Finite Group Databasebeta

← #12243

Subgroups — #12243

Order 256.

232 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 37 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#2693C211yesmaximal, characteristic
2#2062C211yesmaximal, characteristic, complemented
3#1608C211yesmaximal, characteristic, complemented
4#1596C211yesmaximal, characteristic
5#1344C211yesmaximal, characteristic, complemented
6#1343C211yesmaximal, characteristic
7#1326C211yesmaximal, characteristic, complemented
8#543C2211yescharacteristic
9C2×#123—21no—
10#483—21no—
11#450—21no—
12#400—21no—
13#397C2211yescharacteristic
14#396—21no—
15#392C2211yescharacteristic
16#386—21no—
17#386—21no—
18#385—21no—
19#385C2211yescharacteristic
20#380—21no—
21#342—21no—
22#342C2211yescharacteristic
23#328—21no—
24#328—21no—
25#323C2211yescharacteristic
26#323C2211yescharacteristic
27C22×D4—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C23×C4D411yescharacteristic
31C2×SD16—41no—
32#123—42no—
33#123—42no—
34C22⋊Q8—21no—
35C22⋊Q8—21no—
36C22⋊Q8—41no—
37C22⋊Q8—41no—
38C2×C4⋊C4—21no—
39C2×C4⋊C4—21no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—41no—
42C2×C4⋊C4D411yescharacteristic
43C2×C22⋊C4—21no—
44C2×C22⋊C4—21no—
45C2×C22⋊C4—21no—
46C2×C22⋊C4—21no—
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4D411yescharacteristic
51C2×C22⋊C4C2311yescharacteristic
52#106—41no—
53#103—41no—
54#102—41no—
55C22⋊C8—21no—
56C22⋊C8—21no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61#95—41no—
62#95D411yescharacteristic
63#95D411yescharacteristic
64C24—21no—
65C24C2×D411yescharacteristic
66C2×Q8—21no—
67C2×Q8—21no—
68C2×D4—41no—
69C2×D4—42no—
70C2×D4—42no—
71C2×D4—41no—
72C2×D4—41no—
73C2×D4—81no—
74C22×C4—21no—
75C22×C4—21no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—42no—
82C22×C4—41no—
83C22×C4—41no—
84C22×C4—42no—
85C22×C4—81no—
86C22×C4—81no—
87C22×C4C2×D411yescharacteristic
88C22×C4C2×D411yescharacteristic
89SD16—82no—
90C2×C8—41no—
91C2×C8—41no—
92C4⋊C4—21no—
93C4⋊C4—21no—
94C4⋊C4—41no—
95C4⋊C4—42no—
96C4⋊C4—42no—
97C4⋊C4—41no—
98C4⋊C4—41no—
99C4⋊C4—81no—
100C4⋊C4—82no—
101C22⋊C4—21no—
102C22⋊C4—22no—
103C22⋊C4—41no—
104C22⋊C4—42no—
105C22⋊C4—41no—
106C22⋊C4—41no—
107C22⋊C4—41no—
108C22⋊C4—41no—
109C22⋊C4—41no—
110C22⋊C4—81no—
111C22⋊C4—81no—
112C22⋊C4—81no—
113C22⋊C4SD1612yescomplemented
114C23—21no—
115C23—21no—
116C23—22no—
117C23—41no—
118C23—41no—
119C23—41no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23#12011yescharacteristic
126C23C2×SD1611yescharacteristic
127C23AGL(1,ℤ/8)11yescharacteristic
128Q8—81no—
129Q8—81no—
130D4—42no—
131D4—81no—
132D4—82no—
133D4—82no—
134C2×C4—22no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—84no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4—81no—
161C2×C4—81no—
162C2×C4—81no—
163C8—81no—
164C8—81no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—41no—
174C22—41no—
175C22—42no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22#45011yescharacteristic
182C22#45611yescharacteristic
183C22B(4,𝔽2)11yescharacteristic
184C4—42no—
185C4—41no—
186C4—41no—
187C4—81no—
188C4—81no—
189C4—82no—
190C4—81no—
191C4—81no—
192C4—81no—
193C4—81no—
194C2—21no—
195C2—21no—
196C2—41no—
197C2—41no—
198C2—82no—
199C2#160411yescharacteristic
200C2#219711yescharacteristic
201C2#220011yescharacteristic
subgroupG/H
center, socleC22B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23
prime radical#392C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2062C2semidirect
3. #1608C2semidirect
5. #1344C2semidirect
7. #1326C2semidirect
113. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#26932
#20622
#16082
#15962
#13442
#13432
#13262