Finite Group Databasebeta

← #26803

Subgroups — #26803

Order 256.

449 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#474C211yesmaximal, characteristic, complemented
3#2492C211yesmaximal, characteristic, complemented
4#2361C211yesmaximal, characteristic
5#2066C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2000C211yesmaximal, characteristic
9#1973C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1925C211yesmaximal, characteristic, complemented
12#1924C211yesmaximal, characteristic, complemented
13#1877C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#484—24no—
24#474—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×#97C2211yescharacteristic
40#400—21no—
41#393—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#390C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#388—21no—
49#386—21no—
50#385—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—41no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77C22⋊Q8—22no—
78C22⋊Q8—41no—
79#121—22no—
80#121—22no—
81C4×D4—22no—
82C4×D4—41no—
83C4×D4—44no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#107—22no—
109#107—22no—
110#107—41no—
111#107D414yescomplemented
112#106—22no—
113C4⋊C8—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122#97—22no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95D411yescharacteristic
132#95D411yescharacteristic
133C24—21no—
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—81no—
144C2×D4—24no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—42no—
151C2×D4—42no—
152C2×D4#4112yes—
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—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2411yescharacteristic
178SD16—44no—
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—
188C2×C8—21no—
189C2×C8C2×D412yes—
190C2×C8#4112yes—
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—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4C2×D412yes—
205C4⋊C4C2×D412yes—
206C4⋊C4#4112yes—
207C22⋊C4—24no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—42no—
220C42—81no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#13712yes—
235C23C22×D411yescharacteristic
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240Q8—84no—
241D4—22no—
242D4—22no—
243D4—48no—
244D4—41no—
245D4—41no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
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—84no—
286C2×C4—84no—
287C2×C4AGL(1,ℤ/8)12yes—
288C2×C4C22×D411yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4H(5,𝔽2)11yescharacteristic
293C2×C42−1+411yescharacteristic
294C8—22no—
295C8—22no—
296C8—41no—
297C8—41no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22#52511yescharacteristic
313C22D4211yescharacteristic
314C22#54711yescharacteristic
315C22C2×AGL(1,ℤ/8)11yescharacteristic
316C22C2×#13711yescharacteristic
317C22#57611yescharacteristic
318C22#57811yescharacteristic
319C4—22no—
320C4—41no—
321C4—41no—
322C4—42no—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—41no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4#53211yescharacteristic
332C4#54211yescharacteristic
333C4#54811yescharacteristic
334C4#55111yescharacteristic
335C2—42no—
336C2—82no—
337C2#246111yescharacteristic
338C2#311011yescharacteristic
339C2#312411yescharacteristic
340C2#326411yescharacteristic
341C2#327011yescharacteristic
342C2#334611yescharacteristic
343C2#335311yescharacteristic
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. C2×#484C2semidirect
2. C2×#474C2semidirect
3. #2492C2semidirect
5. #2066C2semidirect
6. #2057C2semidirect
9. #1973C2semidirect
10. #1928C2semidirect
11. #1925C2semidirect
12. #1924C2semidirect
13. #1877C2semidirect
14. #1852C2semidirect
15. #1776C2semidirect
111. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4742
#24922
#23612
#20662
#20572
#20342
#20002
#19732
#19282
#19252
#19242
#18772
#18522
#17762