Finite Group Databasebeta

← #33080

Subgroups — #33080

Order 256.

453 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2860C211yesmaximal, characteristic, complemented
2C42×D4C211yesmaximal, characteristic, complemented
3#1926C211yesmaximal, characteristic, complemented
4#1926C211yesmaximal, characteristic, complemented
5#1925C211yesmaximal, characteristic, complemented
6#1921C211yesmaximal, characteristic, complemented
7#1920C211yesmaximal, characteristic, complemented
8#1919C211yesmaximal, characteristic, complemented
9#1879C211yesmaximal, characteristic
10#1879C211yesmaximal, characteristic
11#1879C211yesmaximal, characteristic
12#1864C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1839C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#126—21no—
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C22×C42—21no—
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#388—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×D4C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C2×D8—22no—
65C2×D8—22no—
66C2×D8—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#126—44no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—44no—
100C2×C42D411yescharacteristic
101C2×C42D411yescharacteristic
102C2×C42D411yescharacteristic
103C2×C42D411yescharacteristic
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109#106—22no—
110#102—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114#102—22no—
115#102—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122C24—21no—
123C24—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4#4112yes—
134C2×D4#4112yes—
135C2×D4#4112yes—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—24no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2411yescharacteristic
158D8—44no—
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—
170C2×C8—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—21no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4#4112yes—
184C4⋊C4#4112yes—
185C4⋊C4#4112yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C22⋊C4—42no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—42no—
206C42—42no—
207C42—42no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23C22×D411yescharacteristic
223D4—22no—
224D4—22no—
225D4—22no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4#12614yescomplemented
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—48no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4#13512yes—
268C2×C4AGL(1,ℤ/8)12yes—
269C2×C4AGL(1,ℤ/8)12yes—
270C2×C4C2×#4111yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C8—41no—
277C8—41no—
278C8—41no—
279C8—41no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22—81no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22#52511yescharacteristic
295C22#52511yescharacteristic
296C22#52511yescharacteristic
297C22C2×#13511yescharacteristic
298C22C2×AGL(1,ℤ/8)11yescharacteristic
299C22C2×AGL(1,ℤ/8)11yescharacteristic
300C22#57511yescharacteristic
301C4—22no—
302C4—22no—
303C4—22no—
304C4—41no—
305C4—42no—
306C4—41no—
307C4—42no—
308C4—42no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4C2×#12611yescharacteristic
314C4#52911yescharacteristic
315C4#52911yescharacteristic
316C4#52911yescharacteristic
317C2—44no—
318C2—161no—
319C2#285211yescharacteristic
320C2#310211yescharacteristic
321C2#310511yescharacteristic
322C2#310611yescharacteristic
323C2#310911yescharacteristic
324C2#311411yescharacteristic
325C2#311511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
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. #2860C2semidirect
2. C42×D4C2semidirect
3. #1926C2semidirect
4. #1926C2semidirect
5. #1925C2semidirect
6. #1921C2semidirect
7. #1920C2semidirect
8. #1919C2semidirect
12. #1864C2semidirect
13. #1864C2semidirect
14. #1862C2semidirect
15. #1839C2semidirect
230. D4#126semidirect

Maximal subgroups

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

subgroupindex
#28602
C42×D42
#19262
#19252
#19212
#19202
#19192
#18792
#18642
#18622
#18392