Finite Group Databasebeta

← #382116

Subgroups — #382116

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2023C211yesmaximal, characteristic
2#14714C211yesmaximal, characteristic, complemented
3C4×#390C211yesmaximal, characteristic
4#11384C211yesmaximal, characteristic, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11315C212yesmaximal, complemented
7#11123C212yesmaximal
8#11049C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10795C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391—21no—
18C2×#391—21no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C22⋊C4—21no—
24C2×C4×C22⋊C4—21no—
25C2×C43C2211yescharacteristic
26#2023C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1437—21no—
37#1437—21no—
38C4⋊#95—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C23×Q8—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—42no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8C2×C414yescomplemented
53C2×C4×D4—22no—
54C2×C4×D4—42no—
55C2×C4×D4—42no—
56C2×C4×D4Q814yescomplemented
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—42no—
80C2×#107—42no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102C2×C418yes—
87#391—44no—
88#391—44no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390C2×C414yes—
93#384—42no—
94C4×C4⋊C4—42no—
95C4×C22⋊C4—44no—
96C4×C22⋊C4—44no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—21no—
106C43—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340C2×C418yescomplemented
112#336—22no—
113#336—24no—
114#336—42no—
115#336C2×C414yescomplemented
116C25—21no—
117C22×Q8—24no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4C22×C412yes—
132C22×D4C2×Q812yes—
133C22×D4#4112yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C414yes—
167C4⋊Q8—48no—
168C4⋊Q8—48no—
169C4×D4—48no—
170C4×D4—48no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C22×C414yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4#4112yes—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—41no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42D814yes—
242C2×C42C22×C412yes—
243C2×C42C2×Q812yes—
244C2×C42#4112yes—
245C2×C42#4112yes—
246#107—48no—
247#102—48no—
248#102—48no—
249#95—42no—
250#95—42no—
251#95—44no—
252#95—42no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—44no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24C22×D411yescharacteristic
274C2×Q8—48no—
275C2×Q8—48no—
276C2×Q8—48no—
277C2×Q8—84no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C4×Q814yes—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—48no—
332C22×C4—48no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—48no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4C22⋊Q812yes—
372C22×C4C22⋊Q812yes—
373C22×C4C2×D812yes—
374C22×C4C2×D812yes—
375C22×C4C2×D812yes—
376C22×C4#13512yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4#13712yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+412yes—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—44no—
432C42—44no—
433C42—44no—
434C42—44no—
435C42—42no—
436C42—42no—
437C42—84no—
438C23—24no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23C2×C4×D411yescharacteristic
474C23C2×C22⋊Q811yescharacteristic
475C23#52511yescharacteristic
476C23#53411yescharacteristic
477C23#53511yescharacteristic
478C23#53511yescharacteristic
479C23#53611yescharacteristic
480C23C22×D811yescharacteristic
481C23C2×#13511yescharacteristic
482C23C2×AGL(1,ℤ/8)11yescharacteristic
483C23C2×AGL(1,ℤ/8)11yescharacteristic
484C23C2×#13711yescharacteristic
485C23C2×#13711yescharacteristic
486C23#57511yescharacteristic
487C23#57511yescharacteristic
488Q8—816no—
489D4—28no—
490D4—44no—
491D4—44no—
492D4—44no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—816no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—81no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—82no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4C4×D814yes—
569C2×C4#44014yes—
570C2×C4#47414yes—
571C2×C4C2×C4×Q811yescharacteristic
572C2×C4C4×#4111yescharacteristic
573C2×C4#51812yes—
574C2×C4#52012yes—
575C2×C4#52911yescharacteristic
576C2×C4#53311yescharacteristic
577C2×C4#53812yes—
578C2×C4#53812yes—
579C2×C4#53812yes—
580C2×C4#53812yes—
581C2×C4#53912yes—
582C2×C4#54212yes—
583C2×C4#54212yes—
584C2×C4#54212yes—
585C2×C4#54312yes—
586C2×C4#54412yes—
587C8—44no—
588C8—82no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22C4×C22⋊Q811yescharacteristic
605C22#232111yescharacteristic
606C22#232611yescharacteristic
607C22#262611yescharacteristic
608C22#262811yescharacteristic
609C22#263111yescharacteristic
610C22#263811yescharacteristic
611C22C2×C4×D811yescharacteristic
612C22C2×#44011yescharacteristic
613C22C4×AGL(1,ℤ/8)11yescharacteristic
614C22C4×#13711yescharacteristic
615C22C2×#47411yescharacteristic
616C22#307611yescharacteristic
617C22#307711yescharacteristic
618C22#307911yescharacteristic
619C22#310011yescharacteristic
620C22#310711yescharacteristic
621C22#310911yescharacteristic
622C22#311611yescharacteristic
623C22#318611yescharacteristic
624C22#318811yescharacteristic
625C22#319311yescharacteristic
626C22#319611yescharacteristic
627C22#320511yescharacteristic
628C22#320511yescharacteristic
629C22#321011yescharacteristic
630C22#321111yescharacteristic
631C22#321311yescharacteristic
632C22#321411yescharacteristic
633C22#321511yescharacteristic
634C22#321811yescharacteristic
635C22#322611yescharacteristic
636C22#322811yescharacteristic
637C22#323811yescharacteristic
638C22#324011yescharacteristic
639C4—24no—
640C4—24no—
641C4—24no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—84no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—162no—
650C4#232912yes—
651C4#233712yes—
652C4#265312yes—
653C4#266512yes—
654C2—84no—
655C2#1431911yescharacteristic
656C2C4×#47411yescharacteristic
657C2#2161411yescharacteristic
658C2#2204011yescharacteristic
659C2#2204711yescharacteristic
660C2#2222011yescharacteristic
661C2#2222411yescharacteristic
662C2#2995011yescharacteristic
663C2#2995411yescharacteristic
664C2#2998011yescharacteristic
665C2#2998211yescharacteristic
666C2#3003811yescharacteristic
667C2#3004611yescharacteristic
668C2#3016511yescharacteristic
669C2#3017311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14714C2semidirect
4. #11384C2semidirect
5. #11324C2semidirect
6. #11315C2semidirect
9. #10907C2semidirect
10. #10795C2semidirect
26. #2023C4semidirect
52. C2×C4⋊Q8C2×C4semidirect
56. C2×C4×D4Q8semidirect
111. #340C2×C4semidirect
115. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20232
#147142
C4×#3902
#113842
#113242
#113152
#111232
#110492
#109072
#107952
#107882