Finite Group Databasebeta

← #613651

Subgroups — #613651

Order 512.

5542 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#2536C212yesmaximal, complemented
3C2×#2409C212yesmaximal, complemented
4C2×C4×#121C211yesmaximal, characteristic
5#15176C214yesmaximal, complemented
6#14883C212yesmaximal, complemented
7#14880C212yesmaximal, complemented
8#14821C212yesmaximal, complemented
9#14719C212yesmaximal
10#14696C214yesmaximal, complemented
11#14682C211yesmaximal, characteristic, complemented
12#13969C214yesmaximal, complemented
13#13942C212yesmaximal, complemented
14#13872C212yesmaximal, complemented
15C24×Q8C2211yescharacteristic
16C22×C22⋊Q8C2214yes—
17C22×#121C2211yescharacteristic
18C22×C4×Q8—28no—
19C22×C4×Q8—216no—
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2212yes—
24C22×C4×D4C2212yes—
25C22×C4×D4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C42C2212yes—
29C23×C42C2211yescharacteristic
30#2536—216no—
31#2409—216no—
32C2×#398C2214yes—
33C2×#395C2214yes—
34C2×#394C2214yes—
35C2×#393C2214yes—
36C4×#121—28no—
37C2×#391C2214yes—
38C2×#390C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388C2212yes—
41C2×#388C2218yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#386C2214yes—
45C2×#385C2214yes—
46C2×#382C2212yes—
47C2×#380C2214yes—
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2218yes—
56#1439C2214yes—
57#1439C2218yes—
58#1437C2218yes—
59#1436C2218yes—
60C4⋊#95C2214yes—
61C4⋊#95C2212yes—
62#1434C2212yes—
63#1434C2212yes—
64#1434C2218yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C23×Q8—28no—
69C23×Q8—28no—
70C23×Q8—24no—
71C23×Q8—24no—
72C23×Q8D418yescomplemented
73C23×Q8C2312yes—
74C23×Q8C2312yes—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—216no—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×#121—28no—
87C2×#121—24no—
88C2×C4×Q8—2128no—
89C2×C4×Q8—28no—
90C2×C4×Q8—28no—
91C2×C4×Q8—216no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—216no—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8D4116yescomplemented
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—28no—
103C2×C4×D4—24no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—216no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2318yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C42—28no—
127C22×C42—216no—
128C22×C42—28no—
129C22×C42D418yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#398—216no—
138#395—216no—
139#394—216no—
140#393—216no—
141#391—216no—
142#390—24no—
143#389—24no—
144#388—232no—
145#388—28no—
146#386—216no—
147#386—216no—
148#386—28no—
149#385—216no—
150#382—28no—
151#380—28no—
152#380—24no—
153#380—216no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—28no—
156C2×#95C2314yes—
157C2×#95C2318yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2318yes—
168C2×#95C2314yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×Q8—232no—
172C22×Q8—232no—
173C22×Q8—264no—
174C22×Q8—264no—
175C22×Q8—232no—
176C22×Q8—28no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8C2×D418yes—
185C22×Q8C2×D418yes—
186C22×Q8C2×D418yes—
187C22×Q8#4118yes—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4#4118yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4#4114yes—
226C23×C4C2418yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C22⋊Q8—464no—
239#121—232no—
240C4×Q8—264no—
241C4×Q8—264no—
242C4×Q8—264no—
243C4×Q8—264no—
244C4×D4—432no—
245C4×D4—416no—
246C2×C4⋊C4—264no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—216no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—216no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D4116yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—216no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—216no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—216no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—48no—
296C2×C42—264no—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—232no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D4116yes—
313C2×C42C2×D414yes—
314C2×C42C2×D418yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—216no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—216no—
323#95—216no—
324#95—216no—
325#95—28no—
326#95—44no—
327#95—48no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24C22×D412yes—
343C24C22×D412yes—
344C24C2×#4114yes—
345C24C2×#4112yes—
346C24C2×#4114yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—2256no—
351C2×Q8—232no—
352C2×Q8—216no—
353C2×Q8—216no—
354C2×Q8—232no—
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—216no—
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8#121132yescomplemented
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—416no—
367C2×D4—44no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—44no—
373C22×C4—216no—
374C22×C4—232no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—232no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—416no—
433C22×C4—432no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4#121116yes—
442C22×C4C22×D418yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D418yes—
449C22×C4C22×D412yes—
450C22×C4C22×D418yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4118yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C42−1+414yes—
462C4⋊C4—232no—
463C4⋊C4—216no—
464C4⋊C4—28no—
465C4⋊C4—232no—
466C4⋊C4—216no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—432no—
472C22⋊C4—216no—
473C22⋊C4—432no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—432no—
479C22⋊C4—48no—
480C42—232no—
481C42—28no—
482C42—28no—
483C42—232no—
484C42—28no—
485C42—48no—
486C42#120116yes—
487C23—22no—
488C23—24no—
489C23—22no—
490C23—24no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—42no—
505C23—44no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23C2×#12114yes—
523C23#52514yes—
524C23#53614yes—
525C23#54818yes—
526C23#55218yes—
527C23C23×D411yescharacteristic
528C23C23×D412yes—
529C23C23×D411yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4112yes—
532C23C22×#4112yes—
533C23C2×2−1+412yes—
534C23C2×2−1+412yes—
535C23#58512yes—
536Q8—264no—
537Q8—264no—
538D4—416no—
539D4—432no—
540C2×C4—232no—
541C2×C4—264no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—232no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—216no—
569C2×C4—24no—
570C2×C4—416no—
571C2×C4—432no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—48no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—816no—
598C2×C4C2×#12014yes—
599C2×C4C2×#12014yes—
600C2×C4C2×#12014yes—
601C2×C4C2×#121116yes—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4C2×#12118yes—
605C2×C4#52518yes—
606C2×C4#52514yes—
607C2×C4#53614yes—
608C22—24no—
609C22—216no—
610C22—24no—
611C22—216no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—44no—
617C22—48no—
618C22—48no—
619C22—44no—
620C22—44no—
621C22—82no—
622C22—84no—
623C22—82no—
624C22—84no—
625C22—82no—
626C22—82no—
627C22#253618yescomplemented
628C22C22×#12011yescharacteristic
629C22C22×#12111yescharacteristic
630C22C22×#12112yes—
631C22C2×#52512yes—
632C22C2×#52511yescharacteristic
633C22C2×#52511yescharacteristic
634C22C2×#53611yescharacteristic
635C22C2×#53611yescharacteristic
636C22#344911yescharacteristic
637C22C2×#54812yes—
638C22D4×#4114yes—
639C22C2×#55212yes—
640C22#348214yes—
641C22#348614yes—
642C22#349712yes—
643C22#350712yes—
644C22#351614yes—
645C4—216no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—44no—
651C4—42no—
652C4—48no—
653C4—82no—
654C4—88no—
655C4—82no—
656C4#284718yes—
657C2—28no—
658C2—84no—
659C2#3473611yescharacteristic
660C2C2×#253611yescharacteristic
661C2#3497111yescharacteristic
662C2#3517811yescharacteristic
663C2#3524211yescharacteristic
664C2#3524511yescharacteristic
665C2#3623214yes—
666C2#3643414yes—
667C2C2×#284711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C4×Q8C2semidirect
2. C2×#2536C2semidirect
3. C2×#2409C2semidirect
5. #15176C2semidirect
6. #14883C2semidirect
7. #14880C2semidirect
8. #14821C2semidirect
10. #14696C2semidirect
11. #14682C2semidirect
12. #13969C2semidirect
13. #13942C2semidirect
14. #13872C2semidirect
72. C23×Q8D4semidirect
98. C2×C4×Q8D4semidirect
361. C2×Q8#121semidirect
627. C22#2536semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Q82
C2×#25362
C2×#24092
C2×C4×#1212
#151762
#148832
#148802
#148212
#147192
#146962
#146822
#139692
#139422
#138722