Finite Group Databasebeta

← #9952

Subgroups — #9952

Order 256.

346 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 63 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#3186C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic, complemented
4#1964C211yesmaximal, characteristic, complemented
5#1823C211yesmaximal, characteristic, complemented
6#1519C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#123—22no—
9C2×#121C2211yescharacteristic
10C2×#121C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12#480—22no—
13#480—21no—
14#480—21no—
15#480—22no—
16#466—22no—
17#447—21no—
18#447—21no—
19C2×#107—21no—
20C2×#102—21no—
21C2×#99—22no—
22C2×C22⋊C8C2211yescharacteristic
23C2×C22⋊C8C2211yescharacteristic
24#399—22no—
25#390—21no—
26#384—21no—
27#380C2211yescharacteristic
28C4×C22⋊C4C2211yescharacteristic
29#340—22no—
30#327—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C23×C4C2311yescharacteristic
37C2×D8—22no—
38C22×C8—21no—
39C22×C8D412yes—
40#123—44no—
41#123—44no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#121—21no—
48#121—21no—
49#121—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—24no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4D411yescharacteristic
60C2×C22⋊C4—24no—
61C2×C22⋊C4—21no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4D412yes—
65C2×C42—21no—
66#107—24no—
67#107—22no—
68#102—24no—
69#102—22no—
70#102—22no—
71#99—44no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76#95—21no—
77#95—21no—
78C24—21no—
79C24—21no—
80C24C2×D411yescharacteristic
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—42no—
86C2×D4—42no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—42no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4—42no—
95C22×C4—22no—
96C22×C4—22no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—42no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—41no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4#4111yescharacteristic
111D8—82no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—42no—
115C2×C8—42no—
116C4⋊C4—22no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—48no—
123C4⋊C4—42no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C22⋊C4—22no—
127C22⋊C4—48no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—42no—
133C22⋊C4D814yescomplemented
134C42—42no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23—41no—
140C23—41no—
141C23—42no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23#12011yescharacteristic
146C23#12111yescharacteristic
147C23#12711yescharacteristic
148C23C2×D812yes—
149C23AGL(1,ℤ/8)12yes—
150D4—42no—
151D4—42no—
152D4—42no—
153D4—81no—
154D4—82no—
155D4—82no—
156D4—81no—
157D4—81no—
158D4—81no—
159D4—81no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—44no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—82no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4#12011yescharacteristic
182C2×C4#12011yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C8—42no—
186C8—42no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—42no—
194C22—42no—
195C22—84no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—162no—
203C22#39211yescharacteristic
204C22#44712yes—
205C22#45912yes—
206C22#49311yescharacteristic
207C22#49611yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—44no—
212C4—82no—
213C4—81no—
214C4—81no—
215C4—161no—
216C4#45812yes—
217C2—22no—
218C2—21no—
219C2—22no—
220C2—21no—
221C2—161no—
222C2—161no—
223C2#200911yescharacteristic
224C2#201111yescharacteristic
225C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3186C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#480C2semidirect
3. #2501C2semidirect
4. #1964C2semidirect
5. #1823C2semidirect
6. #1519C2semidirect
7. #1282C2semidirect
133. C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#31862
C2×#4802
#25012
#19642
#18232
#15192
#12822