Finite Group Databasebeta

← #61578

Subgroups — #61578

Order 256.

1057 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3506C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3#3321C214yesmaximal, complemented
4#3313C214yesmaximal, complemented
5#3271C214yesmaximal, complemented
6#3269C214yesmaximal, complemented
7#3215C211yesmaximal, characteristic, complemented
8#3213C211yesmaximal, characteristic, complemented
9#3194C212yesmaximal, complemented
10#3190C212yesmaximal, complemented
11#3113C211yesmaximal, characteristic, complemented
12#3112C211yesmaximal, characteristic, complemented
13#3004C212yesmaximal, complemented
14#3002C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C2×#137C2214yescomplemented
19C2×#135C2214yescomplemented
20#559—22no—
21#556C2212yescomplemented
22#555C2212yescomplemented
23#553—22no—
24#550C2212yescomplemented
25#548—22no—
26#548—22no—
27#548C2214yes—
28#548C2214yes—
29#547C2212yescomplemented
30#538—21no—
31#538—21no—
32#535—21no—
33#533C2211yescharacteristic
34#532C2211yescharacteristic
35C2×#124—21no—
36C2×#124C2211yescharacteristic
37#525C2212yes—
38#525C2212yes—
39C2×C22⋊Q8—21no—
40C2×C22⋊Q8C2211yescharacteristic
41C2×#121—21no—
42C2×#121C2211yescharacteristic
43#520—21no—
44C2×C4×Q8C2211yescharacteristic
45C2×C4×D4—21no—
46C2×C4×D4C2211yescharacteristic
47#497C2212yescomplemented
48#496C2212yescomplemented
49#489C2212yescomplemented
50#489C2212yescomplemented
51#485C2214yescomplemented
52#483C2212yes—
53#482C2214yescomplemented
54#480C2212yes—
55#477C2212yescomplemented
56#475C2212yescomplemented
57#469C2214yescomplemented
58#468C2214yescomplemented
59#467C2214yescomplemented
60#465C2214yescomplemented
61#464C2214yescomplemented
62#463C2214yescomplemented
63#462C2212yescomplemented
64#460C2212yescomplemented
65#452C2214yescomplemented
66#452C2214yescomplemented
67#451C2212yes—
68#451C2212yes—
69#451C2214yescomplemented
70#450C2212yes—
71#449C2214yescomplemented
72#449C2214yescomplemented
73#448C2212yes—
74#448C2212yes—
75#448C2214yescomplemented
76#447C2212yes—
77#441C2214yescomplemented
78#440C2214yescomplemented
79#435C2214yescomplemented
80#435C2214yescomplemented
81#432C2211yescharacteristic
82#432C2211yescharacteristic
83#419C2214yescomplemented
84#416C2214yescomplemented
85#407C2212yes—
86C2×C22⋊C8C2212yes—
87C2×#41—21no—
88C2×#41—22no—
89C2×#41—44no—
90C2×#41C2314yes—
91C2×#41C2312yes—
92C2×#41C2312yes—
93C22×Q8—21no—
94C22×Q8—22no—
95C22×Q8—44no—
96C22×Q8C2311yescharacteristic
97C22×Q8C2314yes—
98C22×Q8C2311yescharacteristic
99C22×D4—21no—
100C22×D4—21no—
101C22×D4D411yescharacteristic
102C22×D4C2311yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4C2311yescharacteristic
106C23×C4C2311yescharacteristic
107#137—48no—
108#135—48no—
109C2×Q16C2314yes—
110C2×Q16C2314yes—
111C2×SD16C2314yes—
112C2×SD16C2314yes—
113C2×SD16C2314yes—
114C2×D8C2314yes—
115C2×M16C2314yes—
116C22×C8C2314yes—
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2311yescharacteristic
119C4⋊Q8C2311yescharacteristic
120#127—22no—
121#127C2311yescharacteristic
122#126—22no—
123#126—22no—
124#125—21no—
125#124—22no—
126#124—24no—
127#124—24no—
128#124—22no—
129#124—22no—
130#124—21no—
131#124—21no—
132#124—21no—
133#124C2312yes—
134#124C2312yes—
135#123—22no—
136#123—22no—
137#123—22no—
138#123—22no—
139C22⋊Q8—22no—
140C22⋊Q8—21no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8C2314yes—
144C22⋊Q8C2312yes—
145C22⋊Q8C2312yes—
146C22⋊Q8C2312yes—
147#121—24no—
148#121—24no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—21no—
153#121—22no—
154#121—22no—
155#121C2314yes—
156#121C2312yes—
157#121C2312yes—
158#121C2312yes—
159#120—22no—
160#120—22no—
161C4×Q8—22no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—21no—
165C4×Q8C2312yes—
166C4×Q8C2312yes—
167C4×Q8C2312yes—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—24no—
174C4×D4—24no—
175C4×D4—21no—
176C4×D4D412yes—
177C4×D4D412yes—
178C4×D4C2312yes—
179C4×D4C2312yes—
180C4×D4C2312yes—
181#117—21no—
182#117—22no—
183#117C2312yes—
184#117C2312yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4D411yescharacteristic
189C2×C4⋊C4C2311yescharacteristic
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4D411yescharacteristic
195C2×C22⋊C4D411yescharacteristic
196C2×C42—21no—
197C2×C42C2311yescharacteristic
198#107C2314yes—
199#106C2314yes—
200C4⋊C8C2312yes—
201C4⋊C8C2312yes—
202#103C2314yes—
203#103C2314yes—
204#103C2314yes—
205#103C2314yes—
206#103C2314yes—
207#102C2314yes—
208#102C2314yes—
209#102C2314yes—
210C22⋊C8C2312yes—
211C22⋊C8C2312yes—
212C22⋊C8C2314yes—
213C22⋊C8C2314yes—
214#97C2312yes—
215#97C2312yes—
216C24—21no—
217C24C2×D411yescharacteristic
218C24C2×D411yescharacteristic
219#41—28no—
220#41—44no—
221#41—44no—
222#41—44no—
223#41—44no—
224#41—48no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—24no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—28no—
232C2×Q8—44no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—48no—
236C2×Q8C2412yes—
237C2×Q8C2412yes—
238C2×Q8C2412yes—
239C2×D4—22no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4C2×D412yes—
254C2×D4C2×D412yes—
255C2×D4C2412yes—
256C2×D4C2412yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D411yescharacteristic
272C22×C4C2×D411yescharacteristic
273C22×C4C2×D412yes—
274C22×C4C2×D412yes—
275C22×C4C2411yescharacteristic
276C22×C4C2412yes—
277C22×C4C2412yes—
278Q16—28no—
279Q16—44no—
280SD16—28no—
281SD16—44no—
282SD16—44no—
283D8—44no—
284M16—44no—
285C2×C8—44no—
286C2×C8C2414yes—
287C2×C8C2414yes—
288C4⋊C4—21no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—22no—
300C4⋊C4C2×D412yes—
301C4⋊C4C2×D412yes—
302C4⋊C4C2412yes—
303C4⋊C4C2412yes—
304C4⋊C4C2412yes—
305C22⋊C4—24no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—22no—
315C22⋊C4—24no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4C2×D412yes—
321C22⋊C4C2×D412yes—
322C22⋊C4C2×D412yes—
323C22⋊C4C2×D412yes—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42C2×D411yescharacteristic
330C42C2×D411yescharacteristic
331C42C2411yescharacteristic
332C42C2411yescharacteristic
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C22×D412yes—
345C23C22×D412yes—
346C23C22×D411yescharacteristic
347Q8—24no—
348Q8—22no—
349Q8—24no—
350Q8—24no—
351Q8—48no—
352Q8—42no—
353Q8—42no—
354Q8—81no—
355Q8H(5,𝔽2)14yes—
356D4—24no—
357D4—24no—
358D4—42no—
359D4—42no—
360D4—48no—
361D4—42no—
362D4—42no—
363D4—48no—
364D4—44no—
365D4—81no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—82no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4C22×D411yescharacteristic
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D412yes—
397C2×C4C22×D412yes—
398C2×C4C22×D411yescharacteristic
399C2×C4C22×D411yescharacteristic
400C2×C4C2511yescharacteristic
401C8—24no—
402C8—24no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—44no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—82no—
419C22—84no—
420C22#57512yes—
421C22C23×D411yescharacteristic
422C4—21no—
423C4—21no—
424C4—24no—
425C4—22no—
426C4—21no—
427C4—22no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4C2×H(5,𝔽2)12yes—
436C2—22no—
437C2—42no—
438C2—42no—
439C2—82no—
440C2—82no—
441C2#348411yescharacteristic
442C2C2×#57511yescharacteristic
443C2#358611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3506C2semidirect
2. C2×#548C2semidirect
3. #3321C2semidirect
4. #3313C2semidirect
5. #3271C2semidirect
6. #3269C2semidirect
7. #3215C2semidirect
8. #3213C2semidirect
9. #3194C2semidirect
10. #3190C2semidirect
11. #3113C2semidirect
12. #3112C2semidirect
13. #3004C2semidirect
14. #3002C2semidirect
15. #2973C2semidirect
18. C2×#137C22semidirect
19. C2×#135C22semidirect
21. #556C22semidirect
22. #555C22semidirect
24. #550C22semidirect
29. #547C22semidirect
47. #497C22semidirect
48. #496C22semidirect
49. #489C22semidirect
50. #489C22semidirect
51. #485C22semidirect
53. #482C22semidirect
55. #477C22semidirect
56. #475C22semidirect
57. #469C22semidirect
58. #468C22semidirect
59. #467C22semidirect
60. #465C22semidirect
61. #464C22semidirect
62. #463C22semidirect
63. #462C22semidirect
64. #460C22semidirect
65. #452C22semidirect
66. #452C22semidirect
69. #451C22semidirect
71. #449C22semidirect
72. #449C22semidirect
75. #448C22semidirect
77. #441C22semidirect
78. #440C22semidirect
79. #435C22semidirect
80. #435C22semidirect
83. #419C22semidirect
84. #416C22semidirect

Maximal subgroups

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

subgroupindex
#35062
C2×#5482
#33212
#33132
#32712
#32692
#32152
#32132
#31942
#31902
#31132
#31122
#30042
#30022
#29732