Finite Group Databasebeta

← #11877

Subgroups — #11877

Order 256.

322 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 59 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#2541C211yesmaximal, characteristic
2#1799C211yesmaximal, characteristic, complemented
3#1459C211yesmaximal, characteristic, complemented
4#1343C214yesmaximal, complemented
5C2×#127—21no—
6C2×#127C411yescharacteristic, complemented
7C2×#121—21no—
8C2×#121—21no—
9C2×C4×D4C2211yescharacteristic
10C2×#99—21no—
11C2×C22⋊C8—21no—
12#392C412yescomplemented
13#392C2212yes—
14#390—21no—
15#380—21no—
16C4×C4⋊C4C411yescharacteristic, complemented
17#342C2212yes—
18#324—21no—
19#323C2212yes—
20C22×D4—22no—
21C22×D4—22no—
22C22×D4—21no—
23C22×D4—21no—
24C22×D4C2×C412yes—
25C22×D4D411yescharacteristic
26C23×C4D411yescharacteristic
27C23×C4D411yescharacteristic
28C22×C8—41no—
29#127—41no—
30#127—41no—
31#127—42no—
32#127—42no—
33#121—44no—
34#121—44no—
35C4×D4—41no—
36C4×D4—41no—
37C2×C4⋊C4—21no—
38C2×C4⋊C4—41no—
39C2×C4⋊C4D411yescharacteristic
40C2×C22⋊C4—22no—
41C2×C22⋊C4—22no—
42C2×C22⋊C4—22no—
43C2×C22⋊C4C2×C411yescharacteristic
44C2×C22⋊C4C2311yescharacteristic
45C2×C42—21no—
46C2×C42C2×C411yescharacteristic
47#99—42no—
48C22⋊C8—22no—
49C22⋊C8—41no—
50#95—41no—
51#95C2×C412yes—
52C24—22no—
53C24C22⋊C411yescharacteristic
54C24C2×D411yescharacteristic
55C2×D4—41no—
56C2×D4—41no—
57C2×D4—42no—
58C2×D4—42no—
59C2×D4—42no—
60C2×D4—44no—
61C2×D4—44no—
62C2×D4—42no—
63C2×D4—42no—
64C2×D4—44no—
65C2×D4—42no—
66C2×D4—42no—
67C2×D4—41no—
68C2×D4—41no—
69C2×D4—42no—
70C2×D4—82no—
71C2×D4—82no—
72C2×D4—82no—
73C2×D4D812yes—
74C2×D4SD1612yes—
75C22×C4—22no—
76C22×C4—21no—
77C22×C4—21no—
78C22×C4—21no—
79C22×C4—21no—
80C22×C4—42no—
81C22×C4—42no—
82C22×C4—42no—
83C22×C4C22⋊C411yescharacteristic
84C22×C4C22⋊C411yescharacteristic
85C22×C4C22⋊C411yescharacteristic
86C22×C4C22×C411yescharacteristic
87C22×C4C2×D411yescharacteristic
88C2×C8—42no—
89C2×C8—82no—
90C4⋊C4—24no—
91C4⋊C4—41no—
92C4⋊C4—82no—
93C22⋊C4—41no—
94C22⋊C4—41no—
95C22⋊C4—44no—
96C22⋊C4—44no—
97C22⋊C4—42no—
98C22⋊C4—42no—
99C42—21no—
100C42—21no—
101C42—42no—
102C23—21no—
103C23—21no—
104C23—21no—
105C23—21no—
106C23—41no—
107C23—41no—
108C23—41no—
109C23—41no—
110C23—42no—
111C23—42no—
112C23—42no—
113C23—42no—
114C23—42no—
115C23—42no—
116C23—42no—
117C23#10212yes—
118C23C2×C22⋊C411yescharacteristic
119C23C2×D811yescharacteristic
120C23C2×SD1611yescharacteristic
121D4—42no—
122D4—42no—
123D4—82no—
124D4—84no—
125D4—88no—
126D4—84no—
127D4—84no—
128D4—84no—
129D4—82no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—24no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—81no—
145C2×C4—81no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—82no—
149C2×C4#9911yescharacteristic
150C2×C4#9911yescharacteristic
151C2×C4#10212yes—
152C8—82no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—22no—
157C22—21no—
158C22—42no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22—81no—
166C22—81no—
167C22—82no—
168C22—82no—
169C22—82no—
170C22—82no—
171C22—82no—
172C22—82no—
173C22—82no—
174C22C2×#9911yescharacteristic
175C22C2×#10211yescharacteristic
176C22#42111yescharacteristic
177C4—21no—
178C4—22no—
179C4—44no—
180C4—41no—
181C4—41no—
182C4—41no—
183C4—162no—
184C4#35312yes—
185C2—21no—
186C2—21no—
187C2—41no—
188C2—42no—
189C2—41no—
190C2—162no—
191C2#151911yescharacteristic
192C2#212211yescharacteristic
193C2C2×#35311yescharacteristic
subgroupG/H
center, socleC22C2×#99
commutatorC22×C4C22×C4
Frattini, cosocleC2×C22⋊C4C23
prime radical#2541C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1799C2semidirect
3. #1459C2semidirect
4. #1343C2semidirect
6. C2×#127C4semidirect
12. #392C4semidirect
16. C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#25412
#17992
#14592
#13432