Finite Group Databasebeta

← #9917

Subgroups — #9917

Order 256.

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

normal subgroups only

This group has 436 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×#448C211yesmaximal, characteristic
2#2618C211yesmaximal, characteristic
3C2×#395C211yesmaximal, characteristic, complemented
4#1787C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic
6C2×#324C211yesmaximal, characteristic, complemented
7#1294C211yesmaximal, characteristic, complemented
8C23×Q8C2211yescharacteristic
9C22×SD16—21no—
10C2×#123—21no—
11C2×#121C2211yescharacteristic
12C22×C4⋊C4—21no—
13C22×C4⋊C4C2211yescharacteristic
14#448—22no—
15#448—22no—
16C2×#103—21no—
17C2×#103—22no—
18C2×C22⋊C8C2211yescharacteristic
19C2×C22⋊C8C2211yescharacteristic
20#398—21no—
21#396—22no—
22#395—22no—
23#395—22no—
24#394—21no—
25#386—21no—
26#380C2211yescharacteristic
27C2×#95—21no—
28C2×#95C2211yescharacteristic
29#327—22no—
30#324—22no—
31C22×Q8—21no—
32C22×Q8—21no—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—21no—
36C22×Q8—44no—
37C22×Q8D412yes—
38C22×D4—21no—
39C22×D4D411yescharacteristic
40C23×C4—21no—
41C23×C4—21no—
42C23×C4D411yescharacteristic
43C23×C4D411yescharacteristic
44C23×C4C2311yescharacteristic
45C2×SD16—24no—
46C2×SD16—44no—
47C22×C8—21no—
48C22×C8—21no—
49#123—42no—
50#123—42no—
51#121—21no—
52#121—21no—
53#121—21no—
54#121—21no—
55C2×C4⋊C4—24no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—44no—
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4D412yes—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70#103—24no—
71#103—44no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8D412yes—
76#95—24no—
77#95—22no—
78#95—21no—
79#95—21no—
80#95D412yes—
81C24—21no—
82C24C2×D411yescharacteristic
83C2×Q8—24no—
84C2×Q8—24no—
85C2×Q8—22no—
86C2×Q8—22no—
87C2×Q8—44no—
88C2×Q8—42no—
89C2×Q8—42no—
90C2×Q8—44no—
91C2×Q8—44no—
92C2×Q8—44no—
93C2×Q8—41no—
94C2×Q8—41no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—42no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C22×C4—24no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4SD1612yescomplemented
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4#4111yescharacteristic
134SD16—48no—
135C2×C8—22no—
136C2×C8—42no—
137C2×C8—42no—
138C2×C8—41no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—42no—
142C4⋊C4—48no—
143C4⋊C4—44no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C22⋊C4—44no—
149C22⋊C4—41no—
150C22⋊C4—41no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—82no—
154C23—22no—
155C23—22no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23#12011yescharacteristic
162C23#12011yescharacteristic
163C23#12111yescharacteristic
164C23C2×SD1611yescharacteristic
165C23#13511yescharacteristic
166C23AGL(1,ℤ/8)11yescharacteristic
167C23#13711yescharacteristic
168Q8—24no—
169Q8—24no—
170Q8—42no—
171Q8—44no—
172Q8—44no—
173Q8—42no—
174D4—42no—
175D4—81no—
176D4—81no—
177D4—81no—
178D4—81no—
179D4—81no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—81no—
213C2×C4#12011yescharacteristic
214C2×C4#12111yescharacteristic
215C2×C4#12111yescharacteristic
216C2×C4#12711yescharacteristic
217C8—42no—
218C8—81no—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—82no—
228C22—82no—
229C22—81no—
230C22—81no—
231C22—81no—
232C22#39211yescharacteristic
233C22#44811yescharacteristic
234C22#44911yescharacteristic
235C22#45011yescharacteristic
236C22#45211yescharacteristic
237C22#45612yes—
238C22B(4,𝔽2)12yes—
239C22#46511yescharacteristic
240C22#46811yescharacteristic
241C4—21no—
242C4—21no—
243C4—21no—
244C4—21no—
245C4—44no—
246C4—42no—
247C4—44no—
248C4—82no—
249C4—81no—
250C4—81no—
251C2—21no—
252C2—21no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#160312yes—
257C2#160412yes—
258C2#200011yescharacteristic
259C2#200511yescharacteristic
260C2#202111yescharacteristic
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×#395C2semidirect
4. #1787C2semidirect
6. C2×#324C2semidirect
7. #1294C2semidirect
127. C22×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#4482
#26182
C2×#3952
#17872
C2×#3272
C2×#3242
#12942