Finite Group Databasebeta

← #6371018

Subgroups — #6371018

Order 512.

4368 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#2924C211yesmaximal, characteristic, complemented
3#34180C211yesmaximal, characteristic, complemented
4#20746C218yesmaximal, complemented
5#20740C212yesmaximal, complemented
6#20739C212yesmaximal, complemented
7#20735C212yesmaximal, complemented
8#20732C212yesmaximal, complemented
9#20700C211yesmaximal, characteristic, complemented
10C2×#1823C211yesmaximal, characteristic, complemented
11#20652C214yesmaximal, complemented
12#20601C211yesmaximal, characteristic
13#20597C211yesmaximal, characteristic
14#20415C214yesmaximal, complemented
15C23×#41C2211yescharacteristic
16C2×#520—28no—
17C2×#520—22no—
18C2×#520—21no—
19C2×#520—21no—
20C2×#520C4116yescomplemented
21C2×#520C2212yes—
22C2×#520C2212yes—
23C2×#520C2214yes—
24C22×C4×Q8C2212yes—
25C22×C4×D4—22no—
26C22×C4×D4C2212yes—
27C22×#117—21no—
28C22×#117C2211yescharacteristic
29C23×C4⋊C4—21no—
30C23×C4⋊C4C2211yescharacteristic
31#2924—28no—
32C2×#433C2214yescomplemented
33C2×#432C2212yes—
34C2×#431C2212yes—
35C2×#424C2212yescomplemented
36C2×#417C2214yes—
37C2×#416C2214yes—
38C2×#407C2211yescharacteristic
39C22×C22⋊C8C2211yescharacteristic
40C2×#405C2212yescomplemented
41#2347—21no—
42#2346—21no—
43#2345—21no—
44#2344—21no—
45#2343—22no—
46#2342—22no—
47#2317C2212yescomplemented
48#2314C2214yescomplemented
49#2285C2212yes—
50#2283C2212yes—
51#2280C2211yescharacteristic
52C2×#380C2211yescharacteristic
53#2274C2212yescomplemented
54#1824C2214yescomplemented
55#1823C2212yes—
56#1823C2212yes—
57#1823C2212yes—
58#1823C2212yes—
59#1823C2214yescomplemented
60#1805C2218yescomplemented
61#1804C2218yescomplemented
62#1803C2218yescomplemented
63#1802C2218yescomplemented
64#1788C2214yes—
65#1788C2214yes—
66#1787C2211yescharacteristic
67#1787C2212yes—
68#1787C2211yescharacteristic
69#1786C2211yescharacteristic
70#1786C2212yes—
71#1786C2211yescharacteristic
72#1763C2218yescomplemented
73#1762C2218yescomplemented
74C4×#103C2218yescomplemented
75C4×#102C2218yescomplemented
76#1736C2214yescomplemented
77C2×#340C2214yescomplemented
78C22×#41—216no—
79C22×#41—24no—
80C22×#41—21no—
81C22×#41—21no—
82C22×#41C2×C418yes—
83C22×#41C2312yes—
84C22×#41C2312yes—
85C23×Q8C2311yescharacteristic
86C23×D4—21no—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C24×C4—21no—
90C24×C4—22no—
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C23×C8C2312yes—
94#520—216no—
95#520—216no—
96#520—232no—
97#520—28no—
98#520—28no—
99#520—216no—
100#520C2×C4116yes—
101#520C2×C4116yes—
102#520C2×C4132yescomplemented
103C2×C4×Q8—28no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8C2×C4116yescomplemented
107C2×C4×Q8C2314yes—
108C2×C4×Q8C2314yes—
109C2×C4×D4—232no—
110C2×C4×D4—24no—
111C2×C4×D4—24no—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4—22no—
115C2×C4×D4—22no—
116C2×C4×D4—28no—
117C2×C4×D4C2×C4116yescomplemented
118C2×C4×D4C2314yes—
119C2×C4×D4C2314yes—
120C2×#117—22no—
121C2×#117—216no—
122C2×#117—22no—
123C2×#117—24no—
124C2×#117—22no—
125C2×#117C2×C418yes—
126C2×#117D418yes—
127C2×#117D412yes—
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C2×#117C2312yes—
131C2×#117C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—21no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—24no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—216no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4—22no—
142C22×C4⋊C4—24no—
143C22×C4⋊C4C2×C418yes—
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4D411yescharacteristic
146C22×C4⋊C4C2312yes—
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2312yes—
150C22×C22⋊C4—22no—
151C22×C22⋊C4—21no—
152C22×C22⋊C4D412yes—
153C22×C42—22no—
154C22×C42C2312yes—
155#433—216no—
156#432—28no—
157#431—28no—
158#424—28no—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161#417—216no—
162#416—216no—
163C2×#103C2314yes—
164C2×#103C2318yes—
165C2×#103C2314yes—
166C2×#102C2314yes—
167C2×#102C2318yes—
168C2×#102C2314yes—
169#407—24no—
170C2×C22⋊C8—24no—
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2311yescharacteristic
173C2×C22⋊C8C2312yes—
174C2×C22⋊C8C2312yes—
175C2×C22⋊C8C2312yes—
176#405—28no—
177C2×#97C2314yes—
178C2×C4×C8C2314yes—
179#389—22no—
180#389—22no—
181#388—24no—
182#388—24no—
183#387—24no—
184#387—24no—
185#385—24no—
186#384C2314yes—
187#383—22no—
188#382—22no—
189#382C2314yes—
190#381—22no—
191#381—21no—
192#380—22no—
193#380—21no—
194#380—21no—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2311yescharacteristic
199#380C2311yescharacteristic
200#379—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2314yes—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—21no—
208C4×C22⋊C4—21no—
209C4×C22⋊C4—24no—
210#376—22no—
211C2×#95—21no—
212#340C2318yes—
213#340C2318yes—
214#340C2318yes—
215#340C2318yes—
216C25—21no—
217C25C2×D411yescharacteristic
218C2×#41—216no—
219C2×#41—264no—
220C2×#41—216no—
221C2×#41—264no—
222C2×#41—28no—
223C2×#41—28no—
224C2×#41—28no—
225C2×#41—28no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—28no—
229C2×#41C22×C418yes—
230C2×#41C22×C418yes—
231C2×#41#41116yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8C22×C414yes—
236C22×Q8C2412yes—
237C22×D4—216no—
238C22×D4—28no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—28no—
242C22×D4—21no—
243C22×D4—22no—
244C22×D4C22×C414yes—
245C22×D4C2412yes—
246C23×C4—216no—
247C23×C4—21no—
248C23×C4—28no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—28no—
252C23×C4—22no—
253C23×C4—28no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—216no—
257C23×C4—24no—
258C23×C4—24no—
259C23×C4—22no—
260C23×C4—84no—
261C23×C4—84no—
262C23×C4—82no—
263C23×C4—82no—
264C23×C4C22×C414yes—
265C23×C4C2×D412yes—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D414yes—
270C23×C4C2×D412yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C2×M16—24no—
275C2×M16—28no—
276C22×C8—24no—
277C22×C8—28no—
278C22×C8C2412yes—
279C22×C8C2412yes—
280C22×C8C2414yes—
281C4×Q8—216no—
282C4×Q8—216no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8C22×C4116yes—
286C4×Q8C22×C4116yes—
287C4×D4—232no—
288C4×D4—232no—
289C4×D4—232no—
290C4×D4—232no—
291C4×D4—216no—
292C4×D4—216no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4C22×C4116yes—
296C4×D4C22×C4116yes—
297#117—216no—
298#117—216no—
299#117—232no—
300#117—28no—
301#117—84no—
302#117C22⋊C4116yes—
303#117C22⋊C418yes—
304#117C22⋊C418yes—
305#117C22×C414yes—
306#117C22×C414yes—
307#117C22×C418yes—
308#117C2×D414yes—
309#117C2×D414yes—
310#117C2×D418yes—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—216no—
313C2×C4⋊C4—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—216no—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—232no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4C22⋊C418yes—
331C2×C4⋊C4C22⋊C418yes—
332C2×C4⋊C4C22×C418yes—
333C2×C4⋊C4C22×C418yes—
334C2×C4⋊C4C22×C418yes—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2414yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2414yes—
345C2×C22⋊C4—216no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—82no—
355C2×C22⋊C4C22⋊C4116yes—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—216no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—22no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42C22×C418yes—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D418yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C4⋊C8—28no—
373C4⋊C8—28no—
374#103—28no—
375#103—28no—
376#103—216no—
377#102—28no—
378#102—28no—
379#102—216no—
380C22⋊C8—22no—
381C22⋊C8—22no—
382C22⋊C8—24no—
383C22⋊C8—24no—
384C22⋊C8—24no—
385#97—28no—
386C4×C8—28no—
387#95—22no—
388#95—22no—
389#95—21no—
390#95—24no—
391#95—21no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—24no—
396C24—28no—
397C24—24no—
398C24—24no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—42no—
403C24—44no—
404C24—42no—
405C24—42no—
406C24C2×C22⋊C414yes—
407C24C22×D411yescharacteristic
408C24C22×D411yescharacteristic
409C24C22×D411yescharacteristic
410#41—264no—
411#41—232no—
412#41—264no—
413#41—216no—
414#41—216no—
415#41—216no—
416#41—232no—
417#41#117132yescomplemented
418C2×Q8—216no—
419C2×Q8—28no—
420C2×Q8—216no—
421C2×Q8—24no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8C23×C414yes—
425C2×Q8C2×#4118yes—
426C2×D4—264no—
427C2×D4—216no—
428C2×D4—28no—
429C2×D4—216no—
430C2×D4—28no—
431C2×D4—28no—
432C2×D4—216no—
433C2×D4—216no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4C23×C414yes—
438C2×D4C2×#4118yes—
439C22×C4—28no—
440C22×C4—264no—
441C22×C4—216no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—216no—
450C22×C4—24no—
451C22×C4—216no—
452C22×C4—24no—
453C22×C4—216no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—216no—
461C22×C4—216no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—832no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—816no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4C2×C22⋊C418yes—
482C22×C4C2×C22⋊C414yes—
483C22×C4C2×C22⋊C414yes—
484C22×C4C2×C22⋊C414yes—
485C22×C4C2×C22⋊C418yes—
486C22×C4C23×C412yes—
487C22×C4C23×C412yes—
488C22×C4C23×C412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D412yes—
499C22×C4C2×#4118yes—
500C22×C4C2511yescharacteristic
501M16—48no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—28no—
508C2×C8—48no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—28no—
514C4⋊C4—24no—
515C4⋊C4—28no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—44no—
519C4⋊C4—84no—
520C4⋊C4—84no—
521C4⋊C4C2×C22⋊C418yes—
522C4⋊C4C2×C22⋊C414yes—
523C4⋊C4C2×C22⋊C414yes—
524C4⋊C4C2×C22⋊C418yes—
525C4⋊C4C2×C22⋊C414yes—
526C4⋊C4C2×C22⋊C414yes—
527C4⋊C4C23×C418yes—
528C22⋊C4—216no—
529C22⋊C4—216no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4C2×C22⋊C418yes—
537C22⋊C4C2×C22⋊C418yes—
538C22⋊C4C2×C22⋊C418yes—
539C22⋊C4C2×C22⋊C418yes—
540C42—216no—
541C42—216no—
542C42—44no—
543C42—84no—
544C42—84no—
545C42C2×C22⋊C418yes—
546C42C2×C22⋊C418yes—
547C42C23×C414yes—
548C42C23×C414yes—
549C42C22×D414yes—
550C42C22×D414yes—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—21no—
555C23—21no—
556C23—216no—
557C23—22no—
558C23—28no—
559C23—24no—
560C23—22no—
561C23—28no—
562C23—28no—
563C23—21no—
564C23—21no—
565C23—44no—
566C23—48no—
567C23—48no—
568C23—416no—
569C23—416no—
570C23—48no—
571C23—44no—
572C23—48no—
573C23—42no—
574C23—42no—
575C23C22×C22⋊C412yes—
576C23C22×C22⋊C412yes—
577C23C22×C22⋊C412yes—
578C23#52514yes—
579C23#57612yes—
580C23#57712yes—
581C23C23×D411yescharacteristic
582Q8—216no—
583Q8—24no—
584Q8—28no—
585Q8C2×#11718yes—
586D4—232no—
587D4—216no—
588D4—232no—
589D4—24no—
590D4—28no—
591D4C2×#11718yes—
592C2×C4—216no—
593C2×C4—232no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—216no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—232no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—48no—
615C2×C4—416no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—164no—
632C2×C4C22×C22⋊C414yes—
633C2×C4C22×C22⋊C414yes—
634C2×C4C22×C22⋊C412yes—
635C2×C4C22×C22⋊C414yes—
636C2×C4C22×C22⋊C414yes—
637C2×C4C22×C22⋊C412yes—
638C2×C4C22×C22⋊C412yes—
639C2×C4C2×#11718yes—
640C2×C4#52518yes—
641C2×C4#52514yes—
642C2×C4C24×C411yescharacteristic
643C2×C4C23×D411yescharacteristic
644C2×C4C22×#4114yes—
645C8—42no—
646C8—42no—
647C8—44no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—28no—
660C22—416no—
661C22—432no—
662C22—42no—
663C22—48no—
664C22—42no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22#228014yes—
669C22#289714yes—
670C22C23×C22⋊C411yescharacteristic
671C22C2×#52512yes—
672C22C2×#57611yescharacteristic
673C22C2×#57711yescharacteristic
674C22#358612yes—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—22no—
680C4—28no—
681C4—22no—
682C4—48no—
683C4—82no—
684C4—84no—
685C4—82no—
686C4#228014yes—
687C4C22×#11712yes—
688C4C2×#52512yes—
689C2—24no—
690C2—24no—
691C2—42no—
692C2—48no—
693C2C2×#228011yescharacteristic
694C2C2×#289711yescharacteristic
695C2#6008811yescharacteristic
696C2#6058412yes—
697C2#6058512yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2924C2semidirect
3. #34180C2semidirect
4. #20746C2semidirect
5. #20740C2semidirect
6. #20739C2semidirect
7. #20735C2semidirect
8. #20732C2semidirect
9. #20700C2semidirect
10. C2×#1823C2semidirect
11. #20652C2semidirect
14. #20415C2semidirect
20. C2×#520C4semidirect
32. C2×#433C22semidirect
35. C2×#424C22semidirect
40. C2×#405C22semidirect
47. #2317C22semidirect
48. #2314C22semidirect
53. #2274C22semidirect
54. #1824C22semidirect
59. #1823C22semidirect
60. #1805C22semidirect
61. #1804C22semidirect
62. #1803C22semidirect
63. #1802C22semidirect
72. #1763C22semidirect
73. #1762C22semidirect
74. C4×#103C22semidirect
75. C4×#102C22semidirect
76. #1736C22semidirect
77. C2×#340C22semidirect
102. #520C2×C4semidirect
106. C2×C4×Q8C2×C4semidirect
117. C2×C4×D4C2×C4semidirect
417. #41#117semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#29242
#341802
#207462
#207402
#207392
#207352
#207322
#207002
C2×#18232
#206522
#206012
#205972
#204152