Finite Group Databasebeta

← #393544

Subgroups — #393544

Order 512.

1806 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#2274C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5#14258C211yesmaximal, characteristic
6#10932C212yesmaximal, complemented
7#10932C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#485—24no—
17C2×#485—24no—
18C2×#485C418yescomplemented
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×#391C2212yes—
27C2×#380C2211yescharacteristic
28#2274—24no—
29#2274—24no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4—21no—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4—22no—
35#1872—24no—
36#1825—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1441—21no—
46#1440—22no—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8C2×C414yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—24no—
66C2×#117—24no—
67C2×#117C2×C414yes—
68C2×#117D414yes—
69C2×#117D414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D414yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D412yes—
87C22×C42—21no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42C2312yes—
91#485—232no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107C2×C414yescomplemented
95C2×#106—22no—
96C2×#106—22no—
97C2×#106C2×C414yescomplemented
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—24no—
105C2×#103C2×C414yescomplemented
106C2×#103C2×C414yescomplemented
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yes—
113#391—24no—
114#391—24no—
115#380—22no—
116#380—22no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—28no—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—24no—
123C4×C22⋊C4—216no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—28no—
157C23×C4—24no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4—81no—
162C23×C4—82no—
163C23×C4—81no—
164C23×C4C22×C412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D412yes—
171C23×C4#4112yes—
172C23×C4#4112yes—
173C23×C4C2411yescharacteristic
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8C22×C412yes—
190C22×C8C22×C412yes—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C22⋊Q8—44no—
194C22⋊Q8—44no—
195#117—28no—
196#117—28no—
197#117—216no—
198#117—28no—
199#117—28no—
200#117—28no—
201#117C22⋊C4116yes—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—216no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4#4114yes—
243C2×C4⋊C4#4114yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—81no—
255C2×C22⋊C4—81no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—82no—
266C2×C42—82no—
267C2×C42—82no—
268C2×C42—82no—
269C2×C42#4114yes—
270C2×C42#4114yes—
271#107—28no—
272#106—28no—
273#103—28no—
274#103—28no—
275#103—48no—
276C22⋊C8—28no—
277C22⋊C8—44no—
278#95—28no—
279#95—28no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—82no—
284#95—82no—
285C24—22no—
286C24—22no—
287C24—21no—
288C24—21no—
289C24—22no—
290C24—21no—
291C24—22no—
292C24—21no—
293C24C4×D412yes—
294C24#12012yes—
295C24#12112yes—
296C24C22×D411yescharacteristic
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—216no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4C2×C22⋊C414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D412yes—
389C22×C4#12012yes—
390C22×C4#12114yes—
391C22×C4#12112yes—
392C22×C4#12314yes—
393C22×C4#12314yes—
394C22×C4#12314yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4111yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—216no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—48no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4C2×C22⋊C414yes—
444C4⋊C4C2×C22⋊C414yes—
445C4⋊C4#12318yes—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—84no—
450C42—24no—
451C42—24no—
452C42—24no—
453C42—24no—
454C42—216no—
455C42—88no—
456C42—88no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—22no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23#38614yes—
480C23C2×C4×D411yescharacteristic
481C23C2×#12011yescharacteristic
482C23C2×#12111yescharacteristic
483C23C2×#12312yes—
484C23#52512yes—
485C23#57611yescharacteristic
486C23#57711yescharacteristic
487C23#57712yes—
488C23#57711yescharacteristic
489C23#57811yescharacteristic
490C23#57812yes—
491Q8—42no—
492Q8—42no—
493Q8—44no—
494Q8—81no—
495Q8—81no—
496Q8—82no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—816no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4#38614yes—
561C2×C4C22×C22⋊C411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C4×#4112yes—
564C2×C4C4×#4112yes—
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12311yescharacteristic
567C2×C4C2×#12312yes—
568C2×C4#52511yescharacteristic
569C2×C4#52511yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52512yes—
573C2×C4#52912yes—
574C2×C4#52912yes—
575C2×C4#52912yes—
576C2×C4#52912yes—
577C2×C4#53212yes—
578C2×C4#53212yes—
579C8—42no—
580C8—42no—
581C8—82no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22C2×#38611yescharacteristic
594C22C4×#12312yes—
595C22#284911yescharacteristic
596C22#284911yescharacteristic
597C22#285412yes—
598C22#294712yes—
599C22#295111yescharacteristic
600C22#295211yescharacteristic
601C22#300911yescharacteristic
602C22#301111yescharacteristic
603C22#301211yescharacteristic
604C22#301311yescharacteristic
605C22#304511yescharacteristic
606C22#304612yes—
607C22#304611yescharacteristic
608C22#309712yes—
609C22#309712yes—
610C22#309812yes—
611C22#309912yes—
612C22#311812yes—
613C22#312112yes—
614C22#312312yes—
615C22#312712yes—
616C4—24no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—48no—
622C4—82no—
623C4—82no—
624C4—162no—
625C4#229711yescharacteristic
626C4#229711yescharacteristic
627C4C4×#12312yes—
628C4#285311yescharacteristic
629C4#285311yescharacteristic
630C4#285412yes—
631C2—44no—
632C2#1470611yescharacteristic
633C2#2093111yescharacteristic
634C2#2093711yescharacteristic
635C2#2158912yes—
636C2#2159112yes—
637C2#3298811yescharacteristic
638C2#3298911yescharacteristic
639C2#3301811yescharacteristic
640C2#3301911yescharacteristic
641C2#3311512yes—
642C2#3311712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10932C2semidirect
7. #10932C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
18. C2×#485C4semidirect
94. C2×#107C2×C4semidirect
97. C2×#106C2×C4semidirect
105. C2×#103C2×C4semidirect
106. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4852
C2×#22742
C2×#18722
C2×#18252
#142582
#109322
#109082
#109072
#107302
#107132