Finite Group Databasebeta

← #7774

Subgroups — #7774

Order 256.

236 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 51 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#2732C211yesmaximal, characteristic
2#2043C211yesmaximal, characteristic, complemented
3#2024C211yesmaximal, characteristic, complemented
4#2023C211yesmaximal, characteristic
5#1272C211yesmaximal, characteristic, complemented
6#1271C211yesmaximal, characteristic, complemented
7#1270C211yesmaximal, characteristic
8C2×C4⋊Q8C2211yescharacteristic
9C2×C22⋊Q8—21no—
10C2×#103—21no—
11C2×#103—21no—
12C2×#103—21no—
13C2×#102—21no—
14#399—21no—
15#398—21no—
16#396—21no—
17#391C2211yescharacteristic
18#390C2211yescharacteristic
19#389C2211yescharacteristic
20#386—21no—
21#385—21no—
22#340—21no—
23#340—21no—
24#336C2211yescharacteristic
25#336C2211yescharacteristic
26#336C2211yescharacteristic
27C22×Q8—21no—
28C22×Q8D411yescharacteristic
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34C4⋊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⋊C4D411yescharacteristic
44C2×C4⋊C4D411yescharacteristic
45C2×C4⋊C4D411yescharacteristic
46C2×C4⋊C4D411yescharacteristic
47C2×C22⋊C4—21no—
48C2×C22⋊C4—21no—
49C2×C22⋊C4—21no—
50C2×C22⋊C4—21no—
51C2×C22⋊C4—21no—
52C2×C42C2311yescharacteristic
53#103—42no—
54#103—42no—
55#103—42no—
56#102—42no—
57#95—21no—
58#95—21no—
59#95—21no—
60#95—21no—
61C24—21no—
62C2×Q8—21no—
63C2×Q8—22no—
64C2×Q8—41no—
65C2×Q8—41no—
66C2×Q8—41no—
67C2×Q8—41no—
68C2×Q8—41no—
69C2×Q8—41no—
70C2×Q8—41no—
71C2×Q8SD1612yes—
72C2×D4—21no—
73C2×D4—41no—
74C2×D4—41no—
75C2×D4Q1612yes—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—21no—
81C22×C4—42no—
82C22×C4—41no—
83C22×C4—81no—
84C22×C4—81no—
85C22×C4C2×D411yescharacteristic
86C22×C4C2×D411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—41no—
89C2×C8—41no—
90C2×C8—41no—
91C2×C8—41no—
92C2×C8—41no—
93C2×C8—41no—
94C2×C8—41no—
95C2×C8—41no—
96C2×C8—41no—
97C4⋊C4—22no—
98C4⋊C4—22no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—41no—
102C4⋊C4—41no—
103C4⋊C4—41no—
104C4⋊C4—42no—
105C4⋊C4—42no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—81no—
111C22⋊C4—42no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C22⋊C4—81no—
117C22⋊C4—81no—
118C42—41no—
119C23—22no—
120C23—41no—
121C23—41no—
122C23—41no—
123C23—41no—
124C23—41no—
125C23—41no—
126C23#12011yescharacteristic
127Q8—42no—
128Q8—42no—
129Q8—42no—
130Q8—81no—
131Q8—81no—
132Q8—81no—
133D4—42no—
134D4—81no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—84no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4C2×SD1611yescharacteristic
161C2×C4C2×Q1611yescharacteristic
162C2×C4#13511yescharacteristic
163C2×C4AGL(1,ℤ/8)11yescharacteristic
164C2×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C4#13711yescharacteristic
166C8—81no—
167C8—81no—
168C8—81no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#44911yescharacteristic
177C22#45011yescharacteristic
178C22#45111yescharacteristic
179C22#45411yescharacteristic
180C22#45511yescharacteristic
181C22#45511yescharacteristic
182C22B(4,𝔽2)11yescharacteristic
183C4—21no—
184C4—21no—
185C4—21no—
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—82no—
191C4—81no—
192C4—81no—
193C4—81no—
194C4—81no—
195C2—82no—
196C2#159711yescharacteristic
197C2#160111yescharacteristic
198C2#160211yescharacteristic
199C2#163711yescharacteristic
200C2#164211yescharacteristic
201C2#164411yescharacteristic
202C2#166211yescharacteristic
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. #2043C2semidirect
3. #2024C2semidirect
5. #1272C2semidirect
6. #1271C2semidirect

Maximal subgroups

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

subgroupindex
#27322
#20432
#20242
#20232
#12722
#12712
#12702