Finite Group Databasebeta

← #33084

Subgroups — #33084

Order 256.

426 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2863C211yesmaximal, characteristic, complemented
2C42×D4C211yesmaximal, characteristic
3#1928C211yesmaximal, characteristic, complemented
4#1928C211yesmaximal, characteristic, complemented
5#1927C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1919C211yesmaximal, characteristic, complemented
8#1918C211yesmaximal, characteristic, complemented
9#1877C211yesmaximal, characteristic
10#1877C211yesmaximal, characteristic
11#1877C211yesmaximal, characteristic
12#1864C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic
14#1862C211yesmaximal, characteristic
15#1839C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C42—21no—
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30#391C2211yescharacteristic
31#391C2211yescharacteristic
32#391C2211yescharacteristic
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#387—21no—
37#387—21no—
38#387—21no—
39#383—21no—
40C4×C4⋊C4C2211yescharacteristic
41C4×C4⋊C4C2211yescharacteristic
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—21no—
44C4×C22⋊C4—21no—
45C4×C22⋊C4—21no—
46C43C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C2×SD16—22no—
65C2×SD16—22no—
66C2×SD16—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—44no—
93C2×C42D411yescharacteristic
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42D411yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—22no—
101#107—22no—
102#106—22no—
103#103—22no—
104#103—22no—
105#103—22no—
106#102—22no—
107#102—22no—
108#102—22no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121C24—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4#4112yes—
132C2×D4#4112yes—
133C2×D4#4112yes—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157SD16—44no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4#4112yes—
183C4⋊C4#4112yes—
184C4⋊C4#4112yes—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C42—22no—
189C42—22no—
190C42—22no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—41no—
198C42—42no—
199C42—42no—
200C42—42no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23C22×D411yescharacteristic
209Q8—41no—
210Q8—41no—
211Q8—41no—
212Q8—41no—
213D4—22no—
214D4—22no—
215D4—22no—
216D4#12614yes—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—48no—
236C2×C4—48no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4#13512yes—
257C2×C4AGL(1,ℤ/8)12yes—
258C2×C4#13712yes—
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C22—24no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22#52511yescharacteristic
277C22#52511yescharacteristic
278C22#52511yescharacteristic
279C22C2×#13511yescharacteristic
280C22C2×AGL(1,ℤ/8)11yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57511yescharacteristic
283C4—22no—
284C4—22no—
285C4—22no—
286C4—41no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—42no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4—81no—
296C4C2×#12611yescharacteristic
297C4#52911yescharacteristic
298C4#52911yescharacteristic
299C4#52911yescharacteristic
300C2—44no—
301C2#285211yescharacteristic
302C2#310211yescharacteristic
303C2#310511yescharacteristic
304C2#310711yescharacteristic
305C2#311011yescharacteristic
306C2#311411yescharacteristic
307C2#311611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2863C2semidirect
3. #1928C2semidirect
4. #1928C2semidirect
5. #1927C2semidirect
6. #1921C2semidirect
7. #1919C2semidirect
8. #1918C2semidirect
15. #1839C2semidirect

Maximal subgroups

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

subgroupindex
#28632
C42×D42
#19282
#19272
#19212
#19192
#19182
#18772
#18642
#18622
#18392