Finite Group Databasebeta

← #61726

Subgroups — #61726

Order 256.

930 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#3519C211yesmaximal, characteristic
2C2×#554C211yesmaximal, characteristic, complemented
3#3357C212yesmaximal, complemented
4#3346C214yesmaximal, complemented
5#3338C212yesmaximal, complemented
6#3333C212yesmaximal, complemented
7#3314C214yesmaximal, complemented
8#3295C212yesmaximal, complemented
9#3238C211yesmaximal, characteristic, complemented
10#3226C211yesmaximal, characteristic, complemented
11#3193C212yesmaximal, complemented
12#3190C212yesmaximal, complemented
13#3112C211yesmaximal, characteristic, complemented
14#3108C211yesmaximal, characteristic, complemented
15#3092C212yesmaximal, complemented
16C2×#480C212yesmaximal, complemented
17#2974C211yesmaximal, characteristic, complemented
18#555C2211yescharacteristic
19#554—21no—
20#554—22no—
21#554—21no—
22#554C2211yescharacteristic
23#554C2212yes—
24#554C2212yescomplemented
25#554C2214yescomplemented
26#553—21no—
27#553C2212yes—
28#550C2211yescharacteristic
29#546—22no—
30#546C2212yes—
31D42—21no—
32D42C2211yescharacteristic
33#540—21no—
34#538—22no—
35#532C2211yescharacteristic
36C2×C4⋊Q8C2211yescharacteristic, complemented
37#529C2211yescharacteristic
38C2×#125—21no—
39#525C2212yes—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8C2212yescomplemented
42C2×#121C2212yes—
43#518—21no—
44C2×C4×Q8C2211yescharacteristic, complemented
45C2×C4×D4—21no—
46C2×C4×D4C2211yescharacteristic
47C22×C4⋊C4C2212yescomplemented
48#501C2212yescomplemented
49#500C2212yescomplemented
50#489C2212yescomplemented
51#486C2212yescomplemented
52#485C2214yescomplemented
53#484C2214yescomplemented
54#484C2214yescomplemented
55#483C2211yescharacteristic, complemented
56#483C2212yescomplemented
57#481C2214yescomplemented
58#480C2211yescharacteristic
59#480C2212yes—
60#480C2214yescomplemented
61#480C2212yescomplemented
62#480C2214yescomplemented
63#480C2212yescomplemented
64#480C2212yescomplemented
65#479C2212yescomplemented
66#477C2212yescomplemented
67#477C2212yescomplemented
68#476C2212yescomplemented
69#474C2214yescomplemented
70#468C2214yescomplemented
71#468C2214yescomplemented
72#467C2214yescomplemented
73#465C2214yescomplemented
74#464C2212yescomplemented
75#460C2212yescomplemented
76#452C2214yescomplemented
77#451C2211yescharacteristic, complemented
78#450C2214yescomplemented
79#448C2211yescharacteristic
80#448C2212yescomplemented
81#440C2214yescomplemented
82C4×Q16C2212yescomplemented
83C4×SD16C2212yescomplemented
84#436C2212yescomplemented
85#435C2214yescomplemented
86C8×D4C2212yes—
87#432C2211yescharacteristic, complemented
88#431C2211yescharacteristic, complemented
89#428C2214yescomplemented
90C2×#107C2214yescomplemented
91#417C2214yescomplemented
92C2×#102C2214yescomplemented
93#407C2212yescomplemented
94C2×C22⋊C8C2212yescomplemented
95C2×#41—21no—
96C2×#41C2312yes—
97C22×Q8—21no—
98C22×Q8C2311yescharacteristic
99C22×D4—22no—
100C22×D4—21no—
101C22×D4D411yescharacteristic
102C22×D4C2312yes—
103C23×C4—22no—
104C23×C4C2312yes—
105C2×Q16C2312yes—
106C2×SD16C2312yes—
107C2×SD16C2314yes—
108C2×M16C2314yes—
109C22×C8C2314yes—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C2312yes—
113C4⋊Q8C2312yes—
114C4⋊Q8C2311yescharacteristic
115#127—21no—
116#127C2311yescharacteristic
117#126—22no—
118#126—21no—
119#125—22no—
120#125—22no—
121#125—24no—
122#125—21no—
123#125C2311yescharacteristic
124#124—22no—
125#124—21no—
126#124C2311yescharacteristic
127#123—22no—
128#123—21no—
129#123—22no—
130#123—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—24no—
134C22⋊Q8—24no—
135C22⋊Q8—24no—
136C22⋊Q8—24no—
137C22⋊Q8—22no—
138C22⋊Q8C2314yes—
139C22⋊Q8C2314yes—
140C22⋊Q8C2311yescharacteristic
141C22⋊Q8C2312yes—
142#121—22no—
143#121—21no—
144#121—22no—
145#121—22no—
146#121—21no—
147#121—21no—
148#121—22no—
149#121—22no—
150#121C2311yescharacteristic
151#121C2312yes—
152#121C2312yes—
153#121C2312yes—
154#121C2312yes—
155#120—22no—
156#120—22no—
157#120—22no—
158C4×Q8—21no—
159C4×Q8—21no—
160C4×Q8C2312yes—
161C4×Q8C2312yes—
162C4×Q8C2311yescharacteristic
163C4×D4—22no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—21no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—22no—
172C4×D4—21no—
173C4×D4D411yescharacteristic
174C4×D4D412yes—
175C4×D4D411yescharacteristic
176C4×D4C2311yescharacteristic
177C4×D4C2312yes—
178C4×D4C2311yescharacteristic
179C4×D4C2312yes—
180C4×D4C2311yescharacteristic
181#117—21no—
182#117—21no—
183#117C2312yes—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4D411yescharacteristic
191C2×C4⋊C4C2311yescharacteristic
192C2×C4⋊C4C2312yes—
193C2×C4⋊C4C2314yes—
194C2×C4⋊C4C2314yes—
195C2×C4⋊C4C2312yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4D412yes—
200C2×C42—21no—
201C2×C42C2311yescharacteristic
202#107C2312yes—
203#107C2314yes—
204#107C2314yes—
205#107C2314yes—
206#107C2314yes—
207#106C2312yes—
208#106C2314yes—
209C4⋊C8C2312yes—
210C4⋊C8C2312yes—
211#103C2312yes—
212#103C2314yes—
213#103C2312yes—
214#103C2312yes—
215#103C2314yes—
216#102C2312yes—
217#102C2314yes—
218#102C2314yes—
219#102C2314yes—
220#102C2314yes—
221C22⋊C8C2311yescharacteristic
222C22⋊C8C2312yes—
223C22⋊C8C2314yes—
224C22⋊C8C2311yescharacteristic
225C22⋊C8C2314yes—
226#97C2312yes—
227C4×C8C2312yes—
228C24—22no—
229C24C2×D412yes—
230#41—44no—
231#41—82no—
232C2×Q8—21no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—21no—
236C2×Q8—42no—
237C2×Q8—82no—
238C2×Q8C2411yescharacteristic
239C2×Q8C2412yes—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—21no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—84no—
253C2×D4C2×D412yes—
254C2×D4C2×D411yescharacteristic
255C2×D4C2×D411yescharacteristic
256C2×D4C2411yescharacteristic
257C2×D4C2412yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—44no—
268C22×C4—84no—
269C22×C4C2×D411yescharacteristic
270C22×C4C2×D411yescharacteristic
271C22×C4C2×D412yes—
272C22×C4C2×D412yes—
273C22×C4C2×D412yes—
274C22×C4C2411yescharacteristic
275C22×C4C2412yes—
276C22×C4C2412yes—
277Q16—42no—
278SD16—44no—
279SD16—42no—
280M16—44no—
281C2×C8—44no—
282C2×C8C2412yes—
283C2×C8C2412yes—
284C2×C8C2414yes—
285C4⋊C4—21no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4C2×D411yescharacteristic
303C4⋊C4C2×D411yescharacteristic
304C4⋊C4C2×D412yes—
305C4⋊C4C2412yes—
306C4⋊C4C2414yes—
307C4⋊C4C2411yescharacteristic
308C4⋊C4C2411yescharacteristic
309C4⋊C4C2412yes—
310C22⋊C4—22no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—22no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—21no—
321C22⋊C4—22no—
322C22⋊C4—21no—
323C22⋊C4—22no—
324C22⋊C4C2×D412yes—
325C22⋊C4C2×D412yes—
326C22⋊C4C2×D412yes—
327C22⋊C4C2×D412yes—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42C2×D411yescharacteristic
333C42C2×D411yescharacteristic
334C42C2411yescharacteristic
335C42C2411yescharacteristic
336C23—24no—
337C23—24no—
338C23—21no—
339C23—22no—
340C23—42no—
341C23—44no—
342C23—82no—
343C23—82no—
344C23C22×D412yes—
345C23C22×D412yes—
346C23C22×D411yescharacteristic
347Q8—22no—
348Q8—22no—
349Q8—22no—
350Q8—42no—
351Q8—82no—
352Q8—82no—
353Q8—81no—
354D4—24no—
355D4—22no—
356D4—22no—
357D4—24no—
358D4—44no—
359D4—42no—
360D4—44no—
361D4—42no—
362D4—82no—
363D4—81no—
364D4—82no—
365D4—82no—
366D4—82no—
367D4#13714yes—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4C22×D411yescharacteristic
393C2×C4C22×D412yes—
394C2×C4C22×D411yescharacteristic
395C2×C4C22×D411yescharacteristic
396C2×C4C22×D411yescharacteristic
397C2×C4C22×D412yes—
398C2×C4C22×D411yescharacteristic
399C2×C4C2511yescharacteristic
400C8—22no—
401C8—24no—
402C8—22no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—48no—
411C22—42no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—164no—
418C22C2×#13714yes—
419C22#57612yes—
420C22C23×D411yescharacteristic
421C4—21no—
422C4—22no—
423C4—22no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—42no—
428C4—41no—
429C4—41no—
430C4—42no—
431C4—44no—
432C4—41no—
433C4—42no—
434C4C2×#13712yes—
435C4#58512yes—
436C2—22no—
437C2—24no—
438C2—42no—
439C2—82no—
440C2—81no—
441C2#348511yescharacteristic
442C2C22×#13711yescharacteristic
443C2C2×#57611yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
prime radical#3519C2
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
2. C2×#554C2semidirect
3. #3357C2semidirect
4. #3346C2semidirect
5. #3338C2semidirect
6. #3333C2semidirect
7. #3314C2semidirect
8. #3295C2semidirect
9. #3238C2semidirect
10. #3226C2semidirect
11. #3193C2semidirect
12. #3190C2semidirect
13. #3112C2semidirect
14. #3108C2semidirect
15. #3092C2semidirect
16. C2×#480C2semidirect
17. #2974C2semidirect
24. #554C22semidirect
25. #554C22semidirect
36. C2×C4⋊Q8C22semidirect
41. C2×C22⋊Q8C22semidirect
44. C2×C4×Q8C22semidirect
47. C22×C4⋊C4C22semidirect
48. #501C22semidirect
49. #500C22semidirect
50. #489C22semidirect
51. #486C22semidirect
52. #485C22semidirect
53. #484C22semidirect
54. #484C22semidirect
55. #483C22semidirect
56. #483C22semidirect
57. #481C22semidirect
60. #480C22semidirect
61. #480C22semidirect
62. #480C22semidirect
63. #480C22semidirect
64. #480C22semidirect
65. #479C22semidirect
66. #477C22semidirect
67. #477C22semidirect
68. #476C22semidirect
69. #474C22semidirect
70. #468C22semidirect
71. #468C22semidirect
72. #467C22semidirect
73. #465C22semidirect
74. #464C22semidirect
75. #460C22semidirect
76. #452C22semidirect
77. #451C22semidirect
78. #450C22semidirect
80. #448C22semidirect
81. #440C22semidirect
82. C4×Q16C22semidirect
83. C4×SD16C22semidirect
84. #436C22semidirect
85. #435C22semidirect
87. #432C22semidirect
88. #431C22semidirect
89. #428C22semidirect
90. C2×#107C22semidirect
91. #417C22semidirect
92. C2×#102C22semidirect
93. #407C22semidirect
94. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35192
C2×#5542
#33572
#33462
#33382
#33332
#33142
#32952
#32382
#32262
#31932
#31902
#31122
#31082
#30922
C2×#4802
#29742