Finite Group Databasebeta

← #414664

Subgroups — #414664

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic, complemented
2C22×#440C211yesmaximal, characteristic
3C2×#2542C211yesmaximal, characteristic
4C2×#1993C211yesmaximal, characteristic, complemented
5C2×#1804C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11123C212yesmaximal
9#11050C212yesmaximal, complemented
10#11023C212yesmaximal
11#10835C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—22no—
15C22×C4×Q8C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#477—24no—
18C2×#477—24no—
19C2×#477—24no—
20C2×#440—24no—
21C2×#440—24no—
22C2×#440—24no—
23C22×#107C2211yescharacteristic
24C22×#107C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2212yes—
30C22×#97C2211yescharacteristic
31#2542—24no—
32#2542—24no—
33C2×#399—22no—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#384C2212yes—
37C2×#382C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39#1993—24no—
40#1804—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C2×#336C2212yes—
46#1441—22no—
47C4⋊#95C2212yes—
48#1434—22no—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—22no—
56C22×SD16—22no—
57C22×SD16—22no—
58C22×SD16—22no—
59C22×SD16Q814yescomplemented
60C23×C8C2311yescharacteristic
61C23×C8C2312yes—
62C23×C8C2311yescharacteristic
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8D414yes—
69C2×C22⋊Q8—216no—
70C2×C22⋊Q8—416no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8D414yes—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—22no—
80C2×C4×D4—22no—
81C2×C4×D4—22no—
82C2×C4×D4D414yes—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D412yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4—21no—
92C22×C22⋊C4—24no—
93C22×C42—21no—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#477—232no—
97#440—232no—
98C2×#107—24no—
99C2×#107—22no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107—22no—
103C2×#107—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—22no—
106C2×C4⋊C8D414yes—
107C2×#103—24no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—22no—
113C2×#102—22no—
114C2×#102—22no—
115C2×#102—22no—
116C2×#102—24no—
117C2×#102—24no—
118C2×#102—42no—
119C2×#97—22no—
120C2×#97—22no—
121C2×#97D414yes—
122#399—216no—
123#391—24no—
124#391—42no—
125#390—24no—
126#390—24no—
127#384—24no—
128#384—24no—
129#382—22no—
130#382—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C2×#95—21no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—21no—
137C2×#95D412yes—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—24no—
142#336—42no—
143C25—21no—
144C22×Q8—24no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—48no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C2×D412yes—
160C22×Q8C2×Q812yes—
161C22×Q8#4112yes—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×D412yes—
174C22×D4C2×Q812yes—
175C22×D4#4112yes—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—48no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C2×D414yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C2×SD16—28no—
190C2×SD16—28no—
191C2×SD16—28no—
192C2×SD16—28no—
193C2×SD16—28no—
194C2×SD16—28no—
195C2×SD16—28no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—22no—
212C22×C8C2×D412yes—
213C22×C8C2×D414yes—
214C22×C8C2×D412yes—
215C22×C8C2×Q812yes—
216C22×C8#4112yes—
217C4⋊Q8—28no—
218C4⋊Q8—28no—
219C4⋊Q8—44no—
220C4⋊Q8—44no—
221C22⋊Q8—464no—
222C4×Q8—28no—
223C4×Q8—28no—
224C4×Q8—44no—
225C4×Q8—44no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—44no—
229C4×D4—44no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—216no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—48no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D412yes—
285C2×C42#4112yes—
286#107—28no—
287#107—216no—
288C4⋊C8—28no—
289#103—28no—
290#103—216no—
291#102—28no—
292#102—48no—
293#97—28no—
294#95—24no—
295#95—48no—
296#95—42no—
297#95—42no—
298#95—48no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—41no—
303C24—41no—
304C24—41no—
305C24—41no—
306C24—44no—
307C24—44no—
308C24C22×D411yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—416no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—48no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×Q8C22⋊Q814yescomplemented
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4C22⋊Q814yes—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—416no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4#12012yes—
407C22×C4#12012yes—
408C22×C4#12112yes—
409C22×C4#12112yes—
410C22×C4#12712yes—
411C22×C4#12712yes—
412C22×C4AGL(1,ℤ/8)18yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×Q811yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C22×C42−1+411yescharacteristic
427SD16—232no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—24no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—24no—
439C2×C8—24no—
440C2×C8—42no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8C22⋊Q814yescomplemented
449C4⋊C4—216no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—432no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C22⋊C4—432no—
470C22⋊C4—48no—
471C42—42no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—48no—
476C42#12014yes—
477C42#12014yes—
478C42#12114yes—
479C42#12114yes—
480C23—24no—
481C23—48no—
482C23—48no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—84no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—84no—
497C23C2×#12011yescharacteristic
498C23C2×#12111yescharacteristic
499C23C2×#12711yescharacteristic
500C23#54611yescharacteristic
501C23#54711yescharacteristic
502C23#54811yescharacteristic
503C23D4×Q811yescharacteristic
504C23C2×AGL(1,ℤ/8)14yes—
505C23#57814yes—
506Q8—24no—
507Q8—24no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—816no—
515D4—24no—
516D4—24no—
517D4—42no—
518D4—42no—
519D4—42no—
520D4—42no—
521D4—42no—
522D4—42no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—816no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—84no—
562C2×C4#39214yes—
563C2×C4#39214yes—
564C2×C4C2×#12011yescharacteristic
565C2×C4C2×#12011yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4#52511yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53711yescharacteristic
578C2×C4#53711yescharacteristic
579C2×C4#53711yescharacteristic
580C2×C4#54611yescharacteristic
581C2×C4#54711yescharacteristic
582C2×C4#54811yescharacteristic
583C2×C4#54811yescharacteristic
584C2×C4#54811yescharacteristic
585C2×C4D4×Q811yescharacteristic
586C2×C4D4×Q811yescharacteristic
587C2×C4#55011yescharacteristic
588C2×C4#55211yescharacteristic
589C2×C4#55211yescharacteristic
590C2×C4#55411yescharacteristic
591C2×C4#55611yescharacteristic
592C8—24no—
593C8—42no—
594C8—44no—
595C22—48no—
596C22—44no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—84no—
604C22C2×#39211yescharacteristic
605C22#240711yescharacteristic
606C22#240911yescharacteristic
607C22#241511yescharacteristic
608C22#241711yescharacteristic
609C22#254011yescharacteristic
610C22#254211yescharacteristic
611C22#300312yes—
612C22#301112yes—
613C22#303812yes—
614C22#304412yes—
615C22#315112yes—
616C22#315612yes—
617C22#330814yes—
618C22#334314yes—
619C22#336714yes—
620C22#338614yes—
621C4—24no—
622C4—24no—
623C4—48no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—84no—
628C4—162no—
629C4#240711yescharacteristic
630C4#240911yescharacteristic
631C4#241511yescharacteristic
632C4#241711yescharacteristic
633C4#253611yescharacteristic
634C4#253811yescharacteristic
635C4#261611yescharacteristic
636C4#284811yescharacteristic
637C2—84no—
638C2#1517711yescharacteristic
639C2#2342511yescharacteristic
640C2#2343011yescharacteristic
641C2#2413212yes—
642C2#2434212yes—
643C2#2469412yes—
644C2#2482712yes—
645C2#2842612yes—
646C2#2854512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#477C2semidirect
4. C2×#1993C2semidirect
6. #13942C2semidirect
7. #11324C2semidirect
9. #11050C2semidirect
11. #10835C2semidirect
59. C22×SD16Q8semidirect
340. C2×Q8C22⋊Q8semidirect
448. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C22×#4402
C2×#25422
C2×#19932
C2×#18042
#139422
#113242
#111232
#110502
#110232
#108352