Finite Group Databasebeta

← #6806089

Subgroups — #6806089

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 756 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic, complemented
3#33226C211yesmaximal, characteristic, complemented
4#31646C214yesmaximal, complemented
5#31643C214yesmaximal, complemented
6#30205C211yesmaximal, characteristic, complemented
7#30204C211yesmaximal, characteristic, complemented
8#29220C211yesmaximal, characteristic, complemented
9#29144C211yesmaximal, characteristic, complemented
10#29139C211yesmaximal, characteristic, complemented
11#28615C211yesmaximal, characteristic
12#28613C211yesmaximal, characteristic, complemented
13#28443C211yesmaximal, characteristic
14#25938C214yesmaximal, complemented
15#25327C214yesmaximal, complemented
16#22269C211yesmaximal, characteristic
17#22268C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic, complemented
19#21339C211yesmaximal, characteristic, complemented
20C2×#554C2212yes—
21C2×#493C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#488C2214yes—
24C2×#486C2211yescharacteristic
25C2×#463C2214yes—
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2592—21no—
33#2592C2212yes—
34#2580—21no—
35#2566C2212yes—
36#2566C2212yes—
37#2563C2212yes—
38#2538—21no—
39#2538—21no—
40#2513—21no—
41#2513—21no—
42#2452—21no—
43#2452—21no—
44#2452—21no—
45#2452—21no—
46#2448—21no—
47#2448C2212yes—
48#2447—21no—
49#2447C2212yes—
50#2367—21no—
51#2367C2212yes—
52#2357—21no—
53D4×C4⋊C4—21no—
54D4×C4⋊C4—21no—
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×#127C2211yescharacteristic
58C4×#127C2211yescharacteristic
59C4×#127C2211yescharacteristic
60C4×#127C2211yescharacteristic
61C42×Q8C2211yescharacteristic
62#2101C2214yes—
63#2073C2214yes—
64#2072C2214yes—
65#2055C2214yes—
66#2054C2214yes—
67#2026C2214yes—
68#2025C2214yes—
69#2024C2214yes—
70#2023C2214yes—
71#2012C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1989C2214yes—
76#1984C2212yes—
77#1958C2214yes—
78#1957C2211yescharacteristic
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1957C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1939C2212yes—
86#1934C2214yes—
87#1916C2212yes—
88#1916C2212yes—
89#1880C2214yes—
90#1879C2214yes—
91#1879C2214yes—
92#1879C2214yes—
93#1877C2214yes—
94#1877C2214yes—
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1805C2214yes—
99#1802C2214yes—
100C4×C4⋊C8C2211yescharacteristic
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×#102C2214yes—
104#1744C2211yescharacteristic
105#1744C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×D8C2314yes—
109#554—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#127C2311yescharacteristic
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×#124—21no—
120C2×#124—21no—
121C2×#124C2312yes—
122C2×#124C2312yes—
123C2×C22⋊Q8—22no—
124C2×C22⋊Q8—22no—
125C2×C22⋊Q8—22no—
126C2×C4×Q8—21no—
127C2×C4×Q8D411yescharacteristic
128C2×C4×Q8D411yescharacteristic
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C22×C4⋊C4—22no—
144#493—24no—
145#492—24no—
146#488—48no—
147#486—24no—
148#463—48no—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2312yes—
166#399—22no—
167#397—22no—
168#397—22no—
169#397—22no—
170#397—22no—
171#396—22no—
172#396—22no—
173#396—22no—
174#396—22no—
175#393—22no—
176#393—22no—
177#393—22no—
178#391—21no—
179#391—21no—
180#391—21no—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#390—21no—
185#390—21no—
186#390—21no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390C2312yes—
199#390C2312yes—
200#390C2312yes—
201#390C2312yes—
202#390C2312yes—
203#390C2312yes—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389C2312yes—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#388—22no—
221#388—22no—
222#385—22no—
223#385—22no—
224#385—22no—
225#385—22no—
226#385—22no—
227#385—22no—
228#384—21no—
229#384—21no—
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2311yescharacteristic
234#384C2312yes—
235#382—21no—
236#382—21no—
237#382C2312yes—
238#382C2312yes—
239#381—22no—
240#380—22no—
241#380—22no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4D411yescharacteristic
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4C2312yes—
252C4×C4⋊C4C2311yescharacteristic
253C4×C4⋊C4C2311yescharacteristic
254C4×C4⋊C4C2311yescharacteristic
255C4×C22⋊C4—22no—
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258C43C2311yescharacteristic
259#340C2314yes—
260#340C2314yes—
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×D4—21no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C22×D4C2412yes—
282C23×C4—22no—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4—22no—
286C2×SD16—28no—
287C2×SD16—44no—
288C2×SD16—44no—
289C2×D8—28no—
290C2×D8—44no—
291C2×D8—44no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—44no—
301C4⋊Q8—82no—
302C4⋊Q8—82no—
303#127—22no—
304#127—22no—
305#127—22no—
306#127#4114yes—
307#125—28no—
308#125—44no—
309#124—28no—
310#124—44no—
311#124—44no—
312#124—82no—
313C22⋊Q8—48no—
314C22⋊Q8—48no—
315C22⋊Q8—48no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—44no—
320C4×Q8—82no—
321C4×D4—28no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—82no—
329C4×D4—82no—
330C4×D4—82no—
331C4×D4—82no—
332C4×D4—82no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
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—48no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
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⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C4⋊C8—24no—
410C4⋊C8—42no—
411C4⋊C8—42no—
412#103—44no—
413#103—44no—
414#103—44no—
415#102—28no—
416#102—44no—
417#102—44no—
418#102—44no—
419#102—44no—
420C4×C8—22no—
421C4×C8—22no—
422C4×C8—22no—
423C4×C8—24no—
424C4×C8—22no—
425C4×C8—42no—
426#95—22no—
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#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444C24—22no—
445C24—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—41no—
453C2×Q8—44no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4C2×#4114yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
512SD16—88no—
513D8—88no—
514C2×C8—22no—
515C2×C8—22no—
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×C8—22no—
525C2×C8—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—82no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—22no—
596C42—22no—
597C42—82no—
598C42—82no—
599C42#13514yes—
600C42C2×#4112yes—
601C42H(5,𝔽2)14yes—
602C422−1+412yes—
603C23—24no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23C23×D411yescharacteristic
618Q8—44no—
619Q8—44no—
620Q8—44no—
621Q8—84no—
622Q8—164no—
623D4—28no—
624D4—48no—
625D4—48no—
626D4—44no—
627D4—44no—
628D4—44no—
629D4—84no—
630D4—84no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
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—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—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
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—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—88no—
682C2×C4—164no—
683C2×C4—164no—
684C2×C4—164no—
685C2×C4#53512yes—
686C2×C4#53512yes—
687C2×C4#54812yes—
688C2×C4#54812yes—
689C2×C4C2×#13512yes—
690C2×C4C2×#13512yes—
691C2×C4C2×#13512yes—
692C2×C4#57512yes—
693C2×C4#57612yes—
694C2×C4#57712yes—
695C2×C4C22×#4111yescharacteristic
696C2×C4C2×H(5,𝔽2)12yes—
697C2×C4C2×2−1+411yescharacteristic
698C2×C4#58512yes—
699C8—42no—
700C8—42no—
701C8—42no—
702C8—42no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22C2×#53511yescharacteristic
718C22C2×#54811yescharacteristic
719C22#350011yescharacteristic
720C22C22×#13511yescharacteristic
721C22C2×#57511yescharacteristic
722C22C2×#57611yescharacteristic
723C22C2×#57711yescharacteristic
724C4—22no—
725C4—22no—
726C4—22no—
727C4—22no—
728C4—41no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4#336014yes—
743C4#337014yes—
744C4#351412yes—
745C4#355412yes—
746C2—84no—
747C2—162no—
748C2#3917111yescharacteristic
749C2#6078211yescharacteristic
750C2#6082211yescharacteristic
751C2C2×#336011yescharacteristic
752C2C2×#337011yescharacteristic
753C2#6252711yescharacteristic
754C2#6254311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
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. #36609C2semidirect
2. #36559C2semidirect
3. #33226C2semidirect
4. #31646C2semidirect
5. #31643C2semidirect
6. #30205C2semidirect
7. #30204C2semidirect
8. #29220C2semidirect
9. #29144C2semidirect
10. #29139C2semidirect
12. #28613C2semidirect
14. #25938C2semidirect
15. #25327C2semidirect
17. #22268C2semidirect
18. C4×#486C2semidirect
19. #21339C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#365592
#332262
#316462
#316432
#302052
#302042
#292202
#291442
#291392
#286152
#286132
#284432
#259382
#253272
#222692
#222682
C4×#4862
#213392