Finite Group Databasebeta

← #6468547

Subgroups — #6468547

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#29324C214yesmaximal, complemented
4#29309C214yesmaximal
5#29207C211yesmaximal, characteristic, complemented
6#29207C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#28687C211yesmaximal, characteristic, complemented
9#27950C211yesmaximal, characteristic
10#27948C211yesmaximal, characteristic
11#25812C211yesmaximal, characteristic
12#25812C211yesmaximal, characteristic
13#23843C214yesmaximal, complemented
14#21727C211yesmaximal, characteristic, complemented
15#21726C211yesmaximal, characteristic, complemented
16C4×#500C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18#21367C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#498C2211yescharacteristic
25C2×#433C2212yes—
26#2868C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2211yescharacteristic
29#2857C2211yescharacteristic
30#2647C2212yes—
31#2629C2212yes—
32#2507—21no—
33#2507—21no—
34#2497—21no—
35#2497—21no—
36#2446—21no—
37#2446—21no—
38#2429—21no—
39#2429—21no—
40#2360—21no—
41#2360—21no—
42#2360—21no—
43#2360—21no—
44#2355—21no—
45#2355—21no—
46Q8×C4⋊C4—21no—
47Q8×C4⋊C4—21no—
48#2330C2212yes—
49#2330C2212yes—
50#2330C2212yes—
51#2317C2212yes—
52#2314C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C2×#384C2211yescharacteristic
60#2274C2212yes—
61C4×#117C2211yescharacteristic
62#2084C2214yes—
63#2084C2214yes—
64#2083C2214yes—
65#2083C2214yes—
66#2059C2214yes—
67#2059C2214yes—
68#2059C2214yes—
69#2059C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1949C2214yes—
79#1949C2214yes—
80#1948C2214yes—
81#1947C2214yes—
82#1944C2214yes—
83#1943C2214yes—
84#1942C2214yes—
85#1941C2214yes—
86#1854C2214yes—
87#1854C2214yes—
88#1853C2214yes—
89#1852C2214yes—
90#1845C2211yescharacteristic
91#1845C2211yescharacteristic
92#1844C2211yescharacteristic
93#1844C2211yescharacteristic
94#1776C2214yes—
95#1776C2214yes—
96C4×#107C2214yes—
97C4×#106C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#533—24no—
108#533—24no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×#117C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4C2312yes—
135C22×C42C2311yescharacteristic
136#500—24no—
137#500—24no—
138#498—24no—
139#498—24no—
140#433Q8116yescomplemented
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2312yes—
154C2×#97C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161#400—22no—
162#400—22no—
163#398—22no—
164#398—22no—
165#398—22no—
166#398—22no—
167#396—22no—
168#396—22no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#385—22no—
214#385—22no—
215#384—22no—
216#384—21no—
217#384—21no—
218#384—21no—
219#384—21no—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2312yes—
225#384C2312yes—
226#384C2312yes—
227#382—22no—
228#382C2312yes—
229#380—21no—
230#380—21no—
231#380C2312yes—
232#380C2312yes—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—22no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4D411yescharacteristic
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C22⋊C4—22no—
252C4×C22⋊C4—22no—
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257#376C2311yescharacteristic
258#376C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—21no—
277C23×C4—22no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—82no—
292C4⋊Q8—82no—
293C4⋊Q8#4114yes—
294C4⋊Q8#4114yes—
295#125—24no—
296#125—24no—
297#125—44no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300C22⋊Q8—82no—
301C22⋊Q8—82no—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—82no—
305C4×Q8—82no—
306C4×Q8—82no—
307C4×Q8—82no—
308C4×Q8—82no—
309C4×Q8—82no—
310#117—24no—
311#117—24no—
312#117—42no—
313#117C2×Q818yes—
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⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4—82no—
347C2×C4⋊C4—82no—
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4C2×D411yescharacteristic
364C2×C22⋊C4C2×D411yescharacteristic
365C2×C22⋊C4C2×D411yescharacteristic
366C2×C22⋊C4C2×D411yescharacteristic
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#107—28no—
394#107—44no—
395#107—44no—
396#106—28no—
397#106—28no—
398#106—44no—
399#106—44no—
400C4⋊C8C2×Q818yes—
401C4⋊C8C2×Q818yes—
402C22⋊C8C2×Q818yes—
403C22⋊C8C2×Q818yes—
404#97—42no—
405#97C2×Q818yes—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—42no—
411C4×C8C2×Q818yes—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438C24C22×D411yescharacteristic
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×Q814yes—
485C22×C4C2511yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8C22×Q814yes—
495C2×C8C22×Q814yes—
496C2×C8C22×Q814yes—
497C2×C8C22×Q814yes—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4C4⋊Q814yes—
542C4⋊C4C4⋊Q814yes—
543C4⋊C4C2×#4114yes—
544C4⋊C4C2×#4114yes—
545C22⋊C4—28no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4C4⋊Q814yes—
551C22⋊C4C4⋊Q814yes—
552C42—28no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—21no—
564C42—21no—
565C42—22no—
566C42—22no—
567C42—82no—
568C42—82no—
569C42—82no—
570C42—82no—
571C42—82no—
572C42—82no—
573C42C22×Q814yes—
574C42C22×Q814yes—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23C2×C4⋊Q812yes—
584C23C23×D411yescharacteristic
585Q8—48no—
586Q8—48no—
587Q8—84no—
588Q8—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—164no—
646C2×C4#52512yes—
647C2×C4#52512yes—
648C2×C4C2×C4⋊Q812yes—
649C2×C4C2×C4⋊Q812yes—
650C2×C4C2×C4⋊Q812yes—
651C2×C4C2×C4⋊Q812yes—
652C2×C4C2×C4⋊Q812yes—
653C2×C4#57612yes—
654C2×C4#57712yes—
655C2×C4#57712yes—
656C2×C4#57812yes—
657C2×C4C23×Q812yes—
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4111yescharacteristic
660C2×C4#58512yes—
661C8—24no—
662C8—24no—
663C8—24no—
664C8—24no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22C2×#52511yescharacteristic
673C22C22×C4⋊Q811yescharacteristic
674C22#344911yescharacteristic
675C22C2×#57611yescharacteristic
676C22C2×#57711yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57811yescharacteristic
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—22no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—44no—
690C4—41no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#311914yes—
699C4#312014yes—
700C4Q8×#4112yes—
701C4Q8×#4112yes—
702C2—42no—
703C2#3495311yescharacteristic
704C2C2×#311911yescharacteristic
705C2C2×#312011yescharacteristic
706C2#6066511yescharacteristic
707C2#6066711yescharacteristic
708C2#6100611yescharacteristic
709C2#6100811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29324C2semidirect
5. #29207C2semidirect
6. #29207C2semidirect
7. #28689C2semidirect
8. #28687C2semidirect
13. #23843C2semidirect
14. #21727C2semidirect
15. #21726C2semidirect
16. C4×#500C2semidirect
17. C4×#498C2semidirect
140. #433Q8semidirect

Maximal subgroups

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

subgroupindex
#352842
#342142
#293242
#293092
#292072
#286892
#286872
#279502
#279482
#258122
#238432
#217272
#217262
C4×#5002
C4×#4982
#213672
C4×#4332