Finite Group Databasebeta

← #26922

Subgroups — #26922

Order 256.

366 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3082C211yesmaximal, characteristic, complemented
2#3081C211yesmaximal, characteristic
3#2950C211yesmaximal, characteristic, complemented
4#2949C211yesmaximal, characteristic
5#2047C211yesmaximal, characteristic, complemented
6#2046C211yesmaximal, characteristic
7#2028C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic
9#1931C211yesmaximal, characteristic
10#1930C211yesmaximal, characteristic, complemented
11#1922C211yesmaximal, characteristic, complemented
12#1922C211yesmaximal, characteristic, complemented
13#1888C211yesmaximal, characteristic
14#1885C211yesmaximal, characteristic
15#1857C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18#544C2211yescharacteristic
19C2×#124C2211yescharacteristic
20#519C2211yescharacteristic
21#519C2211yescharacteristic
22#518C2211yescharacteristic
23#479—21no—
24#479—21no—
25#478—21no—
26#478—21no—
27#477—21no—
28#476—21no—
29#475—21no—
30#474—21no—
31#441—21no—
32#441—21no—
33#440—21no—
34#440—21no—
35C4×Q16—21no—
36C4×Q16—21no—
37C4×SD16—21no—
38C4×SD16—21no—
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#427C2211yescharacteristic
42#427C2211yescharacteristic
43#424C2211yescharacteristic
44#424C2211yescharacteristic
45#419C2211yescharacteristic
46#419C2211yescharacteristic
47#419C2211yescharacteristic
48#419C2211yescharacteristic
49#418C2211yescharacteristic
50#418C2211yescharacteristic
51C2×#103C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#101C2211yescharacteristic
54C2×#101C2211yescharacteristic
55C2×#100C2211yescharacteristic
56#405C2211yescharacteristic
57#405C2211yescharacteristic
58#391C2211yescharacteristic
59#343C2211yescharacteristic
60#341C2211yescharacteristic
61#341C2211yescharacteristic
62#339C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86#125—21no—
87#125—21no—
88#124—22no—
89#124—22no—
90C22⋊Q8—41no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103#117—41no—
104#117—41no—
105#117C2311yescharacteristic
106#117C2311yescharacteristic
107#117C2311yescharacteristic
108#117C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C42C2311yescharacteristic
113#107—21no—
114#107—21no—
115#107D411yescharacteristic
116#107D411yescharacteristic
117#107D411yescharacteristic
118#107D411yescharacteristic
119#106—21no—
120#106—21no—
121#106D411yescharacteristic
122#106D411yescharacteristic
123#106D411yescharacteristic
124#106D411yescharacteristic
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103—21no—
136#103—21no—
137#103—21no—
138#103—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#101—22no—
146#101—22no—
147#100—22no—
148#97—21no—
149#97—21no—
150C4×C8—21no—
151C4×C8—21no—
152#95—41no—
153C24—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8#4112yes—
166C2×Q8#4112yes—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4#4112yes—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4C2411yescharacteristic
180Q16—24no—
181Q16—24no—
182SD16—24no—
183SD16—24no—
184M16—21no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8C2×D411yescharacteristic
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C2×C8C2×D411yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4C2×D411yescharacteristic
214C4⋊C4C2×D411yescharacteristic
215C4⋊C4C2×D411yescharacteristic
216C4⋊C4C2×D411yescharacteristic
217C4⋊C4C2×D411yescharacteristic
218C4⋊C4C2×D411yescharacteristic
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—44no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C42—41no—
237C42—41no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—81no—
242C23—81no—
243C23H(5,𝔽2)11yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—22no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—161no—
253D4—22no—
254D4—22no—
255D4—41no—
256D4—41no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
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—81no—
276C2×C4—161no—
277C2×C4—161no—
278C2×C4C22×D411yescharacteristic
279C2×C4C22×D411yescharacteristic
280C2×C4C2×#4111yescharacteristic
281C2×C4C2×#4111yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C42−1+411yescharacteristic
284C8—21no—
285C8—21no—
286C8—21no—
287C8—21no—
288C8—41no—
289C8—41no—
290C8—41no—
291C8—41no—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—82no—
296C22—81no—
297C22—81no—
298C22—161no—
299C22#53911yescharacteristic
300C22D4211yescharacteristic
301C22#55111yescharacteristic
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4#52511yescharacteristic
315C4#52511yescharacteristic
316C4#54811yescharacteristic
317C4#54811yescharacteristic
318C2—21no—
319C2—21no—
320C2—82no—
321C2#246311yescharacteristic
322C2#328911yescharacteristic
323C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3082C2semidirect
3. #2950C2semidirect
5. #2047C2semidirect
7. #2028C2semidirect
10. #1930C2semidirect
11. #1922C2semidirect
12. #1922C2semidirect
15. #1857C2semidirect

Maximal subgroups

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

subgroupindex
#30822
#30812
#29502
#29492
#20472
#20462
#20282
#20152
#19312
#19302
#19222
#18882
#18852
#18572