Finite Group Databasebeta

← #6578972

Subgroups — #6578972

Order 512.

2105 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C212yesmaximal, complemented
2#36117C211yesmaximal, characteristic
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#33005C212yesmaximal, complemented
5#32903C211yesmaximal, characteristic
6#27906C211yesmaximal, characteristic
7#27648C211yesmaximal, characteristic, complemented
8#26387C214yesmaximal, complemented
9#26185C214yesmaximal, complemented
10#26043C212yesmaximal, complemented
11#24733C212yesmaximal, complemented
12#24607C212yesmaximal, complemented
13#24067C212yesmaximal, complemented
14#23003C212yesmaximal, complemented
15#22674C212yesmaximal, complemented
16#22603C211yesmaximal, characteristic, complemented
17C4×#451C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#3117—216no—
22C2×#484C2212yes—
23C2×#481C2212yes—
24C2×#481C2212yes—
25C2×#479C2214yes—
26C2×#464C2214yes—
27C2×#460C2212yes—
28C2×#451C2211yescharacteristic
29C2×#448C2212yes—
30C2×#441C2214yes—
31C2×#440C2214yes—
32C2×C4×Q16C2212yes—
33C2×#432C2212yes—
34#2856C2211yescharacteristic
35#2854C2212yes—
36#2552C2211yescharacteristic
37#2548C2211yescharacteristic
38#2512—21no—
39#2501—21no—
40#2479—22no—
41#2454C2212yes—
42#2448C2212yes—
43#2425—22no—
44#2366C2211yescharacteristic
45#2361—21no—
46#2354—21no—
47Q8×C4⋊C4—21no—
48Q8×C4⋊C4—22no—
49Q8×C4⋊C4—21no—
50Q8×C4⋊C4C2214yes—
51Q8×C4⋊C4C2211yescharacteristic
52Q8×C4⋊C4C2212yes—
53Q8×C4⋊C4C2212yes—
54C4×C22⋊Q8C2211yescharacteristic
55C2×#389—22no—
56C2×#389C2212yes—
57C2×#384—21no—
58C2×#384C2211yescharacteristic
59C2×C4×C4⋊C4—21no—
60C2×C4×C4⋊C4C2211yescharacteristic
61#2094C2214yes—
62#2088C2214yes—
63#2083C2212yes—
64#2072C2212yes—
65#2067C2214yes—
66#2059C2214yes—
67#2042C2214yes—
68#1957C2212yes—
69#1947C2212yes—
70#1943C2214yes—
71#1939C2212yes—
72#1936C2214yes—
73#1917C2214yes—
74#1880C2214yes—
75#1880C2214yes—
76#1880C2214yes—
77#1878C2212yes—
78#1874C2212yes—
79#1872C2211yescharacteristic
80#1865C2214yes—
81#1865C2214yes—
82#1865C2214yes—
83#1863C2212yes—
84#1862C2214yes—
85#1852C2211yescharacteristic
86#1844C2212yes—
87#1824C2212yes—
88#1817C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1797C2211yescharacteristic
92#1787C2211yescharacteristic
93#1770C2212yes—
94C4×#103C2212yes—
95C4×C22⋊C8C2211yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C2×#127C2311yescharacteristic
100#529—28no—
101#529—44no—
102C2×#126—22no—
103C2×#125C2312yes—
104C2×#124C2312yes—
105C2×#123—22no—
106C2×#123—22no—
107C2×C22⋊Q8C2311yescharacteristic
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×C4×Q8—28no—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8—28no—
117C2×C4×Q8—44no—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×#117—22no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D411yescharacteristic
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C42—21no—
138C22×C42—22no—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141#484—28no—
142#481—28no—
143#481—44no—
144#479—216no—
145#464—216no—
146#460—28no—
147#451D418yescomplemented
148#448—28no—
149#441—216no—
150#440—216no—
151C4×Q16—28no—
152#432—28no—
153C2×#107C2314yes—
154C2×#107C2312yes—
155C2×#106C2314yes—
156C2×#106C2314yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2312yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2314yes—
168C2×C22⋊C8C2312yes—
169C2×C22⋊C8C2311yescharacteristic
170C2×C22⋊C8C2311yescharacteristic
171C2×#97C2314yes—
172C2×C4×C8C2312yes—
173#399—22no—
174#399—22no—
175#398—22no—
176#397—22no—
177#396—22no—
178#394—22no—
179#391C2311yescharacteristic
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390C2312yes—
186#390C2312yes—
187#390C2312yes—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—24no—
192#389—24no—
193#389—24no—
194#389—24no—
195#389—22no—
196#389C2314yes—
197#389C2311yescharacteristic
198#389C2312yes—
199#389C2312yes—
200#389C2314yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—21no—
206#386—22no—
207#386—22no—
208#385—21no—
209#385—22no—
210#384—22no—
211#384—22no—
212#384—24no—
213#384—21no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384C2311yescharacteristic
218#384C2312yes—
219#384C2312yes—
220#380—22no—
221#380C2311yescharacteristic
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—24no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D412yes—
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—21no—
235C2×#95—22no—
236C2×#95D412yes—
237#340C2314yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#336C2312yes—
243#336C2314yes—
244#336C2314yes—
245#336C2312yes—
246C22×Q8—24no—
247C22×Q8—24no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8C2×D414yes—
251C22×Q8C2411yescharacteristic
252C22×Q8C2412yes—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—21no—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2411yescharacteristic
266C2×Q16—24no—
267C2×Q16—24no—
268C2×Q16C2×D418yes—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—28no—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#127—24no—
276#126—48no—
277#125—24no—
278#125—42no—
279#124—24no—
280#124—42no—
281#123—48no—
282#123—84no—
283C22⋊Q8—22no—
284C22⋊Q8C2×D414yes—
285#121—24no—
286#121—24no—
287#121—44no—
288#121—82no—
289C4×Q8—28no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—28no—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—44no—
299C4×Q8—44no—
300C4×Q8—42no—
301C4×Q8—48no—
302C4×Q8—48no—
303C4×Q8#4118yes—
304C4×D4—24no—
305C4×D4—42no—
306C4×D4—48no—
307C4×D4—44no—
308#117—48no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—44no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4—84no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
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×C42—28no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—21no—
376C2×C42—48no—
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2412yes—
382C2×C42C2412yes—
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—24no—
387#106—28no—
388#106—44no—
389C4⋊C8—28no—
390C4⋊C8—24no—
391#103—28no—
392#103—28no—
393#103—24no—
394#103—24no—
395#103—24no—
396#103C2×D418yes—
397#102—28no—
398#102—28no—
399#102—44no—
400C22⋊C8—24no—
401C22⋊C8—22no—
402C22⋊C8C2×D414yes—
403#97—28no—
404C4×C8—24no—
405#95—24no—
406#95—24no—
407#95—21no—
408#95—21no—
409#95—22no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—22no—
419C24C22×D411yescharacteristic
420C2×Q8—24no—
421C2×Q8—216no—
422C2×Q8—24no—
423C2×Q8—216no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8C22×D412yes—
432C2×Q8C22×D414yes—
433C2×Q8C2×#4114yes—
434C2×Q82−1+414yes—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—44no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—81no—
447C2×D4—81no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—82no—
481C22×C4#13714yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4H(5,𝔽2)12yes—
493C22×C4C2511yescharacteristic
494Q16—216no—
495SD16—416no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8C22×D414yes—
505C4⋊C4—28no—
506C4⋊C4—24no—
507C4⋊C4—28no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—42no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—84no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4—81no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4C22×D412yes—
542C4⋊C4C2×#4114yes—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—42no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C42—28no—
556C42—28no—
557C42—22no—
558C42—22no—
559C42—24no—
560C42—24no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—48no—
567C42—48no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42C2×#4114yes—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23D4212yes—
584C23C2×#13712yes—
585C23#57712yes—
586C23C23×D411yescharacteristic
587Q8—216no—
588Q8—24no—
589Q8—24no—
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8#54718yes—
594D4—44no—
595D4—44no—
596D4—88no—
597D4—84no—
598D4—162no—
599C2×C4—216no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—21no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—48no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—41no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4—88no—
654C2×C4#52512yes—
655C2×C4#52512yes—
656C2×C4D4212yes—
657C2×C4C2×#13712yes—
658C2×C4C2×#13712yes—
659C2×C4#57512yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C22×#4112yes—
662C2×C4C2×H(5,𝔽2)11yescharacteristic
663C2×C4C2×2−1+412yes—
664C8—24no—
665C8—44no—
666C8—42no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22#327814yes—
681C22C2×#52511yescharacteristic
682C22C2×D4211yescharacteristic
683C22#348611yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57711yescharacteristic
687C22#358511yescharacteristic
688C4—28no—
689C4—22no—
690C4—22no—
691C4—22no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—41no—
699C4—44no—
700C4—82no—
701C4—81no—
702C4—84no—
703C4—81no—
704C4#311714yescomplemented
705C4C2×#54712yes—
706C4#349212yes—
707C2—24no—
708C2—162no—
709C2#3610311yescharacteristic
710C2C2×#311711yescharacteristic
711C2#6058811yescharacteristic
712C2C2×#327811yescharacteristic
713C2#6107011yescharacteristic
714C2#6183011yescharacteristic
715C2#6183611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
1. C2×#3117C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
4. #33005C2semidirect
7. #27648C2semidirect
8. #26387C2semidirect
9. #26185C2semidirect
10. #26043C2semidirect
11. #24733C2semidirect
12. #24607C2semidirect
13. #24067C2semidirect
14. #23003C2semidirect
15. #22674C2semidirect
16. #22603C2semidirect
17. C4×#451C2semidirect
147. #451D4semidirect
704. C4#3117semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
#361172
C2×Q8×C4⋊C42
#330052
#329032
#279062
#276482
#263872
#261852
#260432
#247332
#246072
#240672
#230032
#226742
#226032
C4×#4512