Finite Group Databasebeta

← #26580

Subgroups — #26580

Order 256.

372 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3099C211yesmaximal, characteristic, complemented
2#3098C211yesmaximal, characteristic, complemented
3#2946C211yesmaximal, characteristic, complemented
4#2946C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2051C211yesmaximal, characteristic, complemented
7#2051C211yesmaximal, characteristic, complemented
8#2006C211yesmaximal, characteristic
9#1952C211yesmaximal, characteristic, complemented
10#1951C211yesmaximal, characteristic, complemented
11#1929C211yesmaximal, characteristic, complemented
12#1929C211yesmaximal, characteristic, complemented
13#1860C211yesmaximal, characteristic, complemented
14#1860C211yesmaximal, characteristic, complemented
15#1856C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18#537C2211yescharacteristic
19#520C2211yescharacteristic
20#520C2211yescharacteristic
21#518C2211yescharacteristic
22#485—21no—
23#485—21no—
24#484—21no—
25#483—21no—
26#482—21no—
27#482—21no—
28#481—21no—
29#480—21no—
30#442—21no—
31#442—21no—
32#440—21no—
33#440—21no—
34C4×SD16—21no—
35C4×SD16—21no—
36C4×D8—21no—
37C4×D8—21no—
38C2×#108C2211yescharacteristic
39C2×#108C2211yescharacteristic
40#428C2211yescharacteristic
41#428C2211yescharacteristic
42#428C2211yescharacteristic
43#428C2211yescharacteristic
44C2×#107C2211yescharacteristic
45C2×#106C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#416C2211yescharacteristic
51#416C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic
54C2×#100C2211yescharacteristic
55#408C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic
58#405C2211yescharacteristic
59#390C2211yescharacteristic
60#382C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C22×D4C2311yescharacteristic
70AGL(1,ℤ/8)—22no—
71AGL(1,ℤ/8)—22no—
72AGL(1,ℤ/8)—22no—
73AGL(1,ℤ/8)—22no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86#121—21no—
87#121—21no—
88#121—41no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×D4—21no—
92C4×D4—21no—
93C4×D4—21no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4—21no—
97C4×D4—41no—
98C4×D4—41no—
99#117—21no—
100#117—21no—
101#117C2311yescharacteristic
102#117C2311yescharacteristic
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C42C2311yescharacteristic
111#108—22no—
112#108—22no—
113#107—21no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—21no—
118#107—21no—
119#106—21no—
120#106—21no—
121#106—21no—
122#106—21no—
123#106—21no—
124#106—21no—
125#104—22no—
126#104—22no—
127#103—21no—
128#103—21no—
129#103D411yescharacteristic
130#103D411yescharacteristic
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102—21no—
138#102—21no—
139#102D411yescharacteristic
140#102D411yescharacteristic
141#102D411yescharacteristic
142#102D411yescharacteristic
143#100—22no—
144C22⋊C8—21no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148#97—21no—
149#97—21no—
150C4×C8—21no—
151C4×C8—21no—
152#95—41no—
153C24—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158#41—21no—
159#41—21no—
160#41—21no—
161#41—21no—
162C2×Q8C2×D411yescharacteristic
163C2×Q8C2×D411yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—41no—
172C2×D4C2×D411yescharacteristic
173C2×D4C2×D411yescharacteristic
174C2×D4#4112yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4C2411yescharacteristic
186SD16—24no—
187SD16—24no—
188D8—24no—
189D8—24no—
190M16—21no—
191M16—21no—
192M16—21no—
193M16—21no—
194M16—21no—
195M16—21no—
196M16#4112yes—
197M16#4112yes—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8C2×D411yescharacteristic
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D411yescharacteristic
209C2×C8C2×D411yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C4⋊C4C2×D411yescharacteristic
222C4⋊C4C2×D411yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—21no—
225C22⋊C4—41no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—82no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—41no—
237C42—41no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—81no—
244C23—81no—
245C23H(5,𝔽2)11yescharacteristic
246Q8—21no—
247Q8—21no—
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—22no—
253D4—22no—
254D4—21no—
255D4—21no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—161no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
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—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—161no—
283C2×C4—161no—
284C2×C4—161no—
285C2×C4—161no—
286C2×C4C22×D411yescharacteristic
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C42−1+411yescharacteristic
292C8—22no—
293C8—22no—
294C8—21no—
295C8—21no—
296C8—21no—
297C8—21no—
298C8—41no—
299C8—41no—
300C22—21no—
301C22—21no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—82no—
310C22—81no—
311C22—81no—
312C22—161no—
313C22#53911yescharacteristic
314C22D4211yescharacteristic
315C22#55911yescharacteristic
316C4—41no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4#52511yescharacteristic
327C4#52511yescharacteristic
328C4#54711yescharacteristic
329C4#54711yescharacteristic
330C2—21no—
331C2—21no—
332C2—81no—
333C2—81no—
334C2—82no—
335C2#245511yescharacteristic
336C2#328811yescharacteristic
337C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#2006C2
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. #3099C2semidirect
2. #3098C2semidirect
3. #2946C2semidirect
4. #2946C2semidirect
5. #2092C2semidirect
6. #2051C2semidirect
7. #2051C2semidirect
9. #1952C2semidirect
10. #1951C2semidirect
11. #1929C2semidirect
12. #1929C2semidirect
13. #1860C2semidirect
14. #1860C2semidirect
15. #1856C2semidirect

Maximal subgroups

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

subgroupindex
#30992
#30982
#29462
#20922
#20512
#20062
#19522
#19512
#19292
#18602
#18562