Finite Group Databasebeta

← #495417

Subgroups — #495417

Order 512.

1229 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
1#35172C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13300C214yesmaximal, complemented
5#13299C212yesmaximal, complemented
6#13298C212yesmaximal, complemented
7#12582C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#99C2212yes—
11C22×#99C2212yes—
12#2848—21no—
13#2848—21no—
14#2626—21no—
15#2626—21no—
16#2580—21no—
17#2580—21no—
18#2566C412yescomplemented
19#2566C2212yes—
20#2563C411yescharacteristic, complemented
21#2563C2211yescharacteristic
22#2542C411yescharacteristic, complemented
23#2542C2211yescharacteristic
24#2538—21no—
25#2538—21no—
26#2317C412yescomplemented
27#2309C2212yes—
28#2285C411yescharacteristic, complemented
29C2×#380—21no—
30C2×C4×C22⋊C4—21no—
31#1799—21no—
32#1799—21no—
33#1799C2212yes—
34#1799C2212yes—
35C4×#99—21no—
36C4×#99—21no—
37C4×#99C2212yes—
38C4×#99C2212yes—
39C2×#342—21no—
40C2×#342—21no—
41#1393C2218yes—
42#1393C2218yes—
43C23×D4C2311yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46#533—41no—
47#533—41no—
48#533—41no—
49#533—41no—
50C2×C4⋊Q8C2×C411yescharacteristic, complemented
51C2×C4⋊Q8C2×C411yescharacteristic, complemented
52C2×#125C2×C412yescomplemented
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4—41no—
64C2×C4×D4—41no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C412yes—
74C2×#117C2×C412yes—
75C22×C4⋊C4—41no—
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4D411yescharacteristic
81C22×C42—41no—
82C22×C42C2×C411yescharacteristic
83C2×#99—22no—
84C2×#99—24no—
85C2×#99—28no—
86C2×#99—22no—
87C2×#99Q818yescomplemented
88C2×#99C2314yes—
89C2×#99C2314yes—
90#399—21no—
91#399—21no—
92#399—21no—
93#399—21no—
94#396—22no—
95#396—22no—
96#396—22no—
97#396—22no—
98#393—21no—
99#393—21no—
100#393—21no—
101#393—21no—
102#390—22no—
103#390—22no—
104#390C2×C412yes—
105#390C2×C412yes—
106#390C2312yes—
107#390C2312yes—
108#389—22no—
109#389—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#384—21no—
117#384C2×C412yescomplemented
118#384C2×C412yescomplemented
119#382—21no—
120#382C2×C412yescomplemented
121#382C2×C412yescomplemented
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380—22no—
129#380—22no—
130#380—41no—
131#380—41no—
132#379—22no—
133#379—22no—
134C4×C4⋊C4C2×C411yescharacteristic, complemented
135C4×C4⋊C4C2×C411yescharacteristic, complemented
136C4×C4⋊C4C2×C412yescomplemented
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—41no—
140C4×C22⋊C4—41no—
141C2×#95—41no—
142C2×#95—41no—
143#342—22no—
144#342—22no—
145#342—22no—
146#342—22no—
147C25C2×D411yescharacteristic
148C25C2×D411yescharacteristic
149C22×Q8—21no—
150C22×Q8—21no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—42no—
154C22×D4C22×C411yescharacteristic
155C22×D4C2×D411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2×Q814yes—
158C22×D4#4114yes—
159C22×D4C2411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—82no—
171C23×C4—82no—
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C4⋊Q8—41no—
186C4⋊Q8—41no—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189#125—42no—
190#125—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C22⋊Q8—41no—
196C22⋊Q8—41no—
197C22⋊Q8—82no—
198C22⋊Q8—82no—
199C22⋊Q8—82no—
200C22⋊Q8—82no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—44no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—84no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—84no—
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C412yes—
290C2×C42C22×C412yes—
291#99—28no—
292#99—24no—
293#99—28no—
294#99—44no—
295#99C2×Q818yes—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24—81no—
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C22⋊Q814yes—
321C24#12412yes—
322C24C4⋊Q812yes—
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—82no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—41no—
343C2×D4C4×Q814yes—
344C2×D4C22⋊Q814yes—
345C2×D4#12412yes—
346C2×D4C4⋊Q812yes—
347C2×D4C22×Q812yes—
348C2×D4C2×#4112yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—44no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—81no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4#9911yescharacteristic
407C22×C4#9911yescharacteristic
408C22×C4#9911yescharacteristic
409C22×C4#9911yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C4×Q814yes—
420C22×C4C23×C411yescharacteristic
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C22⋊C4—28no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—44no—
453C22⋊C4—48no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—42no—
478C42—44no—
479C42—84no—
480C42—84no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—22no—
487C23—22no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—81no—
510C23—81no—
511C23—81no—
512C23—81no—
513C23—84no—
514C23—82no—
515C23—82no—
516C23—84no—
517C23—84no—
518C23#39112yes—
519C23#39112yes—
520C23#39112yes—
521C23C2×#9911yescharacteristic
522C23C2×#9911yescharacteristic
523C23#41012yes—
524C23C22×C22⋊C411yescharacteristic
525C23C2×C22⋊Q812yes—
526C23C2×#12411yescharacteristic
527C23C2×C4⋊Q811yescharacteristic
528Q8—82no—
529Q8—84no—
530Q8—82no—
531D4—44no—
532D4—44no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—42no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—84no—
597C2×C4—816no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—168no—
603C2×C4#39112yes—
604C2×C4C2×#9911yescharacteristic
605C2×C4C2×#9911yescharacteristic
606C2×C4C2×#9911yescharacteristic
607C2×C4C2×#9911yescharacteristic
608C2×C4C2×C4×Q812yes—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—22no—
616C22—21no—
617C22—21no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—41no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22#199612yes—
646C22#199612yes—
647C22C2×#39111yescharacteristic
648C22C22×#9911yescharacteristic
649C22C2×#41011yescharacteristic
650C4—21no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—82no—
662C4—82no—
663C4—88no—
664C4—82no—
665C4—82no—
666C4—164no—
667C4#212712yes—
668C2—21no—
669C2—21no—
670C2—22no—
671C2—22no—
672C2—42no—
673C2—41no—
674C2—41no—
675C2—42no—
676C2—42no—
677C2C2×#199611yescharacteristic
678C2C2×#212711yescharacteristic
679C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13300C2semidirect
5. #13299C2semidirect
6. #13298C2semidirect
8. C2×#533C4semidirect
18. #2566C4semidirect
20. #2563C4semidirect
22. #2542C4semidirect
26. #2317C4semidirect
28. #2285C4semidirect
50. C2×C4⋊Q8C2×C4semidirect
51. C2×C4⋊Q8C2×C4semidirect
52. C2×#125C2×C4semidirect
87. C2×#99Q8semidirect
117. #384C2×C4semidirect
118. #384C2×C4semidirect
120. #382C2×C4semidirect
121. #382C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
135. C4×C4⋊C4C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#17992
C2×C4×#992
#133002
#132992
#132982
#125822