Finite Group Databasebeta

← #370324

Subgroups — #370324

Order 512.

1336 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1922C211yesmaximal, characteristic, complemented
2#20834C211yesmaximal, characteristic
3#20658C211yesmaximal, characteristic
4#20385C211yesmaximal, characteristic
5C22×#336C211yesmaximal, characteristic
6#11376C211yesmaximal, characteristic, complemented
7#11317C211yesmaximal, characteristic, complemented
8#11291C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#10909C211yesmaximal, characteristic, complemented
11#10887C211yesmaximal, characteristic, complemented
12#10809C211yesmaximal, characteristic, complemented
13#10723C211yesmaximal, characteristic
14#10719C211yesmaximal, characteristic
15#10719C211yesmaximal, characteristic
16C24×C8C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C2×#427C2211yescharacteristic
20C2×#427C2211yescharacteristic
21C2×#424C2211yescharacteristic
22C2×#424C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#2280C2211yescharacteristic
28#2280C2211yescharacteristic
29#1922—24no—
30#1850—22no—
31#1850—22no—
32#1808C412yescomplemented
33#1808C412yescomplemented
34#1807C412yescomplemented
35#1807C412yescomplemented
36#1752—22no—
37C8×C22⋊C4—22no—
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56C2×#336—24no—
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—22no—
75C23×C8—41no—
76C23×C8—41no—
77C23×C8—41no—
78C23×C8—41no—
79C23×C8C2×C412yes—
80C23×C8C2×C412yes—
81C23×C8D412yes—
82C23×C8D412yes—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C2×#117C2311yescharacteristic
91C22×C42—21no—
92C22×C42—22no—
93C22×C42—22no—
94C22×C42—22no—
95C22×C42—22no—
96C22×C42—41no—
97C22×C42—41no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#427—22no—
101#427—22no—
102#427—41no—
103#427—41no—
104C2×#107C2×C411yescharacteristic, complemented
105C2×#107C2×C411yescharacteristic, complemented
106C2×#107C2×C412yescomplemented
107C2×#106C2×C411yescharacteristic, complemented
108C2×#106C2×C411yescharacteristic, complemented
109C2×#106C2×C412yescomplemented
110#424—22no—
111#424—22no—
112#424—41no—
113#424—41no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117#407—22no—
118#407—22no—
119#407—22no—
120#407Q814yes—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123#405—22no—
124#405—22no—
125#405—41no—
126#405—41no—
127C2×#97—21no—
128C2×#97—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#381—21no—
132#381—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135#380C2×C412yes—
136#380C2×C412yes—
137#379—21no—
138#379—21no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141#341—22no—
142#341—22no—
143#341—41no—
144#341—41no—
145#340C2×C412yescomplemented
146#340C2×C412yescomplemented
147#340C2×C412yescomplemented
148#340C2×C412yescomplemented
149#339—22no—
150#339—22no—
151#339—22no—
152#339—22no—
153#339—41no—
154#339—41no—
155#339—41no—
156#339—41no—
157#336—22no—
158#336—21no—
159#336—21no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164#336—22no—
165#336D414yes—
166C25#4111yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—28no—
176C23×C4—24no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—44no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C22×C412yes—
183C23×C4C2×D412yes—
184C23×C4C2×Q812yes—
185C23×C4#4112yes—
186C23×C4#4112yes—
187C23×C4#4111yescharacteristic
188C23×C4#4112yes—
189C23×C4C2411yescharacteristic
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16C2×Q812yes—
213C2×M16C2×Q812yes—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—24no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8—24no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—24no—
241C22×C8—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—42no—
250C22×C8—44no—
251C22×C8—44no—
252C22×C8—44no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C22×C412yes—
259C22×C8C22×C411yescharacteristic
260C22×C8C22×C411yescharacteristic
261C22×C8C2×D412yes—
262C22×C8C2×D412yes—
263C22×C8C2×D412yes—
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8#4112yes—
267C22×C8#4112yes—
268#117—21no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—21no—
278#117—21no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—21no—
283#117—21no—
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42C2×D412yes—
320C2×C42#4112yes—
321#107—22no—
322#107—22no—
323#107—42no—
324#106—22no—
325#106—22no—
326#106—42no—
327C4⋊C8—44no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—28no—
331C22⋊C8—44no—
332#97—42no—
333#97—42no—
334C4×C8—42no—
335C4×C8—42no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—24no—
345C24—24no—
346C24C4×D412yes—
347C24#12512yes—
348C24#12612yes—
349C24C2×#4111yescharacteristic
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—216no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—48no—
392C22×C4—416no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4#11714yes—
400C22×C4C4×D412yes—
401C22×C4C4×Q814yes—
402C22×C4C22⋊Q814yes—
403C22×C4#12314yes—
404C22×C4#12512yes—
405C22×C4#12612yes—
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×Q811yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420M16—22no—
421M16—22no—
422M16—22no—
423M16—22no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429M16—41no—
430M16—41no—
431M16—41no—
432M16—41no—
433M16—41no—
434M16—41no—
435M16—41no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—48no—
461C2×C8—48no—
462C2×C8—48no—
463C2×C8—48no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8C4×D412yes—
483C2×C8C4×D412yes—
484C2×C8#12112yes—
485C2×C8#12112yes—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—41no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—48no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—22no—
534C23—28no—
535C23—28no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23#38714yes—
546C23#44314yes—
547C23#44414yes—
548C23C2×C4×D411yescharacteristic
549C23#52511yescharacteristic
550C23C2×#12511yescharacteristic
551C23C2×#12611yescharacteristic
552C23#52911yescharacteristic
553C23#53811yescharacteristic
554C23#53911yescharacteristic
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—432no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4#38714yes—
595C2×C4C2×#11711yescharacteristic
596C2×C4C2×C4×D411yescharacteristic
597C2×C4C2×C4×Q811yescharacteristic
598C2×C4C4×#4111yescharacteristic
599C2×C4#51811yescharacteristic
600C2×C4#52011yescharacteristic
601C2×C4C2×#12111yescharacteristic
602C2×C4C2×C22⋊Q811yescharacteristic
603C2×C4C2×#12311yescharacteristic
604C2×C4#53211yescharacteristic
605C2×C4#53411yescharacteristic
606C2×C4#53711yescharacteristic
607C2×C4#54611yescharacteristic
608C2×C4#54611yescharacteristic
609C2×C4#54611yescharacteristic
610C2×C4D4×Q811yescharacteristic
611C2×C4#55111yescharacteristic
612C2×C4#55111yescharacteristic
613C2×C4#55111yescharacteristic
614C2×C4#55111yescharacteristic
615C2×C4#55111yescharacteristic
616C2×C4#55111yescharacteristic
617C2×C4#55211yescharacteristic
618C2×C4#55211yescharacteristic
619C2×C4#55211yescharacteristic
620C2×C4#55411yescharacteristic
621C2×C4#55411yescharacteristic
622C2×C4#55411yescharacteristic
623C8—22no—
624C8—22no—
625C8—22no—
626C8—22no—
627C8—42no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—81no—
632C8—81no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—21no—
638C22—21no—
639C22—24no—
640C22—21no—
641C22—21no—
642C22—24no—
643C22—22no—
644C22—22no—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22#192218yescomplemented
652C22C2×#38711yescharacteristic
653C22C4×#12111yescharacteristic
654C22#232511yescharacteristic
655C22#247311yescharacteristic
656C22#248011yescharacteristic
657C22#249311yescharacteristic
658C22#250511yescharacteristic
659C22C2×#44312yes—
660C22C2×#44412yes—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—82no—
669C4—82no—
670C4#234211yescharacteristic
671C4#234311yescharacteristic
672C4#236011yescharacteristic
673C4#236811yescharacteristic
674C4#241211yescharacteristic
675C4#241711yescharacteristic
676C4#249811yescharacteristic
677C4#256411yescharacteristic
678C2—21no—
679C2—28no—
680C2—21no—
681C2—21no—
682C2—21no—
683C2#1404711yescharacteristic
684C2C2×#192212yes—
685C2#2154411yescharacteristic
686C2#2154611yescharacteristic
687C2#2220412yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1922C2semidirect
6. #11376C2semidirect
7. #11317C2semidirect
8. #11291C2semidirect
9. #11221C2semidirect
10. #10909C2semidirect
11. #10887C2semidirect
12. #10809C2semidirect
32. #1808C4semidirect
33. #1808C4semidirect
34. #1807C4semidirect
35. #1807C4semidirect
104. C2×#107C2×C4semidirect
105. C2×#107C2×C4semidirect
106. C2×#107C2×C4semidirect
107. C2×#106C2×C4semidirect
108. C2×#106C2×C4semidirect
109. C2×#106C2×C4semidirect
145. #340C2×C4semidirect
146. #340C2×C4semidirect
147. #340C2×C4semidirect
148. #340C2×C4semidirect
651. C22#1922semidirect

Maximal subgroups

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

subgroupindex
C2×#19222
#208342
#206582
#203852
C22×#3362
#113762
#113172
#112912
#112212
#109092
#108872
#108092
#107232
#107192