Finite Group Databasebeta

← #6630108

Subgroups — #6630108

Order 512.

2140 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3082C212yesmaximal
2#35275C211yesmaximal, characteristic
3C4×#519C211yesmaximal, characteristic
4#33206C214yesmaximal, complemented
5#33204C214yesmaximal, complemented
6#33200C212yesmaximal, complemented
7#33200C212yesmaximal, complemented
8#33198C212yesmaximal, complemented
9#33196C212yesmaximal, complemented
10#21315C212yesmaximal
11#21175C212yesmaximal
12#21171C211yesmaximal, characteristic
13#21169C211yesmaximal, characteristic
14#20793C211yesmaximal, characteristic
15#20651C212yesmaximal
16#20647C211yesmaximal, characteristic
17C2×#1803C211yesmaximal, characteristic
18C2×#544C2212yes—
19C2×#519C2211yescharacteristic
20C2×#519C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3082—216no—
23C2×#479C2214yes—
24C2×#479C2214yes—
25C2×#477C2214yes—
26C2×#475C2214yes—
27C2×#428C2212yes—
28C2×#427C2212yes—
29C2×#424C2212yes—
30C2×#424C2212yes—
31C2×#423C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C2×#419C2212yes—
34C22×#103C2212yes—
35#2868C2212yes—
36#2859C2212yes—
37#2660C2214yes—
38#2552—21no—
39#2552—21no—
40#2551—21no—
41#2551—21no—
42#2440—22no—
43#2440—22no—
44#2317C2212yes—
45#2285C2211yescharacteristic
46C2×#384C2211yescharacteristic
47#2276—21no—
48#2276—21no—
49#2276C2212yes—
50#2276C2212yes—
51#2274—22no—
52#2274C2211yescharacteristic
53#2274C2211yescharacteristic
54C42×Q8—21no—
55C42×Q8—21no—
56C42×Q8C2212yes—
57C42×Q8C2212yes—
58C4×#117—22no—
59C4×#117C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#1989C2214yes—
62#1989C2214yes—
63#1988C2214yes—
64#1987C2214yes—
65#1949C2214yes—
66#1949C2214yes—
67#1949C2212yes—
68#1949C2212yes—
69#1948C2211yescharacteristic
70#1948C2212yes—
71#1948C2214yes—
72#1948C2211yescharacteristic
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1947C2212yes—
76#1947C2214yes—
77#1947C2211yescharacteristic
78#1947C2212yes—
79#1843C2214yes—
80#1842C2212yes—
81#1842C2212yes—
82#1805C2214yes—
83#1805C2214yes—
84#1805C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2214yes—
91#1803C2214yes—
92#1736C2212yes—
93#1736C2212yes—
94#1735C2212yes—
95C2×#340C2212yes—
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#544—28no—
100#544—28no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8C2314yes—
105C2×#125—22no—
106C2×#125C2314yes—
107C2×C22⋊Q8—24no—
108#519—24no—
109#519—24no—
110#519D418yes—
111#519Q818yescomplemented
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8—24no—
119C2×C4×Q8Q818yescomplemented
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×Q8C2312yes—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×#117—22no—
127C2×#117—22no—
128C2×#117C2312yes—
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#479—216no—
141#479—216no—
142#477—216no—
143#475—216no—
144#428—28no—
145#427—28no—
146C2×#107C2312yes—
147C2×#107C2314yes—
148C2×#107C2312yes—
149C2×#106C2312yes—
150C2×#106C2314yes—
151C2×#106C2312yes—
152#424—28no—
153#424—28no—
154#423—24no—
155C2×C4⋊C8—24no—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2314yes—
164C2×C4⋊C8C2312yes—
165#419—28no—
166C2×#103—28no—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171#398—24no—
172#398—24no—
173#395—24no—
174#395—24no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#385—24no—
182#384—22no—
183#384—22no—
184#384—21no—
185#384—21no—
186#384C2314yes—
187#384C2311yescharacteristic
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2312yes—
191#382C2314yes—
192#382C2312yes—
193#380—21no—
194#380—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4D411yescharacteristic
214C4×C4⋊C4D411yescharacteristic
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—21no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4D412yes—
230C4×C22⋊C4D412yes—
231#376—21no—
232#376—22no—
233#376—22no—
234#376C2312yes—
235C43—22no—
236C43—22no—
237C43—21no—
238C43C2312yes—
239#340C2312yes—
240#340C2314yes—
241#340C2312yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2312yes—
248#340C2312yes—
249C22×Q8—21no—
250C22×Q8—24no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2×D414yes—
255C22×Q8C2×Q814yes—
256C22×Q8C2×Q814yes—
257C22×Q8C2412yes—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—22no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—24no—
266C2×M16—24no—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C4⋊Q8—28no—
274C4⋊Q8—28no—
275C4⋊Q8—48no—
276C4⋊Q8—84no—
277C4⋊Q8—84no—
278#125—28no—
279#125—28no—
280#125—48no—
281C22⋊Q8—416no—
282C4×Q8—28no—
283C4×Q8—28no—
284C4×Q8—28no—
285C4×Q8—28no—
286C4×Q8—28no—
287C4×Q8—28no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8C2×D418yes—
295C4×Q8C2×D418yes—
296C4×Q8C2×Q818yes—
297C4×Q8C2×Q818yes—
298C4×Q8C2×Q818yes—
299C4×Q8C2×Q818yes—
300#117—22no—
301#117—22no—
302#117—216no—
303#117—216no—
304#117—24no—
305#117—24no—
306#117C2×D414yes—
307#117C2×D414yes—
308#117C2×Q814yes—
309#117C2×Q814yes—
310#117#4114yes—
311#117#4114yes—
312C2×C4⋊C4—28no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—82no—
338C2×C4⋊C4—82no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×Q814yes—
344C2×C4⋊C4C2414yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2411yescharacteristic
351C2×C4⋊C4C2411yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—28no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42C2×D412yes—
376C2×C42C2×D412yes—
377C2×C42C2×D412yes—
378C2×C42C2×D412yes—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×Q814yes—
382C2×C42#4114yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387#107—24no—
388#107—24no—
389#107—28no—
390#106—24no—
391#106—24no—
392#106—28no—
393C4⋊C8—28no—
394C4⋊C8—28no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—28no—
401C4⋊C8—28no—
402#103—28no—
403#103—28no—
404#103—28no—
405#103—28no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—24no—
417#95—24no—
418#95C2×D412yes—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D412yes—
422C24C22×D411yescharacteristic
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—28no—
427C2×Q8—28no—
428C2×Q8—28no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—48no—
433C2×Q8—84no—
434C2×Q8—84no—
435C2×Q8C4⋊Q8116yescomplemented
436C2×Q8C22×D414yes—
437C2×Q8C22×Q814yes—
438C2×Q8C22×Q814yes—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—44no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4C22⋊Q814yes—
462C22×C4C22⋊Q814yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q812yes—
474C22×C4C22×Q812yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2511yescharacteristic
479M16—48no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—48no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—24no—
495C4⋊C4—28no—
496C4⋊C4—28no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4#12114yes—
523C4⋊C4#12114yes—
524C4⋊C4C22⋊Q814yes—
525C4⋊C4C22⋊Q814yes—
526C4⋊C4C22×D412yes—
527C4⋊C4C22×D412yes—
528C4⋊C4C22×Q814yes—
529C4⋊C4C22×Q812yes—
530C4⋊C4C22×Q812yes—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—44no—
534C22⋊C4#12118yes—
535C22⋊C4C22⋊Q818yes—
536C42—28no—
537C42—28no—
538C42—24no—
539C42—28no—
540C42—28no—
541C42—28no—
542C42—28no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—42no—
547C42—42no—
548C42—44no—
549C42—42no—
550C42—42no—
551C42—82no—
552C42—82no—
553C42—81no—
554C42—82no—
555C42—82no—
556C42—81no—
557C42C22⋊Q814yes—
558C42C22⋊Q814yes—
559C42C22×D414yes—
560C42C22×Q812yes—
561C42C22×Q812yes—
562C42C22×Q814yes—
563C42C2×#4114yes—
564C42C2×#4112yes—
565C42C2×#4112yes—
566C42C2×#4114yes—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23C2×#12112yes—
572C23C2×C22⋊Q812yes—
573C23C2×C22⋊Q812yes—
574C23C23×D411yescharacteristic
575Q8—216no—
576Q8—24no—
577Q8—24no—
578Q8—24no—
579Q8—168no—
580Q8C2×C4⋊Q818yes—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—216no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—84no—
626C2×C4—164no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4C2×C22⋊Q812yes—
631C2×C4C2×C22⋊Q814yes—
632C2×C4C2×C22⋊Q814yes—
633C2×C4C2×C22⋊Q812yes—
634C2×C4C2×C22⋊Q812yes—
635C2×C4C2×C22⋊Q812yes—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×C22⋊Q812yes—
638C2×C4C2×C4⋊Q814yes—
639C2×C4#53214yes—
640C2×C4#53314yes—
641C2×C4#53314yes—
642C2×C4#57712yes—
643C2×C4#57812yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C23×Q811yescharacteristic
646C2×C4C23×Q811yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C8—42no—
651C8—42no—
652C8—44no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22#285714yes—
660C22C22×#12111yescharacteristic
661C22C22×C22⋊Q811yescharacteristic
662C22C22×C22⋊Q811yescharacteristic
663C22C2×#57711yescharacteristic
664C22C2×#57811yescharacteristic
665C22#358511yescharacteristic
666C22#358611yescharacteristic
667C4—28no—
668C4—24no—
669C4—24no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—24no—
674C4—24no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4#285714yes—
684C4#308214yescomplemented
685C4C22×C4⋊Q811yescharacteristic
686C4C2×#53211yescharacteristic
687C4C2×#53311yescharacteristic
688C4C2×#53311yescharacteristic
689C2—24no—
690C2C2×#285711yescharacteristic
691C2#6043111yescharacteristic
692C2#6043211yescharacteristic
693C2C2×#308211yescharacteristic
694C2#6049311yescharacteristic
695C2#6049711yescharacteristic
696C2#6049811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #33206C2semidirect
5. #33204C2semidirect
6. #33200C2semidirect
7. #33200C2semidirect
8. #33198C2semidirect
9. #33196C2semidirect
111. #519Q8semidirect
119. C2×C4×Q8Q8semidirect
435. C2×Q8C4⋊Q8semidirect
684. C4#3082semidirect

Maximal subgroups

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

subgroupindex
C2×#30822
#352752
C4×#5192
#332062
#332042
#332002
#331982
#331962
#213152
#211752
#211712
#211692
#207932
#206512
#206472
C2×#18032