Finite Group Databasebeta

← #30431

Subgroups — #30431

Order 256.

456 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3061C211yesmaximal, characteristic, complemented
2#3060C211yesmaximal, characteristic, complemented
3#3012C211yesmaximal, characteristic, complemented
4#3012C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2079C211yesmaximal, characteristic, complemented
7#2078C211yesmaximal, characteristic, complemented
8#2040C211yesmaximal, characteristic, complemented
9#2040C211yesmaximal, characteristic, complemented
10#2006C211yesmaximal, characteristic, complemented
11#1907C211yesmaximal, characteristic, complemented
12#1907C211yesmaximal, characteristic, complemented
13#1876C211yesmaximal, characteristic, complemented
14#1876C211yesmaximal, characteristic, complemented
15#1792C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C2×AGL(1,ℤ/8)C2211yescharacteristic
19C2×#135C2211yescharacteristic
20C2×#135C2211yescharacteristic
21C2×#131C2211yescharacteristic, complemented
22C2×#131C2211yescharacteristic, complemented
23#537C2211yescharacteristic
24#537C2211yescharacteristic
25#537C2211yescharacteristic, complemented
26#470—21no—
27#469—21no—
28#468—21no—
29#468—21no—
30#467—21no—
31#466—21no—
32#465—21no—
33#465—21no—
34#452—21no—
35#452—21no—
36#450—21no—
37#450—21no—
38#449—21no—
39#449—21no—
40#447—21no—
41#447—21no—
42C2×#107C2211yescharacteristic, complemented
43C2×#106C2211yescharacteristic, complemented
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic, complemented
47C2×#104C2211yescharacteristic, complemented
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#417C2211yescharacteristic
51#417C2211yescharacteristic
52#417C2211yescharacteristic, complemented
53#417C2211yescharacteristic, complemented
54C2×#102C2211yescharacteristic
55C2×#102C2211yescharacteristic
56C2×#100C2211yescharacteristic
57#408C2211yescharacteristic
58#408C2211yescharacteristic
59#390C2211yescharacteristic
60#382C2211yescharacteristic, complemented
61#343C2211yescharacteristic, complemented
62#343C2211yescharacteristic, complemented
63#343C2211yescharacteristic, complemented
64#341C2211yescharacteristic, complemented
65#341C2211yescharacteristic, complemented
66#336C2211yescharacteristic, complemented
67H(5,𝔽2)—22no—
68H(5,𝔽2)—41no—
69H(5,𝔽2)—41no—
70H(5,𝔽2)—41no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2311yescharacteristic
74C2×#41C2311yescharacteristic
75C2×#41C2311yescharacteristic
76C2×#41C2311yescharacteristic
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—41no—
80C22×D4C2311yescharacteristic
81AGL(1,ℤ/8)—22no—
82AGL(1,ℤ/8)—22no—
83AGL(1,ℤ/8)—41no—
84AGL(1,ℤ/8)—41no—
85#135—41no—
86#135—41no—
87#135—41no—
88#135—41no—
89C2×Q16D411yescharacteristic
90C2×Q16D411yescharacteristic
91C2×SD16—21no—
92C2×SD16—21no—
93C2×SD16D411yescharacteristic
94C2×SD16D411yescharacteristic
95C2×SD16D411yescharacteristic
96C2×SD16D411yescharacteristic
97C2×D8—21no—
98C2×D8—21no—
99C2×D8D411yescharacteristic
100C2×D8D411yescharacteristic
101#131—41no—
102#131—41no—
103#131—41no—
104#131—41no—
105C2×M16—21no—
106C2×M16—21no—
107C2×M16C2311yescharacteristic
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C22×C8—21no—
112C22×C8—21no—
113C22×C8C2311yescharacteristic
114C22×C8C2311yescharacteristic
115C22⋊Q8—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118C22⋊Q8—21no—
119C22⋊Q8—21no—
120C22⋊Q8—21no—
121#121—21no—
122#121—21no—
123#121—21no—
124#121—21no—
125#121—21no—
126#121—21no—
127#121—41no—
128#121—41no—
129#121—41no—
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—41no—
134C2×C42C2311yescharacteristic
135#107—21no—
136#107—21no—
137#106—21no—
138#106—21no—
139#104—41no—
140#104—41no—
141#104—41no—
142#104—41no—
143#103—21no—
144#103—21no—
145#103—21no—
146#103—21no—
147#103—21no—
148#103—21no—
149#102—21no—
150#102—21no—
151#102—21no—
152#102—21no—
153#102—21no—
154#102—21no—
155#102—21no—
156#102—21no—
157#102—21no—
158#102—21no—
159#100—41no—
160C22⋊C8—21no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—21no—
164#95—41no—
165C24—21no—
166C24—41no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174#41—21no—
175#41—21no—
176#41—21no—
177#41—21no—
178#41—21no—
179#41—21no—
180#41—21no—
181#41—21no—
182#41—21no—
183#41—42no—
184#41—42no—
185#41—42no—
186#41—42no—
187C2×Q8C2×D411yescharacteristic
188C2×Q8C2×D411yescharacteristic
189C2×Q8C2×D411yescharacteristic
190C2×Q8C2×D411yescharacteristic
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4C2×D411yescharacteristic
212C2×D4C2×D411yescharacteristic
213C2×D4C2×D411yescharacteristic
214C2×D4C2×D411yescharacteristic
215C2×D4#4112yes—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4C2411yescharacteristic
226Q16—41no—
227Q16—41no—
228SD16—41no—
229SD16—41no—
230SD16—41no—
231SD16—41no—
232SD16—42no—
233SD16—42no—
234D8—41no—
235D8—41no—
236D8—42no—
237D8—42no—
238M16—21no—
239M16—21no—
240M16—21no—
241M16—21no—
242M16—21no—
243M16—21no—
244M16—21no—
245M16—21no—
246M16—42no—
247M16—42no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8C2×D411yescharacteristic
259C2×C8C2×D411yescharacteristic
260C2×C8C2×D411yescharacteristic
261C2×C8C2×D411yescharacteristic
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—82no—
278C42—21no—
279C42—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—22no—
285C23—44no—
286C23—41no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—161no—
291C23—161no—
292C23H(5,𝔽2)11yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—21no—
300Q8—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—21no—
311D4—44no—
312D4—44no—
313D4—41no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—41no—
322D4—41no—
323D4—161no—
324D4—161no—
325D4—161no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4C22×D411yescharacteristic
349C2×C4C22×D411yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C2×C4H(5,𝔽2)11yescharacteristic
354C8—21no—
355C8—21no—
356C8—21no—
357C8—21no—
358C8—41no—
359C8—41no—
360C8—41no—
361C8—41no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—161no—
381C22—164no—
382C22#53811yescharacteristic
383C22D4211yescharacteristic
384C22#56011yescharacteristic
385C4—41no—
386C4—41no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4#53411yescharacteristic
394C4#53411yescharacteristic
395C4#54611yescharacteristic
396C4#54611yescharacteristic
397C2—21no—
398C2—21no—
399C2—81no—
400C2—81no—
401C2—81no—
402C2—81no—
403C2—82no—
404C2#266811yescharacteristic
405C2#328811yescharacteristic
406C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
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. #3061C2semidirect
2. #3060C2semidirect
3. #3012C2semidirect
4. #3012C2semidirect
5. #2092C2semidirect
6. #2079C2semidirect
7. #2078C2semidirect
8. #2040C2semidirect
9. #2040C2semidirect
10. #2006C2semidirect
11. #1907C2semidirect
12. #1907C2semidirect
13. #1876C2semidirect
14. #1876C2semidirect
15. #1792C2semidirect
21. C2×#131C22semidirect
22. C2×#131C22semidirect
25. #537C22semidirect
42. C2×#107C22semidirect
43. C2×#106C22semidirect
46. C2×#104C22semidirect
47. C2×#104C22semidirect
52. #417C22semidirect
53. #417C22semidirect
60. #382C22semidirect
61. #343C22semidirect
62. #343C22semidirect
63. #343C22semidirect
64. #341C22semidirect
65. #341C22semidirect
66. #336C22semidirect

Maximal subgroups

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

subgroupindex
#30612
#30602
#30122
#20922
#20792
#20782
#20402
#20062
#19072
#18762
#17922