Finite Group Databasebeta

← #6436718

Subgroups — #6436718

Order 512.

2024 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3122C212yesmaximal, complemented
2#35278C211yesmaximal, characteristic, complemented
3#34255C211yesmaximal, characteristic, complemented
4#29870C212yesmaximal, complemented
5#29834C214yesmaximal, complemented
6#29794C212yesmaximal, complemented
7#29765C212yesmaximal, complemented
8#29758C212yesmaximal, complemented
9#29721C212yesmaximal, complemented
10#29712C212yesmaximal, complemented
11#23688C214yesmaximal, complemented
12#23682C212yesmaximal, complemented
13#23676C212yesmaximal, complemented
14#21399C212yesmaximal, complemented
15#20813C211yesmaximal, characteristic, complemented
16C2×#540C2212yes—
17C2×#518C2211yescharacteristic
18#3122—216no—
19C2×#482C2214yes—
20C2×#478C2214yes—
21C2×#459C2214yes—
22C2×#447C2214yes—
23C2×#442C2214yes—
24C2×C4×D8C2214yes—
25C2×#433C2212yes—
26#2867C2211yescharacteristic, complemented
27#2859C2211yescharacteristic, complemented
28#2856C2212yes—
29#2641C2212yescomplemented
30#2619C2212yes—
31#2445—22no—
32#2445—22no—
33#2428—22no—
34#2428—22no—
35#2357—22no—
36#2354—22no—
37#2342—22no—
38#2342—22no—
39#2328C2212yescomplemented
40#2319C2212yes—
41#2319C2212yes—
42#2317C2211yescharacteristic
43#2309C2211yescharacteristic
44C4×#127C2212yescomplemented
45#2296C2211yescharacteristic
46#2293C2212yes—
47#2293C2212yes—
48C2×#390C2211yescharacteristic
49#2274C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2085C2214yescomplemented
52#2084C2212yescomplemented
53#2083C2212yescomplemented
54#2073C2214yes—
55#2071C2212yes—
56#2071C2212yes—
57#2067C2214yescomplemented
58#2067C2214yescomplemented
59#2065C2214yescomplemented
60#2065C2214yescomplemented
61#2054C2214yes—
62#2054C2214yes—
63#2054C2214yes—
64#2054C2214yes—
65#2041C2212yescomplemented
66#2041C2212yescomplemented
67#2041C2214yescomplemented
68#2011C2212yes—
69#2011C2212yes—
70#2011C2214yes—
71#1989C2214yescomplemented
72#1982C2214yescomplemented
73#1875C2214yes—
74#1874C2214yes—
75#1843C2212yescomplemented
76#1831C2212yescomplemented
77#1798C2212yes—
78#1797C2212yes—
79#1768C2212yescomplemented
80#1767C2211yescharacteristic, complemented
81C4×C4⋊C8C2211yescharacteristic, complemented
82#1762C2214yescomplemented
83C4×#102C2214yescomplemented
84C23×D4C2311yescharacteristic
85C22×D8C2314yes—
86C22×D8C2314yes—
87#540—28no—
88#540—28no—
89C2×#127—22no—
90C2×#127—22no—
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#127C2312yes—
94C2×#125C2312yes—
95C2×#123—24no—
96C2×#121—22no—
97C2×#121—22no—
98C2×#121—24no—
99C2×#121C2314yes—
100#518—24no—
101#518D418yescomplemented
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—22no—
105C2×C4×D4C2314yes—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×#117—22no—
109C2×#117—22no—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4—22no—
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#482—216no—
119#478—216no—
120#459—216no—
121#447—216no—
122#442—216no—
123C4×D8—216no—
124#433—28no—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#394—22no—
143#394—22no—
144#394—24no—
145#394—24no—
146#392—22no—
147#392—22no—
148#390—21no—
149#390—21no—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390—22no—
155#390C2314yes—
156#390C2312yes—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#390C2312yes—
161#389—22no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#386—24no—
176#386—24no—
177#385—22no—
178#385—21no—
179#385—21no—
180#384—21no—
181#384C2312yes—
182#382—21no—
183#382C2312yes—
184#381—22no—
185#381—22no—
186#380—22no—
187#380—22no—
188#379—21no—
189#379—21no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4D412yes—
193C4×C4⋊C4D412yes—
194C4×C4⋊C4C2312yes—
195C4×C4⋊C4C2312yes—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212C25—21no—
213C22×D4—24no—
214C22×D4—21no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—24no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4—42no—
222C22×D4—42no—
223C22×D4C2×D414yes—
224C22×D4C2414yes—
225C22×D4C2412yes—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—21no—
229C23×C4—24no—
230C23×C4C2×D412yes—
231C23×C4C2411yescharacteristic
232C2×D8—28no—
233C2×D8—28no—
234C2×D8—44no—
235C2×D8—44no—
236C2×D8—44no—
237C2×D8—44no—
238C22×C8C2414yes—
239C22×C8C2414yes—
240#127—24no—
241#127—24no—
242#127—44no—
243#127—42no—
244#127—42no—
245#127—84no—
246#127—84no—
247#125—24no—
248#125—24no—
249#123—416no—
250#121—28no—
251#121—28no—
252#121—416no—
253#121—84no—
254#121—84no—
255C4×D4—216no—
256C4×D4—28no—
257C4×D4—24no—
258C4×D4—24no—
259C4×D4—28no—
260C4×D4—84no—
261C4×D4—84no—
262C4×D4C2×D418yes—
263C4×D4C2×D418yes—
264#117—22no—
265#117—42no—
266#117—42no—
267#117—84no—
268#117—84no—
269#117C2×D414yes—
270#117C2×D414yes—
271#117#4114yes—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2414yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—84no—
308C2×C22⋊C4C2×D411yescharacteristic
309C2×C22⋊C4C2×D411yescharacteristic
310C2×C22⋊C4C2×D412yes—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327#107—28no—
328#106—28no—
329C4⋊C8—28no—
330C4⋊C8—44no—
331C4⋊C8—42no—
332C4⋊C8—42no—
333#102—28no—
334#102—28no—
335#102—28no—
336#102—28no—
337#102—44no—
338#102—44no—
339#102—44no—
340#102—44no—
341C22⋊C8—28no—
342#97—24no—
343C4×C8—24no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—42no—
360C24—42no—
361C24—82no—
362C24—82no—
363C24C22×D411yescharacteristic
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—22no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—24no—
374C2×D4—48no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—48no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4C22×D414yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—82no—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C2×#4112yes—
426C22×C4C2511yescharacteristic
427D8—416no—
428D8—416no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4#12118yes—
451C4⋊C4C22×D414yes—
452C22⋊C4—24no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4#12118yes—
473C42—24no—
474C42—24no—
475C42—24no—
476C42—22no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—82no—
485C42—82no—
486C42C22×D412yes—
487C42C22×D412yes—
488C42C2×#4112yes—
489C42C2×#4112yes—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—24no—
495C23—48no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—88no—
510C23—164no—
511C23—162no—
512C23—162no—
513C23C2×#12112yes—
514C23C23×D411yescharacteristic
515D4—28no—
516D4—48no—
517D4—44no—
518D4—44no—
519D4—44no—
520D4—44no—
521D4—44no—
522D4—88no—
523D4—88no—
524D4—168no—
525D4—164no—
526D4—164no—
527D4—168no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—84no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4—168no—
565C2×C4—162no—
566C2×C4C2×#12112yes—
567C2×C4C2×#12114yes—
568C2×C4C2×#12114yes—
569C2×C4#57614yes—
570C2×C4C23×D411yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4#58512yes—
573C2×C4#58512yes—
574C8—44no—
575C8—44no—
576C22—24no—
577C22—22no—
578C22—42no—
579C22—41no—
580C22—41no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—168no—
596C22C22×#12111yescharacteristic
597C22#344911yescharacteristic
598C22#344911yescharacteristic
599C22C2×#57612yes—
600C22#358512yes—
601C4—24no—
602C4—24no—
603C4—24no—
604C4—22no—
605C4—44no—
606C4—42no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—84no—
612C4—82no—
613C4—82no—
614C4#344912yes—
615C4#345512yes—
616C2—42no—
617C2—84no—
618C2—164no—
619C2#3463311yescharacteristic
620C2#6042812yes—
621C2#6097812yes—
622C2#6101111yescharacteristic
623C2#6101711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3122C2semidirect
2. #35278C2semidirect
3. #34255C2semidirect
4. #29870C2semidirect
5. #29834C2semidirect
6. #29794C2semidirect
7. #29765C2semidirect
8. #29758C2semidirect
9. #29721C2semidirect
10. #29712C2semidirect
11. #23688C2semidirect
12. #23682C2semidirect
13. #23676C2semidirect
14. #21399C2semidirect
15. #20813C2semidirect
26. #2867C22semidirect
27. #2859C22semidirect
29. #2641C22semidirect
39. #2328C22semidirect
44. C4×#127C22semidirect
51. #2085C22semidirect
52. #2084C22semidirect
53. #2083C22semidirect
57. #2067C22semidirect
58. #2067C22semidirect
59. #2065C22semidirect
60. #2065C22semidirect
65. #2041C22semidirect
66. #2041C22semidirect
67. #2041C22semidirect
71. #1989C22semidirect
72. #1982C22semidirect
75. #1843C22semidirect
76. #1831C22semidirect
79. #1768C22semidirect
80. #1767C22semidirect
81. C4×C4⋊C8C22semidirect
82. #1762C22semidirect
83. C4×#102C22semidirect
101. #518D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31222
#352782
#342552
#298702
#298342
#297942
#297652
#297582
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