Finite Group Databasebeta

← #26672

Subgroups — #26672

Order 256.

443 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2420C211yesmaximal, characteristic, complemented
2#2361C211yesmaximal, characteristic, complemented
3#2101C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#1999C211yesmaximal, characteristic
7#1936C211yesmaximal, characteristic, complemented
8#1924C211yesmaximal, characteristic, complemented
9#1919C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1823C211yesmaximal, characteristic, complemented
14#1804C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×#97C2211yescharacteristic
34#397—21no—
35#394—21no—
36#390C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#388—21no—
41#386—21no—
42#386—21no—
43#385—21no—
44#382C2211yescharacteristic
45#381—21no—
46#380C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×D4—21no—
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C2×D8—22no—
68C2×D8—22no—
69C2×D8—41no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#121—22no—
75#121—22no—
76#121—22no—
77#121—41no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—41no—
82C4×D4—44no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4D411yescharacteristic
104C2×C22⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#106—22no—
109C4⋊C8—41no—
110#102—22no—
111#102—22no—
112#102—22no—
113#102—41no—
114#102—41no—
115#102—41no—
116C22⋊C8—41no—
117C22⋊C8—41no—
118#97—41no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24—21no—
129C24C2×D411yescharacteristic
130C2×D4—24no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—81no—
146C2×D4#4112yes—
147C2×D4#4112yes—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—81no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2411yescharacteristic
175D8—44no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C4⋊C4#4112yes—
202C4⋊C4#4112yes—
203C22⋊C4—24no—
204C22⋊C4—22no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23AGL(1,ℤ/8)12yes—
240C23C22×D411yescharacteristic
241D4—22no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—48no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—84no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—84no—
289C2×C4—84no—
290C2×C4—84no—
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C4H(5,𝔽2)11yescharacteristic
297C8—41no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22#52511yescharacteristic
323C22#54611yescharacteristic
324C22#54611yescharacteristic
325C22C2×AGL(1,ℤ/8)11yescharacteristic
326C22#57511yescharacteristic
327C22#57611yescharacteristic
328C22#57711yescharacteristic
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—42no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#52911yescharacteristic
341C4#54011yescharacteristic
342C4#55111yescharacteristic
343C4#55311yescharacteristic
344C2—42no—
345C2—82no—
346C2—82no—
347C2#245711yescharacteristic
348C2#311411yescharacteristic
349C2#311911yescharacteristic
350C2#330611yescharacteristic
351C2#331811yescharacteristic
352C2#332111yescharacteristic
353C2#332211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
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. #2420C2semidirect
2. #2361C2semidirect
3. #2101C2semidirect
4. #2065C2semidirect
5. #2034C2semidirect
7. #1936C2semidirect
8. #1924C2semidirect
9. #1919C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1874C2semidirect
13. #1823C2semidirect
14. #1804C2semidirect
15. #1762C2semidirect

Maximal subgroups

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

subgroupindex
#24202
#23612
#21012
#20652
#20342
#19992
#19362
#19242
#19192
#18792
#18742
#18232
#18042
#17622