Finite Group Databasebeta

← #364847

Subgroups — #364847

Order 512.

1372 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#21324C211yesmaximal, characteristic, complemented
4#20632C211yesmaximal, characteristic
5C4×#421C211yesmaximal, characteristic
6#11213C212yesmaximal, complemented
7#11057C212yesmaximal
8#10954C212yesmaximal
9#10903C212yesmaximal, complemented
10#10855C211yesmaximal, characteristic, complemented
11#10836C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3079—22no—
15#3079—22no—
16#3079—22no—
17#3079—22no—
18C2×#477C412yescomplemented
19C2×#475C412yescomplemented
20C2×#476C412yescomplemented
21C2×#474C412yescomplemented
22C2×#428C2212yes—
23C2×#423C2211yescharacteristic
24C2×#421C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#417C2212yes—
27C2×#410C2212yes—
28C2×C4×M16C2211yescharacteristic
29#2369—22no—
30#2365—22no—
31#2345—22no—
32Q8×C22⋊C4—22no—
33#2331—22no—
34#2316C2212yes—
35C4×C4⋊Q8—22no—
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38#2274C2212yes—
39C4×#117C2211yescharacteristic
40#1993—22no—
41#1990—22no—
42#1799—22no—
43#1799—22no—
44#1759—22no—
45C4×#104—22no—
46C2×#344C2212yes—
47C2×#344C2212yes—
48#1740C2212yes—
49#1736C2212yes—
50#1736C2212yes—
51#1733C2211yescharacteristic
52#1733C2211yescharacteristic
53C2×H(3,ℤ/4)C2212yes—
54C2×H(3,ℤ/4)C2212yes—
55C22×#41C2311yescharacteristic
56C23×Q8—21no—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2312yes—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8C2×C412yes—
67C2×C4⋊Q8C2×C412yes—
68C2×C4⋊Q8D412yes—
69C2×C4⋊Q8D412yes—
70C2×C4⋊Q8D412yes—
71C2×C4⋊Q8D412yes—
72C4×#41—21no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41—21no—
80C2×C4×Q8—24no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×#117—21no—
88C2×#117C2312yes—
89C2×#117C2311yescharacteristic
90C2×#117C2312yes—
91C2×#117C2311yescharacteristic
92C2×#117C2311yescharacteristic
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2312yes—
95C22×C42—21no—
96C22×C42C2311yescharacteristic
97#477—24no—
98#477—24no—
99#476—24no—
100#476—24no—
101#475—24no—
102#475—24no—
103#474—24no—
104#474—24no—
105#428—24no—
106#428—24no—
107C2×#107C2×C414yes—
108C2×#106C2×C414yes—
109#423—21no—
110#423—21no—
111#423—21no—
112#423—21no—
113C2×C4⋊C8C2×C412yes—
114C2×C4⋊C8C2×C412yes—
115#421—22no—
116#421—22no—
117#421—22no—
118#421Q814yescomplemented
119C2×#104—22no—
120C2×#104—22no—
121#417—24no—
122#417—24no—
123C2×#103C2×C414yes—
124C2×#102C2×C414yes—
125#410—24no—
126#410—42no—
127C2×#99—24no—
128C4×M16—21no—
129C4×M16—21no—
130C4×M16—21no—
131C4×M16—21no—
132C2×#97—21no—
133C2×C4×C8—21no—
134#391—24no—
135#391—28no—
136#389—24no—
137#389—24no—
138#389—24no—
139#387—28no—
140#384—21no—
141#384—21no—
142#382—21no—
143#382—21no—
144#380—24no—
145#380—21no—
146#380—21no—
147#380D412yes—
148#380D412yes—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—24no—
161#376D412yes—
162C43D412yes—
163#344—24no—
164#344—42no—
165#342—24no—
166#340—24no—
167#340—24no—
168#339—22no—
169#339—22no—
170H(3,ℤ/4)—24no—
171H(3,ℤ/4)—42no—
172C2×#41—21no—
173C2×#41—21no—
174C2×#41—21no—
175C2×#41—21no—
176C2×#41—21no—
177C2×#41—21no—
178C2×#41—21no—
179C2×#41—21no—
180C2×#41—21no—
181C2×#41—21no—
182C2×#41C2×Q812yes—
183C2×#41#4112yes—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—84no—
191C22×Q8C22×C411yescharacteristic
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C23×C4—22no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—22no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16C2×Q812yes—
218C2×M16#4112yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8C22×C414yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8C22⋊C414yes—
235C4⋊Q8C22⋊C414yes—
236C4⋊Q8C22⋊C414yes—
237C4⋊Q8C22⋊C414yes—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—88no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263#117—21no—
264#117—21no—
265#117—21no—
266#117—21no—
267#117—21no—
268#117—21no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281#117—41no—
282#117—41no—
283#117—41no—
284#117—41no—
285#117C2×Q812yes—
286#117#4112yes—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—41no—
306C2×C4⋊C4—41no—
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4C2×D414yes—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—28no—
327C2×C42—41no—
328C2×C42—41no—
329C2×C42—41no—
330C2×C42—41no—
331C2×C42C22×C411yescharacteristic
332C2×C42C22×C411yescharacteristic
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42#4111yescharacteristic
342C2×C42#4111yescharacteristic
343C2×C42#4111yescharacteristic
344C2×C42#4111yescharacteristic
345#107—28no—
346#106—28no—
347C4⋊C8—22no—
348C4⋊C8—22no—
349C4⋊C8—22no—
350C4⋊C8—22no—
351#104—28no—
352#104—28no—
353#103—28no—
354#102—28no—
355#99—48no—
356#97—41no—
357#97—41no—
358C4×C8—41no—
359C4×C8—41no—
360#95—28no—
361#95—28no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—41no—
367#95—41no—
368C24—21no—
369C24C22×D411yescharacteristic
370#41—22no—
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386C2×Q8—21no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—48no—
396C2×Q8—84no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8C4×Q812yescomplemented
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4C4×Q812yescomplemented
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—84no—
439C22×C4C4×D411yescharacteristic
440C22×C4C4×D411yescharacteristic
441C22×C4C4×D411yescharacteristic
442C22×C4C4×D411yescharacteristic
443C22×C4#12011yescharacteristic
444C22×C4#12011yescharacteristic
445C22×C4#12011yescharacteristic
446C22×C4#12011yescharacteristic
447C22×C4#12111yescharacteristic
448C22×C4#12111yescharacteristic
449C22×C4#12111yescharacteristic
450C22×C4#12111yescharacteristic
451C22×C4C23×C411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×Q811yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4H(5,𝔽2)11yescharacteristic
462C22×C42−1+411yescharacteristic
463C22×C42−1+411yescharacteristic
464C22×C42−1+411yescharacteristic
465M16—22no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—41no—
470M16—41no—
471M16—42no—
472M16—42no—
473M16—42no—
474M16—42no—
475M16—41no—
476M16—41no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—82no—
512C4⋊C4—82no—
513C4⋊C4—88no—
514C4⋊C4C2×C22⋊C418yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—24no—
518C22⋊C4—24no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—42no—
528C22⋊C4—42no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—44no—
533C42—44no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—42no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—42no—
543C42—42no—
544C42—88no—
545C42C2×C22⋊C411yescharacteristic
546C42C2×C22⋊C411yescharacteristic
547C42C2×C22⋊C411yescharacteristic
548C42C2×C22⋊C411yescharacteristic
549C42C4×D411yescharacteristic
550C42C4×D411yescharacteristic
551C42C4×D411yescharacteristic
552C42C4×D411yescharacteristic
553C42#12111yescharacteristic
554C42#12111yescharacteristic
555C42#12111yescharacteristic
556C42#12111yescharacteristic
557C42C22⋊Q814yes—
558C42#12311yescharacteristic
559C42#12311yescharacteristic
560C42#12311yescharacteristic
561C42#12311yescharacteristic
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—81no—
568C23—81no—
569C23C2×C4×D411yescharacteristic
570C23C2×#12011yescharacteristic
571C23C2×#12111yescharacteristic
572C23#54611yescharacteristic
573C23#54711yescharacteristic
574C23#54811yescharacteristic
575C23D4×Q811yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—48no—
579Q8—41no—
580Q8—41no—
581Q8—48no—
582Q8—84no—
583Q8—84no—
584D4—22no—
585D4—22no—
586D4—42no—
587D4—42no—
588D4—42no—
589D4—42no—
590D4—41no—
591D4—41no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4—82no—
629C2×C4—82no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4#38611yescharacteristic
636C2×C4#38611yescharacteristic
637C2×C4#38611yescharacteristic
638C2×C4#38611yescharacteristic
639C2×C4#38611yescharacteristic
640C2×C4#38611yescharacteristic
641C2×C4#38611yescharacteristic
642C2×C4#38611yescharacteristic
643C2×C4C22×C22⋊C411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×C4×Q811yescharacteristic
646C2×C4C4×#4111yescharacteristic
647C2×C4#51911yescharacteristic
648C2×C4#52011yescharacteristic
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×C22⋊Q811yescharacteristic
651C2×C4C2×#12311yescharacteristic
652C2×C4#52511yescharacteristic
653C2×C4#52911yescharacteristic
654C2×C4#53311yescharacteristic
655C2×C4#53611yescharacteristic
656C2×C4#53711yescharacteristic
657C2×C4#54111yescharacteristic
658C2×C4#54211yescharacteristic
659C2×C4#54811yescharacteristic
660C2×C4#54811yescharacteristic
661C2×C4#54811yescharacteristic
662C2×C4#54811yescharacteristic
663C2×C4D4×Q811yescharacteristic
664C2×C4D4×Q811yescharacteristic
665C2×C4#55011yescharacteristic
666C2×C4#55011yescharacteristic
667C2×C4#55211yescharacteristic
668C2×C4#55211yescharacteristic
669C2×C4#55411yescharacteristic
670C2×C4#55611yescharacteristic
671C8—44no—
672C8—42no—
673C8—42no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—42no—
685C22—82no—
686C22—82no—
687C22—81no—
688C22—81no—
689C22C2×#38611yescharacteristic
690C22#236011yescharacteristic
691C22#236211yescharacteristic
692C22#240711yescharacteristic
693C22#240911yescharacteristic
694C22#242611yescharacteristic
695C22#242911yescharacteristic
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—44no—
703C4—48no—
704C4—42no—
705C4—82no—
706C4—84no—
707C4—81no—
708C4—81no—
709C4—162no—
710C4#190412yes—
711C4#190512yes—
712C4C4×C22⋊Q811yescharacteristic
713C4#232311yescharacteristic
714C4Q8×C22⋊C411yescharacteristic
715C4#234111yescharacteristic
716C4#242111yescharacteristic
717C4#242311yescharacteristic
718C4#253611yescharacteristic
719C4#253811yescharacteristic
720C2—21no—
721C2—21no—
722C2—21no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—82no—
727C2#1396911yescharacteristic
728C2C2×#190411yescharacteristic
729C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #21324C2semidirect
6. #11213C2semidirect
9. #10903C2semidirect
10. #10855C2semidirect
11. #10836C2semidirect
18. C2×#477C4semidirect
19. C2×#475C4semidirect
20. C2×#476C4semidirect
21. C2×#474C4semidirect
118. #421Q8semidirect
401. C2×Q8C4×Q8semidirect
413. C2×D4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#342162
#213242
#206322
C4×#4212
#112132
#110572
#109542
#109032
#108552
#108362