Finite Group Databasebeta

← #6854255

Subgroups — #6854255

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36574C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic
3#33311C211yesmaximal, characteristic, complemented
4#32354C214yesmaximal, complemented
5#32351C214yesmaximal, complemented
6#30307C211yesmaximal, characteristic, complemented
7#30304C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29207C211yesmaximal, characteristic, complemented
10#29206C211yesmaximal, characteristic, complemented
11#29190C211yesmaximal, characteristic
12#29190C211yesmaximal, characteristic
13#28514C211yesmaximal, characteristic
14#25843C214yesmaximal, complemented
15#25458C214yesmaximal, complemented
16#22220C211yesmaximal, characteristic
17#22220C211yesmaximal, characteristic
18C4×#486C211yesmaximal, characteristic
19#21339C211yesmaximal, characteristic, complemented
20C2×#554C2212yes—
21C2×#499C2214yes—
22C2×#486C2211yescharacteristic
23C2×#478C2214yes—
24#2868C2211yescharacteristic
25#2859C2211yescharacteristic
26#2859C2211yescharacteristic
27#2859C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2584—21no—
31#2584C2212yes—
32#2581—21no—
33#2566C2212yes—
34#2566C2212yes—
35#2563C2212yes—
36#2538—21no—
37#2538—21no—
38#2509—21no—
39#2509C2212yes—
40#2508—21no—
41#2508C2212yes—
42#2453—21no—
43#2453—21no—
44#2452—21no—
45#2452—21no—
46#2452—21no—
47#2452—21no—
48#2370—21no—
49#2370C2212yes—
50#2351—21no—
51D4×C4⋊C4—21no—
52D4×C4⋊C4—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×#127C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C42×Q8C2211yescharacteristic
60#2101C2214yes—
61#2092C2214yes—
62#2085C2214yes—
63#2083C2214yes—
64#2065C2214yes—
65#2065C2214yes—
66#2059C2214yes—
67#2059C2214yes—
68#2058C2214yes—
69#2057C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1989C2214yes—
74#1984C2212yes—
75#1957C2211yescharacteristic
76#1957C2211yescharacteristic
77#1957C2211yescharacteristic
78#1949C2214yes—
79#1948C2214yes—
80#1939C2212yes—
81#1934C2214yes—
82#1925C2214yes—
83#1925C2214yes—
84#1916C2212yes—
85#1916C2212yes—
86#1864C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1862C2214yes—
90#1846C2214yes—
91#1845C2211yescharacteristic
92#1845C2211yescharacteristic
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107#554—48no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2312yes—
112C2×#127C2311yescharacteristic
113C2×#125—21no—
114C2×#125—21no—
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#125C2312yes—
118C2×C22⋊Q8—22no—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8D411yescharacteristic
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8C2311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C22×C4⋊C4—22no—
137#499—48no—
138#486Q818yescomplemented
139#478—48no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106C2314yes—
144C2×#106C2314yes—
145C2×#106C2314yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158C2×C4×C8C2311yescharacteristic
159#400—22no—
160#400—22no—
161#400—22no—
162#398—22no—
163#398—22no—
164#396—22no—
165#396—22no—
166#396—22no—
167#396—22no—
168#395—22no—
169#393—22no—
170#393—22no—
171#391—21no—
172#391—21no—
173#391—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
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#389C2312yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#385—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#384—21no—
222#384—21no—
223#384C2311yescharacteristic
224#384C2311yescharacteristic
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382—22no—
233#382C2312yes—
234#382C2312yes—
235#382C2312yes—
236#380—22no—
237#380—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270C22×Q8—21no—
271C22×Q8—22no—
272C22×Q8C2×D411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2412yes—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—22no—
279C22×C8C2412yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8—44no—
285C4⋊Q8—82no—
286C4⋊Q8—82no—
287C4⋊Q8C2×Q814yes—
288#127—22no—
289#127C2×Q814yes—
290#125—28no—
291#125—28no—
292#125—44no—
293#125—44no—
294#125—82no—
295C22⋊Q8—48no—
296C22⋊Q8—48no—
297C22⋊Q8—48no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×Q8—44no—
302C4×Q8—82no—
303C4×Q8—82no—
304C4×D4—28no—
305C4×D4—44no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—44no—
309C4×D4—82no—
310C4×D4—82no—
311C4×D4—82no—
312C4×D4—82no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
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—21no—
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—48no—
343C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
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×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390#107—44no—
391#107—44no—
392#107—44no—
393#106—44no—
394#106—44no—
395#106—44no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399#102—28no—
400#102—44no—
401#102—44no—
402#102C2×Q8116yes—
403C4×C8—24no—
404C4×C8—24no—
405C4×C8—22no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8C2×Q814yes—
409#95—22no—
410#95—22no—
411#95—22no—
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#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431C24—22no—
432C2×Q8—22no—
433C2×Q8—24no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—44no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—82no—
440C2×Q8—82no—
441C2×D4—24no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4C22×Q814yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8C22×Q814yes—
493C2×C82−1+414yes—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
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⋊C4C22×Q814yes—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—82no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—82no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—44no—
566C42—82no—
567C42—82no—
568C42#13514yes—
569C42C22×Q812yes—
570C42H(5,𝔽2)14yes—
571C422−1+412yes—
572C23—24no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23C23×D411yescharacteristic
580Q8—44no—
581Q8—48no—
582Q8—164no—
583D4—28no—
584D4—48no—
585D4—48no—
586D4—44no—
587C2×C4—21no—
588C2×C4—21no—
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—24no—
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—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—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
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—88no—
641C2×C4—164no—
642C2×C4—164no—
643C2×C4#53512yes—
644C2×C4#53512yes—
645C2×C4D4×Q812yes—
646C2×C4D4×Q812yes—
647C2×C4C2×#13512yes—
648C2×C4C2×#13512yes—
649C2×C4C2×#13512yes—
650C2×C4#57512yes—
651C2×C4#57612yes—
652C2×C4#57712yes—
653C2×C4C23×Q811yescharacteristic
654C2×C4C2×H(5,𝔽2)12yes—
655C2×C4C2×2−1+412yes—
656C2×C4C2×2−1+411yescharacteristic
657C8—24no—
658C8—24no—
659C8—42no—
660C8—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22C2×#53511yescharacteristic
669C22C2×D4×Q811yescharacteristic
670C22#349911yescharacteristic
671C22C22×#13511yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57611yescharacteristic
674C22C2×#57711yescharacteristic
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—44no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4#338014yes—
695C4#338814yes—
696C4#350312yes—
697C4#355212yes—
698C2—84no—
699C2#3961611yescharacteristic
700C2#6078211yescharacteristic
701C2#6082211yescharacteristic
702C2C2×#338011yescharacteristic
703C2C2×#338811yescharacteristic
704C2#6247011yescharacteristic
705C2#6248111yescharacteristic
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
1. #36574C2semidirect
3. #33311C2semidirect
4. #32354C2semidirect
5. #32351C2semidirect
6. #30307C2semidirect
7. #30304C2semidirect
8. #29220C2semidirect
9. #29207C2semidirect
10. #29206C2semidirect
14. #25843C2semidirect
15. #25458C2semidirect
19. #21339C2semidirect
138. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#365742
#365592
#333112
#323542
#323512
#303072
#303042
#292202
#292072
#292062
#291902
#285142
#258432
#254582
#222202
C4×#4862
#213392