Finite Group Databasebeta

← #25199

Subgroups — #25199

Order 256.

495 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3069C211yesmaximal, characteristic, complemented
2#3069C211yesmaximal, characteristic, complemented
3#3019C211yesmaximal, characteristic
4C2×#455C211yesmaximal, characteristic, complemented
5#2956C211yesmaximal, characteristic, complemented
6#2888C211yesmaximal, characteristic, complemented
7#2069C211yesmaximal, characteristic, complemented
8#2050C211yesmaximal, characteristic, complemented
9#2049C211yesmaximal, characteristic, complemented
10#2004C211yesmaximal, characteristic, complemented
11#1967C211yesmaximal, characteristic, complemented
12#1932C211yesmaximal, characteristic, complemented
13#1914C211yesmaximal, characteristic, complemented
14#1902C211yesmaximal, characteristic, complemented
15#1861C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17#575C2211yescharacteristic
18#575C2211yescharacteristic
19C2×#137C2211yescharacteristic
20C2×#137C2211yescharacteristic
21C2×#137C2211yescharacteristic, complemented
22#568C2211yescharacteristic
23#568C2211yescharacteristic
24#568C2211yescharacteristic, complemented
25#535C2211yescharacteristic
26C2×#124C2211yescharacteristic
27#473—22no—
28#473—22no—
29#471—22no—
30#471—22no—
31#455—22no—
32#455—22no—
33#455—22no—
34#453—22no—
35#444—22no—
36#443—22no—
37#430C2211yescharacteristic, complemented
38#430C2211yescharacteristic, complemented
39#430C2211yescharacteristic, complemented
40#423C2211yescharacteristic, complemented
41#421C2211yescharacteristic
42#421C2211yescharacteristic
43C2×#104C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic, complemented
46#419C2211yescharacteristic, complemented
47#413—22no—
48#413C2211yescharacteristic
49#413C2211yescharacteristic
50#413C2211yescharacteristic, complemented
51C2×#101C2211yescharacteristic, complemented
52C2×#101C2211yescharacteristic, complemented
53C2×#100—22no—
54C2×#100C2211yescharacteristic
55C2×#100C2211yescharacteristic
56C2×#100C2211yescharacteristic
57C2×#100C2211yescharacteristic
58C4×M16C2211yescharacteristic, complemented
59#344C2211yescharacteristic, complemented
60#344C2211yescharacteristic, complemented
61#339C2211yescharacteristic, complemented
62#338C2211yescharacteristic, complemented
63H(5,𝔽2)—22no—
64H(5,𝔽2)—22no—
65H(5,𝔽2)—41no—
66H(5,𝔽2)—41no—
67C2×#41—21no—
68C2×#41—21no—
69C2×#41C2311yescharacteristic
70C2×#41C2311yescharacteristic
71C2×#41C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×D4—22no—
74C22×D4—21no—
75C22×D4—21no—
76C22×D4C2311yescharacteristic
77#137—21no—
78#137—22no—
79#137—22no—
80#137—21no—
81#137—21no—
82#137—22no—
83#137—22no—
84#137—41no—
85#137—41no—
86#137D412yes—
87AGL(1,ℤ/8)—21no—
88AGL(1,ℤ/8)—21no—
89AGL(1,ℤ/8)—21no—
90AGL(1,ℤ/8)D412yes—
91#135—22no—
92#135—41no—
93C2×Q16—21no—
94C2×Q16—21no—
95C2×Q16—21no—
96C2×SD16—21no—
97C2×SD16—21no—
98C2×SD16—21no—
99#131—22no—
100#131—22no—
101#131—22no—
102#131—22no—
103#131—41no—
104#131—41no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C2×M16C2311yescharacteristic
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C2×M16C2311yescharacteristic
112C2×M16C2311yescharacteristic
113C2×M16C2311yescharacteristic
114C2×M16C2311yescharacteristic
115C2×M16C2311yescharacteristic
116#127—21no—
117#124—21no—
118#124—21no—
119#124—21no—
120#124—41no—
121#121—41no—
122#120—22no—
123#117C2311yescharacteristic
124C2×C22⋊C4—22no—
125C2×C42C2311yescharacteristic
126#108—22no—
127#108—22no—
128#108—22no—
129C4⋊C8—41no—
130#104—22no—
131#104—22no—
132#104—21no—
133#104—21no—
134#104—21no—
135#104—21no—
136#104D412yescomplemented
137#104D412yescomplemented
138#103—41no—
139#101—21no—
140#101—21no—
141#101—22no—
142#101—22no—
143#101D412yescomplemented
144#100—21no—
145#100—21no—
146#100—22no—
147#100—22no—
148#100—22no—
149#100—22no—
150#100D412yescomplemented
151#97—21no—
152C4×C8—21no—
153C24—22no—
154C24#4112yes—
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—
165#41—42no—
166#41—42no—
167#41—42no—
168#41—42no—
169#41C2×D412yes—
170#41C2×D412yes—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8C2×D411yescharacteristic
178C2×Q8C2×D411yescharacteristic
179C2×Q8#4111yescharacteristic
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
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—42no—
193C2×D4—44no—
194C2×D4—44no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4C2×D411yescharacteristic
199C2×D4C2×D411yescharacteristic
200C2×D4#4111yescharacteristic
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4C2411yescharacteristic
208Q16—22no—
209Q16—42no—
210Q16—42no—
211Q16—41no—
212Q16—42no—
213SD16—22no—
214SD16—22no—
215SD16—42no—
216SD16—42no—
217SD16—41no—
218SD16—41no—
219SD16—42no—
220D8—22no—
221D8—41no—
222M16—21no—
223M16—21no—
224M16—21no—
225M16—21no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—21no—
230M16—21no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16C2×D412yes—
238M16C2×D412yes—
239M16C2×D412yes—
240M16C2×D412yes—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—21no—
258C42—21no—
259C42C2×D411yescharacteristic
260C42C2×D411yescharacteristic
261C23—21no—
262C23—22no—
263C23—21no—
264C23—22no—
265C23—22no—
266C23—21no—
267C23—44no—
268C23—82no—
269C23—162no—
270C23C2×#4111yescharacteristic
271Q8—21no—
272Q8—21no—
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8#12112yes—
281D4—22no—
282D4—22no—
283D4—21no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—44no—
289D4—44no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—162no—
299D4#12112yes—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4C22×D411yescharacteristic
318C2×C4C22×D411yescharacteristic
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—22no—
324C8—22no—
325C8—22no—
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22—81no—
342C22—164no—
343C22#52511yescharacteristic
344C22#54611yescharacteristic
345C22#54611yescharacteristic
346C4—22no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4C2×#12111yescharacteristic
354C4#53811yescharacteristic
355C4D4211yescharacteristic
356C4D4211yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—81no—
362C2—82no—
363C2—81no—
364C2#242411yescharacteristic
subgroupG/H
center, socleC2#2424
commutator, Frattini, cosocleC22×C4C24
prime radical#3019C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3069C2semidirect
2. #3069C2semidirect
4. C2×#455C2semidirect
5. #2956C2semidirect
6. #2888C2semidirect
7. #2069C2semidirect
8. #2050C2semidirect
9. #2049C2semidirect
10. #2004C2semidirect
11. #1967C2semidirect
12. #1932C2semidirect
13. #1914C2semidirect
14. #1902C2semidirect
15. #1861C2semidirect
21. C2×#137C22semidirect
24. #568C22semidirect
37. #430C22semidirect
38. #430C22semidirect
39. #430C22semidirect
40. #423C22semidirect
45. C2×#104C22semidirect
46. #419C22semidirect
50. #413C22semidirect
51. C2×#101C22semidirect
52. C2×#101C22semidirect
58. C4×M16C22semidirect
59. #344C22semidirect
60. #344C22semidirect
61. #339C22semidirect
62. #338C22semidirect
136. #104D4semidirect
137. #104D4semidirect
143. #101D4semidirect
150. #100D4semidirect

Maximal subgroups

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

subgroupindex
#30692
#30192
C2×#4552
#29562
#28882
#20692
#20502
#20492
#20042
#19672
#19322
#19142
#19022
#18612