Finite Group Databasebeta

← #60919

Subgroups — #60919

Order 256.

911 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3455C211yesmaximal, characteristic, complemented
2C2×#542C211yesmaximal, characteristic, complemented
3#3263C214yesmaximal, complemented
4#3262C214yesmaximal, complemented
5#3258C214yesmaximal, complemented
6#3257C214yesmaximal, complemented
7#3079C212yesmaximal, complemented
8#3078C211yesmaximal, characteristic, complemented
9#3077C211yesmaximal, characteristic, complemented
10#3056C212yesmaximal, complemented
11#3055C212yesmaximal, complemented
12#3040C212yesmaximal, complemented
13#3037C211yesmaximal, characteristic, complemented
14#3036C211yesmaximal, characteristic, complemented
15#2905C211yesmaximal, characteristic, complemented
16C2×#137C2212yes—
17C2×AGL(1,ℤ/8)C2212yes—
18C22×M16C2212yes—
19#542—22no—
20#542—22no—
21#542C2214yes—
22#542C2214yes—
23#542C2212yescomplemented
24#542C2212yescomplemented
25#541C2212yescomplemented
26#540C2212yescomplemented
27#539—21no—
28#539—21no—
29#538—21no—
30#538—21no—
31#533C2211yescharacteristic
32#532C2211yescharacteristic
33C2×C4⋊Q8C2211yescharacteristic
34#529—22no—
35#529—22no—
36C2×#126—21no—
37C2×#124—21no—
38C2×#124C2211yescharacteristic
39#525C2212yes—
40C2×#123—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8C2211yescharacteristic
43C2×#121C2211yescharacteristic
44C4×#41C2212yes—
45C2×C4×Q8C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic
47C2×#117C2211yescharacteristic
48#479C2214yescomplemented
49#478C2214yescomplemented
50#477C2212yescomplemented
51#477C2214yescomplemented
52#476C2212yescomplemented
53#476C2214yescomplemented
54#475C2212yescomplemented
55#475C2214yescomplemented
56#474C2212yescomplemented
57#474C2214yescomplemented
58#470C2214yescomplemented
59#469C2214yes—
60#468C2214yes—
61#468C2214yescomplemented
62#467C2214yescomplemented
63#466C2214yes—
64#465C2214yes—
65#465C2214yescomplemented
66#464C2214yescomplemented
67#464C2214yescomplemented
68#464C2214yescomplemented
69#463C2214yescomplemented
70#463C2214yescomplemented
71#463C2214yescomplemented
72#462C2212yescomplemented
73#461C2212yescomplemented
74#460C2212yescomplemented
75#459C2212yescomplemented
76#428C2212yes—
77#428C2212yes—
78#424C2214yescomplemented
79#424C2214yescomplemented
80#423C2211yescharacteristic
81#423C2211yescharacteristic
82#423C2212yes—
83#419C2212yes—
84#418C2212yes—
85#417C2212yes—
86#417C2212yes—
87C2×#41C2312yes—
88C22×Q8—21no—
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4C2311yescharacteristic
92C23×C4—21no—
93C23×C4C2311yescharacteristic
94#137—44no—
95AGL(1,ℤ/8)—44no—
96C2×Q16C2314yes—
97C2×SD16C2314yes—
98C2×SD16C2314yes—
99C2×D8C2314yes—
100C2×M16—44no—
101C2×M16C2312yes—
102C2×M16C2312yes—
103C2×M16C2314yes—
104C22×C8C2314yes—
105C4⋊Q8—22no—
106C4⋊Q8C2311yescharacteristic
107C4⋊Q8C2314yes—
108C4⋊Q8C2311yescharacteristic
109C4⋊Q8C2311yescharacteristic
110#127C2311yescharacteristic
111#126—24no—
112#126—24no—
113#126—22no—
114#126—22no—
115#126—22no—
116#126—22no—
117#125—22no—
118#125—22no—
119#125C2312yes—
120#124—24no—
121#124—24no—
122#124—22no—
123#124—22no—
124#124—22no—
125#124C2314yes—
126#124C2312yes—
127#123—22no—
128#123—22no—
129#123—22no—
130#123—24no—
131#123—24no—
132#123—22no—
133#123—22no—
134C22⋊Q8—22no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8C2314yes—
142C22⋊Q8C2312yes—
143C22⋊Q8C2312yes—
144#121—22no—
145#121—21no—
146#121—21no—
147#121—22no—
148#121—22no—
149#121C2314yes—
150#121C2312yes—
151#121C2312yes—
152#120—22no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—21no—
156C4×Q8—21no—
157C4×Q8C2312yes—
158C4×Q8C2312yes—
159C4×Q8C2312yes—
160C4×Q8C2312yes—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—21no—
166C4×D4—21no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4C2312yes—
170C4×D4C2312yes—
171C4×D4C2312yes—
172C4×D4C2312yes—
173#117—22no—
174#117D412yes—
175#117D412yes—
176#117C2312yes—
177#117C2312yes—
178#117C2312yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4D411yescharacteristic
183C2×C4⋊C4D411yescharacteristic
184C2×C4⋊C4C2311yescharacteristic
185C2×C4⋊C4C2311yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4D411yescharacteristic
192C2×C22⋊C4D411yescharacteristic
193C2×C42—21no—
194C2×C42—22no—
195C2×C42C2311yescharacteristic
196C2×C42C2311yescharacteristic
197#107C2314yes—
198#107C2314yes—
199#106C2314yes—
200#106C2314yes—
201C4⋊C8C2312yes—
202C4⋊C8C2312yes—
203C4⋊C8C2312yes—
204C4⋊C8C2314yes—
205C4⋊C8C2314yes—
206C4⋊C8C2312yes—
207#103C2314yes—
208#103C2314yes—
209#103C2314yes—
210#103C2314yes—
211#102C2314yes—
212#102C2314yes—
213#102C2314yes—
214#102C2314yes—
215C24—21no—
216C24C2×D411yescharacteristic
217#41—24no—
218#41—24no—
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8—21no—
222C2×Q8—24no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—82no—
226C2×Q8C2412yes—
227C2×Q8C2412yes—
228C2×D4—22no—
229C2×D4—22no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—82no—
238C2×D4C2412yes—
239C2×D4C2412yes—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—82no—
251C22×C4C2×D411yescharacteristic
252C22×C4C2×D412yes—
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2412yes—
258C22×C4C2411yescharacteristic
259Q16—44no—
260SD16—44no—
261SD16—44no—
262D8—44no—
263M16—24no—
264M16—24no—
265M16—44no—
266C2×C8—44no—
267C2×C8C2414yes—
268C2×C8C2414yes—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4C2×D412yes—
286C4⋊C4C2×D412yes—
287C4⋊C4C2×D412yes—
288C4⋊C4C2×D412yes—
289C4⋊C4C2412yes—
290C4⋊C4C2412yes—
291C4⋊C4C2412yes—
292C4⋊C4C2412yes—
293C22⋊C4—24no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4C2×D412yes—
309C22⋊C4C2×D412yes—
310C22⋊C4C2×D412yes—
311C22⋊C4C2×D412yes—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42C2×D412yes—
318C42C2×D411yescharacteristic
319C42C2×D411yescharacteristic
320C42C2411yescharacteristic
321C42C2412yes—
322C42C2411yescharacteristic
323C23—22no—
324C23—22no—
325C23—22no—
326C23—41no—
327C23—41no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23C22×D412yes—
332C23C22×D411yescharacteristic
333Q8—22no—
334Q8—22no—
335Q8—22no—
336Q8—22no—
337Q8—42no—
338Q8—81no—
339Q8—81no—
340Q8—84no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—22no—
345D4—44no—
346D4—42no—
347D4—81no—
348D4—81no—
349D4—82no—
350D4—82no—
351D4—84no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4C22×D411yescharacteristic
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D412yes—
377C2×C4C22×D412yes—
378C2×C4C22×D411yescharacteristic
379C2×C4C22×D411yescharacteristic
380C2×C4C22×D412yes—
381C2×C4C22×D411yescharacteristic
382C2×C4H(5,𝔽2)12yes—
383C2×C42−1+412yes—
384C2×C4C2511yescharacteristic
385C8—24no—
386C8—24no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—42no—
398C22—82no—
399C22—84no—
400C22#53712yes—
401C22C23×D411yescharacteristic
402C4—21no—
403C4—21no—
404C4—22no—
405C4—22no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—42no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4#53712yes—
416C4C2×H(5,𝔽2)11yescharacteristic
417C4C2×2−1+411yescharacteristic
418C2—22no—
419C2—42no—
420C2—82no—
421C2—82no—
422C2C2×#53711yescharacteristic
423C2#358511yescharacteristic
424C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3455C2semidirect
2. C2×#542C2semidirect
3. #3263C2semidirect
4. #3262C2semidirect
5. #3258C2semidirect
6. #3257C2semidirect
7. #3079C2semidirect
8. #3078C2semidirect
9. #3077C2semidirect
10. #3056C2semidirect
11. #3055C2semidirect
12. #3040C2semidirect
13. #3037C2semidirect
14. #3036C2semidirect
15. #2905C2semidirect
23. #542C22semidirect
24. #542C22semidirect
25. #541C22semidirect
26. #540C22semidirect
48. #479C22semidirect
49. #478C22semidirect
50. #477C22semidirect
51. #477C22semidirect
52. #476C22semidirect
53. #476C22semidirect
54. #475C22semidirect
55. #475C22semidirect
56. #474C22semidirect
57. #474C22semidirect
58. #470C22semidirect
61. #468C22semidirect
62. #467C22semidirect
65. #465C22semidirect
66. #464C22semidirect
67. #464C22semidirect
68. #464C22semidirect
69. #463C22semidirect
70. #463C22semidirect
71. #463C22semidirect
72. #462C22semidirect
73. #461C22semidirect
74. #460C22semidirect
75. #459C22semidirect
78. #424C22semidirect
79. #424C22semidirect

Maximal subgroups

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

subgroupindex
#34552
C2×#5422
#32632
#32622
#32582
#32572
#30792
#30782
#30772
#30562
#30552
#30402
#30372
#30362
#29052