Finite Group Databasebeta

← #364809

Subgroups — #364809

Order 512.

1424 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3C2×#1997C211yesmaximal, characteristic, complemented
4#20654C211yesmaximal, characteristic
5C4×#421C211yesmaximal, characteristic
6#11224C212yesmaximal, complemented
7#11072C212yesmaximal
8#10949C212yesmaximal
9#10888C212yesmaximal, complemented
10#10837C211yesmaximal, characteristic, complemented
11#10836C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#421C2211yescharacteristic
16C2×#421C2212yes—
17C2×#421C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19#2857—22no—
20#2628—22no—
21#2568C412yescomplemented
22#2567C412yescomplemented
23#2563C412yescomplemented
24#2542C412yescomplemented
25#2487—22no—
26#2409—22no—
27#2369—22no—
28#2365—22no—
29#2345—22no—
30Q8×C22⋊C4—22no—
31#2331—22no—
32#2317C2212yes—
33#2316C2212yes—
34#2311C2212yes—
35C4×C4⋊Q8—22no—
36#2285C2211yescharacteristic
37#2285C2211yescharacteristic
38#2274C2212yes—
39C4×#117C2211yescharacteristic
40#1997—22no—
41#1997—22no—
42#1806—22no—
43#1806—22no—
44#1759—22no—
45C4×#104—22no—
46C2×#343C2212yes—
47C2×#343C2212yes—
48#1733C2211yescharacteristic
49#1733C2212yes—
50#1733C2211yescharacteristic
51C2×H(3,ℤ/4)C2212yes—
52C2×H(3,ℤ/4)C2212yes—
53C2×H(3,ℤ/4)C2212yes—
54C2×H(3,ℤ/4)C2212yes—
55C22×#41C2311yescharacteristic
56C23×Q8D412yes—
57C22×M16C2312yes—
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
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⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2×C412yes—
69C2×C4⋊Q8C2×C412yes—
70C2×C4⋊Q8D412yes—
71C2×C4⋊Q8D412yes—
72C2×C22⋊Q8—24no—
73C4×#41—21no—
74C4×#41—21no—
75C4×#41—21no—
76C4×#41—21no—
77C4×#41—21no—
78C4×#41—21no—
79C4×#41—21no—
80C4×#41—21no—
81C2×C4×Q8—24no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×#117—21no—
89C2×#117C2311yescharacteristic
90C2×#117C2312yes—
91C2×#117C2311yescharacteristic
92C2×#117C2312yes—
93C2×#117C2311yescharacteristic
94C22×C4⋊C4D412yes—
95C22×C4⋊C4C2312yes—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98#423—21no—
99#423—21no—
100#423—21no—
101#423—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104#421—22no—
105#421—22no—
106#421—22no—
107#421—24no—
108#421—42no—
109#421Q814yescomplemented
110C2×#104—22no—
111C2×#104—22no—
112C2×#104—24no—
113C4×M16—21no—
114C4×M16—21no—
115C4×M16—21no—
116C4×M16—21no—
117C2×#97—21no—
118C2×C4×C8—21no—
119#400—24no—
120#399—24no—
121#395—24no—
122#393—24no—
123#391—24no—
124#391—28no—
125#391—24no—
126#391C2×C414yes—
127#390C2×C414yes—
128#389—24no—
129#389—24no—
130#389—24no—
131#389—24no—
132#389—24no—
133#387—28no—
134#386—24no—
135#385—24no—
136#384—21no—
137#384—21no—
138#384C2×C414yes—
139#382—21no—
140#382—21no—
141#382C2×C414yes—
142#380—24no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—24no—
148#380—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C412yes—
155C4×C4⋊C4C2×C412yes—
156C4×C4⋊C4D412yes—
157C4×C4⋊C4D412yes—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—21no—
161#376D412yes—
162C43D412yes—
163#343—24no—
164#343—24no—
165#339—24no—
166#339—22no—
167#339—22no—
168H(3,ℤ/4)—24no—
169H(3,ℤ/4)—24no—
170H(3,ℤ/4)—42no—
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—84no—
189C22×Q8C22⋊C414yes—
190C22×Q8C22⋊C414yes—
191C22×Q8C22×C411yescharacteristic
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C2×D414yes—
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2411yescharacteristic
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16C2×Q812yes—
218C2×M16#4112yes—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—22no—
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⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C22⋊Q8—88no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—88no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—41no—
263C4×D4—41no—
264#117—21no—
265#117—21no—
266#117—22no—
267#117—22no—
268#117—22no—
269#117—21no—
270#117—21no—
271#117—22no—
272#117—22no—
273#117—21no—
274#117—22no—
275#117—22no—
276#117—21no—
277#117—22no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—41no—
283#117—41no—
284#117—41no—
285#117—41no—
286#117C2×Q812yes—
287#117#4112yes—
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—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4C22⋊C414yes—
309C2×C4⋊C4C22⋊C414yes—
310C2×C4⋊C4C22×C411yescharacteristic
311C2×C4⋊C4C22×C411yescharacteristic
312C2×C4⋊C4C22×C414yes—
313C2×C4⋊C4C2×D414yes—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—28no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42—28no—
328C2×C42—41no—
329C2×C42—41no—
330C2×C42—41no—
331C2×C42—41no—
332C2×C42C22×C414yes—
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D414yes—
337C2×C42C2×D412yes—
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42#4111yescharacteristic
345C2×C42#4111yescharacteristic
346C2×C42#4111yescharacteristic
347C2×C42#4111yescharacteristic
348C4⋊C8—41no—
349C4⋊C8—41no—
350C4⋊C8—41no—
351C4⋊C8—41no—
352#104—28no—
353#104—28no—
354#104—48no—
355#97—41no—
356#97—41no—
357C4×C8—41no—
358C4×C8—41no—
359#95—28no—
360#95—28no—
361#95—22no—
362#95—22no—
363#95—28no—
364#95—28no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369C24—21no—
370C24C22×D411yescharacteristic
371#41—22no—
372#41—22no—
373#41—22no—
374#41—22no—
375#41—22no—
376#41—22no—
377#41—22no—
378#41—22no—
379#41—41no—
380#41—41no—
381#41—41no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41—41no—
387C2×Q8—21no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×Q8C4×Q812yescomplemented
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—41no—
414C2×D4C4×Q812yescomplemented
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—84no—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C418yes—
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4#12011yescharacteristic
445C22×C4#12011yescharacteristic
446C22×C4#12011yescharacteristic
447C22×C4#12011yescharacteristic
448C22×C4#12311yescharacteristic
449C22×C4#12311yescharacteristic
450C22×C4#12311yescharacteristic
451C22×C4#12311yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×Q811yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4C2×#4111yescharacteristic
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4H(5,𝔽2)11yescharacteristic
463C22×C42−1+411yescharacteristic
464C22×C42−1+411yescharacteristic
465C22×C42−1+411yescharacteristic
466M16—22no—
467M16—22no—
468M16—22no—
469M16—22no—
470M16—41no—
471M16—41no—
472M16—41no—
473M16—41no—
474M16—42no—
475M16—42no—
476M16—42no—
477M16—42no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—42no—
485C2×C8—42no—
486C2×C8—42no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—42no—
522C22⋊C4—41no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—88no—
532C42—22no—
533C42—24no—
534C42—24no—
535C42—22no—
536C42—22no—
537C42—24no—
538C42—24no—
539C42—44no—
540C42—44no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—88no—
548C42C4×D411yescharacteristic
549C42C4×D411yescharacteristic
550C42C4×D411yescharacteristic
551C42C4×D411yescharacteristic
552C42C4×D411yescharacteristic
553C42C4×D411yescharacteristic
554C42C4×D411yescharacteristic
555C42C4×D411yescharacteristic
556C42#12111yescharacteristic
557C42#12111yescharacteristic
558C42#12111yescharacteristic
559C42#12111yescharacteristic
560C42#12111yescharacteristic
561C42#12111yescharacteristic
562C42#12111yescharacteristic
563C42#12111yescharacteristic
564C42C22⋊Q814yescomplemented
565C23—22no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23C22×C22⋊C411yescharacteristic
573C23C2×#12011yescharacteristic
574C23C2×#12311yescharacteristic
575C23#54811yescharacteristic
576C23#54811yescharacteristic
577C23D4×Q811yescharacteristic
578C23#55011yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—48no—
582Q8—41no—
583Q8—41no—
584Q8—48no—
585Q8—84no—
586Q8—84no—
587D4—22no—
588D4—22no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—41no—
594D4—41no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4#38611yescharacteristic
639C2×C4#38611yescharacteristic
640C2×C4#38611yescharacteristic
641C2×C4#38611yescharacteristic
642C2×C4#38611yescharacteristic
643C2×C4#38611yescharacteristic
644C2×C4#38611yescharacteristic
645C2×C4#38611yescharacteristic
646C2×C4C2×C4×D411yescharacteristic
647C2×C4C2×C4×D411yescharacteristic
648C2×C4C2×C4×Q811yescharacteristic
649C2×C4C4×#4111yescharacteristic
650C2×C4#51911yescharacteristic
651C2×C4#52011yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×C22⋊Q811yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4#52911yescharacteristic
657C2×C4#53311yescharacteristic
658C2×C4#53611yescharacteristic
659C2×C4#53711yescharacteristic
660C2×C4#54111yescharacteristic
661C2×C4#54211yescharacteristic
662C2×C4#54611yescharacteristic
663C2×C4#54711yescharacteristic
664C2×C4#54811yescharacteristic
665C2×C4#54811yescharacteristic
666C2×C4#54811yescharacteristic
667C2×C4D4×Q811yescharacteristic
668C2×C4D4×Q811yescharacteristic
669C2×C4#55011yescharacteristic
670C2×C4#55211yescharacteristic
671C2×C4#55211yescharacteristic
672C2×C4#55411yescharacteristic
673C2×C4#55611yescharacteristic
674C8—42no—
675C8—42no—
676C8—82no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—44no—
688C22—42no—
689C22—82no—
690C22—81no—
691C22—81no—
692C22C2×#38611yescharacteristic
693C22Q8×C22⋊C411yescharacteristic
694C22#234111yescharacteristic
695C22#240711yescharacteristic
696C22#240911yescharacteristic
697C22#242111yescharacteristic
698C22#242311yescharacteristic
699C4—21no—
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—21no—
705C4—44no—
706C4—48no—
707C4—42no—
708C4—82no—
709C4—84no—
710C4—84no—
711C4—81no—
712C4—81no—
713C4#191112yes—
714C4#191212yes—
715C4C4×C22⋊Q811yescharacteristic
716C4#232311yescharacteristic
717C4#236011yescharacteristic
718C4#236211yescharacteristic
719C4#242611yescharacteristic
720C4#242911yescharacteristic
721C4#253611yescharacteristic
722C4#253811yescharacteristic
723C2—21no—
724C2—21no—
725C2—21no—
726C2—21no—
727C2—21no—
728C2—21no—
729C2—82no—
730C2#1396911yescharacteristic
731C2C2×#191111yescharacteristic
732C2C2×#191211yescharacteristic
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. C2×#1997C2semidirect
6. #11224C2semidirect
9. #10888C2semidirect
10. #10837C2semidirect
11. #10836C2semidirect
21. #2568C4semidirect
22. #2567C4semidirect
23. #2563C4semidirect
24. #2542C4semidirect
109. #421Q8semidirect
402. C2×Q8C4×Q8semidirect
414. C2×D4C4×Q8semidirect
564. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
#352792
#342162
C2×#19972
#206542
C4×#4212
#112242
#110722
#109492
#108882
#108372
#108362