Finite Group Databasebeta

← #495397

Subgroups — #495397

Order 512.

1191 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3C2×C4×#100C211yesmaximal, characteristic
4#13294C214yesmaximal, complemented
5#13293C212yesmaximal, complemented
6#13292C212yesmaximal, complemented
7#12574C214yesmaximal
8C22×C22⋊Q8C411yescharacteristic, complemented
9C22×C4×D4C2211yescharacteristic
10C22×#100C2212yes—
11C22×#99C2212yes—
12C2×C4×M16—21no—
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×#380—21no—
30C2×#380C411yescharacteristic, complemented
31#1799—21no—
32#1799—21no—
33#1799C2212yes—
34#1799C2212yes—
35C4×#100—21no—
36C4×#100—21no—
37C4×#100C2212yes—
38C4×#100C2212yes—
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⋊C4—41no—
72C22×C4⋊C4C2×C412yescomplemented
73C22×C4⋊C4C2×C411yescharacteristic, complemented
74C22×C4⋊C4D411yescharacteristic
75C22×C22⋊C4—22no—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42D411yescharacteristic
81C2×#100—22no—
82C2×#100—24no—
83C2×#100Q818yescomplemented
84C2×#100C2314yes—
85C2×#99—28no—
86C2×#99—22no—
87C2×#99C2314yes—
88C4×M16—22no—
89C4×M16—22no—
90C4×M16—41no—
91C4×M16—41no—
92C2×#97—41no—
93C2×C4×C8—41no—
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#380—22no—
134#380—22no—
135#380—41no—
136#380—41no—
137#380C2×C412yescomplemented
138#380C2×C411yescharacteristic, complemented
139#380C2×C411yescharacteristic, complemented
140#379—22no—
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—28no—
191C2×M16—24no—
192C2×M16—44no—
193C2×M16—42no—
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—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—22no—
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—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22×C412yes—
271C2×C22⋊C4C22×C412yes—
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284#100—28no—
285#100—24no—
286#100C2×Q818yes—
287#99—28no—
288#99—44no—
289#97—42no—
290#97—42no—
291C4×C8—42no—
292C4×C8—42no—
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—
320C24#12412yes—
321C24C4⋊Q812yes—
322C24C23×C411yescharacteristic
323C24C22×D411yescharacteristic
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—82no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4C4×Q814yes—
345C2×D4C22⋊Q814yes—
346C2×D4#12412yes—
347C2×D4C4⋊Q812yes—
348C2×D4C22×Q812yes—
349C2×D4C2×#4112yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
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—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—44no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
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—84no—
410C22×C4#9911yescharacteristic
411C22×C4#9911yescharacteristic
412C22×C4#9911yescharacteristic
413C22×C4#9911yescharacteristic
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×C4C2×C22⋊C411yescharacteristic
423C22×C4C22⋊Q814yes—
424C22×C4C22×D411yescharacteristic
425M16—28no—
426M16—44no—
427M16—48no—
428C2×C8—44no—
429C2×C8—48no—
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—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—44no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—44no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—82no—
468C22⋊C4—82no—
469C22⋊C4—84no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—81no—
505C23—81no—
506C23—81no—
507C23—81no—
508C23—81no—
509C23—81no—
510C23—82no—
511C23—82no—
512C23—84no—
513C23—84no—
514C23—84no—
515C23#39112yes—
516C23#39112yes—
517C23#39112yes—
518C23C2×#9911yescharacteristic
519C23C2×#9911yescharacteristic
520C23#41012yes—
521C23C22×C22⋊C411yescharacteristic
522C23C2×C4×Q812yes—
523C23C2×#12411yescharacteristic
524C23C2×C4⋊Q811yescharacteristic
525Q8—82no—
526Q8—84no—
527Q8—82no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
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—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
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—42no—
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—81no—
592C2×C4—84no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—164no—
598C2×C4—168no—
599C2×C4#39112yes—
600C2×C4C2×#9911yescharacteristic
601C2×C4C2×#9911yescharacteristic
602C2×C4C2×#9911yescharacteristic
603C2×C4C2×#9911yescharacteristic
604C2×C4C2×C22⋊Q812yes—
605C8—48no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—22no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—81no—
637C22—81no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22#199612yes—
643C22#199612yes—
644C22C2×#39111yescharacteristic
645C22C22×#9911yescharacteristic
646C22C2×#41011yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—164no—
663C4#212112yes—
664C2—21no—
665C2—21no—
666C2—22no—
667C2—22no—
668C2—41no—
669C2—41no—
670C2—42no—
671C2—42no—
672C2—42no—
673C2C2×#199611yescharacteristic
674C2C2×#212111yescharacteristic
675C2#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. #13294C2semidirect
5. #13293C2semidirect
6. #13292C2semidirect
8. C22×C22⋊Q8C4semidirect
17. #2579C4semidirect
21. #2487C4semidirect
25. #2409C4semidirect
27. #2315C4semidirect
30. C2×#380C4semidirect
43. C23×Q8C2×C4semidirect
72. C22×C4⋊C4C2×C4semidirect
73. C22×C4⋊C4C2×C4semidirect
83. C2×#100Q8semidirect
122. #385C2×C4semidirect
123. #385C2×C4semidirect
124. #385C2×C4semidirect
125. #385C2×C4semidirect
137. #380C2×C4semidirect
138. #380C2×C4semidirect
139. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#17992
C2×C4×#1002
#132942
#132932
#132922
#125742