Finite Group Databasebeta

← #8987

Subgroups — #8987

Order 256.

288 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 93 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×#462C211yesmaximal, characteristic
2Q8×C4⋊C4C211yesmaximal, characteristic
3#1990C211yesmaximal, characteristic
4#1988C211yesmaximal, characteristic
5C2×Q8⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic
7#1279C211yesmaximal, characteristic
8C22×Q16—21no—
9C2×C4⋊Q8C2211yescharacteristic
10C2×C4×Q8—21no—
11C2×C4×Q8C2211yescharacteristic
12#462—22no—
13#462C414yescomplemented
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103—21no—
17C2×C4×C8C2211yescharacteristic
18#389—21no—
19#389—22no—
20#384—21no—
21#384C2211yescharacteristic
22C4×C4⋊C4—21no—
23C4×C4⋊C4C2211yescharacteristic
24#340—21no—
25#340—21no—
26#340—22no—
27Q8⋊C8—22no—
28C22×Q8—21no—
29C22×Q8D411yescharacteristic
30C2×Q16—24no—
31C2×Q16—42no—
32C2×Q16—42no—
33C22×C8—21no—
34C22×C8—21no—
35C22×C8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8C2×C412yes—
40C4×Q8—21no—
41C4×Q8—21no—
42C4×Q8—42no—
43C4×Q8—42no—
44C4×Q8C2×C412yes—
45C4×Q8D412yes—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D412yes—
57C2×C42—21no—
58C2×C42—22no—
59C2×C42—21no—
60C2×C42C2311yescharacteristic
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8C2×C412yes—
65#103—24no—
66C4×C8—21no—
67C4×C8—21no—
68#95—22no—
69#95—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—22no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×Q8C22⋊C412yes—
80C2×Q8D812yes—
81C2×Q8SD1612yes—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91Q16—48no—
92C2×C8—22no—
93C2×C8—22no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C4⋊C4—22no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—42no—
110C4⋊C4—44no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—82no—
118C4⋊C4C22⋊C412yes—
119C4⋊C4Q1612yescomplemented
120C42—22no—
121C42—22no—
122C42—44no—
123C42—41no—
124C42—41no—
125C42C22×C411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C42#4111yescharacteristic
129C23#12011yescharacteristic
130Q8—22no—
131Q8—22no—
132Q8—22no—
133Q8—42no—
134Q8—42no—
135Q8—81no—
136Q8—81no—
137Q8#10214yes—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4C2×C22⋊C411yescharacteristic
159C2×C4C4×D411yescharacteristic
160C2×C4C4×D411yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12311yescharacteristic
164C2×C4C2×D811yescharacteristic
165C2×C4C2×SD1611yescharacteristic
166C2×C4C2×Q1611yescharacteristic
167C2×C4#13511yescharacteristic
168C2×C4AGL(1,ℤ/8)11yescharacteristic
169C2×C4#13711yescharacteristic
170C2×C4#13711yescharacteristic
171C2×C4#13711yescharacteristic
172C8—42no—
173C8—42no—
174C8—81no—
175C22#38611yescharacteristic
176C22#44711yescharacteristic
177C22#44811yescharacteristic
178C22#45111yescharacteristic
179C22#45211yescharacteristic
180C22#45511yescharacteristic
181C22#45611yescharacteristic
182C4—24no—
183C4—21no—
184C4—21no—
185C4—42no—
186C4—42no—
187C4—82no—
188C4—81no—
189C4—82no—
190C4C2×#10211yescharacteristic
191C4#41711yescharacteristic
192C4C4×Q1611yescharacteristic
193C4#44111yescharacteristic
194C4#46211yescharacteristic
195C4#46411yescharacteristic
196C4#48011yescharacteristic
197C4#48111yescharacteristic
198C2#164111yescharacteristic
199C2#164511yescharacteristic
200C2#164911yescharacteristic
201C2#165311yescharacteristic
202C2#187411yescharacteristic
203C2#187811yescharacteristic
204C2#188211yescharacteristic
subgroupG/H
center, socle, prime radicalC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
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
13. #462C4semidirect
119. C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
Q8×C4⋊C42
#19902
#19882
C2×Q8⋊C82
#12962
#12792