Finite Group Databasebeta

← #6381760

Subgroups — #6381760

Order 512.

2120 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2#34213C211yesmaximal, characteristic, complemented
3#34183C211yesmaximal, characteristic
4#22980C214yesmaximal, complemented
5#22974C214yesmaximal, complemented
6#22970C212yesmaximal, complemented
7#22969C212yesmaximal, complemented
8#22965C212yesmaximal, complemented
9#22962C212yesmaximal, complemented
10#21322C211yesmaximal, characteristic, complemented
11C2×#1947C211yesmaximal, characteristic, complemented
12#21156C212yesmaximal, complemented
13#21039C212yesmaximal, complemented
14#20881C212yesmaximal
15#20441C211yesmaximal, characteristic, complemented
16#20415C212yesmaximal
17C4×#417C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#520C2212yes—
20C2×C4×#41C2211yescharacteristic
21#3076C4116yescomplemented
22C2×#479C2212yes—
23C2×#477C2211yescharacteristic
24C2×#475C2211yescharacteristic
25C2×#478C2212yes—
26C2×#476C2211yescharacteristic
27C2×#474C2211yescharacteristic
28C2×#427C2212yes—
29C22×#107C2212yes—
30C22×C4⋊C8C2211yescharacteristic
31C2×#417C2211yescharacteristic
32C2×#417C2211yescharacteristic
33C2×#416C2212yes—
34C2×#405C2212yes—
35C2×C4×M16C2211yescharacteristic
36#2370C2214yes—
37#2369C2212yes—
38#2365C2212yes—
39#2355—21no—
40#2355—22no—
41Q8×C4⋊C4—21no—
42Q8×C4⋊C4C2212yes—
43Q8×C4⋊C4C2212yes—
44D4×C4⋊C4—22no—
45D4×C4⋊C4—22no—
46D4×C4⋊C4C2212yes—
47D4×C4⋊C4C2212yes—
48#2329—22no—
49#2317—22no—
50#2317C2212yes—
51#2317C2212yes—
52C4×C22⋊Q8—22no—
53#2285—21no—
54#2285C2211yescharacteristic
55C2×#384—21no—
56C2×#384C2211yescharacteristic
57#2274C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59#1992C2214yes—
60#1990C2212yes—
61#1990C2212yes—
62#1947C2212yes—
63#1947C2212yes—
64#1947C2214yes—
65#1944C2214yes—
66#1944C2214yes—
67#1942C2214yes—
68#1942C2214yes—
69#1921C2214yes—
70#1920C2214yes—
71#1919C2214yes—
72#1918C2214yes—
73#1863C2214yes—
74#1863C2214yes—
75#1862C2214yes—
76#1862C2214yes—
77#1771C2214yes—
78#1770C2212yes—
79C8×C4⋊C4C2212yes—
80#1763C2214yes—
81#1763C2212yes—
82#1762C2214yes—
83#1762C2212yes—
84C4×#103C2212yes—
85C4×#103C2214yes—
86C4×#102C2212yes—
87C4×#102C2214yes—
88#1736C2211yescharacteristic
89#1736C2212yes—
90#1736C2212yes—
91#1736C2211yescharacteristic
92C2×#340C2212yes—
93#1729C2212yes—
94C22×#41C2311yescharacteristic
95C22×M16C2311yescharacteristic
96C22×M16C2311yescharacteristic
97C23×C8C2312yes—
98#533—24no—
99#533C2×C418yescomplemented
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#125C2312yes—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105#520—28no—
106#520—44no—
107C4×#41—24no—
108C4×#41C2×C418yes—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2314yes—
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×D4—21no—
116C2×C4×D4—21no—
117C2×C4×D4—24no—
118C2×C4×D4C2314yes—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×#117—22no—
122C2×#117—21no—
123C2×#117C2311yescharacteristic
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—22no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#479C2×C4116yescomplemented
138#478C2×C4116yescomplemented
139#477C2×C418yescomplemented
140#476C2×C418yescomplemented
141#475C2×C418yescomplemented
142#474C2×C418yescomplemented
143#427C2×C4116yescomplemented
144C2×#107—44no—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#107C2312yes—
148C2×#107C2312yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2×C418yescomplemented
152C2×C4⋊C8C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155#417—24no—
156#417Q818yescomplemented
157#416C2×C4116yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166#405—44no—
167C4×M16—24no—
168C2×#97C2312yes—
169C2×#97C2314yes—
170C2×C4×C8C2312yes—
171C2×C4×C8C2314yes—
172#391—22no—
173#391—24no—
174#389—24no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#387—24no—
183#387—24no—
184#387—24no—
185#385—24no—
186#385—24no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384—22no—
191#384—22no—
192#384—22no—
193#384—21no—
194#384—21no—
195#384D411yescharacteristic
196#384D411yescharacteristic
197#384C2312yes—
198#384C2312yes—
199#382—22no—
200#382—22no—
201#382D412yes—
202#380—22no—
203#380—22no—
204#380D412yes—
205#380D412yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2312yes—
220C4×C22⋊C4—24no—
221C2×#95—21no—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2314yes—
232#336C2314yes—
233#336C2314yes—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41C22×C414yes—
238C2×#41C2×Q814yes—
239C2×#41#4114yes—
240C22×Q8—21no—
241C22×Q8—21no—
242C22×Q8C2411yescharacteristic
243C22×D4—21no—
244C22×D4C2411yescharacteristic
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—24no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×M16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C2×M16—22no—
259C2×M16—22no—
260C2×M16C2×Q814yes—
261C22×C8—24no—
262C22×C8—24no—
263C22×C8C22×C418yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22×C8C2412yes—
267C22×C8C2412yes—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8C22×C414yes—
271C4⋊Q8C22×C414yes—
272#125—24no—
273#125C22×C418yes—
274C22⋊Q8—28no—
275C22⋊Q8—28no—
276C4×Q8—28no—
277C4×Q8—28no—
278C4×Q8—22no—
279C4×Q8—22no—
280C4×Q8—28no—
281C4×Q8—44no—
282C4×Q8—84no—
283C4×Q8C22×C414yes—
284C4×Q8C22×C414yes—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—22no—
289C4×D4—22no—
290C4×D4—44no—
291C4×D4—416no—
292C4×D4C22×C414yes—
293C4×D4C22×C414yes—
294#117—22no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117—28no—
299#117—48no—
300#117C22×C414yes—
301#117C22×C418yes—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—48no—
326C2×C4⋊C4—84no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×Q814yes—
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C2×D414yes—
350C2×C42—21no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—82no—
361C2×C42C22×C414yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42#4114yes—
365C2×C42C2414yes—
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368#107—44no—
369#107—44no—
370#107C22×C418yes—
371#107C22×C418yes—
372#106C22×C418yes—
373#106C22×C418yes—
374C4⋊C8C22×C414yes—
375C4⋊C8C22×C414yes—
376C4⋊C8C22×C418yes—
377#103—24no—
378#103C22×C418yes—
379#103C22×C418yes—
380#103C2×Q818yes—
381#102—24no—
382#102C22×C418yes—
383#102C22×C418yes—
384#102C2×Q818yes—
385#97—24no—
386#97—44no—
387C4×C8—24no—
388C4×C8—44no—
389#95—22no—
390#95—22no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95C2×D414yes—
398#95C2×D414yes—
399C24—21no—
400C24C22×D411yescharacteristic
401#41—28no—
402#41—28no—
403#41—28no—
404#41C4×Q8116yescomplemented
405C2×Q8—24no—
406C2×Q8—24no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—42no—
411C2×Q8—42no—
412C2×Q8—81no—
413C2×Q8—81no—
414C2×Q8C23×C412yes—
415C2×Q8C22×Q812yes—
416C2×Q8C2×#4112yes—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4C23×C412yes—
424C2×D4C22×Q812yes—
425C2×D4C2×#4112yes—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—48no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—44no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—81no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4C4×D414yes—
460C22×C4C4×D414yes—
461C22×C4C23×C412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×Q812yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475M16—28no—
476M16—44no—
477C2×C8—216no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8C23×C414yes—
488C2×C8C23×C414yes—
489C2×C8C22×Q814yes—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—48no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4C23×C414yes—
524C4⋊C4C23×C414yes—
525C4⋊C4C23×C412yes—
526C4⋊C4C23×C412yes—
527C4⋊C4C22×Q814yes—
528C22⋊C4—24no—
529C22⋊C4—28no—
530C22⋊C4—28no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—48no—
534C42—24no—
535C42—24no—
536C42—24no—
537C42—28no—
538C42—24no—
539C42—24no—
540C42—48no—
541C42—44no—
542C42—84no—
543C42—82no—
544C42—84no—
545C42C4×D414yes—
546C42C4×D414yes—
547C42C23×C412yes—
548C42C23×C412yes—
549C42C2×#4112yes—
550C42C2×#4112yes—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×C4×D412yes—
563C23#54612yes—
564C23D4×Q812yes—
565C23C23×D411yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—48no—
570Q8—84no—
571Q8C2×C4×Q814yes—
572D4—216no—
573D4—22no—
574D4—22no—
575D4—22no—
576D4C2×C4×Q814yes—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—216no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—44no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—84no—
619C2×C4—84no—
620C2×C4—168no—
621C2×C4—164no—
622C2×C4C2×C4×D412yes—
623C2×C4C2×C4×D412yes—
624C2×C4C2×C4×D412yes—
625C2×C4C2×C4×D412yes—
626C2×C4C2×C4×D412yes—
627C2×C4C2×C4×Q814yes—
628C2×C4#52014yes—
629C2×C4#54612yes—
630C2×C4D4×Q812yes—
631C2×C4#55214yes—
632C2×C4#55414yes—
633C2×C4#57612yes—
634C2×C4#57812yes—
635C2×C4C24×C411yescharacteristic
636C2×C4C23×Q811yescharacteristic
637C2×C4C22×#4111yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4C2×H(5,𝔽2)11yescharacteristic
640C2×C4C2×2−1+411yescharacteristic
641C8—24no—
642C8—24no—
643C8—24no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22#236014yes—
659C22C22×C4×D411yescharacteristic
660C22C2×#54611yescharacteristic
661C22C2×D4×Q811yescharacteristic
662C22C2×#57611yescharacteristic
663C22C2×#57811yescharacteristic
664C22#358511yescharacteristic
665C22#358611yescharacteristic
666C4—22no—
667C4—22no—
668C4—22no—
669C4—24no—
670C4—42no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—82no—
678C4—84no—
679C4#236014yes—
680C4#295114yes—
681C4C22×C4×Q811yescharacteristic
682C4C2×#52011yescharacteristic
683C4C2×#55211yescharacteristic
684C4C2×#55411yescharacteristic
685C2—24no—
686C2—44no—
687C2C2×#236011yescharacteristic
688C2C2×#295111yescharacteristic
689C2#6017011yescharacteristic
690C2#6120511yescharacteristic
691C2#6121011yescharacteristic
692C2#6138111yescharacteristic
693C2#6138411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. #22980C2semidirect
5. #22974C2semidirect
6. #22970C2semidirect
7. #22969C2semidirect
8. #22965C2semidirect
9. #22962C2semidirect
10. #21322C2semidirect
11. C2×#1947C2semidirect
12. #21156C2semidirect
13. #21039C2semidirect
15. #20441C2semidirect
21. #3076C4semidirect
99. #533C2×C4semidirect
137. #479C2×C4semidirect
138. #478C2×C4semidirect
139. #477C2×C4semidirect
140. #476C2×C4semidirect
141. #475C2×C4semidirect
142. #474C2×C4semidirect
143. #427C2×C4semidirect
151. C2×C4⋊C8C2×C4semidirect
156. #417Q8semidirect
404. #41C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#342132
#341832
#229802
#229742
#229702
#229692
#229652
#229622
#213222
C2×#19472
#211562
#210392
#208812
#204412
#204152
C4×#4172