Finite Group Databasebeta

← #8856

Subgroups — #8856

Order 256.

306 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 83 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#2316C211yesmaximal, characteristic
2#1852C212yesmaximal
3C2×#324C211yesmaximal, characteristic
4#1441C211yesmaximal, characteristic
5#1294C212yesmaximal
6C2×#117—21no—
7C22×C4⋊C4C2211yescharacteristic
8C22×C4⋊C4C2211yescharacteristic
9C2×#107—22no—
10C2×C22⋊C8C2212yes—
11#389—22no—
12#389—22no—
13#380—21no—
14#380C2212yes—
15C2×#95—21no—
16C2×#95—21no—
17C2×#95C2211yescharacteristic
18#340—22no—
19#324C414yescomplemented
20C23×C4—21no—
21C23×C4—21no—
22C23×C4D411yescharacteristic
23C23×C4D411yescharacteristic
24C23×C4C2311yescharacteristic
25C22×C8—22no—
26#117—42no—
27#117—42no—
28C2×C4⋊C4—22no—
29C2×C4⋊C4—22no—
30C2×C4⋊C4—22no—
31C2×C4⋊C4—21no—
32C2×C4⋊C4—21no—
33C2×C4⋊C4—21no—
34C2×C4⋊C4—21no—
35C2×C4⋊C4—42no—
36C2×C4⋊C4C2×C412yes—
37C2×C4⋊C4D412yes—
38C2×C4⋊C4D412yes—
39C2×C4⋊C4Q812yes—
40C2×C22⋊C4—22no—
41C2×C42—22no—
42#107—44no—
43C22⋊C8—22no—
44C22⋊C8C2×C414yes—
45#95—22no—
46#95—22no—
47#95—22no—
48#95—42no—
49#95—42no—
50C24C2×D411yescharacteristic
51C22×C4—21no—
52C22×C4—21no—
53C22×C4—21no—
54C22×C4—21no—
55C22×C4—21no—
56C22×C4—21no—
57C22×C4—22no—
58C22×C4—22no—
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—41no—
62C22×C4—41no—
63C22×C4—41no—
64C22×C4—41no—
65C22×C4—41no—
66C22×C4—41no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—41no—
72C22×C4—41no—
73C22×C4—41no—
74C22×C4—41no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4C4⋊C412yes—
78C22×C4C4⋊C412yes—
79C22×C4D812yes—
80C22×C4Q1612yes—
81C22×C4C22×C411yescharacteristic
82C22×C4C2×D412yes—
83C22×C4C2×Q811yescharacteristic
84C22×C4#4112yes—
85C2×C8—24no—
86C2×C8—42no—
87C2×C8—42no—
88C4⋊C4—24no—
89C4⋊C4—24no—
90C4⋊C4—24no—
91C4⋊C4—42no—
92C4⋊C4—42no—
93C4⋊C4—42no—
94C4⋊C4—42no—
95C4⋊C4—42no—
96C4⋊C4—81no—
97C4⋊C4—81no—
98C22⋊C4—42no—
99C22⋊C4—42no—
100C42—44no—
101C23—21no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23C2×C4⋊C411yescharacteristic
106C23#12011yescharacteristic
107C23#12311yescharacteristic
108C23C2×D811yescharacteristic
109C23C2×Q1611yescharacteristic
110C23AGL(1,ℤ/8)11yescharacteristic
111C23#13711yescharacteristic
112C2×C4—22no—
113C2×C4—22no—
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—22no—
119C2×C4—22no—
120C2×C4—22no—
121C2×C4—41no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—81no—
140C2×C4—81no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4#10714yescomplemented
149C2×C4C4×D412yes—
150C2×C4C22⋊Q812yes—
151C8—44no—
152C22—21no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—21no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22#38511yescharacteristic
165C22C2×#10711yescharacteristic
166C22#42811yescharacteristic
167C22#44711yescharacteristic
168C22#45111yescharacteristic
169C22#45411yescharacteristic
170C22#45611yescharacteristic
171C22B(4,𝔽2)11yescharacteristic
172C22#45811yescharacteristic
173C22#48011yescharacteristic
174C22#48411yescharacteristic
175C4—24no—
176C4—22no—
177C4—22no—
178C4—42no—
179C4—82no—
180C4—82no—
181C4—82no—
182C4—82no—
183C2—21no—
184C2—21no—
185C2—21no—
186C2—21no—
187C2#159811yescharacteristic
188C2#160511yescharacteristic
189C2#161011yescharacteristic
190C2#161711yescharacteristic
191C2#185211yescharacteristic
192C2#185711yescharacteristic
193C2#186711yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
19. #324C4semidirect
148. C2×C4#107semidirect

Maximal subgroups

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

subgroupindex
#23162
#18522
C2×#3242
#14412
#12942