Finite Group Databasebeta

← #374422

Subgroups — #374422

Order 512.

2180 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C23×#107C211yesmaximal, characteristic, complemented
3C2×#1973C212yesmaximal, complemented
4C2×#1874C211yesmaximal, characteristic
5C2×#1872C211yesmaximal, characteristic, complemented
6C2×#1752C211yesmaximal, characteristic, complemented
7#14139C211yesmaximal, characteristic, complemented
8#14137C211yesmaximal, characteristic
9#11124C212yesmaximal, complemented
10#11123C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#107—28no—
22C22×#107—24no—
23C22×#107C2212yes—
24C22×#107C2212yes—
25C22×#106C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×#102C2212yes—
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×#97C2212yes—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#386—22no—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1973—28no—
37#1874—24no—
38#1872—24no—
39#1752—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44#1439—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4—21no—
52C24×C4—21no—
53C24×C4C2311yescharacteristic
54C22×SD16—24no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16—24no—
58C22×SD16C2×C414yescomplemented
59C23×C8—24no—
60C23×C8—24no—
61C23×C8C2×C414yes—
62C23×C8D414yes—
63C23×C8D414yes—
64C23×C8C2312yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8C2×C414yes—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121C2×C414yes—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—42no—
89C22×C4⋊C4—42no—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4D412yes—
98C22×C42C2312yes—
99#465—232no—
100C2×#107—232no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#107—24no—
104C2×#107—24no—
105C2×#107—24no—
106C2×#107D418yescomplemented
107C2×#106—22no—
108C2×#106—22no—
109C2×#106C2×C414yescomplemented
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#103C2×C414yescomplemented
116C2×#102—22no—
117C2×#102—22no—
118C2×#102—24no—
119C2×#102—24no—
120C2×#102—42no—
121C2×#102C2×C414yescomplemented
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8D414yes—
125C2×#97—24no—
126C2×#97—24no—
127C2×#97—24no—
128#391—24no—
129#391—42no—
130#390—24no—
131#390—42no—
132#386—48no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95D412yes—
139#340—24no—
140#340—24no—
141#336—24no—
142C25—21no—
143C25C2×D411yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—22no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C412yes—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C22×C412yes—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—82no—
185C23×C4—82no—
186C23×C4C22×C412yes—
187C23×C4C2×D414yes—
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D412yes—
192C23×C4#4112yes—
193C23×C4#4112yes—
194C23×C4C2411yescharacteristic
195C2×SD16—28no—
196C2×SD16—216no—
197C2×SD16—44no—
198C2×SD16—48no—
199C2×SD16—44no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—216no—
203C22×C8—28no—
204C22×C8—216no—
205C22×C8—28no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8C22⋊C4116yescomplemented
213C22×C8C22×C412yes—
214C22×C8C22×C412yes—
215C22×C8C2×D414yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8C2×D412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D414yes—
221C22×C8#4114yes—
222C22⋊Q8—28no—
223C22⋊Q8—28no—
224C22⋊Q8—44no—
225C22⋊Q8—44no—
226#121—28no—
227#121—28no—
228#121—44no—
229#121—44no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—84no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285#107—216no—
286#107—216no—
287#106—28no—
288#103—28no—
289#103—48no—
290#102—28no—
291#102—48no—
292C22⋊C8—28no—
293#97—48no—
294#95—24no—
295#95—48no—
296#95—48no—
297#95—84no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24C4×D412yes—
312C24#12112yes—
313C24#12712yes—
314C24AGL(1,ℤ/8)14yes—
315C24#13714yes—
316C24C22×D411yescharacteristic
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—24no—
338C2×D4—24no—
339C2×D4—48no—
340C2×D4—416no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—88no—
356C2×D4—84no—
357C2×D4—84no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—42no—
408C22×C4—816no—
409C22×C4—816no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4C2×C22⋊C414yes—
415C22×C4C4×D414yes—
416C22×C4C4×D412yes—
417C22×C4#12114yes—
418C22×C4#12112yes—
419C22×C4#12414yes—
420C22×C4#12712yes—
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4112yes—
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+412yes—
431SD16—416no—
432C2×C8—24no—
433C2×C8—22no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—232no—
437C2×C8—24no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8C2×C22⋊C414yes—
445C2×C8C2×C22⋊C414yes—
446C2×C8C4×D418yes—
447C2×C8#12014yes—
448C2×C8#12118yes—
449C2×C8#12314yes—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—816no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—42no—
477C23—28no—
478C23—28no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—24no—
485C23—28no—
486C23—22no—
487C23—28no—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—84no—
494C23—84no—
495C23—81no—
496C23—81no—
497C23—81no—
498C23—81no—
499C23—82no—
500C23—81no—
501C23—84no—
502C23—81no—
503C23—84no—
504C23—82no—
505C23—82no—
506C23#39014yes—
507C23#44018yes—
508C23#46818yes—
509C23#49614yes—
510C23#49714yes—
511C23C2×C4×D411yescharacteristic
512C23C2×#12111yescharacteristic
513C23C2×#12711yescharacteristic
514C23D4212yes—
515C23#54712yes—
516C23C2×AGL(1,ℤ/8)12yes—
517C23C2×#13712yes—
518C23#57612yes—
519C23#57812yes—
520Q8—42no—
521Q8—44no—
522Q8—42no—
523Q8—82no—
524Q8—81no—
525Q8—81no—
526D4—42no—
527D4—44no—
528D4—42no—
529D4—816no—
530D4—82no—
531D4—81no—
532D4—81no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—816no—
561C2×C4—816no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—81no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—81no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4#39014yes—
578C2×C4C22×C22⋊C411yescharacteristic
579C2×C4C2×C4×D411yescharacteristic
580C2×C4C2×C4×D412yes—
581C2×C4#52012yes—
582C2×C4C2×#12011yescharacteristic
583C2×C4C2×#12111yescharacteristic
584C2×C4C2×#12112yes—
585C2×C4C2×#12311yescharacteristic
586C2×C4C2×#12411yescharacteristic
587C2×C4#53511yescharacteristic
588C2×C4#53611yescharacteristic
589C2×C4#53712yes—
590C2×C4D4212yes—
591C2×C4#54712yes—
592C2×C4#54812yes—
593C2×C4#54812yes—
594C2×C4#55312yes—
595C2×C4#55312yes—
596C8—24no—
597C8—44no—
598C8#38618yescomplemented
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—216no—
605C22—24no—
606C22—216no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—88no—
611C22—161no—
612C22—161no—
613C22—161no—
614C22—161no—
615C22—164no—
616C22#197318yescomplemented
617C22C2×#39011yescharacteristic
618C22#236112yes—
619C22#241612yes—
620C22#243311yescharacteristic
621C22#250111yescharacteristic
622C22C2×#44012yes—
623C22#295112yes—
624C22C2×#46812yes—
625C22#306112yes—
626C22C2×#49611yescharacteristic
627C22C2×#49711yescharacteristic
628C22#315511yescharacteristic
629C22#315611yescharacteristic
630C22#326414yes—
631C22#327014yes—
632C22#334314yes—
633C22#334614yes—
634C4—24no—
635C4—82no—
636C4—84no—
637C4—84no—
638C4—84no—
639C4—82no—
640C4—82no—
641C4C2×#38611yescharacteristic
642C4#231111yescharacteristic
643C4#236112yes—
644C4#241612yes—
645C4#243611yescharacteristic
646C4#250411yescharacteristic
647C2—28no—
648C2—162no—
649C2#1413711yescharacteristic
650C2C2×#197311yescharacteristic
651C2#2123011yescharacteristic
652C2#2303214yes—
653C2#2478714yes—
654C2#2578811yescharacteristic
655C2#2578911yescharacteristic
656C2#2793811yescharacteristic
657C2#2793911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
2. C23×#107C2semidirect
3. C2×#1973C2semidirect
5. C2×#1872C2semidirect
6. C2×#1752C2semidirect
7. #14139C2semidirect
9. #11124C2semidirect
10. #11123C2semidirect
11. #10863C2semidirect
20. C2×#465C4semidirect
58. C22×SD16C2×C4semidirect
106. C2×#107D4semidirect
109. C2×#106C2×C4semidirect
115. C2×#103C2×C4semidirect
121. C2×#102C2×C4semidirect
212. C22×C8C22⋊C4semidirect
598. C8#386semidirect
616. C22#1973semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C23×#1072
C2×#19732
C2×#18742
C2×#18722
C2×#17522
#141392
#141372
#111242
#111232
#108632