Finite Group Databasebeta

← #6614699

Subgroups — #6614699

Order 512.

1561 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36189C211yesmaximal, characteristic
2#36151C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic, complemented
4#33212C211yesmaximal, characteristic, complemented
5#33049C211yesmaximal, characteristic, complemented
6#29176C211yesmaximal, characteristic, complemented
7#28953C214yesmaximal, complemented
8#28656C211yesmaximal, characteristic, complemented
9#28075C214yesmaximal, complemented
10#27905C211yesmaximal, characteristic, complemented
11#27875C211yesmaximal, characteristic, complemented
12#25812C211yesmaximal, characteristic, complemented
13#25672C211yesmaximal, characteristic, complemented
14#25262C214yesmaximal
15#22945C214yesmaximal, complemented
16C4×#498C211yesmaximal, characteristic, complemented
17#21409C211yesmaximal, characteristic, complemented
18#21322C211yesmaximal, characteristic, complemented
19C4×#436C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#483C2214yes—
26C2×#436C2212yes—
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2862C2211yescharacteristic
30#2859C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2570—21no—
36#2536—21no—
37#2511—21no—
38#2507C2212yes—
39#2497C2212yes—
40#2489—21no—
41#2489C2212yes—
42#2487C2212yes—
43#2453—21no—
44#2453C2212yes—
45#2442—21no—
46#2441—21no—
47#2423—21no—
48#2421—21no—
49#2421—21no—
50#2417—21no—
51#2415—21no—
52#2415C2212yes—
53#2362—21no—
54#2362C2212yes—
55#2360—21no—
56#2352—21no—
57#2351—21no—
58#2345C2212yes—
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61#2285C2211yescharacteristic
62C2×#384C2211yescharacteristic
63C42×D4C2211yescharacteristic
64#2096C2214yes—
65#2088C2214yes—
66#2077C2214yes—
67#2068C2214yes—
68#2059C2214yes—
69#2035C2214yes—
70#1992C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1944C2214yes—
82#1941C2214yes—
83#1935C2214yes—
84#1933C2214yes—
85#1933C2214yes—
86#1928C2214yes—
87#1920C2214yes—
88#1920C2214yes—
89#1894C2214yes—
90#1894C2214yes—
91#1865C2214yes—
92#1854C2214yes—
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1834C2211yescharacteristic
96#1809C2214yes—
97C4×#106C2214yes—
98#1770C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100#1763C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1736C2214yes—
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108C22×M16C2312yes—
109#533—24no—
110#533—24no—
111#533—42no—
112#533—42no—
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—21no—
118C2×#125—21no—
119C2×#125C2312yes—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8—21no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×D4D411yescharacteristic
135C2×C4×D4D411yescharacteristic
136C2×C4×D4C2311yescharacteristic
137C2×#117—21no—
138C2×#117—21no—
139C2×#117C2312yes—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2312yes—
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#500—42no—
146#498—24no—
147#487—42no—
148#483—216no—
149#436—28no—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×C22⋊C8C2312yes—
160C2×C22⋊C8C2312yes—
161C4×M16—42no—
162C4×M16—42no—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#400—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#399—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#397—22no—
180#397—22no—
181#396—22no—
182#394—22no—
183#391—21no—
184#391—21no—
185#391—21no—
186#391—21no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391C2312yes—
191#391C2312yes—
192#391C2312yes—
193#389—22no—
194#389—22no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—22no—
205#389C2312yes—
206#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#388—22no—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#386—22no—
224#385—22no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2312yes—
232#382—21no—
233#382—21no—
234#382—21no—
235#382—21no—
236#382—22no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380—21no—
243#380—22no—
244#380C2312yes—
245#380C2312yes—
246#380C2312yes—
247#380C2312yes—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—22no—
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C4⋊C4C2312yes—
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C22⋊C4—22no—
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263C4×C22⋊C4C2311yescharacteristic
264C4×C22⋊C4C2311yescharacteristic
265C43C2311yescharacteristic
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277C22×Q8—21no—
278C22×Q8—21no—
279C22×Q8—21no—
280C22×Q8C2412yes—
281C22×D4C2×D411yescharacteristic
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×M16—24no—
290C2×M16—24no—
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—42no—
294C2×M16—42no—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8#4114yes—
307#125—24no—
308#125—42no—
309#125—44no—
310#125—82no—
311C22⋊Q8—28no—
312C22⋊Q8—24no—
313C22⋊Q8—42no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319C22⋊Q8—82no—
320C4×Q8—24no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×Q8—44no—
326C4×Q8—82no—
327C4×D4—24no—
328C4×D4—24no—
329C4×D4—42no—
330#117—24no—
331#117—42no—
332#117—44no—
333#117—82no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—28no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—82no—
364C2×C4⋊C4—82no—
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—44no—
411#106—28no—
412#106—28no—
413#106—28no—
414C4⋊C8—24no—
415#103—28no—
416#103—28no—
417#103—28no—
418#103—44no—
419C22⋊C8—24no—
420C22⋊C8#4118yes—
421#97—42no—
422#97—42no—
423C4×C8—24no—
424C4×C8—24no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—22no—
428C4×C8—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455C24C22×D411yescharacteristic
456C24C22×D411yescharacteristic
457C2×Q8—24no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—42no—
463C2×Q8—42no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4114yes—
516C22×C4C2511yescharacteristic
517M16—48no—
518M16—48no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8C2×#4114yes—
529C2×C8C2×#4114yes—
530C2×C82−1+414yes—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—41no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4C2×#4114yes—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—41no—
577C22⋊C4—41no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—44no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—21no—
594C42—21no—
595C42—21no—
596C42—21no—
597C42—22no—
598C42—44no—
599C42—44no—
600C42—44no—
601C42—82no—
602C42—82no—
603C42#13714yes—
604C42C2×#4112yes—
605C42H(5,𝔽2)12yes—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—21no—
619C23#53212yes—
620C23C23×D411yescharacteristic
621Q8—48no—
622Q8—44no—
623Q8—44no—
624Q8—84no—
625Q8—84no—
626D4—44no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—24no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—164no—
687C2×C4#53212yes—
688C2×C4#54612yes—
689C2×C4#54612yes—
690C2×C4C2×#13712yes—
691C2×C4C2×#13712yes—
692C2×C4C2×#13712yes—
693C2×C4#57512yes—
694C2×C4#57712yes—
695C2×C4#57712yes—
696C2×C4C22×#4111yescharacteristic
697C2×C4C22×#4112yes—
698C2×C4C2×H(5,𝔽2)11yescharacteristic
699C2×C4C2×2−1+412yes—
700C8—24no—
701C8—24no—
702C8—24no—
703C8—42no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22C2×#53211yescharacteristic
719C22C2×#54611yescharacteristic
720C22#349911yescharacteristic
721C22C22×#13711yescharacteristic
722C22C2×#57511yescharacteristic
723C22C2×#57711yescharacteristic
724C22C2×#57711yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4#332614yes—
744C4#333114yes—
745C4#348212yes—
746C4#352012yes—
747C2—42no—
748C2—81no—
749C2#3637611yescharacteristic
750C2#6070111yescharacteristic
751C2#6071211yescharacteristic
752C2C2×#332611yescharacteristic
753C2C2×#333111yescharacteristic
754C2#6247811yescharacteristic
755C2#6248311yescharacteristic
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
3. #33278C2semidirect
4. #33212C2semidirect
5. #33049C2semidirect
6. #29176C2semidirect
7. #28953C2semidirect
8. #28656C2semidirect
9. #28075C2semidirect
10. #27905C2semidirect
11. #27875C2semidirect
12. #25812C2semidirect
13. #25672C2semidirect
15. #22945C2semidirect
16. C4×#498C2semidirect
17. #21409C2semidirect
18. #21322C2semidirect

Maximal subgroups

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

subgroupindex
#361892
#361512
#332782
#332122
#330492
#291762
#289532
#286562
#280752
#279052
#278752
#258122
#256722
#252622
#229452
C4×#4982
#214092
#213222
C4×#4362