Finite Group Databasebeta

← #495407

Subgroups — #495407

Order 512.

1203 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3C2×C4×#99C211yesmaximal, characteristic
4#13297C214yesmaximal, complemented
5#13296C212yesmaximal, complemented
6#13295C212yesmaximal, complemented
7#12574C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#423—21no—
11C22×#100C2212yes—
12C22×#99C2212yes—
13#2855—21no—
14#2848—21no—
15#2627—21no—
16#2626—21no—
17#2579C412yescomplemented
18#2579C2212yes—
19#2564—21no—
20#2564—21no—
21#2487C411yescharacteristic, complemented
22#2487C2211yescharacteristic
23#2417—21no—
24#2417—21no—
25#2409C411yescharacteristic, complemented
26#2409C2211yescharacteristic
27#2315C412yescomplemented
28#2308C2212yes—
29C2×#380C411yescharacteristic, complemented
30C2×C4×C22⋊C4—21no—
31#1800—21no—
32#1800—21no—
33#1800C2212yes—
34#1800C2212yes—
35C4×#99—21no—
36C4×#99—21no—
37C4×#99C2212yes—
38C4×#99C2212yes—
39C2×#343—21no—
40C2×#342—21no—
41#1390C2218yes—
42#1390C2218yes—
43C23×Q8C2×C411yescharacteristic, complemented
44C23×D4C2311yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8—41no—
57C2×C22⋊Q8—41no—
58C2×C4×D4—41no—
59C2×C4×D4—41no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C412yescomplemented
72C22×C4⋊C4C2×C411yescharacteristic, complemented
73C22×C4⋊C4D411yescharacteristic
74C22×C22⋊C4—22no—
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—41no—
80C22×C42D411yescharacteristic
81#423—22no—
82#423—22no—
83#423—41no—
84#423—41no—
85C2×C4⋊C8—41no—
86C2×C4⋊C8—41no—
87C2×#100—28no—
88C2×#100—22no—
89C2×#100C2314yes—
90C2×#99—22no—
91C2×#99—24no—
92C2×#99Q818yescomplemented
93C2×#99C2314yes—
94#400—21no—
95#400—21no—
96#399—21no—
97#399—21no—
98#398—22no—
99#398—22no—
100#396—22no—
101#396—22no—
102#395—21no—
103#395—21no—
104#393—21no—
105#393—21no—
106#391—21no—
107#391—21no—
108#389—21no—
109#389—21no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#386C2×C412yes—
115#386C2×C412yes—
116#386C2312yes—
117#386C2312yes—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385C2×C412yescomplemented
123#385C2×C412yescomplemented
124#385C2×C412yescomplemented
125#385C2×C412yescomplemented
126#381—22no—
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131#380—22no—
132#380—22no—
133#380C2×C412yescomplemented
134#380C2×C411yescharacteristic, complemented
135#380C2×C411yescharacteristic, complemented
136#379—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C4×C22⋊C4—41no—
140C4×C22⋊C4—41no—
141C2×#95—21no—
142C2×#95—41no—
143#343—22no—
144#343—22no—
145#342—22no—
146#342—22no—
147C25C22×C411yescharacteristic
148C25C2×D411yescharacteristic
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8—82no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4C22×C411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C22×D4C2×Q814yes—
161C22×D4#4114yes—
162C22×D4C2411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—82no—
174C23×C4—82no—
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C412yes—
186C23×C4C22×C412yes—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—44no—
193C2×M16—44no—
194C22×C8—42no—
195C22⋊Q8—42no—
196C22⋊Q8—42no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C22⋊Q8—42no—
200C22⋊Q8—42no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205C22⋊Q8—82no—
206C22⋊Q8—82no—
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—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22×C412yes—
268C2×C22⋊C4C22×C412yes—
269C2×C22⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C22×C411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—84no—
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288#100—28no—
289#100—44no—
290#99—28no—
291#99—24no—
292#99C2×Q818yes—
293#95—22no—
294#95—21no—
295#95—21no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—42no—
305#95—42no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24—81no—
313C24—81no—
314C24—81no—
315C24—81no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C4×Q814yes—
320C24C22⋊Q814yes—
321C24C23×C411yescharacteristic
322C24C22×D411yescharacteristic
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—82no—
329C2×Q8—84no—
330C2×Q8—84no—
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—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
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—81no—
402C22×C4—81no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4#9911yescharacteristic
410C22×C4#9911yescharacteristic
411C22×C4#9911yescharacteristic
412C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4#12412yes—
423C22×C4C4⋊Q812yes—
424C22×C4C22×D411yescharacteristic
425M16—44no—
426M16—44no—
427M16—84no—
428C2×C8—44no—
429C2×C8—84no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—82no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C22⋊C4—28no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—44no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—84no—
477C42—84no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—24no—
485C23—21no—
486C23—21no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—84no—
509C23—84no—
510C23—84no—
511C23—84no—
512C23#39112yes—
513C23#39112yes—
514C23#39112yes—
515C23C2×#9911yescharacteristic
516C23C2×#9911yescharacteristic
517C23#41012yes—
518C23C22×C22⋊C411yescharacteristic
519C23C2×C4×Q812yes—
520C23C2×C22⋊Q812yes—
521Q8—82no—
522Q8—84no—
523Q8—82no—
524D4—44no—
525D4—44no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—42no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—48no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—84no—
592C2×C4—816no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—164no—
597C2×C4#39112yes—
598C2×C4C2×#9911yescharacteristic
599C2×C4C2×#9911yescharacteristic
600C2×C4C2×#9911yescharacteristic
601C2×C4C2×#9911yescharacteristic
602C2×C4C2×#12411yescharacteristic
603C2×C4C2×C4⋊Q811yescharacteristic
604C8—84no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—48no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—81no—
633C22—81no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22#199612yes—
638C22#199612yes—
639C22C2×#39111yescharacteristic
640C22C22×#9911yescharacteristic
641C22C2×#41011yescharacteristic
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—41no—
654C4—82no—
655C4—82no—
656C4—88no—
657C4—82no—
658C4—82no—
659C4#212112yes—
660C2—21no—
661C2—21no—
662C2—22no—
663C2—22no—
664C2—42no—
665C2—42no—
666C2—42no—
667C2—42no—
668C2C2×#199611yescharacteristic
669C2C2×#212111yescharacteristic
670C2#3334711yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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. #13297C2semidirect
5. #13296C2semidirect
6. #13295C2semidirect
8. C22×C22⋊Q8C4semidirect
17. #2579C4semidirect
21. #2487C4semidirect
25. #2409C4semidirect
27. #2315C4semidirect
29. C2×#380C4semidirect
43. C23×Q8C2×C4semidirect
71. C22×C4⋊C4C2×C4semidirect
72. C22×C4⋊C4C2×C4semidirect
92. C2×#99Q8semidirect
122. #385C2×C4semidirect
123. #385C2×C4semidirect
124. #385C2×C4semidirect
125. #385C2×C4semidirect
133. #380C2×C4semidirect
134. #380C2×C4semidirect
135. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#18002
C2×C4×#992
#132972
#132962
#132952
#125742