Finite Group Databasebeta

← #362333

Subgroups — #362333

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C211yesmaximal, characteristic
2#21049C212yesmaximal
3C2×#1822C211yesmaximal, characteristic
4C2×#1801C211yesmaximal, characteristic, complemented
5#11204C212yesmaximal, complemented
6#10968C212yesmaximal
7#10963C212yesmaximal, complemented
8#10802C211yesmaximal, characteristic, complemented
9#10763C211yesmaximal, characteristic, complemented
10#10728C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#423C2212yes—
16C22×#101C2212yes—
17C22×#101C2212yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20C2×C4×M16C2212yes—
21#2362—22no—
22#2362—22no—
23#2362—22no—
24#2362—22no—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388C412yes—
28C2×#388C412yes—
29C2×#387C412yes—
30C2×C4×C4⋊C4C412yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1924—22no—
34#1924—22no—
35#1923—22no—
36#1923—22no—
37#1822—24no—
38#1801—24no—
39C2×#343C2211yescharacteristic
40C2×#343C2211yescharacteristic
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#343C2212yes—
46C2×#343C2211yescharacteristic
47#1728C2212yes—
48#1728C2212yes—
49C23×Q8C2311yescharacteristic
50C24×C4C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2312yes—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—24no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—42no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—42no—
70C22×C4⋊C4C2×C412yes—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42C2×C412yes—
82C22×C42C2×C412yes—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#423—24no—
86#423—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×#101—24no—
90C2×#101—24no—
91C2×#101—24no—
92C2×#101—24no—
93C2×#101—42no—
94C2×#101—42no—
95#407—24no—
96#407—24no—
97#407—42no—
98#407—42no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C4×M16—24no—
104C4×M16—42no—
105C2×#97—22no—
106C2×C4×C8—22no—
107#391—22no—
108#391—22no—
109#389—22no—
110#389—22no—
111#388—24no—
112#388—24no—
113#388—24no—
114#388—24no—
115#387—24no—
116#387—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126C4×C22⋊C4D412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C411yescharacteristic
132C2×#95C2×C411yescharacteristic
133#343—24no—
134#343—22no—
135#343—22no—
136#343—24no—
137#343—24no—
138#343—22no—
139#343—42no—
140#343D414yes—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145C25C22×C411yescharacteristic
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8#4112yes—
155C22×Q8#4112yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C412yes—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2411yescharacteristic
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16C2×D414yes—
200C2×M16#4114yes—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22⋊Q8—28no—
206C22⋊Q8—28no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C4×Q8—44no—
210C4×Q8—44no—
211C4×Q8—44no—
212C4×Q8—44no—
213#117—44no—
214#117—44no—
215#117—44no—
216#117—44no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C4⋊C4#4112yes—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C22⋊C4C22⋊C412yes—
250C2×C22⋊C4C22⋊C412yes—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C4⋊C8—44no—
273C4⋊C8—44no—
274#101—48no—
275#101—48no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280#97—44no—
281C4×C8—44no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95C22⋊C411yescharacteristic
288#95C22⋊C411yescharacteristic
289#95C22⋊C411yescharacteristic
290#95C22⋊C411yescharacteristic
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—41no—
300C24—41no—
301C24—41no—
302C24—41no—
303C24C2×C22⋊C412yes—
304C24C23×C411yescharacteristic
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—42no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C412yes—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4#11714yes—
393C22×C4#11714yes—
394C22×C4C4×D414yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4112yes—
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)11yescharacteristic
403C22×C4H(5,𝔽2)12yes—
404C22×C42−1+411yescharacteristic
405M16—24no—
406M16—24no—
407M16—24no—
408M16—24no—
409M16—42no—
410M16—42no—
411M16—42no—
412M16—42no—
413M16—42no—
414M16—42no—
415M16—42no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C22⋊C4—216no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—48no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—44no—
459C42—44no—
460C42—44no—
461C42—44no—
462C42—44no—
463C42—44no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23—81no—
487C23—81no—
488C23—81no—
489C23—81no—
490C23—81no—
491C23C22×C22⋊C411yescharacteristic
492C23C2×#11711yescharacteristic
493C23C2×#11711yescharacteristic
494C23C2×C4×D411yescharacteristic
495C23C4×#4111yescharacteristic
496C23#51811yescharacteristic
497C23#52011yescharacteristic
498Q8—44no—
499Q8—44no—
500Q8—44no—
501Q8—44no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4#41312yes—
567C2×C4#41312yes—
568C2×C4#52512yes—
569C2×C4#52512yes—
570C2×C4#52912yes—
571C2×C4#53212yes—
572C2×C4#54012yes—
573C2×C4#54212yes—
574C2×C4D4212yes—
575C2×C4#54612yes—
576C2×C4#54612yes—
577C2×C4#54611yescharacteristic
578C2×C4#54712yes—
579C2×C4#54811yescharacteristic
580C2×C4#55112yes—
581C2×C4#55211yescharacteristic
582C2×C4#55311yescharacteristic
583C2×C4#55312yes—
584C8—44no—
585C8—44no—
586C8—82no—
587C8—82no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22#228011yescharacteristic
612C22#229311yescharacteristic
613C22#231911yescharacteristic
614C22D4×C22⋊C411yescharacteristic
615C22#233911yescharacteristic
616C22#235211yescharacteristic
617C22#235311yescharacteristic
618C22C2×#41311yescharacteristic
619C22C2×#41311yescharacteristic
620C22#288911yescharacteristic
621C22#288911yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—44no—
627C4—41no—
628C4—42no—
629C4—44no—
630C4—41no—
631C4—42no—
632C4—44no—
633C4—82no—
634C4#245712yes—
635C4#246112yes—
636C4#246412yes—
637C4#246812yes—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—21no—
642C2—82no—
643C2#1391211yescharacteristic
644C2#2072811yescharacteristic
645C2#2073011yescharacteristic
646C2#2241011yescharacteristic
647C2#2241111yescharacteristic
648C2#2248011yescharacteristic
649C2#2248411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1801C2semidirect
5. #11204C2semidirect
7. #10963C2semidirect
8. #10802C2semidirect
9. #10763C2semidirect
11. C4×#343C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
#210492
C2×#18222
C2×#18012
#112042
#109682
#109632
#108022
#107632
#107282
C4×#3432