Finite Group Databasebeta

← #26137

Subgroups — #26137

Order 256.

447 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#482C211yesmaximal, characteristic, complemented
2C2×#427C211yesmaximal, characteristic, complemented
3#2414C211yesmaximal, characteristic
4#2354C211yesmaximal, characteristic, complemented
5#2093C211yesmaximal, characteristic, complemented
6#2084C211yesmaximal, characteristic, complemented
7#2075C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1921C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1869C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1854C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23#482—24no—
24#427—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#394—21no—
39#392—21no—
40#390—21no—
41#390—21no—
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#388—21no—
45#388—21no—
46#388—21no—
47#384C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4—21no—
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4D411yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
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—
80#117—22no—
81#117—22no—
82#117—22no—
83#117—41no—
84#117—41no—
85#117—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42—21no—
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—22no—
115#107—41no—
116#107D414yescomplemented
117#106—22no—
118#106—22no—
119#106—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—41no—
124C22⋊C8—22no—
125C22⋊C8—22no—
126C22⋊C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—81no—
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—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4#4112yes—
172C22×C4#4112yes—
173C22×C4C2411yescharacteristic
174C2×C8—28no—
175C2×C8—21no—
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×C8C2×D412yes—
186C2×C8#4112yes—
187C4⋊C4—22no—
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—21no—
199C4⋊C4—21no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4C2×D412yes—
204C4⋊C4C2×D412yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—42no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#12312yes—
244C23C22×D411yescharacteristic
245D4—44no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4—41no—
250D4—84no—
251D4—84no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4#12312yes—
292C2×C4#13512yes—
293C2×C4C22×D411yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4H(5,𝔽2)11yescharacteristic
298C2×C42−1+411yescharacteristic
299C8—22no—
300C8—22no—
301C8—41no—
302C8—41no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×#12311yescharacteristic
325C22D4211yescharacteristic
326C22#54711yescharacteristic
327C22C2×#13511yescharacteristic
328C22#57511yescharacteristic
329C22#57611yescharacteristic
330C22#57711yescharacteristic
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—42no—
336C4—42no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4#52511yescharacteristic
343C4#53911yescharacteristic
344C4#54811yescharacteristic
345C4#55311yescharacteristic
346C2—42no—
347C2—81no—
348C2—161no—
349C2#244311yescharacteristic
350C2#309111yescharacteristic
351C2#309611yescharacteristic
352C2#326711yescharacteristic
353C2#328711yescharacteristic
354C2#334011yescharacteristic
355C2#334711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. C2×#427C2semidirect
4. #2354C2semidirect
5. #2093C2semidirect
6. #2084C2semidirect
7. #2075C2semidirect
8. #2034C2semidirect
9. #1926C2semidirect
10. #1921C2semidirect
11. #1893C2semidirect
12. #1875C2semidirect
13. #1869C2semidirect
14. #1854C2semidirect
15. #1854C2semidirect
116. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
C2×#4272
#24142
#23542
#20932
#20842
#20752
#20342
#19262
#19212
#18932
#18752
#18692
#18542