Finite Group Databasebeta

← #72256124

Subgroups — #72256124

Order 2048.

At least 9436 conjugacy classes of subgroups; only the 1100 normal subgroups are shown.

The subgroup lattice has 428 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72256114C212yesmaximal, complemented
2#72256110C212yesmaximal, complemented
3#72256102C212yesmaximal, complemented
4#72256098C212yesmaximal, complemented
5#72255895C214yesmaximal, complemented
6#72255895C214yesmaximal, complemented
7#72255848C212yesmaximal, complemented
8#72251739C211yesmaximal, characteristic, complemented
9#72249540C211yesmaximal, characteristic, complemented
10#72238525C211yesmaximal, characteristic, complemented
11#72235178C214yesmaximal, complemented
12#72235174C212yesmaximal, complemented
13#72218455C212yesmaximal, complemented
14#72210511C212yesmaximal, complemented
15C22×#3484C2211yescharacteristic
16#7713911C2212yes—
17#7713863C2212yes—
18#7710478C2212yescomplemented
19#7710478C2212yescomplemented
20#7643564C2211yescharacteristic, complemented
21#7643563C2211yescharacteristic, complemented
22#7630939C2212yes—
23#7626386C2212yescomplemented
24C2×#34171C2211yescharacteristic
25#6628867C2214yescomplemented
26#6437445C2214yes—
27C2×#20585C2211yescharacteristic, complemented
28C2×#20585C2211yescharacteristic, complemented
29#517411C2211yescharacteristic
30#393988C2218yescomplemented
31#392696C2214yescomplemented
32#392113C2212yes—
33#359509C2214yes—
34#359433C2212yes—
35#359433C2212yes—
36#359429C2212yescomplemented
37#359429C2212yescomplemented
38#353314C2211yescharacteristic, complemented
39#353314C2211yescharacteristic, complemented
40#238058C2214yescomplemented
41#238058C2214yescomplemented
42#232528C2214yescomplemented
43#232528C2214yescomplemented
44#202015C2218yescomplemented
45#202015C2218yescomplemented
46#201322C2214yescomplemented
47#201322C2214yescomplemented
48C2×#8752C2212yescomplemented
49C2×#8751C2212yescomplemented
50#186835C2214yescomplemented
51#122111C2214yescomplemented
52#122111C2214yescomplemented
53#112857C2218yescomplemented
54#111321C2214yescomplemented
55#111321C2214yescomplemented
56#108238C2214yescomplemented
57#108036C2218yescomplemented
58#108036C2218yescomplemented
59#105817C2212yescomplemented
60#105817C2212yescomplemented
61#105717C2212yescomplemented
62#105717C2212yescomplemented
63C23×H(5,𝔽2)C2311yescharacteristic
64C25×D4C2311yescharacteristic
65C2×#3484D412yes—
66C2×#3484D412yes—
67C2×#3484C2312yes—
68C22×#535C2312yes—
69C22×#525C2312yes—
70C22×#518D411yescharacteristic
71C22×#518D411yescharacteristic
72C22×#518C2311yescharacteristic
73#60359C2318yes—
74#47035D412yescomplemented
75#47035D412yescomplemented
76#47033D414yescomplemented
77#47033D414yescomplemented
78#47031D412yescomplemented
79#47031D412yescomplemented
80#46457D414yescomplemented
81#46454D414yescomplemented
82#46452D414yescomplemented
83#46452D414yescomplemented
84#37479C2312yes—
85#37479C2312yes—
86#37454C2312yes—
87#37454C2312yes—
88#36614C2314yes—
89#35222D414yes—
90#35222D414yes—
91#35222C2314yes—
92#34396C2314yes—
93#34396C2314yes—
94#34171D412yes—
95#34171D412yes—
96#34171C2312yes—
97C2×#2309D411yescharacteristic
98C2×#2309D411yescharacteristic
99C2×#2308C2311yescharacteristic
100C2×#2308C2311yescharacteristic
101C2×#2308C2311yescharacteristic
102C2×#2308C2311yescharacteristic
103C2×#2308C2311yescharacteristic
104C2×#2308C2311yescharacteristic
105C2×#2280C2311yescharacteristic
106C2×#2280C2311yescharacteristic
107C22×#379D411yescharacteristic
108C22×#379D411yescharacteristic
109C22×#379D411yescharacteristic
110C22×#379D411yescharacteristic
111#20979C2318yes—
112#20605C2318yes—
113C2×#1783C2314yes—
114C2×#1783C2314yes—
115C2×#1782C2312yes—
116C2×#1782C2312yes—
117C2×#1782C2312yes—
118C2×#1782C2312yes—
119#20579C2314yes—
120#17587C2318yes—
121#17587C2318yes—
122#14681C2312yes—
123#14335C2312yes—
124#10726C2312yes—
125#10726C2312yes—
126#10726C2312yes—
127#10726C2312yes—
128#10716C2318yes—
129C2×#1306C2314yes—
130C2×#1306C2314yes—
131C2×#1306C2314yes—
132C2×#1306C2314yes—
133#7943C2318yes—
134#7943C2318yes—
135C27C2×D411yescharacteristic
136C27C2×D411yescharacteristic
137C22×H(5,𝔽2)C2×D412yes—
138C23×#41C2412yes—
139C24×D4C2×D411yescharacteristic
140C24×D4C2×D412yes—
141C24×D4C2×D412yes—
142C24×D4C2×D412yes—
143C24×D4C2×D412yes—
144C24×D4C2×D411yescharacteristic
145C24×D4C2411yescharacteristic
146C24×D4C2411yescharacteristic
147C25×C4C2411yescharacteristic
148C2×#535C2×D414yes—
149C2×#535C2×D414yes—
150C2×#535C2×D414yes—
151C22×#127#4114yescomplemented
152C22×#124#4114yescomplemented
153C2×#525C2×D414yes—
154C2×#525C2×D414yes—
155C2×#525C2414yes—
156C2×#518C2×D412yes—
157C2×#518C2×D412yes—
158C2×#518C2×D412yes—
159C22×#117C2×D411yescharacteristic
160C22×#117C2×D411yescharacteristic
161C22×#117C2×D411yescharacteristic
162C22×#117C2×D411yescharacteristic
163C22×#117C2412yes—
164C23×C22⋊C4C2×D411yescharacteristic
165C23×C22⋊C4C2×D411yescharacteristic
166C23×C22⋊C4C2×D411yescharacteristic
167C23×C22⋊C4C2×D411yescharacteristic
168C23×C22⋊C4C2×D411yescharacteristic
169C23×C22⋊C4C2×D411yescharacteristic
170C23×C22⋊C4C2×D411yescharacteristic
171C23×C22⋊C4C2×D411yescharacteristic
172C2×#407C2418yes—
173#2817C2×D412yescomplemented
174#2817C2×D412yescomplemented
175#2816C2×D414yescomplemented
176#2816C2×D414yescomplemented
177#2815C2×D412yescomplemented
178#2815C2×D412yescomplemented
179#2608C2×D414yescomplemented
180#2608C2×D414yescomplemented
181#2607C2×D412yescomplemented
182#2607C2×D412yescomplemented
183#2607C2×D412yescomplemented
184#2607C2×D412yescomplemented
185#2435C2×D414yes—
186#2435C2×D414yes—
187#2435C2×D414yes—
188#2435C2×D414yes—
189#2433C2×D414yes—
190#2433C2×D414yes—
191#2433C2×D414yes—
192#2433C2×D414yes—
193#2333C2×D414yescomplemented
194#2333C2×D414yescomplemented
195#2333C2×D414yescomplemented
196#2333C2×D414yescomplemented
197#2332C2×D414yescomplemented
198#2332C2×D414yescomplemented
199#2332C2×D414yescomplemented
200#2332C2×D414yescomplemented
201#2309C2×D414yes—
202#2309C2×D414yes—
203#2309C2×D414yes—
204#2308C2×D412yes—
205#2308C2×D412yes—
206#2308C2×D412yes—
207#2308C2×D412yes—
208#2308C2×D414yes—
209#2308C2×D414yes—
210#2280C2×D414yes—
211#2280C2×D414yes—
212#2280C2414yes—
213C2×#381C2412yes—
214C2×#381C2412yes—
215C2×#381C2411yescharacteristic
216C2×#381C2411yescharacteristic
217C2×#381C2411yescharacteristic
218C2×#381C2411yescharacteristic
219C2×#379C2×D411yescharacteristic
220C2×#379C2×D411yescharacteristic
221C2×#379C2×D411yescharacteristic
222C2×#379C2×D411yescharacteristic
223C2×#379C2×D411yescharacteristic
224C2×#379C2×D411yescharacteristic
225C2×#379C2×D411yescharacteristic
226C2×#379C2×D411yescharacteristic
227C2×#379C2×D411yescharacteristic
228C2×#379C2×D411yescharacteristic
229C2×#379C2×D411yescharacteristic
230C2×#379C2×D411yescharacteristic
231C26#12012yes—
232C26#12012yes—
233C26#12012yes—
234C26#12012yes—
235C26C22×D411yescharacteristic
236C26C22×D411yescharacteristic
237C26C22×D411yescharacteristic
238C26C22×D411yescharacteristic
239C26C22×D411yescharacteristic
240C22×#41C22×D414yes—
241C23×Q8C2×#4112yes—
242C23×D4#12011yescharacteristic
243C23×D4#12011yescharacteristic
244C23×D4#12011yescharacteristic
245C23×D4#12011yescharacteristic
246C23×D4#12012yes—
247C23×D4#12012yes—
248C23×D4C22×D412yes—
249C23×D4C22×D412yes—
250C23×D4C22×D412yes—
251C23×D4C22×D414yes—
252C23×D4C22×D412yes—
253C23×D4C22×D412yes—
254C23×D4C22×D412yes—
255C23×D4C2×#4112yes—
256C23×D4C2×#4114yes—
257C23×D4H(5,𝔽2)14yes—
258C24×C4C22×D411yescharacteristic
259C24×C4C22×D412yes—
260C24×C4C22×D412yes—
261C24×C4C2511yescharacteristic
262C2×#127#12014yes—
263C2×#127#12014yes—
264C2×#127#12014yes—
265C2×#127#12014yes—
266C2×#124#12014yes—
267C2×#124#12014yes—
268C2×#124#12014yes—
269C2×#124#12014yes—
270C2×C22⋊Q8C22×D414yes—
271C2×C22⋊Q8C22×D414yes—
272C2×#121C22×D414yes—
273C2×#121C22×D414yes—
274C2×#117C22×D414yes—
275C2×#117C22×D414yes—
276C2×#117C22×D414yes—
277C22×C22⋊C4#12012yes—
278C22×C22⋊C4#12012yes—
279C22×C22⋊C4#12012yes—
280C22×C22⋊C4#12012yes—
281C22×C22⋊C4#12012yes—
282C22×C22⋊C4#12014yes—
283C22×C22⋊C4#12012yes—
284C22×C22⋊C4C22×D411yescharacteristic
285C22×C22⋊C4C22×D411yescharacteristic
286C22×C22⋊C4C22×D411yescharacteristic
287C22×C22⋊C4C22×D411yescharacteristic
288C22×C22⋊C4C22×D411yescharacteristic
289C22×C22⋊C4C22×D411yescharacteristic
290C22×C22⋊C4C22×D411yescharacteristic
291C22×C22⋊C4C22×D411yescharacteristic
292C22×C42C2×#4114yes—
293#390#12014yes—
294#390#12014yes—
295#390#12014yes—
296#390#12014yes—
297#390#12014yes—
298#390#12014yes—
299#390#12014yes—
300#390#12014yes—
301#381C22×D412yes—
302#381C22×D412yes—
303#381C22×D412yes—
304#381C22×D412yes—
305#381C22×D412yes—
306#381C22×D414yes—
307#381C22×D412yes—
308#379#12011yescharacteristic
309#379#12012yes—
310#379#12011yescharacteristic
311#379#12012yes—
312#379#12012yes—
313#379#12014yes—
314#379#12012yes—
315#379#12011yescharacteristic
316#379#12011yescharacteristic
317C4×C22⋊C4C22×D414yes—
318C4×C22⋊C4C22×D414yes—
319C4×C22⋊C4C22×D414yes—
320C4×C22⋊C4C22×D414yes—
321C25C2×#12012yes—
322C25C2×#12012yes—
323C25C2×#12012yes—
324C25C2×#12012yes—
325C25C2×#12012yes—
326C25C2×#12011yescharacteristic
327C25C2×#12011yescharacteristic
328C25C2×#12011yescharacteristic
329C25C2×#12011yescharacteristic
330C25C2×#12011yescharacteristic
331C25C2×#12011yescharacteristic
332C25C2×#12012yes—
333C25#52512yes—
334C25#52512yes—
335C25#53512yes—
336C25#53512yes—
337C25C23×D411yescharacteristic
338C22×Q8C2×#12014yes—
339C22×D4B(4,𝔽2)14yes—
340C22×D4B(4,𝔽2)14yes—
341C22×D4C2×#12014yes—
342C22×D4C2×#12014yes—
343C22×D4C2×#12014yes—
344C22×D4C2×#12014yes—
345C22×D4C2×#12014yes—
346C22×D4#54618yes—
347C22×D4#54618yes—
348C23×C4C2×#12011yescharacteristic
349C23×C4C2×#12011yescharacteristic
350C23×C4C23×D412yes—
351C23×C4C23×D412yes—
352C23×C4C23×D412yes—
353C23×C4C22×#4112yes—
354C23×C4C2×H(5,𝔽2)12yes—
355C23×C4#58512yes—
356C23×C4#58512yes—
357C2×C4⋊C4C2×#12014yes—
358C2×C4⋊C4C2×#12014yes—
359C2×C22⋊C4C2×#12011yescharacteristic
360C2×C22⋊C4C2×#12012yes—
361C2×C22⋊C4C2×#12012yes—
362C2×C22⋊C4C2×#12012yes—
363C2×C22⋊C4C2×#12011yescharacteristic
364C2×C22⋊C4C2×#12011yescharacteristic
365C2×C22⋊C4C2×#12011yescharacteristic
366C2×C22⋊C4C2×#12012yes—
367C2×C22⋊C4C2×#12012yes—
368C2×C22⋊C4C2×#12011yescharacteristic
369C2×C22⋊C4C2×#12011yescharacteristic
370C2×C42C2×#12014yes—
371C2×C42C2×#12014yes—
372C2×C42C2×#12014yes—
373C2×C42C2×#12014yes—
374C24#240414yes—
375C24#240412yes—
376C24#240412yes—
377C24#284612yes—
378C24#284612yes—
379C24#284612yes—
380C24#284612yes—
381C24C2×B(4,𝔽2)12yes—
382C24C2×B(4,𝔽2)12yes—
383C24C2×B(4,𝔽2)12yes—
384C24C2×B(4,𝔽2)12yes—
385C24#302612yes—
386C24#302612yes—
387C24C22×#12011yescharacteristic
388C24C22×#12011yescharacteristic
389C24C22×#12011yescharacteristic
390C24C2×#52511yescharacteristic
391C24C2×#53511yescharacteristic
392C24#344911yescharacteristic
393C24#344911yescharacteristic
394C22×C4C2×B(4,𝔽2)12yes—
395C22×C4C2×B(4,𝔽2)12yes—
396C22×C4C22×#12014yes—
397C22×C4C2×#54614yes—
398C22×C4D4×#4114yes—
399C22×C4D4×#4114yes—
400C22×C4#348514yes—
401C22×C4#348814yes—
402C22×C4#348814yes—
403C23#2398414yes—
404C23#3289814yes—
405C23#3289814yes—
406C23C2×#240411yescharacteristic
407C23#3473611yescharacteristic
408C23#3473611yescharacteristic
409C23#3508811yescharacteristic
410C23#3525011yescharacteristic
411C23#3525011yescharacteristic
412C23C2×#284611yescharacteristic
413C23C22×B(4,𝔽2)11yescharacteristic
414C23C22×B(4,𝔽2)11yescharacteristic
415C23C2×#302611yescharacteristic
416C23C2×#302611yescharacteristic
417C2×C4#3621218yes—
418C22#61318811yescharacteristic
419C22C2×#2398411yescharacteristic
420C22#645187411yescharacteristic
421C22#647758011yescharacteristic
422C22#647758011yescharacteristic
423C22C2×#3289811yescharacteristic
424C22C2×#3289811yescharacteristic
425C2#7225118011yescharacteristic
426C2#7225141912yes—
subgroupG/H
center, socleC22#613188
commutator, Frattini, cosocleC24×C4C25
LCSG ▷ C24×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72256114C2semidirect
2. #72256110C2semidirect
3. #72256102C2semidirect
4. #72256098C2semidirect
5. #72255895C2semidirect
6. #72255895C2semidirect
7. #72255848C2semidirect
8. #72251739C2semidirect
9. #72249540C2semidirect
10. #72238525C2semidirect
11. #72235178C2semidirect
12. #72235174C2semidirect
13. #72218455C2semidirect
14. #72210511C2semidirect
18. #7710478C22semidirect
19. #7710478C22semidirect
20. #7643564C22semidirect
21. #7643563C22semidirect
23. #7626386C22semidirect
25. #6628867C22semidirect
27. C2×#20585C22semidirect
28. C2×#20585C22semidirect
30. #393988C22semidirect
31. #392696C22semidirect
36. #359429C22semidirect
37. #359429C22semidirect
38. #353314C22semidirect
39. #353314C22semidirect
40. #238058C22semidirect
41. #238058C22semidirect
42. #232528C22semidirect
43. #232528C22semidirect
44. #202015C22semidirect
45. #202015C22semidirect
46. #201322C22semidirect
47. #201322C22semidirect
48. C2×#8752C22semidirect
49. C2×#8751C22semidirect
50. #186835C22semidirect
51. #122111C22semidirect
52. #122111C22semidirect
53. #112857C22semidirect
54. #111321C22semidirect
55. #111321C22semidirect
56. #108238C22semidirect
57. #108036C22semidirect
58. #108036C22semidirect
59. #105817C22semidirect
60. #105817C22semidirect
61. #105717C22semidirect
62. #105717C22semidirect
74. #47035D4semidirect
75. #47035D4semidirect
76. #47033D4semidirect
77. #47033D4semidirect
78. #47031D4semidirect
79. #47031D4semidirect
80. #46457D4semidirect
81. #46454D4semidirect
82. #46452D4semidirect
83. #46452D4semidirect
151. C22×#127#41semidirect
152. C22×#124#41semidirect
173. #2817C2×D4semidirect
174. #2817C2×D4semidirect
175. #2816C2×D4semidirect
176. #2816C2×D4semidirect
177. #2815C2×D4semidirect
178. #2815C2×D4semidirect
179. #2608C2×D4semidirect
180. #2608C2×D4semidirect
181. #2607C2×D4semidirect
182. #2607C2×D4semidirect
183. #2607C2×D4semidirect
184. #2607C2×D4semidirect
193. #2333C2×D4semidirect
194. #2333C2×D4semidirect
195. #2333C2×D4semidirect
196. #2333C2×D4semidirect
197. #2332C2×D4semidirect
198. #2332C2×D4semidirect
199. #2332C2×D4semidirect
200. #2332C2×D4semidirect

Maximal subgroups

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

subgroupindex
#722561142
#722561102
#722561022
#722560982
#722558952
#722558482
#722517392
#722495402
#722385252
#722351782
#722351742
#722184552
#722105112