Finite Group Databasebeta

← #6461723

Subgroups — #6461723

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34255C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#29406C214yesmaximal, complemented
4#29404C214yesmaximal, complemented
5#29393C212yesmaximal, complemented
6#29392C212yesmaximal, complemented
7#29389C212yesmaximal, complemented
8#29388C212yesmaximal, complemented
9#23658C211yesmaximal, characteristic, complemented
10C2×#2071C211yesmaximal, characteristic
11#23551C211yesmaximal, characteristic, complemented
12#23454C211yesmaximal, characteristic
13#21065C212yesmaximal
14#21039C212yesmaximal, complemented
15#21035C212yesmaximal, complemented
16#20876C212yesmaximal
17#20774C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C2×#428C2212yes—
21C2×#427C2212yes—
22C2×#418C2212yes—
23C22×#102C2212yes—
24#2360—22no—
25#2360—22no—
26#2357—21no—
27#2357—21no—
28#2355—21no—
29#2355—21no—
30#2354—21no—
31#2354—21no—
32Q8×C4⋊C4—21no—
33Q8×C4⋊C4—21no—
34#2342—22no—
35#2342—22no—
36#2330C2214yescomplemented
37#2319C2212yes—
38#2319C2212yes—
39#2317C2211yescharacteristic
40#2317C2212yes—
41#2309C2211yescharacteristic
42C4×C4⋊Q8C2212yescomplemented
43C4×C4⋊Q8C2212yescomplemented
44#2296C2211yescharacteristic
45#2293C2212yes—
46#2293C2212yes—
47C2×#390C2211yescharacteristic
48C2×#384C2211yescharacteristic
49#2274C2212yes—
50#2274C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2084C2212yescomplemented
53#2084C2212yescomplemented
54#2083C2212yescomplemented
55#2083C2212yescomplemented
56#2071C2212yes—
57#2071C2212yes—
58#2071C2214yes—
59#2059C2212yescomplemented
60#2059C2212yescomplemented
61#2058C2211yescharacteristic, complemented
62#2058C2211yescharacteristic, complemented
63#2057C2211yescharacteristic, complemented
64#2057C2211yescharacteristic, complemented
65#2025C2214yes—
66#2023C2212yes—
67#2023C2212yes—
68#1926C2214yes—
69#1926C2214yes—
70#1925C2214yes—
71#1925C2214yes—
72#1921C2214yescomplemented
73#1921C2214yescomplemented
74#1920C2214yescomplemented
75#1920C2214yescomplemented
76#1919C2214yescomplemented
77#1919C2214yescomplemented
78#1918C2214yescomplemented
79#1918C2214yescomplemented
80#1864C2214yes—
81#1864C2214yes—
82#1862C2214yes—
83#1862C2214yes—
84#1840C2214yescomplemented
85#1839C2212yescomplemented
86#1839C2212yescomplemented
87#1736C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×D4C2311yescharacteristic
96C22×M16C2312yes—
97C23×C8C2312yes—
98#533—28no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104#518—24no—
105#518—24no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×#117—22no—
115C2×#117—22no—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C22⋊C4—22no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C42C2311yescharacteristic
129#428—28no—
130#427—28no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2314yes—
134C2×#106C2312yes—
135C2×#106C2312yes—
136C2×#106C2314yes—
137#418—28no—
138C2×#102—28no—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143#391—22no—
144#391—22no—
145#390—21no—
146#390—21no—
147#390—21no—
148#390—21no—
149#390C2312yes—
150#390C2312yes—
151#390C2312yes—
152#390C2312yes—
153#389—22no—
154#389—22no—
155#389—24no—
156#389—24no—
157#389C2314yes—
158#389C2314yes—
159#389C2311yescharacteristic
160#389C2311yescharacteristic
161#389C2311yescharacteristic
162#389C2311yescharacteristic
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#388—22no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#387—24no—
174#387—24no—
175#385—21no—
176#385—21no—
177#385—22no—
178#385—24no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2312yes—
185#382—22no—
186#382—21no—
187#381—22no—
188#381—22no—
189#380—22no—
190#380—22no—
191#379—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214#376C2312yes—
215C43C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2312yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2311yescharacteristic
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2312yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234#336C2314yes—
235C25—21no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×D4—21no—
239C22×D4—24no—
240C22×D4—22no—
241C22×D4—24no—
242C22×D4#4114yes—
243C22×D4C2412yes—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—21no—
247C23×C4—22no—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—24no—
254C22×C8—24no—
255C22×C8—24no—
256C22×C8—24no—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C22×C8C2414yes—
260C4⋊Q8—24no—
261C4⋊Q8—24no—
262#125—24no—
263#125—24no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C4×Q8—28no—
267C4×Q8—28no—
268C4×Q8—84no—
269C4×Q8—84no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—24no—
280#117—24no—
281#117—24no—
282#117—84no—
283#117—84no—
284#117#4114yes—
285#117#4118yes—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C22⋊C4—216no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4C2×D412yes—
338C2×C22⋊C4C2×D412yes—
339C2×C42—21no—
340C2×C42—22no—
341C2×C42—24no—
342C2×C42—21no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#107—24no—
358#107—24no—
359#107—28no—
360#106—24no—
361#106—24no—
362#106—28no—
363#102—28no—
364#102—28no—
365#102—28no—
366#102—28no—
367#95—21no—
368#95—21no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—21no—
376#95—21no—
377#95—24no—
378#95—24no—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382#95C2×D412yes—
383C24—22no—
384C24—24no—
385C24—42no—
386C24—42no—
387C24—81no—
388C24—81no—
389C24—81no—
390C24—81no—
391C24C22×D411yescharacteristic
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×D4—24no—
399C2×D4—216no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—48no—
406C2×D4C2×#4114yes—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—44no—
424C22×C4—42no—
425C22×C4—82no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—81no—
430C22×C4—81no—
431C22×C4—82no—
432C22×C4—82no—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4112yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C42−1+414yes—
446C22×C4C2511yescharacteristic
447M16—48no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—24no—
451C2×C8—22no—
452C2×C8—24no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—24no—
457C2×C8—48no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—24no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—44no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—82no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—82no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4C2×#4114yes—
490C4⋊C4C2×#4114yes—
491C4⋊C4C2×#4112yes—
492C4⋊C4C2×#4112yes—
493C22⋊C4—28no—
494C22⋊C4—28no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—24no—
499C22⋊C4—42no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—82no—
503C22⋊C4—82no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—82no—
508C42—22no—
509C42—22no—
510C42—24no—
511C42—28no—
512C42—24no—
513C42—24no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—24no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42—84no—
525C42—84no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—24no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—48no—
538C23—84no—
539C23—84no—
540C23—162no—
541C23—162no—
542C23—162no—
543C23—162no—
544C23#52512yes—
545C23#52512yes—
546C23#53612yes—
547C23C23×D411yescharacteristic
548Q8—48no—
549Q8—48no—
550D4—28no—
551D4—48no—
552D4—44no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—88no—
602C2×C4—88no—
603C2×C4—164no—
604C2×C4—162no—
605C2×C4#52512yes—
606C2×C4#52512yes—
607C2×C4#53612yes—
608C2×C4#55218yes—
609C2×C4#55618yes—
610C2×C4#57612yes—
611C2×C4#57712yes—
612C2×C4C22×#4111yescharacteristic
613C2×C4C22×#4112yes—
614C2×C4C22×#4111yescharacteristic
615C2×C4C2×2−1+412yes—
616C8—42no—
617C8—42no—
618C8—44no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—168no—
633C22#259914yes—
634C22C2×#52511yescharacteristic
635C22C2×#52511yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×#57611yescharacteristic
638C22C2×#57711yescharacteristic
639C22#358511yescharacteristic
640C22#358611yescharacteristic
641C4—22no—
642C4—22no—
643C4—41no—
644C4—44no—
645C4—41no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—81no—
654C4—81no—
655C4—84no—
656C4—82no—
657C4#259914yes—
658C4C2×#55212yes—
659C4C2×#55612yes—
660C2—24no—
661C2—84no—
662C2C2×#259911yescharacteristic
663C2#6058411yescharacteristic
664C2#6058811yescharacteristic
665C2#6059011yescharacteristic
666C2#6059111yescharacteristic
667C2#6087711yescharacteristic
668C2#6088511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. #34214C2semidirect
3. #29406C2semidirect
4. #29404C2semidirect
5. #29393C2semidirect
6. #29392C2semidirect
7. #29389C2semidirect
8. #29388C2semidirect
9. #23658C2semidirect
11. #23551C2semidirect
14. #21039C2semidirect
15. #21035C2semidirect
17. #20774C2semidirect
36. #2330C22semidirect
42. C4×C4⋊Q8C22semidirect
43. C4×C4⋊Q8C22semidirect
52. #2084C22semidirect
53. #2084C22semidirect
54. #2083C22semidirect
55. #2083C22semidirect
59. #2059C22semidirect
60. #2059C22semidirect
61. #2058C22semidirect
62. #2058C22semidirect
63. #2057C22semidirect
64. #2057C22semidirect
72. #1921C22semidirect
73. #1921C22semidirect
74. #1920C22semidirect
75. #1920C22semidirect
76. #1919C22semidirect
77. #1919C22semidirect
78. #1918C22semidirect
79. #1918C22semidirect
84. #1840C22semidirect
85. #1839C22semidirect
86. #1839C22semidirect

Maximal subgroups

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

subgroupindex
#342552
#342142
#294062
#294042
#293932
#293922
#293892
#293882
#236582
C2×#20712
#235512
#234542
#210652
#210392
#210352
#208762
#207742