Finite Group Databasebeta

← #9921

Subgroups — #9921

Order 256.

336 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#481C211yesmaximal, characteristic
2#2618C211yesmaximal, characteristic
3C2×#400C211yesmaximal, characteristic, complemented
4#1824C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic
6C2×#324C211yesmaximal, characteristic, complemented
7#1294C211yesmaximal, characteristic, complemented
8C2×#123—21no—
9C2×#121C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12#481—22no—
13#481—22no—
14C2×#106—21no—
15C2×#102—21no—
16C2×C22⋊C8C2211yescharacteristic
17C2×C22⋊C8C2211yescharacteristic
18#400—22no—
19#400—22no—
20#398—21no—
21#396—22no—
22#394—21no—
23#386—21no—
24#380C2211yescharacteristic
25C2×#95—21no—
26C2×#95—21no—
27C2×#95C2211yescharacteristic
28#340—22no—
29#327—22no—
30#324—22no—
31C22×D4—21no—
32C22×D4D411yescharacteristic
33C23×C4—21no—
34C23×C4—21no—
35C23×C4D411yescharacteristic
36C23×C4D411yescharacteristic
37C23×C4C2311yescharacteristic
38C22×C8—21no—
39C22×C8—21no—
40#123—42no—
41#123—42no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D412yes—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—22no—
65#106—24no—
66#102—24no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8D412yes—
71#95—24no—
72#95—24no—
73#95—22no—
74#95—21no—
75#95—21no—
76#95D412yes—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—42no—
82C2×D4—41no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—42no—
102C22×C4—42no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4—44no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4SD1612yes—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4#4111yescharacteristic
117C2×C8—22no—
118C2×C8—42no—
119C2×C8—42no—
120C2×C8—41no—
121C4⋊C4—22no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—41no—
133C22⋊C4—44no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—82no—
139C23—22no—
140C23—22no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23—42no—
145C23—42no—
146C23#12011yescharacteristic
147C23#12011yescharacteristic
148C23#12111yescharacteristic
149C23C2×SD1611yescharacteristic
150C23#13511yescharacteristic
151C23AGL(1,ℤ/8)11yescharacteristic
152C23#13711yescharacteristic
153D4—42no—
154D4—81no—
155D4—81no—
156D4—81no—
157D4—81no—
158D4—81no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—82no—
190C2×C4—81no—
191C2×C4#12011yescharacteristic
192C2×C4#12111yescharacteristic
193C2×C4#12111yescharacteristic
194C2×C4#12711yescharacteristic
195C8—42no—
196C8—81no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—21no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—82no—
206C22—82no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22#39211yescharacteristic
211C22#44811yescharacteristic
212C22#44911yescharacteristic
213C22#45011yescharacteristic
214C22#45211yescharacteristic
215C22#45412yes—
216C22#45812yes—
217C22#46511yescharacteristic
218C22#46811yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—82no—
225C4—82no—
226C4—81no—
227C4—82no—
228C4—81no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—161no—
234C2#161512yes—
235C2#161612yes—
236C2#200011yescharacteristic
237C2#200311yescharacteristic
238C2#202211yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×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
3. C2×#400C2semidirect
4. #1824C2semidirect
6. C2×#324C2semidirect
7. #1294C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
#26182
C2×#4002
#18242
C2×#3272
C2×#3242
#12942