Finite Group Databasebeta

← #6495047

Subgroups — #6495047

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35283C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#33314C211yesmaximal, characteristic, complemented
4#33313C211yesmaximal, characteristic, complemented
5#33293C211yesmaximal, characteristic, complemented
6#33289C211yesmaximal, characteristic, complemented
7#30315C214yesmaximal, complemented
8#29217C214yesmaximal, complemented
9#28689C211yesmaximal, characteristic, complemented
10#28688C211yesmaximal, characteristic, complemented
11#28688C211yesmaximal, characteristic, complemented
12#28687C211yesmaximal, characteristic, complemented
13#27959C214yesmaximal
14#25813C211yesmaximal, characteristic
15#25812C211yesmaximal, characteristic
16#25812C211yesmaximal, characteristic
17#25811C211yesmaximal, characteristic
18#21992C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#500C2211yescharacteristic
24C2×#500C2211yescharacteristic
25C2×#500C2211yescharacteristic
26C2×#498C2211yescharacteristic
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#498C2211yescharacteristic
30C2×#433C2212yes—
31#2868C2211yescharacteristic
32#2868C2211yescharacteristic
33#2868C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2211yescharacteristic
36#2862C2211yescharacteristic
37#2857C2211yescharacteristic
38#2857C2211yescharacteristic
39#2660C2212yes—
40#2660C2212yes—
41#2647C2212yes—
42#2628C2212yes—
43#2567—21no—
44#2567—21no—
45#2546—21no—
46#2546—21no—
47#2511—21no—
48#2511—21no—
49#2502—21no—
50#2502—21no—
51#2492—21no—
52#2492—21no—
53#2451—21no—
54#2451—21no—
55#2434—21no—
56#2434—21no—
57#2423—21no—
58#2423—21no—
59#2331C2212yes—
60#2317C2212yes—
61#2317C2212yes—
62#2314C2212yes—
63C4×C4⋊Q8C2211yescharacteristic
64C4×C4⋊Q8C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2085C2214yes—
67#2085C2214yes—
68#2084C2214yes—
69#2083C2214yes—
70#2059C2214yes—
71#2059C2214yes—
72#2058C2214yes—
73#2057C2214yes—
74#1993C2214yes—
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1990C2211yescharacteristic
79#1990C2214yes—
80#1990C2211yescharacteristic
81#1990C2211yescharacteristic
82#1990C2211yescharacteristic
83#1990C2211yescharacteristic
84#1949C2214yes—
85#1949C2214yes—
86#1948C2214yes—
87#1947C2214yes—
88#1854C2214yes—
89#1854C2214yes—
90#1853C2214yes—
91#1852C2214yes—
92#1847C2214yes—
93#1847C2214yes—
94#1846C2214yes—
95#1846C2214yes—
96#1836C2214yes—
97#1834C2214yes—
98#1768C2212yes—
99C4×C4⋊C8C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108#533—42no—
109#533—42no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—42no—
119C2×C4⋊Q8—42no—
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2311yescharacteristic
127C2×C4⋊Q8C2312yes—
128C2×#125C2312yes—
129C2×C22⋊Q8—21no—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×Q8—21no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×#117C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C42C2311yescharacteristic
143#500—42no—
144#500—42no—
145#500—42no—
146#500—42no—
147#498—42no—
148#498—42no—
149#498—42no—
150#498—42no—
151#433Q8116yescomplemented
152C2×#107C2314yes—
153C2×#107C2314yes—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×#106C2314yes—
158C2×#106C2314yes—
159C2×#106C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C2×C22⋊C8C2312yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2312yes—
172#400—22no—
173#400—22no—
174#400—22no—
175#399—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#398—22no—
180#398—22no—
181#398—22no—
182#395—22no—
183#395—22no—
184#395—22no—
185#395—22no—
186#395—22no—
187#393—22no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391—21no—
192#391—21no—
193#391—21no—
194#391—21no—
195#391—21no—
196#391—21no—
197#391—21no—
198#391—21no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—22no—
205#391—22no—
206#389—22no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#387—22no—
224#385—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384—21no—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2312yes—
237#384C2312yes—
238#384C2312yes—
239#384C2312yes—
240#384C2312yes—
241#382C2312yes—
242#382C2312yes—
243#382C2312yes—
244#382C2312yes—
245#382C2312yes—
246#382C2312yes—
247#380—21no—
248#380—21no—
249#380C2312yes—
250#380C2312yes—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4D411yescharacteristic
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2312yes—
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4D411yescharacteristic
263#376C2311yescharacteristic
264#376C2311yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#340C2314yes—
273#340C2314yes—
274#340C2314yes—
275#336C2312yes—
276#336C2312yes—
277#336C2312yes—
278#336C2312yes—
279C22×Q8—21no—
280C22×Q8—21no—
281C22×Q8—21no—
282C22×Q8—21no—
283C22×Q8—21no—
284C22×Q8—21no—
285C22×Q8—22no—
286C22×Q8—22no—
287C22×Q8—44no—
288C22×Q8—82no—
289C22×Q8—82no—
290C23×C4—22no—
291C23×C4C2×D411yescharacteristic
292C23×C4C2×D411yescharacteristic
293C23×C4C2411yescharacteristic
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—28no—
311C4⋊Q8—42no—
312C4⋊Q8—42no—
313C4⋊Q8—44no—
314C4⋊Q8—44no—
315C4⋊Q8—82no—
316C4⋊Q8—82no—
317C4⋊Q8—82no—
318C4⋊Q8—82no—
319C4⋊Q8—82no—
320C4⋊Q8—82no—
321#125—42no—
322#125—42no—
323C22⋊Q8—44no—
324C22⋊Q8—44no—
325C22⋊Q8—82no—
326C22⋊Q8—82no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—82no—
330C4×Q8—82no—
331#117—24no—
332#117—42no—
333#117—42no—
334#117C2×Q818yes—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
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—22no—
364C2×C4⋊C4—42no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
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×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—44no—
408#107—44no—
409#107—44no—
410#107—44no—
411#106—44no—
412#106—44no—
413#106—44no—
414#106—44no—
415C4⋊C8C2×Q818yescomplemented
416C4⋊C8C2×Q818yescomplemented
417C22⋊C8C2×Q818yes—
418C22⋊C8C2×Q818yes—
419#97—24no—
420#97C2×Q818yescomplemented
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—24no—
426C4×C8C2×Q818yescomplemented
427#95—22no—
428#95—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#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455C24C22×D411yescharacteristic
456C2×Q8—44no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8—44no—
464C2×Q8—42no—
465C2×Q8—42no—
466C2×Q8—42no—
467C2×Q8—42no—
468C2×Q8—42no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—42no—
476C2×Q8—82no—
477C2×Q8—82no—
478C2×Q8—82no—
479C2×Q8—82no—
480C2×Q8—816no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—44no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
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×Q814yes—
515C22×C4C2511yescharacteristic
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8C22×Q814yes—
525C2×C8C22×Q814yes—
526C2×C8C22×Q814yes—
527C2×C8C22×Q814yes—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C4⋊C4—82no—
573C4⋊C4C4⋊Q814yes—
574C4⋊C4C4⋊Q814yes—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—44no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4C4⋊Q814yes—
581C22⋊C4C4⋊Q814yes—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—21no—
593C42—21no—
594C42—22no—
595C42—44no—
596C42—82no—
597C42—82no—
598C42C22×Q814yes—
599C42C22×Q814yes—
600C42H(5,𝔽2)12yes—
601C42H(5,𝔽2)12yes—
602C422−1+414yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23C2×C4⋊Q812yes—
610C23C23×D411yescharacteristic
611Q8—88no—
612Q8—84no—
613Q8—84no—
614Q8—84no—
615Q8—84no—
616Q8—84no—
617Q8—84no—
618Q8—164no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
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—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
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—88no—
674C2×C4C2×C4⋊Q812yes—
675C2×C4C2×C4⋊Q812yes—
676C2×C4C2×C4⋊Q812yes—
677C2×C4C2×C4⋊Q812yes—
678C2×C4C2×C4⋊Q812yes—
679C2×C4#53512yes—
680C2×C4#53512yes—
681C2×C4#53612yes—
682C2×C4#53612yes—
683C2×C4#57612yes—
684C2×C4#57712yes—
685C2×C4#57712yes—
686C2×C4#57812yes—
687C2×C4C23×Q812yes—
688C2×C4C2×H(5,𝔽2)11yescharacteristic
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C2×C4C2×2−1+412yes—
691C8—24no—
692C8—24no—
693C8—24no—
694C8—24no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22C22×C4⋊Q811yescharacteristic
703C22C2×#53511yescharacteristic
704C22C2×#53611yescharacteristic
705C22C2×#57611yescharacteristic
706C22C2×#57711yescharacteristic
707C22C2×#57711yescharacteristic
708C22C2×#57811yescharacteristic
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—22no—
715C4—22no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#322414yes—
729C4#322514yes—
730C4#350312yes—
731C4#350312yes—
732C2—42no—
733C2#3528111yescharacteristic
734C2#6066511yescharacteristic
735C2#6066711yescharacteristic
736C2C2×#322411yescharacteristic
737C2C2×#322511yescharacteristic
738C2#6087711yescharacteristic
739C2#6087811yescharacteristic
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. #33314C2semidirect
4. #33313C2semidirect
5. #33293C2semidirect
6. #33289C2semidirect
7. #30315C2semidirect
8. #29217C2semidirect
9. #28689C2semidirect
10. #28688C2semidirect
11. #28688C2semidirect
12. #28687C2semidirect
151. #433Q8semidirect
415. C4⋊C8C2×Q8semidirect
416. C4⋊C8C2×Q8semidirect
420. #97C2×Q8semidirect
426. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#352832
#352812
#333142
#333132
#332932
#332892
#303152
#292172
#286892
#286882
#286872
#279592
#258132
#258122
#258112
#219922
C4×#4332