Finite Group Databasebeta

← #6510715

Subgroups — #6510715

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37480C211yesmaximal, characteristic, complemented
2#35262C211yesmaximal, characteristic
3#29116C214yesmaximal, complemented
4#27971C214yesmaximal, complemented
5#27719C214yesmaximal, complemented
6#27712C214yesmaximal, complemented
7#27703C214yesmaximal, complemented
8#27696C214yesmaximal, complemented
9#21767C211yesmaximal, characteristic
10#21761C211yesmaximal, characteristic
11#20701C211yesmaximal, characteristic, complemented
12#20699C211yesmaximal, characteristic, complemented
13#20527C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18#2854—22no—
19#2842—21no—
20#2840—21no—
21#2665—22no—
22#2665—22no—
23#2628C2214yes—
24#2624—22no—
25#2607—22no—
26#2504C2212yes—
27#2501C2212yes—
28#2500C2214yes—
29#2482—22no—
30#2439C2214yescomplemented
31#2439C2214yescomplemented
32#2405—21no—
33#2405—21no—
34#2317C2212yes—
35#2309C2212yes—
36#2280C2211yescharacteristic
37#2280C2211yescharacteristic
38#2280C2211yescharacteristic
39#2101C2214yescomplemented
40#2101C2214yescomplemented
41#2100C2214yescomplemented
42#2099C2214yescomplemented
43#2073C2214yescomplemented
44#2073C2214yescomplemented
45#2023C2214yescomplemented
46#2023C2214yescomplemented
47#1993C2214yescomplemented
48#1990C2214yescomplemented
49#1936C2214yescomplemented
50#1936C2214yescomplemented
51#1934C2214yescomplemented
52#1934C2214yescomplemented
53#1895C2214yes—
54#1893C2212yes—
55#1893C2212yes—
56#1875C2214yes—
57#1875C2214yes—
58#1836C2214yescomplemented
59#1835C2214yescomplemented
60#1825C2214yes—
61#1825C2214yes—
62#1825C2214yescomplemented
63#1824C2212yes—
64#1824C2212yescomplemented
65#1823C2212yes—
66#1823C2212yescomplemented
67#1786C2212yes—
68#1786C2212yes—
69#1753C2214yescomplemented
70#1752C2212yescomplemented
71C8×C22⋊C4C2212yescomplemented
72#1750C2214yes—
73#1736C2212yescomplemented
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76#1735C2212yescomplemented
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C22×M16C2312yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C2×C4⋊Q8C2314yes—
84C2×#123—22no—
85C2×#123—22no—
86C2×#121—21no—
87C2×#121—21no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#121C2312yes—
91C2×#121C2312yes—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117C2312yes—
97C2×#117C2312yes—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C2×#117C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C22⋊C4—21no—
106C22×C42—21no—
107C2×#102C2314yes—
108C2×#102C2314yes—
109C2×#102C2314yes—
110C2×#102C2314yes—
111#407—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8C2312yes—
114C2×C22⋊C8C2312yes—
115C2×C22⋊C8C2312yes—
116C2×C22⋊C8C2312yes—
117C2×#97C2314yes—
118C2×C4×C8C2314yes—
119#400—24no—
120#399—24no—
121#399—24no—
122#397—24no—
123#396—24no—
124#396—24no—
125#395—24no—
126#394—24no—
127#394—24no—
128#393—24no—
129#393—24no—
130#393—24no—
131#391—24no—
132#391—24no—
133#390—24no—
134#390C2314yes—
135#390C2314yes—
136#389—24no—
137#389C2314yes—
138#389C2314yes—
139#388—24no—
140#388—24no—
141#387—24no—
142#387—24no—
143#386—22no—
144#386—22no—
145#385—22no—
146#385—22no—
147#384C2314yes—
148#382C2314yes—
149#382C2314yes—
150#381—21no—
151#381—21no—
152#381D412yes—
153#380D412yes—
154#380D412yes—
155#380C2312yes—
156#380C2312yes—
157#380C2312yes—
158#380C2312yes—
159#379—21no—
160#379—21no—
161#379—22no—
162#379—22no—
163#379D411yescharacteristic
164#379D411yescharacteristic
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4C2312yes—
170C4×C22⋊C4C2312yes—
171#340C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2314yes—
185C25—21no—
186C25C2×D411yescharacteristic
187C22×Q8—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—21no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—84no—
197C22×D4C2412yes—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4#4114yes—
207C23×C4#4114yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C2×M16—24no—
212C2×M16—42no—
213C2×M16—42no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C2412yes—
219C22×C8C2412yes—
220C22×C8C2414yes—
221C4⋊Q8—44no—
222C4⋊Q8—44no—
223#123—84no—
224#123—84no—
225#121—24no—
226#121—24no—
227#121—24no—
228#121—24no—
229#121—44no—
230#121—44no—
231#117—24no—
232#117—24no—
233#117—24no—
234#117—24no—
235#117—42no—
236#117—42no—
237#117—44no—
238#117—44no—
239#117—42no—
240#117—42no—
241#117—42no—
242#117—42no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2414yes—
262C2×C4⋊C4C2414yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D412yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—44no—
290C2×C42C2414yes—
291#102—44no—
292#102—44no—
293#102—44no—
294#102—44no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—24no—
298C22⋊C8—24no—
299#97—44no—
300C4×C8—44no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95C2×D414yes—
312#95C2×D414yes—
313C24—22no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—44no—
318C24—42no—
319C24—82no—
320C24—82no—
321C24#12414yes—
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—42no—
338C2×D4—82no—
339C2×D4—82no—
340C2×D4—84no—
341C2×D4—82no—
342C2×D4—82no—
343C2×D4—816no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4#12414yes—
368C22×C4#12414yes—
369C22×C4#12414yes—
370C22×C4C22×D414yes—
371C22×C4C22×D414yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4112yes—
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2511yescharacteristic
382M16—84no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—82no—
391C2×C8—82no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—84no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C22⋊C4—82no—
431C42—24no—
432C42—24no—
433C42—44no—
434C42—44no—
435C42—82no—
436C42—82no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—84no—
455C23—84no—
456C23—84no—
457C23—88no—
458C23—168no—
459C23C2×#12412yes—
460C23C2×#12412yes—
461C23C2×#12412yes—
462C23C23×D411yescharacteristic
463Q8—168no—
464D4—42no—
465D4—42no—
466D4—44no—
467D4—84no—
468D4—88no—
469D4—88no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4C2×#12412yes—
508C2×C4C2×#12412yes—
509C2×C4C2×#12412yes—
510C2×C4C22×#4111yescharacteristic
511C2×C4C22×#4111yescharacteristic
512C2×C4#58512yes—
513C2×C4#58512yes—
514C8—42no—
515C8—42no—
516C8—44no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—88no—
530C22—84no—
531C22—84no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—168no—
536C22C22×#12411yescharacteristic
537C22#348512yes—
538C22#358412yes—
539C22#358512yes—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—44no—
545C4—84no—
546C4—84no—
547C4—82no—
548C4—82no—
549C4—82no—
550C4#349111yescharacteristic
551C4#349111yescharacteristic
552C4#349112yes—
553C2—42no—
554C2—42no—
555C2—42no—
556C2—162no—
557C2#3545511yescharacteristic
558C2#6061211yescharacteristic
559C2#6061911yescharacteristic
560C2#6176214yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37480C2semidirect
3. #29116C2semidirect
4. #27971C2semidirect
5. #27719C2semidirect
6. #27712C2semidirect
7. #27703C2semidirect
8. #27696C2semidirect
11. #20701C2semidirect
12. #20699C2semidirect
13. #20527C2semidirect
30. #2439C22semidirect
31. #2439C22semidirect
39. #2101C22semidirect
40. #2101C22semidirect
41. #2100C22semidirect
42. #2099C22semidirect
43. #2073C22semidirect
44. #2073C22semidirect
45. #2023C22semidirect
46. #2023C22semidirect
47. #1993C22semidirect
48. #1990C22semidirect
49. #1936C22semidirect
50. #1936C22semidirect
51. #1934C22semidirect
52. #1934C22semidirect
58. #1836C22semidirect
59. #1835C22semidirect
62. #1825C22semidirect
64. #1824C22semidirect
66. #1823C22semidirect
69. #1753C22semidirect
70. #1752C22semidirect
71. C8×C22⋊C4C22semidirect
73. #1736C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect
76. #1735C22semidirect

Maximal subgroups

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

subgroupindex
#374802
#352622
#291162
#279712
#277192
#277122
#277032
#276962
#217672
#217612
#207012
#206992
#205272