Finite Group Databasebeta

← #6804370

Subgroups — #6804370

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic, complemented
3#33278C211yesmaximal, characteristic, complemented
4#32609C214yesmaximal, complemented
5#31163C214yesmaximal, complemented
6#30219C211yesmaximal, characteristic
7#30203C211yesmaximal, characteristic
8#29190C211yesmaximal, characteristic
9#29177C211yesmaximal, characteristic, complemented
10#28616C211yesmaximal, characteristic, complemented
11#28614C211yesmaximal, characteristic
12#28512C211yesmaximal, characteristic, complemented
13#28444C211yesmaximal, characteristic
14#26898C214yesmaximal, complemented
15#26161C214yesmaximal, complemented
16#22270C211yesmaximal, characteristic, complemented
17#22218C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#494C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#475C2214yes—
24C2×#464C2214yes—
25C2×C4×SD16C2214yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2592—21no—
33#2592C2212yes—
34#2582C2212yes—
35#2580—21no—
36#2567—21no—
37#2552—21no—
38#2550—21no—
39#2550—21no—
40#2513—21no—
41#2513—21no—
42#2507—21no—
43#2507—21no—
44#2451C2212yes—
45#2451C2212yes—
46#2448—21no—
47#2448C2212yes—
48#2447—21no—
49#2447C2212yes—
50#2369—21no—
51#2367—21no—
52#2367C2212yes—
53#2357—21no—
54#2355—21no—
55Q8×C4⋊C4C2212yes—
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C42×Q8C2211yescharacteristic
63#2096C2214yes—
64#2088C2214yes—
65#2071C2214yes—
66#2065C2214yes—
67#2056C2214yes—
68#2055C2214yes—
69#2042C2214yes—
70#2042C2214yes—
71#2023C2214yes—
72#2015C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1984C2212yes—
77#1968C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1939C2212yes—
86#1933C2214yes—
87#1933C2214yes—
88#1927C2214yes—
89#1919C2214yes—
90#1916C2212yes—
91#1916C2212yes—
92#1880C2214yes—
93#1878C2214yes—
94#1878C2214yes—
95#1878C2214yes—
96#1865C2214yes—
97#1864C2214yes—
98#1844C2211yescharacteristic
99#1844C2211yescharacteristic
100#1834C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C4⋊C8C2211yescharacteristic
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125—21no—
117C2×#125—21no—
118C2×#124—21no—
119C2×#124—21no—
120C2×#124C2312yes—
121C2×#124C2312yes—
122C2×C4×Q8—21no—
123C2×C4×Q8D411yescharacteristic
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2311yescharacteristic
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4C2312yes—
133#494—24no—
134#492—24no—
135#487—24no—
136#475—48no—
137#464—48no—
138C4×SD16—48no—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2312yes—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#397—22no—
164#397—22no—
165#397—22no—
166#397—22no—
167#395—22no—
168#395—22no—
169#393—22no—
170#391—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—21no—
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—21no—
205#389—21no—
206#389—21no—
207#389—22no—
208#389—22no—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#385—22no—
223#385—22no—
224#384—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#381—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4C2312yes—
254C4×C4⋊C4C2311yescharacteristic
255C4×C4⋊C4C2311yescharacteristic
256C4×C4⋊C4C2311yescharacteristic
257C4×C22⋊C4—22no—
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—22no—
274C22×Q8C2×D411yescharacteristic
275C22×Q8C2412yes—
276C22×Q8C2412yes—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C23×C4—22no—
281C2×Q16—28no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×SD16—28no—
285C2×SD16—28no—
286C2×SD16—44no—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2412yes—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—44no—
297C4⋊Q8—44no—
298C4⋊Q8—82no—
299C4⋊Q8#4114yes—
300#127—22no—
301#127—22no—
302#125—82no—
303#125—82no—
304#124—44no—
305#124—44no—
306#124—44no—
307#124—82no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—42no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—82no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—82no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
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
392C2×C42C2411yescharacteristic
393#106—28no—
394#106—44no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—42no—
398#103—28no—
399#103—28no—
400#103—44no—
401#103—44no—
402#103—44no—
403#102—44no—
404#102—44no—
405#102—44no—
406C4×C8—24no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—42no—
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
436C24—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—44no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8C2×#4114yes—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—42no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—82no—
473C22×C4—82no—
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×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489Q16—88no—
490SD16—416no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8H(5,𝔽2)14yes—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4—44no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
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⋊C4—82no—
542C4⋊C4—82no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—44no—
571C42—82no—
572C42—82no—
573C42—82no—
574C42—82no—
575C42#13514yes—
576C42#13714yes—
577C42C2×#4112yes—
578C422−1+412yes—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—44no—
592Q8—84no—
593Q8—164no—
594D4—44no—
595D4—44no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—24no—
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—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—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—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—164no—
652C2×C4#54812yes—
653C2×C4#54812yes—
654C2×C4C2×#13512yes—
655C2×C4C2×#13512yes—
656C2×C4C2×#13512yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4#57712yes—
661C2×C4#57712yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C2×H(5,𝔽2)12yes—
664C2×C4C2×2−1+411yescharacteristic
665C2×C4#58512yes—
666C8—24no—
667C8—42no—
668C8—42no—
669C8—42no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22#344911yescharacteristic
678C22C2×#54811yescharacteristic
679C22#349811yescharacteristic
680C22C22×#13511yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57711yescharacteristic
683C22C2×#57711yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#336514yes—
704C4#336914yes—
705C4#351312yes—
706C4#355412yes—
707C2—162no—
708C2#3916111yescharacteristic
709C2#6094711yescharacteristic
710C2#6101511yescharacteristic
711C2C2×#336511yescharacteristic
712C2C2×#336911yescharacteristic
713C2#6241411yescharacteristic
714C2#6242211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #36563C2semidirect
3. #33278C2semidirect
4. #32609C2semidirect
5. #31163C2semidirect
9. #29177C2semidirect
10. #28616C2semidirect
12. #28512C2semidirect
14. #26898C2semidirect
15. #26161C2semidirect
16. #22270C2semidirect
17. #22218C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365632
#332782
#326092
#311632
#302192
#302032
#291902
#291772
#286162
#286142
#285122
#284442
#268982
#261612
#222702
#222182
C4×#4922
#213392