Finite Group Databasebeta

← #26422

Subgroups — #26422

Order 256.

429 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2442C211yesmaximal, characteristic, complemented
2#2361C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2065C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1919C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic
10#1874C211yesmaximal, characteristic
11#1864C211yesmaximal, characteristic, complemented
12#1853C211yesmaximal, characteristic, complemented
13#1850C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32#400—21no—
33#398—21no—
34#390C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#388—21no—
39#386—21no—
40#385—21no—
41#385—21no—
42#382C2211yescharacteristic
43#380—21no—
44#380C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C22⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C22×C8—22no—
66C22×C8—41no—
67C22×C8—41no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72#125—22no—
73#125—41no—
74#121—22no—
75#121Q814yes—
76C4×D4—22no—
77C4×D4—41no—
78C4×D4—44no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42C2311yescharacteristic
106C2×C42C2311yescharacteristic
107#106—22no—
108#106—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#102—22no—
112#102—41no—
113#102—41no—
114#102—41no—
115C22⋊C8—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C24C2×D411yescharacteristic
127C2×D4—24no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×Q812yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—81no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×Q812yes—
162C22×C4#4112yes—
163C22×C4C2411yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—44no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4C2×Q812yes—
196C4⋊C4#4112yes—
197C4⋊C4#4112yes—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23C22⋊Q812yes—
222C23#13512yes—
223C23AGL(1,ℤ/8)12yes—
224C23C22×D411yescharacteristic
225D4—22no—
226D4—22no—
227D4—48no—
228D4—41no—
229D4—41no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—21no—
250C2×C4—21no—
251C2×C4—41no—
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—41no—
268C2×C4—41no—
269C2×C4—84no—
270C2×C4—84no—
271C2×C4—84no—
272C2×C4C22⋊Q812yes—
273C2×C4C22×Q811yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C8—41no—
280C8—41no—
281C8—41no—
282C8—41no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#47814yes—
297C22C2×C22⋊Q811yescharacteristic
298C22#54611yescharacteristic
299C22#54611yescharacteristic
300C22C2×#13511yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22#57711yescharacteristic
303C22#57711yescharacteristic
304C4—22no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#52911yescharacteristic
317C4#53811yescharacteristic
318C4#55411yescharacteristic
319C4#55411yescharacteristic
320C2—24no—
321C2—82no—
322C2#245011yescharacteristic
323C2C2×#47811yescharacteristic
324C2#308411yescharacteristic
325C2#330111yescharacteristic
326C2#330111yescharacteristic
327C2#330611yescharacteristic
328C2#331011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2442C2semidirect
3. #2101C2semidirect
4. #2088C2semidirect
5. #2065C2semidirect
6. #2025C2semidirect
7. #1989C2semidirect
8. #1919C2semidirect
11. #1864C2semidirect
12. #1853C2semidirect
13. #1850C2semidirect
14. #1804C2semidirect
15. C2×#340C2semidirect

Maximal subgroups

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

subgroupindex
#24422
#23612
#21012
#20882
#20652
#20252
#19892
#19192
#18932
#18742
#18642
#18532
#18502
#18042
C2×#3402