Finite Group Databasebeta

← #6457448

Subgroups — #6457448

Order 512.

1682 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35628C211yesmaximal, characteristic
2#35557C211yesmaximal, characteristic
3#28088C214yesmaximal, complemented
4#28081C214yesmaximal, complemented
5#28076C214yesmaximal, complemented
6#28069C214yesmaximal, complemented
7C2×#2095C211yesmaximal, characteristic, complemented
8#23639C211yesmaximal, characteristic
9#23551C211yesmaximal, characteristic, complemented
10#23487C211yesmaximal, characteristic
11#21322C211yesmaximal, characteristic, complemented
12#21094C211yesmaximal, characteristic, complemented
13#21088C211yesmaximal, characteristic, complemented
14#21066C211yesmaximal, characteristic, complemented
15#21037C211yesmaximal, characteristic, complemented
16#20974C211yesmaximal, characteristic
17#20663C211yesmaximal, characteristic
18#20455C211yesmaximal, characteristic, complemented
19#20387C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#533C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C2×#427C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C2×#408C2211yescharacteristic
32C2×#405C2211yescharacteristic
33C2×C4×M16C2211yescharacteristic
34#2592—22no—
35#2568—22no—
36#2512—22no—
37#2511—22no—
38#2497C2214yes—
39#2496C2214yes—
40#2492C2214yes—
41#2489C2214yes—
42#2488—24no—
43#2481—24no—
44#2316C2211yescharacteristic
45#2311C2211yescharacteristic
46#2311C2211yescharacteristic
47#2311C2211yescharacteristic
48#2310C2211yescharacteristic
49#2297C2211yescharacteristic
50#2296C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2274C2211yescharacteristic
54#2274C2211yescharacteristic
55#2095C2212yes—
56#2095C2212yes—
57#2095C2212yes—
58#2095C2212yes—
59#2076C2214yes—
60#2075C2214yes—
61#2059C2212yes—
62#2059C2212yes—
63#2058C2212yes—
64#2057C2212yes—
65#2036C2214yes—
66#2035C2214yes—
67#1992C2214yes—
68#1990C2214yes—
69#1935C2214yes—
70#1935C2214yes—
71#1933C2214yes—
72#1933C2214yes—
73#1928C2214yes—
74#1927C2214yes—
75#1921C2214yes—
76#1918C2214yes—
77#1894C2214yes—
78#1892C2214yes—
79#1809C2214yes—
80#1808C2214yes—
81#1776C2214yes—
82C4×#107C2214yes—
83#1753C2212yes—
84#1753C2212yes—
85#1752C2212yes—
86C8×C22⋊C4C2212yes—
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C2×#131—24no—
100C2×#131—24no—
101C2×#131—42no—
102C22×M16C2311yescharacteristic
103C22×M16C2312yes—
104C22×M16C2311yescharacteristic
105C23×C8C2312yes—
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110#533—24no—
111#533—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C2312yes—
114C2×#126—24no—
115C2×#125C2312yes—
116C2×#124—21no—
117C2×#123—22no—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8C2314yes—
120C2×C4×Q8—22no—
121C2×#117C2311yescharacteristic
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C2×#117C2311yescharacteristic
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—22no—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C42C2311yescharacteristic
131#427—24no—
132#427—42no—
133C2×#107—24no—
134C2×#107—24no—
135C2×#107C2312yes—
136C2×#107C2312yes—
137C2×#107C2312yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#107C2312yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143#419—24no—
144#419—42no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153#408—24no—
154C2×C22⋊C8C2312yes—
155C2×C22⋊C8C2312yes—
156#405—24no—
157C4×M16—42no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2312yes—
162#400—24no—
163#400—24no—
164#400—24no—
165#400—24no—
166#397—24no—
167#397—24no—
168#396—24no—
169#393—24no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—22no—
174#391—22no—
175#391—24no—
176#391D412yes—
177#391C2314yes—
178#391C2312yes—
179#391C2312yes—
180#390—21no—
181#390—21no—
182#390—21no—
183#390D412yes—
184#389—22no—
185#389C2312yes—
186#389C2312yes—
187#389C2314yes—
188#388—24no—
189#388—24no—
190#388—24no—
191#387—24no—
192#387—24no—
193#386—22no—
194#386—22no—
195#386—22no—
196#386D414yes—
197#385—22no—
198#384—22no—
199#384C2312yes—
200#384C2312yes—
201#382—24no—
202#381—22no—
203#380—22no—
204#380C2314yes—
205#380C2314yes—
206#380C2314yes—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—22no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4C2312yes—
215C4×C22⋊C4C2312yes—
216C2×#95—22no—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2311yescharacteristic
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41—22no—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8C2×D411yescharacteristic
248C22×Q8C2412yes—
249C22×D4C2×D412yes—
250C22×D4C2×D411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4C2×D412yes—
256C23×C4C2412yes—
257C23×C4C2411yescharacteristic
258#131—416no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—42no—
267C2×M16#4114yes—
268C22×C8—22no—
269C22×C8—22no—
270C22×C8—22no—
271C22×C8—22no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—42no—
275C22×C8#4114yes—
276C22×C8#4114yes—
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C22×C8C2411yescharacteristic
283C4⋊Q8—24no—
284C4⋊Q8—42no—
285C4⋊Q8—44no—
286#126—88no—
287#125—24no—
288#125—42no—
289#124—44no—
290#123—48no—
291C22⋊Q8—28no—
292C22⋊Q8—48no—
293C22⋊Q8—44no—
294C4×Q8—48no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117—22no—
304#117#4114yes—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—84no—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4C2×D412yes—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42C2×D412yes—
352C2×C42C2412yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—24no—
356#107—24no—
357#107—24no—
358#107—24no—
359#107—24no—
360#107—42no—
361#106—24no—
362#106—42no—
363#103—24no—
364#103—24no—
365#103—24no—
366#103—24no—
367#103—28no—
368#103—44no—
369C22⋊C8—24no—
370C22⋊C8—24no—
371#97—24no—
372#97—42no—
373C4×C8—24no—
374C4×C8—42no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95C2×D414yes—
388#95C2×D414yes—
389C24C22×D412yes—
390C24C22×D411yescharacteristic
391#41—44no—
392#41—44no—
393#41—44no—
394#41—44no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—82no—
414C2×D4—24no—
415C2×D4—24no—
416C2×D4—24no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—82no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458M16—28no—
459M16—48no—
460M16—44no—
461C2×C8—28no—
462C2×C8—28no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—48no—
477C2×C8C2×#4114yes—
478C2×C8C2×#4112yes—
479C2×C8C2×#4112yes—
480C2×C8C2×#4112yes—
481C2×C8C2×#4112yes—
482C4⋊C4—24no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4C2×#4114yes—
509C22⋊C4—24no—
510C22⋊C4—24no—
511C22⋊C4—28no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—42no—
529C42—42no—
530C42—42no—
531C42—84no—
532C42—84no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23#53212yes—
541C23#54612yes—
542C23#54712yes—
543C23C23×D411yescharacteristic
544Q8—44no—
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—88no—
550Q8—84no—
551D4—48no—
552D4—44no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—48no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—162no—
592C2×C4—162no—
593C2×C4—162no—
594C2×C4#52914yes—
595C2×C4#53212yes—
596C2×C4#54214yes—
597C2×C4#54612yes—
598C2×C4#54712yes—
599C2×C4#55114yes—
600C2×C4#55614yes—
601C2×C4#57814yes—
602C2×C4C22×#4111yescharacteristic
603C2×C4C22×#4111yescharacteristic
604C2×C4C22×#4111yescharacteristic
605C2×C4C22×#4111yescharacteristic
606C2×C4C2×H(5,𝔽2)11yescharacteristic
607C2×C4C2×2−1+411yescharacteristic
608C8—22no—
609C8—22no—
610C8—22no—
611C8—22no—
612C8—24no—
613C8—42no—
614C22—24no—
615C22—22no—
616C22—22no—
617C22—24no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22#252114yes—
624C22C2×#53211yescharacteristic
625C22C2×#54611yescharacteristic
626C22C2×#54711yescharacteristic
627C22C2×#57812yes—
628C22#358512yes—
629C4—24no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—42no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4#252114yes—
642C4C2×#52911yescharacteristic
643C4C2×#54211yescharacteristic
644C4C2×#55111yescharacteristic
645C4C2×#55611yescharacteristic
646C2—24no—
647C2—82no—
648C2C2×#252111yescharacteristic
649C2#6071511yescharacteristic
650C2#6071911yescharacteristic
651C2#6119412yes—
652C2#6128212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #28088C2semidirect
4. #28081C2semidirect
5. #28076C2semidirect
6. #28069C2semidirect
7. C2×#2095C2semidirect
9. #23551C2semidirect
11. #21322C2semidirect
12. #21094C2semidirect
13. #21088C2semidirect
14. #21066C2semidirect
15. #21037C2semidirect
18. #20455C2semidirect

Maximal subgroups

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

subgroupindex
#356282
#355572
#280882
#280812
#280762
#280692
C2×#20952
#236392
#235512
#234872
#213222
#210942
#210882
#210662
#210372
#209742
#206632
#204552
#203872