Finite Group Databasebeta

← #6484530

Subgroups — #6484530

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#32960C212yesmaximal
4#32957C211yesmaximal, characteristic
5#32954C211yesmaximal, characteristic
6#29957C212yesmaximal
7#29953C211yesmaximal, characteristic
8#29951C211yesmaximal, characteristic
9#26540C214yesmaximal, complemented
10#26539C214yesmaximal, complemented
11#26474C214yesmaximal, complemented
12#26473C214yesmaximal, complemented
13#21955C211yesmaximal, characteristic
14#21811C212yesmaximal
15#21363C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#479C2214yes—
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#432C2212yes—
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25#2851C2212yes—
26#2627—21no—
27#2627—21no—
28#2606—22no—
29#2605—22no—
30#2570C2212yes—
31#2570C2212yes—
32#2550C2212yes—
33#2550C2212yes—
34#2365—22no—
35#2365—22no—
36#2365C2214yes—
37#2365C2214yes—
38#2329—21no—
39#2329—21no—
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#389—21no—
43C2×#387—21no—
44C2×#387—21no—
45#2283C2212yes—
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49#2097C2214yes—
50#2096C2212yes—
51#2095C2212yes—
52#2068C2214yes—
53#2068C2214yes—
54#2066C2214yes—
55#2066C2214yes—
56#2026C2214yes—
57#2024C2212yes—
58#2024C2212yes—
59#1981C2214yes—
60#1981C2214yes—
61#1960C2214yes—
62#1960C2214yes—
63#1949C2214yes—
64#1949C2214yes—
65#1940C2214yes—
66#1940C2214yes—
67#1865C2214yes—
68#1865C2214yes—
69#1863C2214yes—
70#1863C2214yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1805C2214yes—
76#1803C2212yes—
77#1803C2212yes—
78#1797C2211yescharacteristic
79#1797C2212yes—
80#1797C2211yescharacteristic
81#1787C2212yes—
82#1787C2212yes—
83#1787C2212yes—
84#1787C2212yes—
85#1776C2214yes—
86#1776C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C2×C4⋊Q8—22no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8Q818yescomplemented
92C2×C4⋊Q8C2314yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2311yescharacteristic
95C2×#125C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C4×Q8—22no—
99C2×C4×Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—44no—
102C2×C4×Q8—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C42—21no—
115C22×C42—21no—
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#479—48no—
121#477—44no—
122#475—44no—
123#432—28no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×#97C2314yes—
143#400—22no—
144#400—22no—
145#398—22no—
146#398—22no—
147#398—22no—
148#398—22no—
149#395—22no—
150#395—22no—
151#391—21no—
152#391—21no—
153#391—22no—
154#391—21no—
155#391—21no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2314yes—
160#391C2312yes—
161#389—24no—
162#389—24no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—24no—
178#387—24no—
179#387—24no—
180#387—24no—
181#385—22no—
182#385—22no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#383—22no—
186#382C2312yes—
187#382C2312yes—
188#382C2312yes—
189#381—22no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194C4×C4⋊C4—24no—
195C4×C4⋊C4—24no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C4⋊C4C2312yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4—22no—
205#376—22no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2312yes—
218#336C2312yes—
219#336C2314yes—
220C22×Q8—21no—
221C22×Q8—21no—
222C22×Q8—21no—
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2×Q814yes—
231C22×Q8C2×Q814yes—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2414yes—
246C22×C8C2414yes—
247C4⋊Q8—216no—
248C4⋊Q8—28no—
249C4⋊Q8—42no—
250C4⋊Q8—42no—
251C4⋊Q8C2×Q8116yes—
252#125—44no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—28no—
258C4×Q8—44no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—48no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D412yes—
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×Q814yes—
319C2×C42#4114yes—
320C2×C42C2412yes—
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2412yes—
324#107—28no—
325#107—44no—
326#106—28no—
327#106—44no—
328C4⋊C8—28no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—44no—
332#103—44no—
333#103—44no—
334#103—44no—
335#103—44no—
336#103—44no—
337#103—44no—
338#103—44no—
339#103—44no—
340C22⋊C8—24no—
341C22⋊C8—24no—
342#97—28no—
343#95—24no—
344#95—22no—
345#95—22no—
346#95—24no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24C22×D411yescharacteristic
369C2×Q8—24no—
370C2×Q8—24no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—24no—
376C2×Q8—24no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—81no—
390C2×Q8—81no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×Q8C22×Q814yes—
394C2×Q8C22×Q814yes—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4C22⋊Q814yes—
433C22×C4C22⋊Q814yes—
434C22×C4#13714yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q812yes—
445C22×C4C2×#4112yes—
446C22×C4C2511yescharacteristic
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C8—24no—
451C2×C8—24no—
452C2×C8—24no—
453C4⋊C4—24no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4H(5,𝔽2)14yes—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—41no—
489C42—41no—
490C42—42no—
491C42—44no—
492C42—44no—
493C42—48no—
494C42—48no—
495C42—82no—
496C42—82no—
497C42C22⋊Q818yes—
498C42C22×Q814yes—
499C42C2×#4114yes—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23C2×C22⋊Q812yes—
505C23C2×#13712yes—
506C23#57512yes—
507C23C23×D411yescharacteristic
508Q8—24no—
509Q8—24no—
510Q8—28no—
511Q8—48no—
512Q8—48no—
513Q8—42no—
514Q8—42no—
515Q8—44no—
516Q8—162no—
517Q8—162no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—28no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—48no—
551C2×C4—48no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—84no—
576C2×C4C2×C22⋊Q812yes—
577C2×C4C2×C22⋊Q814yes—
578C2×C4C2×C22⋊Q814yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4C23×Q811yescharacteristic
582C2×C4C22×#4111yescharacteristic
583C2×C4C2×H(5,𝔽2)12yes—
584C2×C4#58512yes—
585C8—44no—
586C8—44no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22#307814yes—
594C22C22×C22⋊Q811yescharacteristic
595C22#348411yescharacteristic
596C22#348711yescharacteristic
597C22C22×#13711yescharacteristic
598C22C2×#57511yescharacteristic
599C22#358611yescharacteristic
600C22#358611yescharacteristic
601C4—24no—
602C4—24no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—44no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4#346012yes—
619C4#350112yes—
620C2—24no—
621C2#3517411yescharacteristic
622C2C2×#307811yescharacteristic
623C2#6049111yescharacteristic
624C2#6157411yescharacteristic
625C2#6157511yescharacteristic
626C2#6189311yescharacteristic
627C2#6189411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #26540C2semidirect
10. #26539C2semidirect
11. #26474C2semidirect
12. #26473C2semidirect
91. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#23652
#329602
#329572
#329542
#299572
#299532
#299512
#265402
#265392
#264742
#264732
#219552
#218112
#213632