Finite Group Databasebeta

← #60727

Subgroups — #60727

Order 256.

951 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3511C211yesmaximal, characteristic, complemented
2#3510C211yesmaximal, characteristic, complemented
3#3181C212yesmaximal, complemented
4#3180C212yesmaximal, complemented
5#3175C214yesmaximal, complemented
6#3173C214yesmaximal, complemented
7#3172C212yesmaximal, complemented
8#3171C212yesmaximal, complemented
9#3165C211yesmaximal, characteristic, complemented
10#3153C211yesmaximal, characteristic, complemented
11#3135C211yesmaximal, characteristic, complemented
12C2×#489C211yesmaximal, characteristic, complemented
13#3063C212yesmaximal, complemented
14#3062C212yesmaximal, complemented
15#2947C212yesmaximal, complemented
16#2946C212yesmaximal, complemented
17#2875C211yesmaximal, characteristic, complemented
18C2×#137C2212yes—
19C2×AGL(1,ℤ/8)C2212yes—
20C2×#131C2212yes—
21#559—21no—
22#559—21no—
23#556—21no—
24#556—21no—
25#556—21no—
26#555—21no—
27#554—21no—
28#553—22no—
29#551—22no—
30#551—22no—
31#548—22no—
32#547—21no—
33#542C2212yes—
34#542C2212yes—
35#542C2214yescomplemented
36#537C2212yes—
37#536C2211yescharacteristic
38#535C2211yescharacteristic
39#533C2211yescharacteristic
40#533C2212yescomplemented
41#532C2211yescharacteristic
42#532C2212yescomplemented
43C2×C4⋊Q8C2211yescharacteristic
44#529C2212yes—
45#529C2212yes—
46C2×#124C2211yescharacteristic
47#520C2212yes—
48#519C2211yescharacteristic
49#518C2211yescharacteristic
50C4×#41C2211yescharacteristic
51#501C2212yescomplemented
52#501C2212yescomplemented
53#500C2212yescomplemented
54#498C2212yescomplemented
55#497C2212yes—
56#496C2212yes—
57#495C2214yes—
58#491C2211yescharacteristic, complemented
59#491C2211yescharacteristic, complemented
60#490C2211yescharacteristic, complemented
61#490C2211yescharacteristic, complemented
62#489C2212yes—
63#489C2214yes—
64#489C2212yes—
65#489C2212yescomplemented
66#489C2212yescomplemented
67#470C2214yescomplemented
68#469C2214yes—
69#468C2214yes—
70#468C2214yescomplemented
71#467C2214yescomplemented
72#466C2214yes—
73#465C2214yes—
74#465C2214yescomplemented
75#442C2214yes—
76#441C2214yescomplemented
77#440C2214yes—
78#440C2214yescomplemented
79C4×Q16C2214yescomplemented
80C4×SD16C2214yes—
81C4×SD16C2214yescomplemented
82C4×D8C2214yes—
83#428C2212yes—
84#428C2212yes—
85#416C2212yes—
86#416C2212yes—
87C2×#103C2212yes—
88C2×#102C2212yes—
89#405C2212yes—
90#405C2214yescomplemented
91C4×M16C2211yescharacteristic
92C4×M16C2212yescomplemented
93C2×#97C2211yescharacteristic
94C2×#97C2212yescomplemented
95C2×#41—22no—
96C2×#41C2312yes—
97C2×#41C2311yescharacteristic
98C22×Q8—21no—
99C22×Q8C2311yescharacteristic
100C22×D4C2311yescharacteristic
101#137—28no—
102AGL(1,ℤ/8)—28no—
103C2×Q16C2314yes—
104C2×SD16C2314yes—
105C2×SD16C2314yes—
106C2×D8C2314yes—
107#131—28no—
108C2×M16C2314yes—
109C2×M16C2312yes—
110C22×C8C2312yes—
111C22×C8C2314yes—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C2311yescharacteristic
116C4⋊Q8C2312yes—
117C4⋊Q8C2312yes—
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2311yescharacteristic
120#127—21no—
121#127C2311yescharacteristic
122#126—22no—
123#126—22no—
124#126—24no—
125#126—22no—
126#126—22no—
127#125—22no—
128#125C2311yescharacteristic
129#125C2311yescharacteristic
130#124—21no—
131#124—21no—
132#124—22no—
133#124—22no—
134#124—24no—
135#124—21no—
136#124—21no—
137#124—22no—
138#124C2312yes—
139#124C2312yes—
140#124C2312yes—
141#123—22no—
142#123—22no—
143#123—22no—
144#123—22no—
145#123—22no—
146#123—24no—
147C22⋊Q8—22no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—24no—
154C22⋊Q8C2312yes—
155C22⋊Q8C2312yes—
156C22⋊Q8C2314yes—
157#121—22no—
158#121—24no—
159#121—21no—
160#121—21no—
161#121—22no—
162#121C2312yes—
163#121C2312yes—
164#121C2314yes—
165#120—22no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—22no—
171C4×Q8D411yescharacteristic
172C4×Q8D411yescharacteristic
173C4×Q8C2312yes—
174C4×Q8C2312yes—
175C4×Q8C2314yes—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—24no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—22no—
184C4×D4D411yescharacteristic
185C4×D4D412yes—
186C4×D4D412yes—
187C4×D4D411yescharacteristic
188C4×D4C2312yes—
189C4×D4C2312yes—
190C4×D4C2314yes—
191#117—22no—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117C2311yescharacteristic
197#117C2311yescharacteristic
198#117C2311yescharacteristic
199#117C2312yes—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C2312yes—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C42C2311yescharacteristic
207C2×C42C2312yes—
208#107C2314yes—
209#107C2314yes—
210#106C2314yes—
211#106C2314yes—
212#103C2314yes—
213#103C2314yes—
214#103C2312yes—
215#103C2312yes—
216#103C2312yes—
217#103C2312yes—
218#102C2314yes—
219#102C2314yes—
220#102C2312yes—
221#102C2312yes—
222#102C2312yes—
223#102C2312yes—
224#97C2311yescharacteristic
225#97C2311yescharacteristic
226#97C2312yes—
227#97C2312yes—
228#97C2314yes—
229C4×C8C2312yes—
230C4×C8C2314yes—
231C24—21no—
232#41—24no—
233#41—24no—
234#41—24no—
235#41—84no—
236C2×Q8—21no—
237C2×Q8—22no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—82no—
243C2×Q8C2×D411yescharacteristic
244C2×Q8C2412yes—
245C2×Q8C2412yes—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—24no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—24no—
254C2×D4C2×D412yes—
255C2×D4C2×D411yescharacteristic
256C2×D4C2412yes—
257C2×D4C2412yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4C2×D411yescharacteristic
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D412yes—
267C22×C4C2×D412yes—
268C22×C4C2412yes—
269C22×C4C2411yescharacteristic
270Q16—28no—
271SD16—28no—
272SD16—28no—
273D8—28no—
274M16—28no—
275M16#4118yescomplemented
276C2×C8—24no—
277C2×C8—28no—
278C2×C8C2412yes—
279C2×C8C2412yes—
280C2×C8C2414yes—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—24no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4C2×D411yescharacteristic
299C4⋊C4C2×D411yescharacteristic
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C4⋊C4C2411yescharacteristic
303C4⋊C4C2411yescharacteristic
304C4⋊C4C2414yes—
305C4⋊C4C2411yescharacteristic
306C4⋊C4C2411yescharacteristic
307C22⋊C4—21no—
308C22⋊C4—21no—
309C22⋊C4—24no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—24no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—24no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C42—22no—
328C42—22no—
329C42—24no—
330C42—21no—
331C42—21no—
332C42C2×D412yes—
333C42C2×D412yes—
334C42C2×D411yescharacteristic
335C42C2×D411yescharacteristic
336C42C2412yes—
337C42C2411yescharacteristic
338C42C2411yescharacteristic
339C23—22no—
340C23—22no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23C22×D412yes—
345C23C22×D411yescharacteristic
346Q8—22no—
347Q8—22no—
348Q8—24no—
349Q8—22no—
350Q8—44no—
351Q8—42no—
352Q8—81no—
353Q8—81no—
354Q8—82no—
355D4—24no—
356D4—22no—
357D4—22no—
358D4—24no—
359D4—22no—
360D4—42no—
361D4—42no—
362D4—44no—
363D4—42no—
364D4—82no—
365D4—82no—
366D4—84no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—84no—
384C2×C4—82no—
385C2×C4C22×D412yes—
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4C22×D411yescharacteristic
389C2×C4C22×D412yes—
390C2×C4C22×D411yescharacteristic
391C2×C4C22×D411yescharacteristic
392C2×C4C22×D411yescharacteristic
393C2×C4C2×#4114yes—
394C2×C4C2511yescharacteristic
395C8—22no—
396C8—22no—
397C8C2×#4118yes—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—162no—
410C22#53212yes—
411C22C23×D411yescharacteristic
412C4—21no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—21no—
418C4—22no—
419C4—44no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—42no—
425C4—42no—
426C4#53212yes—
427C4C22×#4112yes—
428C2—22no—
429C2—42no—
430C2—82no—
431C2—82no—
432C2C2×#53211yescharacteristic
433C2#358511yescharacteristic
434C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3511C2semidirect
2. #3510C2semidirect
3. #3181C2semidirect
4. #3180C2semidirect
5. #3175C2semidirect
6. #3173C2semidirect
7. #3172C2semidirect
8. #3171C2semidirect
9. #3165C2semidirect
10. #3153C2semidirect
11. #3135C2semidirect
12. C2×#489C2semidirect
13. #3063C2semidirect
14. #3062C2semidirect
15. #2947C2semidirect
16. #2946C2semidirect
17. #2875C2semidirect
35. #542C22semidirect
40. #533C22semidirect
42. #532C22semidirect
51. #501C22semidirect
52. #501C22semidirect
53. #500C22semidirect
54. #498C22semidirect
58. #491C22semidirect
59. #491C22semidirect
60. #490C22semidirect
61. #490C22semidirect
65. #489C22semidirect
66. #489C22semidirect
67. #470C22semidirect
70. #468C22semidirect
71. #467C22semidirect
74. #465C22semidirect
76. #441C22semidirect
78. #440C22semidirect
79. C4×Q16C22semidirect
81. C4×SD16C22semidirect
90. #405C22semidirect
92. C4×M16C22semidirect
94. C2×#97C22semidirect
275. M16#41semidirect

Maximal subgroups

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

subgroupindex
#35112
#35102
#31812
#31802
#31752
#31732
#31722
#31712
#31652
#31532
#31352
C2×#4892
#30632
#30622
#29472
#29462
#28752