Finite Group Databasebeta

← #26195

Subgroups — #26195

Order 256.

379 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3096C211yesmaximal, characteristic, complemented
2#3045C211yesmaximal, characteristic
3#2952C211yesmaximal, characteristic, complemented
4#2913C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2090C211yesmaximal, characteristic, complemented
7#2078C211yesmaximal, characteristic, complemented
8#2037C211yesmaximal, characteristic, complemented
9#1931C211yesmaximal, characteristic, complemented
10#1922C211yesmaximal, characteristic, complemented
11#1907C211yesmaximal, characteristic, complemented
12#1883C211yesmaximal, characteristic, complemented
13#1876C211yesmaximal, characteristic, complemented
14#1860C211yesmaximal, characteristic, complemented
15#1860C211yesmaximal, characteristic, complemented
16C2×AGL(1,ℤ/8)C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#131C2211yescharacteristic
19#535C2211yescharacteristic
20#520C2211yescharacteristic
21#520C2211yescharacteristic
22#520C2211yescharacteristic
23#482—21no—
24#482—21no—
25#481—21no—
26#480—21no—
27#464—21no—
28#463—21no—
29#460—21no—
30#459—21no—
31#442—21no—
32#441—21no—
33C4×SD16—21no—
34C4×SD16—21no—
35#428—21no—
36#428—21no—
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#428C2211yescharacteristic
40#427—21no—
41#427C2211yescharacteristic
42C2×#107—21no—
43C2×#107C2211yescharacteristic
44C2×#106C2211yescharacteristic
45#424C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#418C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic
54C2×#100C2211yescharacteristic
55#408C2211yescharacteristic
56#405C2211yescharacteristic
57#390C2211yescharacteristic
58#382C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#341C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C22×D4C2311yescharacteristic
70AGL(1,ℤ/8)—22no—
71AGL(1,ℤ/8)—22no—
72#135—22no—
73#135—22no—
74C2×Q16D411yescharacteristic
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×SD16D411yescharacteristic
78C2×D8—21no—
79C2×D8D411yescharacteristic
80#131—22no—
81#131—22no—
82C2×M16—21no—
83C2×M16C2311yescharacteristic
84C2×M16C2311yescharacteristic
85C2×M16C2311yescharacteristic
86C2×M16C2311yescharacteristic
87C22×C8—21no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#127—21no—
91#124—21no—
92#121—21no—
93#121—21no—
94#120—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—41no—
102C4×D4—41no—
103C4×D4—41no—
104#117—21no—
105#117—21no—
106#117—21no—
107#117C2311yescharacteristic
108#117C2311yescharacteristic
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—41no—
116C2×C42C2311yescharacteristic
117#107—21no—
118#107—21no—
119#107—21no—
120#107—21no—
121#107—21no—
122#107D411yescharacteristic
123#107D411yescharacteristic
124#106—21no—
125#106—21no—
126#106—21no—
127#106—21no—
128#106—21no—
129#106D411yescharacteristic
130#106D411yescharacteristic
131C4⋊C8—21no—
132C4⋊C8—21no—
133#104—22no—
134#104—22no—
135#103—21no—
136#103—21no—
137#103—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102—21no—
144#102—21no—
145#102—21no—
146#102—21no—
147#100—22no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150#97—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#4112yes—
161#41#4112yes—
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—21no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4C2×D411yescharacteristic
175C2×D4C2×D411yescharacteristic
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4C2411yescharacteristic
186Q16—22no—
187SD16—22no—
188SD16—22no—
189SD16—42no—
190D8—22no—
191D8—42no—
192M16—24no—
193M16—21no—
194M16—21no—
195M16—21no—
196M16—21no—
197M16—21no—
198M16—21no—
199M16—21no—
200M16#4112yes—
201C2×C8—24no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8C2×D411yescharacteristic
211C2×C8C2×D411yescharacteristic
212C2×C8C2×D411yescharacteristic
213C2×C8C2×D411yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4C2×D411yescharacteristic
224C4⋊C4C2×D411yescharacteristic
225C4⋊C4C2×D411yescharacteristic
226C4⋊C4C2×D411yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—21no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—41no—
240C42—41no—
241C42—41no—
242C23—21no—
243C23—21no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—81no—
248C23—81no—
249C23H(5,𝔽2)11yescharacteristic
250Q8—21no—
251Q8—21no—
252Q8—21no—
253Q8#12312yes—
254D4—22no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—21no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—161no—
266D4—161no—
267D4#12312yes—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
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—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—161no—
288C2×C4—161no—
289C2×C4—161no—
290C2×C4C22×D411yescharacteristic
291C2×C4C22×D411yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C42−1+411yescharacteristic
296C8—21no—
297C8—21no—
298C8—21no—
299C8—22no—
300C8—21no—
301C8—41no—
302C8—41no—
303C8—41no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—82no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—161no—
317C22#53911yescharacteristic
318C22D4211yescharacteristic
319C22#55311yescharacteristic
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4C2×#12311yescharacteristic
331C4#52511yescharacteristic
332C4#54711yescharacteristic
333C4#54811yescharacteristic
334C2—21no—
335C2—21no—
336C2—42no—
337C2—81no—
338C2—161no—
339C2#244311yescharacteristic
340C2#328811yescharacteristic
341C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#3045C2
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. #3096C2semidirect
3. #2952C2semidirect
4. #2913C2semidirect
5. #2092C2semidirect
6. #2090C2semidirect
7. #2078C2semidirect
8. #2037C2semidirect
9. #1931C2semidirect
10. #1922C2semidirect
11. #1907C2semidirect
12. #1883C2semidirect
13. #1876C2semidirect
14. #1860C2semidirect
15. #1860C2semidirect

Maximal subgroups

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

subgroupindex
#30962
#30452
#29522
#29132
#20922
#20902
#20782
#20372
#19312
#19222
#19072
#18832
#18762
#18602