#107—42no—
53#106—44no—
54C4⋊C8—21no—
55C4⋊C8—21no—
56C4⋊C8—21no—
57C4⋊C8C2×C412yes—
58#102—42no—
59C4×C8—21no—
60C4×C8C2×C412yes—
61#95—22no—
62#95—21no—
63C24—21no—
64C2×D4—21no—
65C2×D4—21no—
66C2×D4—21no—
67C2×D4—21no—
68C2×D4—21no—
69C2×D4C4⋊C412yes—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—22no—
73C22×C4—22no—
74C22×C4—21no—
75C22×C4—22no—
76C22×C4—41no—
77C22×C4—41no—
78C22×C4—41no—
79C22×C4—41no—
80C22×C4—41no—
81C22×C4C2×D411yescharacteristic
82C22×C4C2×D411yescharacteristic
83C22×C4C2×D411yescharacteristic
84C2×C8—22no—
85C2×C8—22no—
86C2×C8—22no—
87C2×C8—22no—
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C4⋊C4—21no—
94C4⋊C4—21no—
95C4⋊C4—21no—
96C4⋊C4—21no—
97C4⋊C4—42no—
98C4⋊C4—42no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4C4⋊C412yes—
108C4⋊C4SD1612yes—
109C22⋊C4—22no—
110C22⋊C4—22no—
111C22⋊C4—44no—
112C22⋊C4—82no—
113C42—44no—
114C42—41no—
115C42—41no—
116C42C22×C411yescharacteristic
117C42C2×Q811yescharacteristic
118C42#4111yescharacteristic
119C42#4111yescharacteristic
120C23—22no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23—41no—
127C23#12011yescharacteristic
128D4—22no—
129D4—22no—
130D4—41no—
131D4—41no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—81no—
151C2×C4—81no—
152C2×C4—81no—
153C2×C4C2×C4⋊C411yescharacteristic
154C2×C4C4×D411yescharacteristic
155C2×C4C4×D411yescharacteristic
156C2×C4C22⋊Q811yescharacteristic
157C2×C4C22⋊Q811yescharacteristic
158C2×C4#12311yescharacteristic
159C2×C4C2×SD1611yescharacteristic
160C2×C4#13511yescharacteristic
161C2×C4#13511yescharacteristic
162C2×C4#13511yescharacteristic
163C2×C4AGL(1,ℤ/8)11yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C4AGL(1,ℤ/8)11yescharacteristic
166C2×C4#13711yescharacteristic
167C8—24no—
168C8—42no—
169C8—81no—
170C22—42no—
171C22—42no—
172C22—42no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22#38511yescharacteristic
178C22#44911yescharacteristic
179C22#44911yescharacteristic
180C22#45011yescharacteristic
181C22#45211yescharacteristic
182C22#45311yescharacteristic
183C22#45611yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—42no—
188C4—81no—
189C4—82no—
190C4—161no—
191C4#42711yescharacteristic
192C4#42811yescharacteristic
193C4C4×SD1611yescharacteristic
194C4#44211yescharacteristic
195C4#47611yescharacteristic
196C4#47811yescharacteristic
197C4#48211yescharacteristic
198C4#48511yescharacteristic
199C2—82no—
200C2#162411yescharacteristic
201C2#163111yescharacteristic
202C2#165211yescharacteristic
203C2#165411yescharacteristic
204C2#185411yescharacteristic
205C2#185911yescharacteristic
206C2#186411yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1947C2semidirect
3. #1844C2semidirect
6. #1296C2semidirect
7. #1279C2semidirect
25. D4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#23542
#19472
#18442
#18022
C2×D4⋊C82
#12962
#12792