Finite Group Databasebeta

← #6381746

Subgroups — #6381746

Order 512.

2220 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#34181C211yesmaximal, characteristic
4#22972C214yesmaximal, complemented
5#22969C212yesmaximal, complemented
6#22968C212yesmaximal, complemented
7#22966C214yesmaximal, complemented
8#22963C212yesmaximal, complemented
9#22962C212yesmaximal, complemented
10#21318C211yesmaximal, characteristic, complemented
11#21170C211yesmaximal, characteristic, complemented
12#21160C212yesmaximal, complemented
13#21036C212yesmaximal, complemented
14#20876C212yesmaximal
15#20437C211yesmaximal, characteristic, complemented
16#20416C212yesmaximal
17C2×C4×#102C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#518C2212yes—
20C22×C4×D4C2211yescharacteristic
21#3077C4116yescomplemented
22C2×#478C2214yes—
23C2×#476C2212yes—
24C2×#474C2212yes—
25C2×#428C2212yes—
26C2×#428C2212yes—
27C2×#423C2211yescharacteristic
28C2×#418C2212yes—
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#405C2212yes—
32C22×C4×C8C2211yescharacteristic
33#2370C2214yescomplemented
34#2369C2212yescomplemented
35#2365C2212yescomplemented
36#2355—21no—
37#2355—22no—
38Q8×C4⋊C4—21no—
39#2349—22no—
40#2349C2214yes—
41D4×C4⋊C4—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4C2212yes—
44D4×C4⋊C4C2212yes—
45#2329—22no—
46#2317C2212yes—
47#2316C2212yes—
48#2315—22no—
49C4×C22⋊Q8—22no—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52C2×#380—22no—
53#2274C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#1991C2212yescomplemented
56#1991C2212yescomplemented
57#1990C2211yescharacteristic, complemented
58#1990C2211yescharacteristic, complemented
59#1990C2211yescharacteristic, complemented
60#1990C2211yescharacteristic, complemented
61#1949C2214yescomplemented
62#1948C2212yescomplemented
63#1947C2212yescomplemented
64#1944C2214yescomplemented
65#1943C2214yescomplemented
66#1942C2214yescomplemented
67#1941C2214yescomplemented
68#1921C2214yescomplemented
69#1920C2214yescomplemented
70#1919C2214yescomplemented
71#1918C2214yescomplemented
72#1864C2214yes—
73#1864C2214yes—
74#1862C2214yes—
75#1862C2214yes—
76#1770C2212yescomplemented
77#1770C2212yescomplemented
78C8×C4⋊C4C2212yescomplemented
79C8×C4⋊C4C2212yescomplemented
80#1762C2214yes—
81#1762C2214yes—
82C4×#102C2212yes—
83C4×#102C2214yes—
84C4×#102C2214yes—
85C4×#102C2212yes—
86C4×#102C2212yes—
87C4×#102C2212yes—
88#1736C2212yes—
89#1735C2212yes—
90#1735C2212yes—
91C2×#340C2211yescharacteristic
92C2×#340C2211yescharacteristic
93#1729C2212yes—
94C23×D4C2311yescharacteristic
95C24×C4—21no—
96C22×M16C2312yes—
97C23×C8C2311yescharacteristic
98C23×C8C2311yescharacteristic
99#533—24no—
100#533C2×C418yescomplemented
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125C2312yes—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8—21no—
106#518—28no—
107#518—44no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—24no—
116C2×C4×D4C2×C418yes—
117C2×C4×D4C2314yes—
118C2×C4×D4C2314yes—
119C2×C4×D4C2312yes—
120C2×C4×D4C2312yes—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C2×#117C2312yes—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4—24no—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133#478C2×C4132yescomplemented
134#476C2×C4116yescomplemented
135#474C2×C4116yescomplemented
136#428—44no—
137#428C2×C4116yescomplemented
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#423C2×C418yescomplemented
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145#418C2×C4116yes—
146C2×#102—24no—
147C2×#102Q818yescomplemented
148C2×#102C2312yes—
149C2×#102C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154#405—44no—
155C2×#97C2314yes—
156C2×C4×C8—24no—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2314yes—
162#391—22no—
163#391—24no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—22no—
168#389C2314yes—
169#389C2314yes—
170#389C2314yes—
171#389C2314yes—
172#387—24no—
173#387—24no—
174#387—24no—
175#385—24no—
176#385—24no—
177#385—24no—
178#385—24no—
179#385—24no—
180#384—22no—
181#384D411yescharacteristic
182#384D411yescharacteristic
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—22no—
186#382D412yes—
187#382C2312yes—
188#380—21no—
189#380—21no—
190#380—22no—
191#380—22no—
192#380—24no—
193#380—24no—
194#380D412yes—
195#380D412yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—24no—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2312yes—
210C4×C22⋊C4—24no—
211C2×#95—21no—
212C2×#95—24no—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229C25—21no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×D4—21no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C22×C414yes—
237C22×D4C2×Q814yes—
238C22×D4#4114yes—
239C22×D4C2412yes—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—82no—
249C23×C4—82no—
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2411yescharacteristic
253C2×M16—24no—
254C2×M16—24no—
255C2×M16C22×C418yes—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8C2×Q814yes—
262C22×C8C2411yescharacteristic
263C22×C8C2411yescharacteristic
264C22×C8C2414yes—
265C22×C8C2411yescharacteristic
266C22×C8C2411yescharacteristic
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8C22×C414yes—
270C4⋊Q8C22×C414yes—
271#125—24no—
272#125C22×C418yes—
273C22⋊Q8—28no—
274C22⋊Q8—28no—
275C4×Q8—28no—
276C4×Q8—28no—
277C4×Q8—84no—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—28no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—44no—
285C4×D4—44no—
286C4×D4—48no—
287C4×D4—48no—
288C4×D4C22×C418yes—
289C4×D4C22×C418yes—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117C22×C418yes—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—84no—
316C2×C4⋊C4—84no—
317C2×C4⋊C4C22×C414yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×Q814yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2412yes—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—28no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—416no—
340C2×C22⋊C4C2×D414yes—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—82no—
348C2×C42C22×C414yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42#4114yes—
352C2×C42C2414yes—
353C2×C42C2411yescharacteristic
354C2×C42C2411yescharacteristic
355#107—44no—
356#107C22×C4116yes—
357#106—44no—
358#106C22×C4116yes—
359C4⋊C8C22×C418yes—
360C4⋊C8C22×C418yes—
361#102—24no—
362#102—24no—
363#102C22×C4116yes—
364#102C22×C4116yes—
365#102C2×Q818yes—
366#102C2×Q818yes—
367#97—44no—
368C4×C8—22no—
369C4×C8—22no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—44no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—22no—
381#95—24no—
382#95—24no—
383#95C2×D414yes—
384#95C2×D414yes—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—81no—
390C24—81no—
391C24—81no—
392C24—81no—
393C24C22×D411yescharacteristic
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—42no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×D4—24no—
401C2×D4—28no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4C4×Q8116yescomplemented
410C2×D4C23×C414yes—
411C2×D4C22×Q814yes—
412C2×D4C2×#4114yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—42no—
433C22×C4—816no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4C4×D414yes—
447C22×C4C4×D414yes—
448C22×C4C23×C412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q812yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C4C2511yescharacteristic
462M16—216no—
463C2×C8—28no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—24no—
468C2×C8—21no—
469C2×C8—24no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—44no—
477C2×C8C23×C418yes—
478C2×C8C22×Q812yes—
479C2×C8C22×Q812yes—
480C4⋊C4—24no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4C23×C414yes—
512C4⋊C4C23×C414yes—
513C4⋊C4C23×C414yes—
514C4⋊C4C22×Q814yes—
515C22⋊C4—28no—
516C22⋊C4—28no—
517C22⋊C4—28no—
518C22⋊C4—28no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C42—28no—
523C42—24no—
524C42—24no—
525C42—44no—
526C42—44no—
527C42—84no—
528C42—82no—
529C42—84no—
530C42C4×D414yes—
531C42C4×D414yes—
532C42C23×C412yes—
533C42C23×C412yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—216no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23C2×C4×D412yes—
555C23#54612yes—
556C23D4×Q812yes—
557C23C23×D411yescharacteristic
558Q8—48no—
559Q8—84no—
560D4—216no—
561D4—24no—
562D4—24no—
563D4—24no—
564D4C2×C4×Q818yes—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—88no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—168no—
606C2×C4—168no—
607C2×C4C2×C4×D412yes—
608C2×C4C2×C4×D412yes—
609C2×C4C2×C4×D412yes—
610C2×C4C2×C4×D412yes—
611C2×C4C2×C4×D412yes—
612C2×C4C2×C4×Q814yes—
613C2×C4#52014yes—
614C2×C4#54612yes—
615C2×C4D4×Q812yes—
616C2×C4#55214yes—
617C2×C4#55414yes—
618C2×C4#57612yes—
619C2×C4#57712yes—
620C2×C4C24×C411yescharacteristic
621C2×C4C23×Q811yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C8—28no—
627C8—22no—
628C8—22no—
629C8—41no—
630C8—41no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—28no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—816no—
645C22#236014yes—
646C22C22×C4×D411yescharacteristic
647C22C2×#54611yescharacteristic
648C22C2×D4×Q811yescharacteristic
649C22C2×#57611yescharacteristic
650C22C2×#57711yescharacteristic
651C22#358411yescharacteristic
652C22#358411yescharacteristic
653C4—22no—
654C4—22no—
655C4—22no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4#236014yes—
666C4#294614yes—
667C4C22×C4×Q811yescharacteristic
668C4C2×#52011yescharacteristic
669C4C2×#55211yescharacteristic
670C4C2×#55411yescharacteristic
671C2—24no—
672C2—48no—
673C2C2×#236011yescharacteristic
674C2C2×#294611yescharacteristic
675C2#6016111yescharacteristic
676C2#6121111yescharacteristic
677C2#6121211yescharacteristic
678C2#6137011yescharacteristic
679C2#6137111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34213C2semidirect
4. #22972C2semidirect
5. #22969C2semidirect
6. #22968C2semidirect
7. #22966C2semidirect
8. #22963C2semidirect
9. #22962C2semidirect
10. #21318C2semidirect
11. #21170C2semidirect
12. #21160C2semidirect
13. #21036C2semidirect
15. #20437C2semidirect
21. #3077C4semidirect
33. #2370C22semidirect
34. #2369C22semidirect
35. #2365C22semidirect
55. #1991C22semidirect
56. #1991C22semidirect
57. #1990C22semidirect
58. #1990C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1949C22semidirect
62. #1948C22semidirect
63. #1947C22semidirect
64. #1944C22semidirect
65. #1943C22semidirect
66. #1942C22semidirect
67. #1941C22semidirect
68. #1921C22semidirect
69. #1920C22semidirect
70. #1919C22semidirect
71. #1918C22semidirect
76. #1770C22semidirect
77. #1770C22semidirect
78. C8×C4⋊C4C22semidirect
79. C8×C4⋊C4C22semidirect
100. #533C2×C4semidirect
133. #478C2×C4semidirect
134. #476C2×C4semidirect
135. #474C2×C4semidirect
137. #428C2×C4semidirect
142. #423C2×C4semidirect
147. C2×#102Q8semidirect
409. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#342132
#341812
#229722
#229692
#229682
#229662
#229632
#229622
#213182
#211702
#211602
#210362
#208762
#204372
#204162
C2×C4×#1022