Finite Group Databasebeta

← #381865

Subgroups — #381865

Order 512.

1614 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×#1999C211yesmaximal, characteristic
2C2×#1970C211yesmaximal, characteristic, complemented
3C2×#1750C211yesmaximal, characteristic, complemented
4#14257C212yesmaximal, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic, complemented
8#11022C211yesmaximal, characteristic, complemented
9#10969C212yesmaximal, complemented
10#10927C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#121C2212yes—
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#380C2212yes—
26#1999C418yescomplemented
27#1970—24no—
28#1750—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1439—22no—
37#1439—22no—
38#1438—22no—
39#1438—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×D4—22no—
43C23×D4C2312yes—
44C24×C4C2311yescharacteristic
45C22×D8—22no—
46C22×D8—22no—
47C22×D8—42no—
48C22×D8—42no—
49C22×D8C2×C414yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×#121—24no—
55C2×#121—42no—
56C2×#121—42no—
57C2×#121—42no—
58C2×#121—42no—
59C2×#121C2×C418yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—42no—
63C2×#117—42no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—22no—
80C2×#107—42no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102—42no—
89C2×#102C2×C418yes—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8C2×C414yes—
96C2×#97—22no—
97C2×#97—42no—
98C2×C4×C8—42no—
99C2×C4×C8D414yescomplemented
100#390—24no—
101#390—24no—
102#390—42no—
103#390—42no—
104#380—42no—
105#380—42no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111#340—22no—
112#340—42no—
113#340—42no—
114#340—42no—
115#340C2×C414yes—
116#336—22no—
117#336—22no—
118C25—22no—
119C25C2×D411yescharacteristic
120C22×D4—28no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—82no—
137C22×D4—82no—
138C22×D4—82no—
139C22×D4—82no—
140C22×D4C22×C414yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×D8—28no—
161C2×D8—48no—
162C2×D8—44no—
163C2×D8—44no—
164C2×D8—48no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D412yes—
183C22×C8#4112yes—
184C22×C8#4112yes—
185#121—48no—
186#121—48no—
187#121—48no—
188#121—48no—
189#117—44no—
190#117—44no—
191#117—48no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C22×C414yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42C2×D412yes—
243C2×C42#4112yes—
244#107—44no—
245#102—48no—
246#102—48no—
247#102—48no—
248C22⋊C8—28no—
249C22⋊C8—28no—
250#97—44no—
251C4×C8—44no—
252#95—44no—
253#95—82no—
254#95—82no—
255#95—82no—
256#95—82no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—22no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—21no—
267C24—24no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24C4×D412yes—
283C24C22×D411yescharacteristic
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—416no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—88no—
318C2×D4—88no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4C4×D412yes—
381C22×C4#12112yes—
382C22×C4#12112yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C4H(5,𝔽2)12yes—
393D8—416no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8C4×D414yes—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—416no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—22no—
437C42—22no—
438C42—24no—
439C42—41no—
440C42—41no—
441C42—42no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23C2×C4×D411yescharacteristic
501C23C2×#12111yescharacteristic
502C23#53211yescharacteristic
503C23#53412yes—
504C23#53512yes—
505C23#57611yescharacteristic
506C23#57611yescharacteristic
507C23#57611yescharacteristic
508C23#57612yes—
509C23#57712yes—
510C23#57811yescharacteristic
511D4—44no—
512D4—44no—
513D4—44no—
514D4—44no—
515D4—832no—
516D4—82no—
517D4—82no—
518D4—82no—
519D4—82no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C4×#4111yescharacteristic
574C2×C4#51812yes—
575C2×C4#51812yes—
576C2×C4#52511yescharacteristic
577C2×C4#52911yescharacteristic
578C2×C4#53411yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53511yescharacteristic
581C2×C4#53511yescharacteristic
582C2×C4#53812yes—
583C2×C4#53812yes—
584C2×C4#53812yes—
585C2×C4#53812yes—
586C2×C4#53812yes—
587C2×C4#54012yes—
588C2×C4#54012yes—
589C2×C4#54012yes—
590C8—24no—
591C8—42no—
592C8—42no—
593C8—42no—
594C22—42no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—82no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22—162no—
613C22—162no—
614C22—162no—
615C22—162no—
616C22—162no—
617C22—162no—
618C22—162no—
619C22—162no—
620C22C4×#12111yescharacteristic
621C22#232112yes—
622C22#261711yescharacteristic
623C22#261911yescharacteristic
624C22#266112yes—
625C22#294612yes—
626C22#294811yescharacteristic
627C22#295111yescharacteristic
628C22#305811yescharacteristic
629C22#305811yescharacteristic
630C22#306011yescharacteristic
631C22#306111yescharacteristic
632C22#317511yescharacteristic
633C22#317611yescharacteristic
634C22#317912yes—
635C22#319812yes—
636C22#319812yes—
637C22#319912yes—
638C22#320312yes—
639C22#322012yes—
640C22#322012yes—
641C22#322412yes—
642C22#322512yes—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—84no—
647C4—81no—
648C4—81no—
649C4—162no—
650C4—162no—
651C4#232111yescharacteristic
652C4#232811yescharacteristic
653C4#233312yes—
654C4#263111yescharacteristic
655C4#263511yescharacteristic
656C4#264812yes—
657C2—82no—
658C2—164no—
659C2#1431111yescharacteristic
660C2#2152411yescharacteristic
661C2#2153611yescharacteristic
662C2#2205512yes—
663C2#2205612yes—
664C2#2973611yescharacteristic
665C2#2973811yescharacteristic
666C2#2981011yescharacteristic
667C2#2981311yescharacteristic
668C2#3032812yes—
669C2#3032912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1970C2semidirect
3. C2×#1750C2semidirect
4. #14257C2semidirect
5. #11256C2semidirect
6. #11189C2semidirect
7. #11096C2semidirect
8. #11022C2semidirect
9. #10969C2semidirect
10. #10927C2semidirect
11. #10820C2semidirect
26. #1999C4semidirect
99. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#19992
C2×#19702
C2×#17502
#142572
#112562
#111892
#110962
#110222
#109692
#109272
#108202