Finite Group Databasebeta

← #61421

Subgroups — #61421

Order 256.

982 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#3508C211yesmaximal, characteristic, complemented
2#3507C211yesmaximal, characteristic, complemented
3#3401C214yesmaximal, complemented
4#3400C212yesmaximal, complemented
5#3399C212yesmaximal, complemented
6#3397C214yesmaximal, complemented
7#3396C212yesmaximal, complemented
8#3392C212yesmaximal, complemented
9#3154C212yesmaximal, complemented
10#3151C211yesmaximal, characteristic, complemented
11#3045C212yesmaximal, complemented
12#3042C212yesmaximal, complemented
13#3035C211yesmaximal, characteristic, complemented
14C2×#460C211yesmaximal, characteristic, complemented
15#2962C211yesmaximal, characteristic, complemented
16#2946C212yesmaximal, complemented
17C4×#137C211yesmaximal, characteristic, complemented
18C2×#137C2211yescharacteristic
19C2×AGL(1,ℤ/8)C2212yes—
20C2×AGL(1,ℤ/8)C2212yes—
21C2×#135C2212yes—
22C22×SD16C2212yes—
23#559C2212yes—
24#559C2212yes—
25#559C2212yescomplemented
26#557C2212yescomplemented
27#556—21no—
28#556C2214yescomplemented
29#554—21no—
30#552—22no—
31#550C2212yes—
32#550C2212yes—
33#546—22no—
34#546—22no—
35#540—22no—
36#539—22no—
37#536C2212yes—
38#535C2212yes—
39#533C2211yescharacteristic
40#532—21no—
41#532—21no—
42#532C2211yescharacteristic
43C2×#127C2211yescharacteristic
44#529—22no—
45#520C2212yes—
46#518C2212yes—
47C4×#41C2211yescharacteristic
48C2×C4×Q8C2211yescharacteristic
49#497C2214yescomplemented
50#496C2214yes—
51#496C2214yescomplemented
52#495C2214yescomplemented
53#493C2212yes—
54#492C2212yes—
55#492C2214yescomplemented
56#464C2214yescomplemented
57#464C2212yescomplemented
58#463C2214yescomplemented
59#463C2214yescomplemented
60#463C2212yescomplemented
61#463C2214yescomplemented
62#462C2211yescharacteristic, complemented
63#461C2211yescharacteristic, complemented
64#461C2214yescomplemented
65#461C2214yescomplemented
66#460C2212yes—
67#460C2214yes—
68#460C2212yes—
69#460C2214yescomplemented
70#460C2211yescharacteristic, complemented
71#459C2214yescomplemented
72#459C2211yescharacteristic, complemented
73#446C2214yescomplemented
74#446C2212yescomplemented
75C8×Q8C2212yescomplemented
76#442C2214yescomplemented
77#441C2212yescomplemented
78#440C2212yescomplemented
79#440C2214yescomplemented
80C4×Q16C2212yescomplemented
81C4×SD16C2212yescomplemented
82C4×SD16C2214yescomplemented
83C4×D8C2214yescomplemented
84#428C2211yescharacteristic
85#424C2212yes—
86C2×C4⋊C8C2211yescharacteristic
87#419C2211yescharacteristic
88#417C2211yescharacteristic
89#416C2212yes—
90C2×#102C2212yes—
91#405C2212yes—
92C4×M16C2211yescharacteristic
93C2×#41—22no—
94C2×#41—22no—
95C2×#41C2311yescharacteristic
96C2×#41C2312yes—
97C22×Q8C2311yescharacteristic
98C22×D4—21no—
99C22×D4—21no—
100C22×D4C2312yes—
101#137—24no—
102AGL(1,ℤ/8)—28no—
103AGL(1,ℤ/8)—44no—
104#135—44no—
105C2×Q16C2312yes—
106C2×Q16C2312yes—
107C2×SD16—44no—
108C2×SD16C2314yes—
109C2×SD16C2312yes—
110C2×SD16C2314yes—
111C2×SD16C2314yes—
112C2×SD16C2312yes—
113C2×SD16C2312yes—
114C2×SD16C2314yes—
115C2×D8C2314yes—
116C2×D8C2312yes—
117C2×D8C2314yes—
118C2×M16C2311yescharacteristic
119C2×M16C2311yescharacteristic
120C22×C8C2312yes—
121C4⋊Q8D411yescharacteristic
122C4⋊Q8D411yescharacteristic
123C4⋊Q8C2311yescharacteristic
124C4⋊Q8C2314yes—
125#127—21no—
126#127—21no—
127#127C2312yes—
128#127C2312yes—
129#127C2311yescharacteristic
130#127C2314yes—
131#126—24no—
132#126—24no—
133#125—22no—
134#125—24no—
135#125D412yes—
136#124—22no—
137#124—22no—
138#124—22no—
139#124C2312yes—
140#124C2314yes—
141#124C2314yes—
142#123—24no—
143#123—24no—
144#123—24no—
145#123—24no—
146C22⋊Q8—24no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8D412yes—
150C22⋊Q8D412yes—
151#121—21no—
152#121—21no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—22no—
157#121—24no—
158#121—24no—
159#120—24no—
160C4×Q8—22no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8C2312yes—
164C4×Q8C2312yes—
165C4×Q8C2311yescharacteristic
166C4×Q8C2311yescharacteristic
167C4×Q8C2314yes—
168C4×Q8C2312yes—
169C4×Q8C2312yes—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—24no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4C2311yescharacteristic
181C4×D4C2311yescharacteristic
182C4×D4C2314yes—
183C4×D4C2314yes—
184C4×D4C2314yes—
185#117—22no—
186#117—24no—
187#117—21no—
188#117C2311yescharacteristic
189#117C2312yes—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C2311yescharacteristic
193C2×C22⋊C4—24no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42C2311yescharacteristic
197#107C2312yes—
198#106C2312yes—
199C4⋊C8C2311yescharacteristic
200C4⋊C8C2311yescharacteristic
201C4⋊C8C2312yes—
202C4⋊C8C2314yes—
203C4⋊C8C2314yes—
204#103C2312yes—
205#103C2312yes—
206#103C2312yes—
207#103C2312yes—
208#103C2312yes—
209#102C2314yes—
210#102C2314yes—
211#102C2312yes—
212#102C2312yes—
213#102C2312yes—
214#97C2312yes—
215#97C2314yes—
216C4×C8C2312yes—
217C4×C8C2314yes—
218C24—22no—
219#41—24no—
220#41—22no—
221#41—22no—
222#41—24no—
223#41—84no—
224#41—84no—
225C2×Q8—21no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8C2×D412yes—
229C2×Q8C2×D412yes—
230C2×Q8C2411yescharacteristic
231C2×Q8C2412yes—
232C2×Q8C2412yes—
233C2×D4—21no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—21no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4C2412yes—
249C2×D4C2414yes—
250C2×D4C2411yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4C2×D411yescharacteristic
261C22×C4C2×D411yescharacteristic
262C22×C4C2411yescharacteristic
263Q16—24no—
264Q16—42no—
265SD16—28no—
266SD16—24no—
267SD16—44no—
268SD16—44no—
269SD16—42no—
270SD16—42no—
271SD16—44no—
272D8—28no—
273D8—42no—
274D8—44no—
275M16—22no—
276M16#4114yescomplemented
277C2×C8—24no—
278C2×C8C2412yes—
279C2×C8C2412yes—
280C2×C8C2412yes—
281C2×C8C2412yes—
282C4⋊C4—21no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—24no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—24no—
292C4⋊C4C2×D412yes—
293C4⋊C4C2×D412yes—
294C4⋊C4C2×D414yes—
295C4⋊C4C2×D412yes—
296C4⋊C4C2×D412yes—
297C4⋊C4C2×D414yes—
298C4⋊C4C2412yes—
299C4⋊C4C2411yescharacteristic
300C4⋊C4C2411yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—22no—
312C22⋊C4—24no—
313C22⋊C4C2×D414yes—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—22no—
319C42—24no—
320C42C2×D411yescharacteristic
321C42C2×D411yescharacteristic
322C42C2411yescharacteristic
323C42C2414yes—
324C42C2411yescharacteristic
325C23—22no—
326C23—24no—
327C23—21no—
328C23—44no—
329C23—82no—
330C23—82no—
331C23C22×D411yescharacteristic
332Q8—24no—
333Q8—22no—
334Q8—22no—
335Q8—21no—
336Q8—22no—
337Q8—22no—
338Q8—21no—
339Q8—42no—
340Q8—42no—
341D4—22no—
342D4—24no—
343D4—21no—
344D4—22no—
345D4—21no—
346D4—24no—
347D4—44no—
348D4—42no—
349D4—44no—
350D4—44no—
351D4—42no—
352D4—44no—
353D4—81no—
354D4—82no—
355D4—82no—
356D4—84no—
357D4—84no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—81no—
375C2×C4C22×D411yescharacteristic
376C2×C4C22×D411yescharacteristic
377C2×C4C22×D414yes—
378C2×C4C22×D411yescharacteristic
379C2×C4C22×D414yes—
380C2×C4C22×D411yescharacteristic
381C2×C4C22×D411yescharacteristic
382C2×C4C2×#4112yes—
383C2×C4H(5,𝔽2)12yes—
384C2×C4C2511yescharacteristic
385C8—22no—
386C8—22no—
387C8—22no—
388C8C2×#4114yes—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—42no—
393C22—44no—
394C22—42no—
395C22—41no—
396C22—44no—
397C22—41no—
398C22—44no—
399C22—42no—
400C22—164no—
401C22#55012yes—
402C22C23×D411yescharacteristic
403C4—21no—
404C4—21no—
405C4—24no—
406C4—21no—
407C4—24no—
408C4—41no—
409C4—42no—
410C4—41no—
411C4—42no—
412C4—42no—
413C4—41no—
414C4#55012yes—
415C4#57712yes—
416C4C22×#4111yescharacteristic
417C4C2×H(5,𝔽2)11yescharacteristic
418C2—22no—
419C2—81no—
420C2—82no—
421C2—84no—
422C2C2×#55011yescharacteristic
423C2C2×#57711yescharacteristic
424C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ #533 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3508C2semidirect
2. #3507C2semidirect
3. #3401C2semidirect
4. #3400C2semidirect
5. #3399C2semidirect
6. #3397C2semidirect
7. #3396C2semidirect
8. #3392C2semidirect
9. #3154C2semidirect
10. #3151C2semidirect
11. #3045C2semidirect
12. #3042C2semidirect
13. #3035C2semidirect
14. C2×#460C2semidirect
15. #2962C2semidirect
16. #2946C2semidirect
17. C4×#137C2semidirect
25. #559C22semidirect
26. #557C22semidirect
28. #556C22semidirect
49. #497C22semidirect
51. #496C22semidirect
52. #495C22semidirect
55. #492C22semidirect
56. #464C22semidirect
57. #464C22semidirect
58. #463C22semidirect
59. #463C22semidirect
60. #463C22semidirect
61. #463C22semidirect
62. #462C22semidirect
63. #461C22semidirect
64. #461C22semidirect
65. #461C22semidirect
69. #460C22semidirect
70. #460C22semidirect
71. #459C22semidirect
72. #459C22semidirect
73. #446C22semidirect
74. #446C22semidirect
75. C8×Q8C22semidirect
76. #442C22semidirect
77. #441C22semidirect
78. #440C22semidirect
79. #440C22semidirect
80. C4×Q16C22semidirect
81. C4×SD16C22semidirect
82. C4×SD16C22semidirect
83. C4×D8C22semidirect
276. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35082
#35072
#34012
#34002
#33992
#33972
#33962
#33922
#31542
#31512
#30452
#30422
#30352
C2×#4602
#29622
#29462
C4×#1372