Finite Group Databasebeta

← #25196

Subgroups — #25196

Order 256.

516 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3068C211yesmaximal, characteristic, complemented
2#3068C211yesmaximal, characteristic, complemented
3#3019C211yesmaximal, characteristic, complemented
4C2×#453C211yesmaximal, characteristic, complemented
5#2956C211yesmaximal, characteristic, complemented
6#2888C211yesmaximal, characteristic, complemented
7#2069C211yesmaximal, characteristic, complemented
8#2050C211yesmaximal, characteristic, complemented
9#2048C211yesmaximal, characteristic, complemented
10#2002C211yesmaximal, characteristic, complemented
11#1965C211yesmaximal, characteristic, complemented
12#1932C211yesmaximal, characteristic, complemented
13#1914C211yesmaximal, characteristic, complemented
14#1903C211yesmaximal, characteristic, complemented
15#1861C211yesmaximal, characteristic, complemented
16C2×H(5,𝔽2)C2211yescharacteristic
17#575C2211yescharacteristic
18#575C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2211yescharacteristic
20C2×AGL(1,ℤ/8)C2211yescharacteristic
21C2×AGL(1,ℤ/8)C2211yescharacteristic, complemented
22#568C2211yescharacteristic
23#568C2211yescharacteristic
24#568C2211yescharacteristic, complemented
25#535C2211yescharacteristic
26C2×#127C2211yescharacteristic
27#472—22no—
28#472—22no—
29#471—22no—
30#471—22no—
31#455—22no—
32#453—22no—
33#453—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#418C2211yescharacteristic, complemented
47#413—22no—
48#413C2211yescharacteristic
49#413C2211yescharacteristic
50#413C2211yescharacteristic, complemented
51C2×#100—22no—
52C2×#100C2211yescharacteristic
53C2×#100C2211yescharacteristic
54C2×#100C2211yescharacteristic
55C2×#100C2211yescharacteristic
56C2×#100C2211yescharacteristic, complemented
57C2×#100C2211yescharacteristic, complemented
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×D4—22no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78#137—21no—
79#137—21no—
80#137—21no—
81#137D412yescomplemented
82AGL(1,ℤ/8)—21no—
83AGL(1,ℤ/8)—22no—
84AGL(1,ℤ/8)—22no—
85AGL(1,ℤ/8)—21no—
86AGL(1,ℤ/8)—21no—
87AGL(1,ℤ/8)—22no—
88AGL(1,ℤ/8)—22no—
89AGL(1,ℤ/8)—41no—
90AGL(1,ℤ/8)—41no—
91AGL(1,ℤ/8)D412yescomplemented
92#135—22no—
93#135—41no—
94C2×SD16—21no—
95C2×SD16—21no—
96C2×SD16—21no—
97C2×D8—21no—
98C2×D8—21no—
99C2×D8—21no—
100#131—22no—
101#131—22no—
102#131—22no—
103#131—22no—
104#131—41no—
105#131—41no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—41no—
109C2×M16C2311yescharacteristic
110C2×M16C2311yescharacteristic
111C2×M16C2311yescharacteristic
112C2×M16C2311yescharacteristic
113C2×M16C2311yescharacteristic
114C2×M16C2311yescharacteristic
115C2×M16C2311yescharacteristic
116C2×M16C2311yescharacteristic
117#127—21no—
118#127—21no—
119#127—21no—
120#127—41no—
121#124—21no—
122#121—41no—
123#120—22no—
124#117C2311yescharacteristic
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#102—41no—
139#101—21no—
140#101—21no—
141#101D412yescomplemented
142#100—21no—
143#100—21no—
144#100—22no—
145#100—22no—
146#100—22no—
147#100—22no—
148#100—22no—
149#100—22no—
150#100D412yescomplemented
151#97—21no—
152C4×C8—21no—
153C24—22no—
154C24—21no—
155C24#4112yes—
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—21no—
166#41—42no—
167#41—42no—
168#41—42no—
169#41—42no—
170#41C2×D412yes—
171#41C2×D412yes—
172C2×Q8C2×D411yescharacteristic
173C2×Q8C2×D411yescharacteristic
174C2×Q8#4111yescharacteristic
175C2×D4—21no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—21no—
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—21no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—84no—
202C2×D4C2×D411yescharacteristic
203C2×D4C2×D411yescharacteristic
204C2×D4#4111yescharacteristic
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4C2411yescharacteristic
211Q16—22no—
212Q16—41no—
213SD16—22no—
214SD16—22no—
215SD16—42no—
216SD16—42no—
217SD16—41no—
218SD16—41no—
219SD16—42no—
220D8—22no—
221D8—42no—
222D8—42no—
223D8—41no—
224D8—42no—
225M16—21no—
226M16—21no—
227M16—21no—
228M16—21no—
229M16—21no—
230M16—21no—
231M16—21no—
232M16—21no—
233M16—21no—
234M16—21no—
235M16—21no—
236M16—21no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16C2×D412yes—
241M16C2×D412yes—
242M16C2×D412yes—
243M16C2×D412yes—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C4⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42C2×D411yescharacteristic
261C42C2×D411yescharacteristic
262C23—21no—
263C23—22no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—44no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—82no—
273C23—82no—
274C23—162no—
275C23C2×#4111yescharacteristic
276Q8—21no—
277Q8—21no—
278Q8—21no—
279Q8—21no—
280Q8—21no—
281Q8#12112yes—
282D4—22no—
283D4—22no—
284D4—21no—
285D4—21no—
286D4—21no—
287D4—21no—
288D4—21no—
289D4—44no—
290D4—44no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—41no—
298D4—41no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—84no—
304D4—84no—
305D4—162no—
306D4#12112yes—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C22×D411yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—22no—
329C8—22no—
330C8—22no—
331C8—41no—
332C8—41no—
333C8—41no—
334C8—41no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—42no—
340C22—81no—
341C22—82no—
342C22—81no—
343C22—82no—
344C22—82no—
345C22—81no—
346C22—82no—
347C22—81no—
348C22—161no—
349C22—161no—
350C22—164no—
351C22#52511yescharacteristic
352C22#54611yescharacteristic
353C22#54611yescharacteristic
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4C2×#12111yescharacteristic
361C4#53811yescharacteristic
362C4D4211yescharacteristic
363C4D4211yescharacteristic
364C2—21no—
365C2—21no—
366C2—21no—
367C2—42no—
368C2—81no—
369C2—82no—
370C2—81no—
371C2—161no—
372C2#242411yescharacteristic
subgroupG/H
center, socleC2#2424
commutator, Frattini, cosocleC22×C4C24
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. #3068C2semidirect
2. #3068C2semidirect
3. #3019C2semidirect
4. C2×#453C2semidirect
5. #2956C2semidirect
6. #2888C2semidirect
7. #2069C2semidirect
8. #2050C2semidirect
9. #2048C2semidirect
10. #2002C2semidirect
11. #1965C2semidirect
12. #1932C2semidirect
13. #1914C2semidirect
14. #1903C2semidirect
15. #1861C2semidirect
21. C2×AGL(1,ℤ/8)C22semidirect
24. #568C22semidirect
37. #430C22semidirect
38. #430C22semidirect
39. #430C22semidirect
40. #423C22semidirect
45. C2×#104C22semidirect
46. #418C22semidirect
50. #413C22semidirect
56. C2×#100C22semidirect
57. C2×#100C22semidirect
58. C4×M16C22semidirect
59. #344C22semidirect
60. #344C22semidirect
61. #339C22semidirect
62. #338C22semidirect
81. #137D4semidirect
91. AGL(1,ℤ/8)D4semidirect
136. #104D4semidirect
137. #104D4semidirect
141. #101D4semidirect
150. #100D4semidirect

Maximal subgroups

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

subgroupindex
#30682
#30192
C2×#4532
#29562
#28882
#20692
#20502
#20482
#20022
#19652
#19322
#19142
#19032
#18612