Finite Group Databasebeta

← #25055

Subgroups — #25055

Order 256.

362 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#3060C211yesmaximal, characteristic
4#2951C211yesmaximal, characteristic, complemented
5#2079C211yesmaximal, characteristic, complemented
6#2078C211yesmaximal, characteristic, complemented
7#2040C211yesmaximal, characteristic, complemented
8#2039C211yesmaximal, characteristic, complemented
9#1975C211yesmaximal, characteristic, complemented
10#1951C211yesmaximal, characteristic, complemented
11#1860C211yesmaximal, characteristic, complemented
12#1859C211yesmaximal, characteristic, complemented
13#1847C211yesmaximal, characteristic, complemented
14#1816C211yesmaximal, characteristic, complemented
15#1765C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C2×#131C2211yescharacteristic
18#537C2211yescharacteristic
19#537C2211yescharacteristic
20#520C2211yescharacteristic
21#520C2211yescharacteristic
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#470—21no—
31#469—21no—
32#465—21no—
33#465—21no—
34#440—21no—
35#440—21no—
36C4×Q16—21no—
37C4×D8—21no—
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41C2×#107C2211yescharacteristic
42C2×#107C2211yescharacteristic
43C2×#106C2211yescharacteristic
44C2×#106C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#417C2211yescharacteristic
53#417C2211yescharacteristic
54#416C2211yescharacteristic
55#408C2211yescharacteristic
56#408C2211yescharacteristic
57#405C2211yescharacteristic
58C2×#97C2211yescharacteristic
59#384C2211yescharacteristic
60#382C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#343C2211yescharacteristic
64#343C2211yescharacteristic
65#341C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C2×#41C2311yescharacteristic
70#135—22no—
71#135—22no—
72C2×Q16D411yescharacteristic
73C2×SD16D411yescharacteristic
74C2×SD16D411yescharacteristic
75C2×D8D411yescharacteristic
76#131—22no—
77#131—22no—
78C2×M16—21no—
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C2×M16C2311yescharacteristic
82C2×M16C2311yescharacteristic
83C22×C8—21no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90#121—21no—
91#121—21no—
92#121—21no—
93#121—21no—
94#121—41no—
95#121—41no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—41no—
101C4×D4—41no—
102#117—21no—
103#117—21no—
104#117C2311yescharacteristic
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#108—22no—
114#107—21no—
115#107—21no—
116#107—21no—
117#107—21no—
118#107—21no—
119#107—21no—
120#106—21no—
121#106—21no—
122#106—21no—
123#106—21no—
124#106—21no—
125#106—21no—
126#104—22no—
127#104—22no—
128#104—22no—
129#103—21no—
130#103—21no—
131#103—21no—
132#103—21no—
133#103—21no—
134#103—21no—
135#103D411yescharacteristic
136#103D411yescharacteristic
137#102—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102—21no—
142#102—21no—
143#102D411yescharacteristic
144#102D411yescharacteristic
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—21no—
149#97—21no—
150#97—22no—
151C4×C8—21no—
152#95—41no—
153#41—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—
162#41—21no—
163#41—21no—
164#41—21no—
165C2×Q8C2×D411yescharacteristic
166C2×Q8C2×D411yescharacteristic
167C2×Q8C2×D411yescharacteristic
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4C2×D411yescharacteristic
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—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4C2411yescharacteristic
187Q16—22no—
188SD16—22no—
189SD16—22no—
190D8—22no—
191M16—21no—
192M16—21no—
193M16—21no—
194M16—21no—
195M16—21no—
196M16—21no—
197M16—21no—
198M16—21no—
199M16—42no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—42no—
207C2×C8C2×D411yescharacteristic
208C2×C8C2×D411yescharacteristic
209C2×C8C2×D411yescharacteristic
210C2×C8C2×D411yescharacteristic
211C2×C8#4112yes—
212C2×C8#4112yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—82no—
226C4⋊C4C2×D411yescharacteristic
227C4⋊C4C2×D411yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C42—21no—
236C42—21no—
237C42—41no—
238C42—41no—
239C42#4112yes—
240C23—21no—
241C23—21no—
242C23—21no—
243C23H(5,𝔽2)11yescharacteristic
244Q8—21no—
245Q8—21no—
246Q8—21no—
247Q8—21no—
248Q8—21no—
249Q8—21no—
250D4—21no—
251D4—21no—
252D4—21no—
253D4—21no—
254D4—21no—
255D4—21no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—161no—
263D4—161no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
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—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—161no—
287C2×C4—161no—
288C2×C4C22×D411yescharacteristic
289C2×C4C22×D411yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C42−1+411yescharacteristic
294C8—21no—
295C8—21no—
296C8—41no—
297C8—41no—
298C8—41no—
299C8—41no—
300C8#12312yes—
301C8#12312yes—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22#54211yescharacteristic
314C22D4211yescharacteristic
315C22#55311yescharacteristic
316C4—42no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4C2×#12311yescharacteristic
327C4#53211yescharacteristic
328C4#54711yescharacteristic
329C4#54711yescharacteristic
330C2—21no—
331C2—21no—
332C2—81no—
333C2—81no—
334C2—81no—
335C2#242211yescharacteristic
336C2#328811yescharacteristic
337C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#3060C2
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
4. #2951C2semidirect
5. #2079C2semidirect
6. #2078C2semidirect
7. #2040C2semidirect
8. #2039C2semidirect
9. #1975C2semidirect
10. #1951C2semidirect
11. #1860C2semidirect
12. #1859C2semidirect
13. #1847C2semidirect
14. #1816C2semidirect
15. #1765C2semidirect

Maximal subgroups

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

subgroupindex
#30992
#30982
#30602
#29512
#20792
#20782
#20402
#20392
#19752
#19512
#18602
#18592
#18472
#18162
#17652