Finite Group Databasebeta

← #414759

Subgroups — #414759

Order 512.

1614 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×#501C211yesmaximal, characteristic, complemented
2C2×#2629C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic
4C2×#1991C211yesmaximal, characteristic, complemented
5C2×#1990C211yesmaximal, characteristic, complemented
6C2×#1750C211yesmaximal, characteristic
7#11328C212yesmaximal, complemented
8#11327C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10843C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#501—24no—
16C2×#501—24no—
17C2×#501—24no—
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×C22⋊C8C2212yes—
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2629—24no—
24#2629—24no—
25#2628—24no—
26#2628—24no—
27C2×#400—21no—
28C2×#399—21no—
29C2×#398—21no—
30C2×#398—21no—
31C2×#396—21no—
32C2×#396—21no—
33C2×#395—21no—
34C2×#393—21no—
35C2×#389C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43#1991—24no—
44#1990—24no—
45#1750—24no—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#340C2212yes—
50C2×#340C2212yes—
51C2×#340C2212yes—
52C2×#336C2212yes—
53C23×Q8—21no—
54C24×C4C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D414yes—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125—22no—
67C2×#125—22no—
68C2×#125D414yes—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—22no—
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4D411yescharacteristic
95C22×C22⋊C4D411yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#501—232no—
99C2×#107—24no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#106—24no—
103C2×#106—24no—
104C2×#106—24no—
105C2×C22⋊C8—24no—
106C2×C22⋊C8—24no—
107C2×C22⋊C8Q818yescomplemented
108C2×#97—22no—
109C2×#97—22no—
110C2×#97D414yes—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114#400—28no—
115#399—28no—
116#398—28no—
117#398—28no—
118#396—28no—
119#396—28no—
120#395—28no—
121#393—28no—
122#389—24no—
123#389—24no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#384—22no—
129#384D414yes—
130#382—22no—
131#382D414yes—
132#380—24no—
133#380—24no—
134#380—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—21no—
146#340—24no—
147#340—24no—
148#340—24no—
149#340—24no—
150#340—24no—
151#340—24no—
152#336—24no—
153C25C2×D411yescharacteristic
154C22×Q8—24no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—82no—
158C22×Q8—82no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×Q814yes—
180C23×C4C2411yescharacteristic
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×C8—22no—
190C22×C8—22no—
191C22×C8C2×D414yes—
192C22×C8C2×Q814yes—
193C22×C8C2×Q814yes—
194C22×C8#4114yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199#125—28no—
200#125—28no—
201#125—44no—
202#125—44no—
203#117—44no—
204#117—44no—
205#117—44no—
206#117—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—82no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—82no—
256C2×C4⋊C4—82no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4—82no—
277C2×C22⋊C4—82no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C2×D412yes—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42#4112yes—
290#107—216no—
291#106—216no—
292C22⋊C8—216no—
293#97—28no—
294C4×C8—44no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24C4⋊Q812yes—
314C24C22×D411yescharacteristic
315C2×Q8—48no—
316C2×Q8—88no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8—88no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4#12012yes—
391C22×C4#12012yes—
392C22×C4#12012yes—
393C22×C4#12012yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4C4⋊Q812yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×Q812yes—
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4112yes—
405C22×C42−1+412yes—
406C22×C42−1+412yes—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8C22⋊Q818yes—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—88no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42#12014yes—
461C42#12114yes—
462C42#12114yes—
463C42#12714yes—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23C2×#12011yescharacteristic
488C23C2×#12011yescharacteristic
489C23C2×#12111yescharacteristic
490C23C2×C4⋊Q811yescharacteristic
491C23#53211yescharacteristic
492C23#53611yescharacteristic
493C23#53611yescharacteristic
494C23#57612yes—
495C23#57712yes—
496C23#57712yes—
497C23#57812yes—
498Q8—816no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4#39214yes—
563C2×C4#39214yes—
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×C22⋊Q812yes—
568C2×C4C2×#12711yescharacteristic
569C2×C4#53312yes—
570C2×C4#53612yes—
571C2×C4#54112yes—
572C2×C4#54712yes—
573C2×C4#54712yes—
574C2×C4#54812yes—
575C2×C4#54812yes—
576C2×C4D4×Q812yes—
577C2×C4D4×Q812yes—
578C2×C4D4×Q812yes—
579C2×C4D4×Q812yes—
580C8—28no—
581C8—44no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—82no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—82no—
593C22C2×#39211yescharacteristic
594C22#243211yescharacteristic
595C22#243211yescharacteristic
596C22#243411yescharacteristic
597C22#243411yescharacteristic
598C22#262111yescharacteristic
599C22#285711yescharacteristic
600C22#300811yescharacteristic
601C22#300911yescharacteristic
602C22#301011yescharacteristic
603C22#301111yescharacteristic
604C22#301212yes—
605C22#301312yes—
606C22#306212yes—
607C22#306312yes—
608C22#316312yes—
609C22#316412yes—
610C22#317812yes—
611C22#317912yes—
612C22#324112yes—
613C22#324212yes—
614C22#324312yes—
615C22#324412yes—
616C4—24no—
617C4—24no—
618C4—42no—
619C4—84no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—82no—
624C4—84no—
625C4—81no—
626C4—82no—
627C4#240912yes—
628C4#242912yes—
629C4#242912yes—
630C4#254212yes—
631C2—82no—
632C2#1517811yescharacteristic
633C2#2339311yescharacteristic
634C2#2339411yescharacteristic
635C2#2568412yes—
636C2#2568511yescharacteristic
637C2#2568611yescharacteristic
638C2#2582011yescharacteristic
639C2#2582111yescharacteristic
640C2#2582212yes—
641C2#2988512yes—
642C2#3321912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
1. C22×#501C2semidirect
4. C2×#1991C2semidirect
5. C2×#1990C2semidirect
7. #11328C2semidirect
8. #11327C2semidirect
11. #10843C2semidirect
107. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#26292
C2×#26282
C2×#19912
C2×#19902
C2×#17502
#113282
#113272
#110962
#110952
#108432