Finite Group Databasebeta

← #7791

Subgroups — #7791

Order 256.

231 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 53 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#2683C211yesmaximal, characteristic
2#2099C211yesmaximal, characteristic, complemented
3#2084C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic
5#1271C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×C22⋊Q8—21no—
9C2×C22⋊Q8—21no—
10C2×#106—21no—
11C2×#102—21no—
12#396—21no—
13#396—21no—
14#395—21no—
15#390C2211yescharacteristic
16#389C2211yescharacteristic
17#389C2211yescharacteristic
18#386—21no—
19#384C2211yescharacteristic
20#340—21no—
21#340—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22⋊Q8—42no—
34C22⋊Q8—42no—
35C22⋊Q8—42no—
36C22⋊Q8—42no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—21no—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C4⋊C4D411yescharacteristic
48C2×C4⋊C4D411yescharacteristic
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C42C2311yescharacteristic
55#106—42no—
56#102—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60C24—21no—
61C2×Q8—42no—
62C2×Q8—42no—
63C2×Q8—81no—
64C2×D4—21no—
65C2×D4—41no—
66C2×D4—41no—
67C2×D4SD1612yes—
68C22×C4—21no—
69C22×C4—21no—
70C22×C4—21no—
71C22×C4—21no—
72C22×C4—21no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—41no—
76C22×C4—81no—
77C22×C4C2×D411yescharacteristic
78C22×C4C2×D411yescharacteristic
79C22×C4C2×D411yescharacteristic
80C2×C8—22no—
81C2×C8—41no—
82C2×C8—41no—
83C2×C8—41no—
84C2×C8—41no—
85C2×C8—41no—
86C2×C8—41no—
87C2×C8—41no—
88C2×C8—41no—
89C4⋊C4—21no—
90C4⋊C4—21no—
91C4⋊C4—41no—
92C4⋊C4—41no—
93C4⋊C4—41no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—41no—
100C4⋊C4—41no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—41no—
105C4⋊C4D812yes—
106C4⋊C4SD1612yes—
107C22⋊C4—42no—
108C22⋊C4—42no—
109C22⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—81no—
114C42—41no—
115C23—22no—
116C23—41no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23#12011yescharacteristic
123Q8—84no—
124D4—42no—
125D4—81no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—84no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4—81no—
151C2×C4C2×D811yescharacteristic
152C2×C4C2×SD1611yescharacteristic
153C2×C4C2×SD1611yescharacteristic
154C2×C4AGL(1,ℤ/8)11yescharacteristic
155C2×C4AGL(1,ℤ/8)11yescharacteristic
156C2×C4#13711yescharacteristic
157C8—42no—
158C8—81no—
159C8—81no—
160C22—42no—
161C22—42no—
162C22—42no—
163C22—81no—
164C22—81no—
165C22—81no—
166C22—81no—
167C22#44711yescharacteristic
168C22#44811yescharacteristic
169C22#45011yescharacteristic
170C22#45411yescharacteristic
171C22#45511yescharacteristic
172C22#45611yescharacteristic
173C22#45811yescharacteristic
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—42no—
181C4—42no—
182C4—81no—
183C4—81no—
184C4—81no—
185C4—161no—
186C2—82no—
187C2#160811yescharacteristic
188C2#161011yescharacteristic
189C2#161611yescharacteristic
190C2#162711yescharacteristic
191C2#163911yescharacteristic
192C2#165111yescharacteristic
193C2#166411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2099C2semidirect
3. #2084C2semidirect
5. #1271C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#26832
#20992
#20842
#20712
#12712
#12702