Finite Group Databasebeta

← #27286

Subgroups — #27286

Order 256.

451 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3136C211yesmaximal, characteristic, complemented
2#3083C211yesmaximal, characteristic, complemented
3#3008C211yesmaximal, characteristic
4#2897C211yesmaximal, characteristic, complemented
5#2078C211yesmaximal, characteristic, complemented
6#2045C211yesmaximal, characteristic, complemented
7#2037C211yesmaximal, characteristic, complemented
8#2027C211yesmaximal, characteristic, complemented
9#1970C211yesmaximal, characteristic, complemented
10#1937C211yesmaximal, characteristic, complemented
11#1931C211yesmaximal, characteristic, complemented
12#1907C211yesmaximal, characteristic, complemented
13#1883C211yesmaximal, characteristic, complemented
14#1792C211yesmaximal, characteristic, complemented
15#1764C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17C2×AGL(1,ℤ/8)C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#131C2211yescharacteristic
20#544C2211yescharacteristic, complemented
21#535C2211yescharacteristic
22#520C2211yescharacteristic
23#490—21no—
24#489—21no—
25#488—21no—
26#486—21no—
27#479—21no—
28#478—21no—
29#477—21no—
30#474—21no—
31#449—21no—
32#449—21no—
33#447—21no—
34#447—21no—
35#428C2211yescharacteristic, complemented
36C2×#107C2211yescharacteristic, complemented
37#424C2211yescharacteristic, complemented
38C2×#104C2211yescharacteristic
39C2×#104C2211yescharacteristic
40#418—21no—
41#418C2211yescharacteristic
42#418C2211yescharacteristic
43#418C2211yescharacteristic, complemented
44#417—21no—
45#417C2211yescharacteristic
46#417C2211yescharacteristic
47#417C2211yescharacteristic
48#416—21no—
49#416C2211yescharacteristic
50#416C2211yescharacteristic
51C2×#102—21no—
52C2×#102C2211yescharacteristic
53C2×#102C2211yescharacteristic, complemented
54C2×#100C2211yescharacteristic
55C2×#100C2211yescharacteristic, complemented
56#408C2211yescharacteristic
57#405C2211yescharacteristic, complemented
58C2×C4×C8C2211yescharacteristic, complemented
59#390C2211yescharacteristic
60#390C2211yescharacteristic, complemented
61#343C2211yescharacteristic, complemented
62#343C2211yescharacteristic, complemented
63#341C2211yescharacteristic, complemented
64#341C2211yescharacteristic, complemented
65#339C2211yescharacteristic, complemented
66#339C2211yescharacteristic, complemented
67H(5,𝔽2)—22no—
68H(5,𝔽2)—22no—
69H(5,𝔽2)—22no—
70H(5,𝔽2)—41no—
71C2×#41—21no—
72C2×#41—21no—
73C2×#41C2311yescharacteristic
74C2×#41C2311yescharacteristic
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—41no—
78C22×D4C2311yescharacteristic
79C22×D4C2311yescharacteristic
80AGL(1,ℤ/8)—22no—
81AGL(1,ℤ/8)—22no—
82C2×SD16—21no—
83C2×D8—21no—
84C2×D8—22no—
85C2×D8—21no—
86C2×D8—21no—
87C2×D8—41no—
88#131—22no—
89#131—41no—
90C2×M16—21no—
91C2×M16C2311yescharacteristic
92C2×M16C2311yescharacteristic
93C2×M16C2311yescharacteristic
94C2×M16C2311yescharacteristic
95C22×C8—21no—
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C4⋊Q8—21no—
99C4⋊Q8—21no—
100#127—21no—
101#125—21no—
102#125—21no—
103#124—21no—
104C22⋊Q8—41no—
105#121—21no—
106#121—21no—
107#120—41no—
108C4×Q8—21no—
109C4×D4—21no—
110C4×D4—41no—
111#117—21no—
112#117C2311yescharacteristic
113#117C2311yescharacteristic
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—41no—
118C2×C22⋊C4—41no—
119C2×C42C2311yescharacteristic
120#107—21no—
121#107—21no—
122#107—21no—
123#106—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126#104—22no—
127#104—41no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103D411yescharacteristic
132#103D411yescharacteristic
133#103D411yescharacteristic
134#103D411yescharacteristic
135#102—21no—
136#102—21no—
137#102—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#102—21no—
148#102D411yescharacteristic
149#102D411yescharacteristic
150#102D411yescharacteristic
151#102D411yescharacteristic
152#100—22no—
153#100—41no—
154C22⋊C8—21no—
155C22⋊C8—21no—
156#97—21no—
157C4×C8—21no—
158C4×C8—41no—
159C24—21no—
160C24—21no—
161C24—41no—
162#41—21no—
163#41—21no—
164#41—24no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—21no—
169#41—42no—
170#41—42no—
171#41—42no—
172#41#4112yes—
173#41#4112yes—
174C2×Q8—41no—
175C2×Q8C2×D411yescharacteristic
176C2×Q8C2×D411yescharacteristic
177C2×D4—21no—
178C2×D4—24no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—42no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4—42no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—42no—
200C2×D4C2×D411yescharacteristic
201C2×D4C2×D411yescharacteristic
202C2×D4#4112yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4C2411yescharacteristic
211SD16—42no—
212D8—42no—
213D8—42no—
214D8—42no—
215M16—21no—
216M16—21no—
217M16—21no—
218M16—21no—
219M16—21no—
220M16—21no—
221M16—21no—
222M16—21no—
223M16—42no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—42no—
233C2×C8C2×D411yescharacteristic
234C2×C8C2×D411yescharacteristic
235C2×C8C2×D411yescharacteristic
236C2×C8C2×D411yescharacteristic
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4C2×D411yescharacteristic
246C4⋊C4C2×D411yescharacteristic
247C4⋊C4C2×D411yescharacteristic
248C4⋊C4C2×D411yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—41no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—44no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23—161no—
275C23—161no—
276C23H(5,𝔽2)11yescharacteristic
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—161no—
281Q8#12412yes—
282D4—22no—
283D4—22no—
284D4—21no—
285D4—22no—
286D4—21no—
287D4—22no—
288D4—21no—
289D4—44no—
290D4—44no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—161no—
300D4—161no—
301D4#12412yes—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—44no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—161no—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—21no—
328C8—21no—
329C8—21no—
330C8—21no—
331C8—41no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—82no—
345C22—82no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—161no—
352C22—161no—
353C22—164no—
354C22#53811yescharacteristic
355C22D4211yescharacteristic
356C22#55111yescharacteristic
357C4—22no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—81no—
365C4—81no—
366C4#52511yescharacteristic
367C4C2×#12411yescharacteristic
368C4#54611yescharacteristic
369C4#54611yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—81no—
374C2—82no—
375C2—161no—
376C2#247211yescharacteristic
377C2#329011yescharacteristic
378C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#3008C2
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. #3136C2semidirect
2. #3083C2semidirect
4. #2897C2semidirect
5. #2078C2semidirect
6. #2045C2semidirect
7. #2037C2semidirect
8. #2027C2semidirect
9. #1970C2semidirect
10. #1937C2semidirect
11. #1931C2semidirect
12. #1907C2semidirect
13. #1883C2semidirect
14. #1792C2semidirect
15. #1764C2semidirect
20. #544C22semidirect
35. #428C22semidirect
36. C2×#107C22semidirect
37. #424C22semidirect
43. #418C22semidirect
53. C2×#102C22semidirect
55. C2×#100C22semidirect
57. #405C22semidirect
58. C2×C4×C8C22semidirect
60. #390C22semidirect
61. #343C22semidirect
62. #343C22semidirect
63. #341C22semidirect
64. #341C22semidirect
65. #339C22semidirect
66. #339C22semidirect

Maximal subgroups

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

subgroupindex
#31362
#30832
#30082
#28972
#20782
#20452
#20372
#20272
#19702
#19372
#19312
#19072
#18832
#17922
#17642