Finite Group Databasebeta

← #6852259

Subgroups — #6852259

Order 512.

1739 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36561C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3#32954C211yesmaximal, characteristic, complemented
4#32346C214yesmaximal, complemented
5#31522C214yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#28530C212yesmaximal, complemented
8#27948C212yesmaximal, complemented
9#27815C212yesmaximal, complemented
10#27678C212yesmaximal
11#25811C211yesmaximal, characteristic, complemented
12#25453C212yesmaximal, complemented
13#24815C212yesmaximal, complemented
14#24299C211yesmaximal, characteristic
15#22625C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C2×#554—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#498C2212yes—
22C2×#475C2212yes—
23C2×#476C2212yes—
24C2×#448C2211yescharacteristic
25C2×C4×SD16C2212yes—
26#2857C2212yes—
27#2857C2211yescharacteristic
28#2567—21no—
29#2567—21no—
30#2567—22no—
31#2567C2214yes—
32#2567C2212yes—
33#2567C2212yes—
34#2564—22no—
35#2556—22no—
36#2551C2211yescharacteristic
37#2542C2211yescharacteristic
38#2506C2212yes—
39#2492—21no—
40#2452C2212yes—
41#2417—21no—
42C2×#395—22no—
43#2365C2211yescharacteristic
44#2360—21no—
45D4×C4⋊C4C2211yescharacteristic
46C4×#121C2212yes—
47C4×#121C2211yescharacteristic
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389—21no—
51C2×#384C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53#2100C2214yes—
54#2096C2214yes—
55#2094C2214yes—
56#2084C2214yes—
57#2084C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2058C2214yes—
61#2058C2212yes—
62#2024C2214yes—
63#2023C2212yes—
64#1990C2214yes—
65#1990C2212yes—
66#1987C2212yes—
67#1968C2214yes—
68#1948C2214yes—
69#1939C2212yes—
70#1933C2214yes—
71#1927C2214yes—
72#1916C2214yes—
73#1880C2212yes—
74#1877C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1865C2214yes—
78#1865C2212yes—
79#1853C2212yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1853C2212yes—
84#1844C2214yes—
85#1803C2212yes—
86#1802C2214yes—
87#1797C2212yes—
88#1797C2211yescharacteristic
89#1787C2212yes—
90#1787C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98#554—48no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—44no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#125—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—21no—
113C2×#121C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2312yes—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C42—21no—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#498—44no—
137#476—44no—
138#475—44no—
139#448Q818yescomplemented
140C4×SD16—44no—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#102C2312yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2311yescharacteristic
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2311yescharacteristic
158C2×C4×C8C2312yes—
159C2×C4×C8C2314yes—
160#400—22no—
161#399—22no—
162#398—22no—
163#396—22no—
164#395—22no—
165#395—24no—
166#395—24no—
167#395—24no—
168#395—24no—
169#393—22no—
170#391—21no—
171#391—22no—
172#391—22no—
173#391—21no—
174#391—22no—
175#391C2314yes—
176#391C2314yes—
177#390C2312yes—
178#390C2311yescharacteristic
179#389—22no—
180#389—24no—
181#389—22no—
182#389—24no—
183#389—22no—
184#389—22no—
185#389—21no—
186#389—22no—
187#389—21no—
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#388—22no—
194#388—21no—
195#387—22no—
196#387—22no—
197#386—21no—
198#386—22no—
199#385—22no—
200#385—22no—
201#384—22no—
202#384—22no—
203#384C2314yes—
204#384C2314yes—
205#384C2311yescharacteristic
206#384C2312yes—
207#380—21no—
208#380—22no—
209#380—21no—
210#380C2312yes—
211#380C2311yescharacteristic
212C4×C4⋊C4D411yescharacteristic
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D412yes—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—21no—
221C4×C22⋊C4—22no—
222C2×#95—22no—
223C2×#95—22no—
224C2×#95D412yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234#336C2312yes—
235#336C2314yes—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—42no—
243C22×Q8—42no—
244C22×Q8—44no—
245C22×Q8—82no—
246C22×Q8—82no—
247C22×Q8C2×Q814yes—
248C22×Q8C2412yes—
249C22×D4—21no—
250C22×D4C2411yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—22no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D412yes—
260C23×C4C2411yescharacteristic
261C2×SD16—42no—
262C2×SD16—42no—
263C2×SD16C2×Q818yescomplemented
264C22×C8C2412yes—
265C22×C8C2414yes—
266C22×C8C2412yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—24no—
270C4⋊Q8—44no—
271C4⋊Q8—44no—
272C4⋊Q8—44no—
273C4⋊Q8—44no—
274C4⋊Q8—44no—
275#125—48no—
276C22⋊Q8—48no—
277C22⋊Q8—44no—
278C22⋊Q8—48no—
279C22⋊Q8—44no—
280C22⋊Q8—44no—
281#121—22no—
282#121C2×Q814yes—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—42no—
286C4×Q8—42no—
287C4×Q8—44no—
288C4×Q8—44no—
289C4×Q8—44no—
290C4×D4—24no—
291C4×D4—44no—
292C4×D4—48no—
293C4×D4—44no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—84no—
322C2×C4⋊C4—84no—
323C2×C4⋊C4SD1618yescomplemented
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C22⋊C4—21no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—44no—
352C2×C42—44no—
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D412yes—
356C2×C42C2×D412yes—
357C2×C42C2412yes—
358C2×C42C2411yescharacteristic
359C2×C42C2412yes—
360C2×C42C2411yescharacteristic
361#106—42no—
362#106—44no—
363#106—44no—
364#106—44no—
365#106—44no—
366C4⋊C8—24no—
367C4⋊C8—44no—
368#103—44no—
369#103—42no—
370#103—42no—
371#103—44no—
372#103C2×Q818yescomplemented
373#102—42no—
374#102—44no—
375C22⋊C8—22no—
376C22⋊C8—24no—
377C22⋊C8C2×Q814yes—
378C4×C8—24no—
379C4×C8—44no—
380#95—24no—
381#95—24no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—21no—
386#95—21no—
387#95—22no—
388#95—21no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95C2×D412yes—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397C24—21no—
398C24C22×D411yescharacteristic
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—48no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—48no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—42no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—82no—
420C2×Q8—82no—
421C2×Q8—816no—
422C2×Q8C22×Q814yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4C22×Q812yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—44no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C2×SD1614yes—
466C22×C4C2×SD1618yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D412yes—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×Q812yes—
477C22×C42−1+412yes—
478C22×C4C2511yescharacteristic
479SD16—48no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—24no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—22no—
488C2×C8C22×Q814yes—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4—82no—
521C4⋊C4C2×SD1614yes—
522C4⋊C4C2×SD1618yes—
523C4⋊C4C2×SD1614yes—
524C4⋊C4#13714yes—
525C4⋊C4C22×Q812yes—
526C22⋊C4—24no—
527C22⋊C4—44no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—42no—
532C22⋊C4—44no—
533C22⋊C4—81no—
534C22⋊C4—82no—
535C42—22no—
536C42—22no—
537C42—22no—
538C42—24no—
539C42—24no—
540C42—42no—
541C42—42no—
542C42—44no—
543C42—44no—
544C42—44no—
545C42—42no—
546C42H(5,𝔽2)14yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23D4×Q812yes—
559C23C22×SD1612yes—
560C23#57712yes—
561C23C23×D411yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—48no—
565Q8—42no—
566Q8—42no—
567Q8—88no—
568Q8—88no—
569Q8—84no—
570Q8—164no—
571D4—24no—
572D4—48no—
573D4—42no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—84no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4#53512yes—
628C2×C4#53512yes—
629C2×C4D4×Q812yes—
630C2×C4C22×SD1614yes—
631C2×C4C22×SD1612yes—
632C2×C4C22×SD1614yes—
633C2×C4C22×SD1612yes—
634C2×C4C2×#13714yes—
635C2×C4C2×#13712yes—
636C2×C4C23×Q811yescharacteristic
637C2×C4C2×H(5,𝔽2)12yes—
638C2×C4C2×2−1+411yescharacteristic
639C2×C4#58512yes—
640C8—24no—
641C8—42no—
642C8—44no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22Q8×SD1614yes—
657C22C2×#53511yescharacteristic
658C22C2×D4×Q811yescharacteristic
659C22#348511yescharacteristic
660C22C23×SD1611yescharacteristic
661C22C22×#13711yescharacteristic
662C22C2×#57711yescharacteristic
663C22#358511yescharacteristic
664C4—24no—
665C4—24no—
666C4—24no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—81no—
676C4—81no—
677C4—82no—
678C4—82no—
679C4—84no—
680C4#320614yes—
681C4#350112yes—
682C4#354112yes—
683C2—24no—
684C2—82no—
685C2#3959411yescharacteristic
686C2C2×#320611yescharacteristic
687C2#6081211yescharacteristic
688C2C2×Q8×SD1611yescharacteristic
689C2Q8×#13711yescharacteristic
690C2#6169711yescharacteristic
691C2#6173511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. C2×#2567C2semidirect
3. #32954C2semidirect
4. #32346C2semidirect
5. #31522C2semidirect
6. #29951C2semidirect
7. #28530C2semidirect
8. #27948C2semidirect
9. #27815C2semidirect
11. #25811C2semidirect
12. #25453C2semidirect
13. #24815C2semidirect
15. #22625C2semidirect
139. #448Q8semidirect
263. C2×SD16C2×Q8semidirect
323. C2×C4⋊C4SD16semidirect
372. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365612
C2×#25672
#329542
#323462
#315222
#299512
#285302
#279482
#278152
#276782
#258112
#254532
#248152
#242992
#226252
#220412
C4×#4482