Finite Group Databasebeta

← #374189

Subgroups — #374189

Order 512.

1722 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#485C211yesmaximal, characteristic
2C2×#2346C211yesmaximal, characteristic
3C2×#1894C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#14139C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11067C212yesmaximal, complemented
8#10932C212yesmaximal, complemented
9#10908C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
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
25#2346—24no—
26#2346—24no—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#388—22no—
30C2×#382—21no—
31C2×#380C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×#376—21no—
35#1894—24no—
36#1736—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—21no—
45#1441—21no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117C2×C414yes—
69C2×#117D414yes—
70C2×#117D414yes—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D412yes—
87C22×C42—21no—
88C22×C42—22no—
89C22×C42C2312yes—
90#485—232no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107C2×C414yescomplemented
94C2×#106—22no—
95C2×#106—22no—
96C2×#106C2×C414yescomplemented
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—42no—
104C2×#103C2×C414yescomplemented
105C2×#103C2×C414yescomplemented
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8C2×C414yes—
109#391—24no—
110#391—42no—
111#389—24no—
112#389—24no—
113#388—216no—
114#382—28no—
115#380—22no—
116#380D414yes—
117C4×C4⋊C4—24no—
118C4×C4⋊C4—24no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121#376—28no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—42no—
136#336—24no—
137#336—42no—
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—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—81no—
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
174C2×M16—28no—
175C2×M16—48no—
176C2×M16—48no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C412yes—
194C22×C8C22×C412yes—
195C22×C8#4114yes—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200#117—28no—
201#117—28no—
202#117—28no—
203#117C22⋊C4116yes—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
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—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4#4114yes—
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—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—81no—
256C2×C22⋊C4—81no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—28no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—44no—
268C2×C42—44no—
269#107—28no—
270#106—28no—
271#103—28no—
272#103—28no—
273#103—48no—
274C22⋊C8—28no—
275#95—28no—
276#95—28no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—44no—
281#95—44no—
282#95—82no—
283#95—82no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—21no—
289C24—21no—
290C24—21no—
291C24—21no—
292C24C4×D412yes—
293C24#12012yes—
294C24#12112yes—
295C24C22×D411yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—48no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—88no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4C2×C22⋊C414yes—
387C22×C4C4×D414yes—
388C22×C4C4×D412yes—
389C22×C4#12012yes—
390C22×C4#12114yes—
391C22×C4#12112yes—
392C22×C4#12314yes—
393C22×C4#13714yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4111yescharacteristic
402C22×C4H(5,𝔽2)12yes—
403C22×C42−1+412yes—
404M16—416no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—48no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—48no—
442C4⋊C4—88no—
443C4⋊C4C2×C22⋊C414yes—
444C4⋊C4C2×C22⋊C414yes—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—84no—
449C42—216no—
450C42—416no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—42no—
455C23—24no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23#38614yes—
478C23C2×C4×D411yescharacteristic
479C23C2×#12011yescharacteristic
480C23C2×#12111yescharacteristic
481C23#54612yes—
482C23#54712yes—
483C23C2×#13712yes—
484C23#57512yes—
485C23#57611yescharacteristic
486C23#57711yescharacteristic
487C23#57711yescharacteristic
488C23#57811yescharacteristic
489Q8—42no—
490Q8—44no—
491Q8—42no—
492Q8—81no—
493Q8—81no—
494Q8—82no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—816no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4#38614yes—
559C2×C4C22×C22⋊C411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C4×#4112yes—
562C2×C4#52012yes—
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12311yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4#52511yescharacteristic
567C2×C4#53212yes—
568C2×C4#53511yescharacteristic
569C2×C4#53611yescharacteristic
570C2×C4#53712yes—
571C2×C4#54612yes—
572C2×C4#54712yes—
573C2×C4#55112yes—
574C2×C4#55112yes—
575C2×C4#55212yes—
576C2×C4#55212yes—
577C8—42no—
578C8—44no—
579C8—42no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—44no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—44no—
591C22C2×#38611yescharacteristic
592C22#235912yes—
593C22#241111yescharacteristic
594C22#241311yescharacteristic
595C22#243012yes—
596C22C4×#13712yes—
597C22#295111yescharacteristic
598C22#295211yescharacteristic
599C22#300712yes—
600C22#301211yescharacteristic
601C22#301311yescharacteristic
602C22#303911yescharacteristic
603C22#304011yescharacteristic
604C22#304511yescharacteristic
605C22#304611yescharacteristic
606C22#331212yes—
607C22#331912yes—
608C22#332312yes—
609C22#332912yes—
610C22#334412yes—
611C22#334812yes—
612C22#335212yes—
613C22#335712yes—
614C4—24no—
615C4—44no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—84no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4#229711yescharacteristic
624C4#231111yescharacteristic
625C4#235912yes—
626C4#241111yescharacteristic
627C4#241311yescharacteristic
628C4#249812yes—
629C2—44no—
630C2#1413511yescharacteristic
631C2#2092711yescharacteristic
632C2#2093711yescharacteristic
633C2#2292812yes—
634C2#2294212yes—
635C2#2449511yescharacteristic
636C2#2449811yescharacteristic
637C2#2459511yescharacteristic
638C2#2459811yescharacteristic
639C2#2555212yes—
640C2#2556212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11219C2semidirect
7. #11067C2semidirect
8. #10932C2semidirect
9. #10908C2semidirect
18. C2×#485C4semidirect
93. C2×#107C2×C4semidirect
96. C2×#106C2×C4semidirect
104. C2×#103C2×C4semidirect
105. 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×#23462
C2×#18942
C2×#17362
#141392
#112192
#110672
#109322
#109082
#107302
#107152