Finite Group Databasebeta

← #6817425

Subgroups — #6817425

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3237C212yesmaximal, complemented
2#36134C211yesmaximal, characteristic
3#36132C211yesmaximal, characteristic
4#33232C211yesmaximal, characteristic, complemented
5#31143C214yesmaximal, complemented
6#31141C214yesmaximal, complemented
7#29852C212yesmaximal, complemented
8#28755C211yesmaximal, characteristic, complemented
9#28693C211yesmaximal, characteristic, complemented
10#26466C212yesmaximal, complemented
11#26462C212yesmaximal, complemented
12#26233C212yesmaximal
13#26231C212yesmaximal
14#25910C212yesmaximal, complemented
15#25677C212yesmaximal, complemented
16C4×#501C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#533C2212yes—
20#3237—216no—
21C2×#501C2212yes—
22C2×#501C2211yescharacteristic
23C2×#491C2212yes—
24C2×#491C2212yes—
25C2×#491C2212yes—
26C2×#485C2214yes—
27C2×#485C2214yes—
28C2×#477C2214yes—
29C2×#475C2214yes—
30C2×#432C2212yes—
31#2868C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2212yes—
34#2851C2212yes—
35#2552—21no—
36#2551C2211yescharacteristic
37#2551C2211yescharacteristic
38#2542C2212yes—
39#2538—21no—
40#2465—22no—
41#2465—22no—
42#2452—21no—
43#2452—21no—
44#2441—22no—
45#2439C2212yes—
46#2439C2212yes—
47#2429C2214yes—
48#2423—22no—
49#2365—21no—
50#2365—21no—
51#2365C2212yes—
52#2354—21no—
53#2354C2211yescharacteristic
54#2354C2211yescharacteristic
55Q8×C4⋊C4—21no—
56C4×C4⋊Q8C2211yescharacteristic
57C4×#125C2211yescharacteristic
58C42×D4C2211yescharacteristic
59#2085C2214yes—
60#2068C2214yes—
61#2066C2214yes—
62#2043C2212yes—
63#2043C2212yes—
64#2036C2214yes—
65#2036C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2024C2214yes—
69#2024C2214yes—
70#1993C2211yescharacteristic
71#1992C2212yes—
72#1992C2211yescharacteristic
73#1981C2212yes—
74#1981C2212yes—
75#1960C2212yes—
76#1960C2214yes—
77#1956C2212yes—
78#1956C2212yes—
79#1954C2212yes—
80#1948C2212yes—
81#1947C2212yes—
82#1944C2212yes—
83#1943C2212yes—
84#1939C2212yes—
85#1924C2214yes—
86#1873C2214yes—
87#1873C2214yes—
88#1865C2214yes—
89#1863C2214yes—
90#1847C2211yescharacteristic
91C4×#107C2212yes—
92C4×#106C2212yes—
93#1767C2211yescharacteristic
94#1749C2212yes—
95C4×#97C2211yescharacteristic
96#533—28no—
97#533—28no—
98#533—44no—
99#533—44no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125—22no—
109C2×#125C2312yes—
110C2×#125C2311yescharacteristic
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2314yes—
116C2×C4×Q8—21no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8—22no—
120C2×C4×Q8C2314yes—
121C2×C4×D4D411yescharacteristic
122C2×C4×D4D411yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×#117—22no—
125C2×#117C2312yes—
126C2×#117C2312yes—
127C22×C42C2312yes—
128#501—28no—
129#501D418yescomplemented
130#491—28no—
131#491—28no—
132#491—44no—
133#485—216no—
134#485—216no—
135#477—216no—
136#475—216no—
137#432—28no—
138C2×#107C2314yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#106C2314yes—
142C2×#106C2312yes—
143C2×#106C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×#97C2311yescharacteristic
157C2×#97C2312yes—
158C2×#97C2312yes—
159C2×#97C2311yescharacteristic
160C2×C4×C8C2312yes—
161#399—24no—
162#398—24no—
163#396—24no—
164#396—22no—
165#396—22no—
166#395—22no—
167#395—22no—
168#393—22no—
169#393—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391C2314yes—
174#390—22no—
175#390—21no—
176#390—21no—
177#389—21no—
178#389—21no—
179#389—22no—
180#389—21no—
181#389—22no—
182#389—22no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—24no—
188#389C2311yescharacteristic
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—24no—
194#388—22no—
195#388—22no—
196#387—21no—
197#387—21no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—24no—
203#385—22no—
204#385—22no—
205#384—22no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384—21no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#382C2311yescharacteristic
214#382C2312yes—
215#380—22no—
216#380—22no—
217#380C2312yes—
218#380C2312yes—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C4⋊C4C2311yescharacteristic
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2311yescharacteristic
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234C43C2311yescharacteristic
235#340C2314yes—
236#340C2314yes—
237#340C2312yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#336C2312yes—
242#336C2314yes—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2414yes—
247C22×D4C2×D411yescharacteristic
248C23×C4C2×D412yes—
249C23×C4C2×D412yes—
250C23×C4C2412yes—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—22no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—24no—
261C4⋊Q8—42no—
262C4⋊Q8—42no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—44no—
266C4⋊Q8C2×D414yes—
267#125—24no—
268#125—22no—
269#125—24no—
270#125—48no—
271#125C2×D414yes—
272C22⋊Q8—28no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275C22⋊Q8—48no—
276C22⋊Q8—48no—
277C22⋊Q8—44no—
278C22⋊Q8—44no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×Q8—48no—
282C4×Q8—48no—
283C4×Q8—48no—
284C4×Q8—44no—
285C4×D4—24no—
286C4×D4—24no—
287C4×D4—42no—
288#117—24no—
289#117—24no—
290#117—42no—
291#117—42no—
292#117—48no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D411yescharacteristic
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C22⋊C4C2×D412yes—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2×D412yes—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—28no—
359#107—42no—
360#107C2×D418yes—
361#106—28no—
362#106—42no—
363#106C2×D418yes—
364C4⋊C8—28no—
365C4⋊C8—24no—
366#103—28no—
367#103—28no—
368#103—28no—
369#103—28no—
370#103—44no—
371#103—44no—
372#103—44no—
373#103—44no—
374C22⋊C8—28no—
375C22⋊C8—44no—
376#97—24no—
377#97—22no—
378#97—42no—
379#97C2×D414yes—
380C4×C8—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—22no—
389#95—21no—
390#95—21no—
391#95—22no—
392#95—24no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95C2×D412yes—
398#95C2×D412yes—
399#95C2×D412yes—
400C24C22×D412yes—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—24no—
404C2×Q8—24no—
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—24no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8—82no—
415C2×Q8—82no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D412yes—
446C22×C4C22×D411yescharacteristic
447C22×C4C2511yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—24no—
456C2×C8C22×D414yes—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4#13714yes—
498C4⋊C4C22×D412yes—
499C4⋊C4C22×D412yes—
500C4⋊C4C22×D414yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—82no—
508C22⋊C4—82no—
509C22⋊C4—84no—
510C42—24no—
511C42—24no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—41no—
518C42—41no—
519C42—44no—
520C42—44no—
521C42—48no—
522C42—42no—
523C42—44no—
524C42—42no—
525C42C22×D412yes—
526C42H(5,𝔽2)12yes—
527C422−1+414yes—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23C23×D411yescharacteristic
536Q8—48no—
537Q8—44no—
538Q8—44no—
539Q8—44no—
540Q8—44no—
541Q8—88no—
542Q8—88no—
543D4—44no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—84no—
595C2×C4#53612yes—
596C2×C4#53612yes—
597C2×C4D4212yes—
598C2×C4D4212yes—
599C2×C4C2×#13714yes—
600C2×C4C2×#13712yes—
601C2×C4#57512yes—
602C2×C4C23×D411yescharacteristic
603C2×C4C2×H(5,𝔽2)11yescharacteristic
604C2×C4C2×2−1+412yes—
605C2×C4#58512yes—
606C8—24no—
607C8—44no—
608C8—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22C2×#53611yescharacteristic
617C22C2×D4211yescharacteristic
618C22#350011yescharacteristic
619C22C22×#13711yescharacteristic
620C22C2×#57511yescharacteristic
621C22#358611yescharacteristic
622C22#358611yescharacteristic
623C4—22no—
624C4—22no—
625C4—24no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—81no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—84no—
640C4#323714yescomplemented
641C4#349712yes—
642C4#355412yes—
643C2—82no—
644C2#3936311yescharacteristic
645C2C2×#323711yescharacteristic
646C2#6088411yescharacteristic
647C2#6106711yescharacteristic
648C2D4×#13711yescharacteristic
649C2#6258111yescharacteristic
650C2#6258911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3237C2semidirect
4. #33232C2semidirect
5. #31143C2semidirect
6. #31141C2semidirect
7. #29852C2semidirect
8. #28755C2semidirect
9. #28693C2semidirect
10. #26466C2semidirect
11. #26462C2semidirect
14. #25910C2semidirect
15. #25677C2semidirect
16. C4×#501C2semidirect
129. #501D4semidirect
640. C4#3237semidirect

Maximal subgroups

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

subgroupindex
C2×#32372
#361342
#361322
#332322
#311432
#311412
#298522
#287552
#286932
#264662
#264622
#262332
#262312
#259102
#256772
C4×#5012
#212702