Finite Group Databasebeta

← #6382086

Subgroups — #6382086

Order 512.

2010 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#34188C211yesmaximal, characteristic
4#23069C212yesmaximal, complemented
5#23068C212yesmaximal, complemented
6#23067C212yesmaximal, complemented
7#23066C212yesmaximal, complemented
8#23063C212yesmaximal, complemented
9#23062C212yesmaximal, complemented
10#23061C212yesmaximal, complemented
11#23060C212yesmaximal, complemented
12C2×#1990C211yesmaximal, characteristic, complemented
13#21171C211yesmaximal, characteristic, complemented
14#21092C212yesmaximal
15#21062C212yesmaximal
16#21036C212yesmaximal, complemented
17#20453C212yesmaximal, complemented
18#20437C211yesmaximal, characteristic, complemented
19C4×#416C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22#3079C4116yescomplemented
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×#476C2212yes—
26C2×#474C2212yes—
27C2×#428C2212yes—
28C2×#428C2212yes—
29C2×#423C2211yescharacteristic
30C2×#417C2212yes—
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C2×#405C2212yes—
34C22×C4×C8C2211yescharacteristic
35#2369C2212yes—
36#2369C2212yes—
37#2365C2212yes—
38#2365C2212yes—
39#2355C2212yes—
40#2355C2212yes—
41#2354C2212yes—
42#2354C2212yes—
43#2352—22no—
44#2352—22no—
45#2345—22no—
46Q8×C22⋊C4—22no—
47#2331—22no—
48#2316C2212yes—
49#2314—22no—
50#2311C2212yes—
51C4×C4⋊Q8—21no—
52C4×C4⋊Q8—21no—
53#2285—21no—
54#2285—21no—
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2274C2212yes—
58#2274C2212yes—
59C4×#117C2211yescharacteristic
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1990C2212yes—
64#1948C2212yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1936C2214yes—
69#1935C2214yes—
70#1934C2214yes—
71#1933C2214yes—
72#1928C2214yes—
73#1927C2214yes—
74#1926C2214yes—
75#1925C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1918C2214yes—
80#1776C2214yes—
81#1776C2214yes—
82C4×#107C2214yes—
83C4×#106C2214yes—
84#1770C2212yes—
85#1770C2212yes—
86C8×C4⋊C4C2212yes—
87C8×C4⋊C4C2212yes—
88C4×#103C2211yescharacteristic
89C4×#103C2211yescharacteristic
90C4×#103C2211yescharacteristic
91C4×#103C2211yescharacteristic
92C4×#102C2211yescharacteristic
93C4×#102C2211yescharacteristic
94C4×#102C2211yescharacteristic
95C4×#102C2211yescharacteristic
96#1736C2212yes—
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99C2×#340C2212yes—
100C2×#340C2212yes—
101#1729C2212yes—
102C22×#41C2311yescharacteristic
103C23×Q8—21no—
104C22×M16C2312yes—
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C2×C4⋊Q8—21no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8C2×C418yes—
111C2×C4⋊Q8C2312yes—
112C2×C4⋊Q8C2312yes—
113C4×#41—24no—
114C4×#41C2×C418yes—
115C2×C4×Q8—22no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—22no—
124C2×#117—22no—
125C2×#117—21no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137#477C2×C4116yescomplemented
138#476C2×C4116yescomplemented
139#475C2×C4116yescomplemented
140#474C2×C4116yescomplemented
141#428—28no—
142#428C2×C4116yes—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#106C2314yes—
146C2×#106C2314yes—
147#423C2×C418yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150#417C2×C4116yes—
151#416—24no—
152#416—24no—
153C2×#103C2311yescharacteristic
154C2×#103C2311yescharacteristic
155C2×#103C2311yescharacteristic
156C2×#103C2311yescharacteristic
157C2×#103C2314yes—
158C2×#102C2311yescharacteristic
159C2×#102C2311yescharacteristic
160C2×#102C2311yescharacteristic
161C2×#102C2311yescharacteristic
162C2×#102C2314yes—
163#405—28no—
164C2×#97C2314yes—
165C2×C4×C8—24no—
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2314yes—
171#391—22no—
172#391—22no—
173#391—24no—
174#391—24no—
175#391C2314yes—
176#390C2314yes—
177#389—22no—
178#389—22no—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#388—24no—
184#388—24no—
185#387—24no—
186#387—24no—
187#387—24no—
188#387—24no—
189#386—24no—
190#384—21no—
191#384—21no—
192#384D411yescharacteristic
193#384D411yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382D412yes—
199#380—22no—
200#380—22no—
201#380—22no—
202#380—22no—
203#380—24no—
204#380D412yes—
205#380D412yes—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4C2314yes—
215C4×C4⋊C4C2314yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—21no—
226C4×C22⋊C4—24no—
227C4×C22⋊C4—24no—
228#376—21no—
229C43—21no—
230#340C2311yescharacteristic
231#340C2314yes—
232#340C2314yes—
233#340C2311yescharacteristic
234#340C2314yes—
235#340C2311yescharacteristic
236#340C2314yes—
237#340C2311yescharacteristic
238#340C2314yes—
239#340C2314yes—
240#340C2311yescharacteristic
241#340C2311yescharacteristic
242#340C2311yescharacteristic
243#340C2311yescharacteristic
244#336C2314yes—
245#336C2314yes—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41C22×C414yes—
252C22×Q8—28no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—82no—
256C22×Q8—82no—
257C22×Q8C2411yescharacteristic
258C22×D4—21no—
259C22×D4C2411yescharacteristic
260C23×C4—22no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—24no—
269C2×M16—24no—
270C2×M16C22×C418yes—
271C22×C8—22no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C22×C8—22no—
276C22×C8#4114yes—
277C22×C8C2411yescharacteristic
278C22×C8C2411yescharacteristic
279C22×C8C2414yes—
280C22×C8C2411yescharacteristic
281C22×C8C2411yescharacteristic
282C4⋊Q8—28no—
283C4⋊Q8—28no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8C22×C418yes—
287C4⋊Q8C22×C418yes—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—84no—
291C4×Q8—84no—
292C4×Q8C22×C414yes—
293C4×Q8C22×C414yes—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4C22×C414yes—
299C4×D4C22×C414yes—
300#117—22no—
301#117—24no—
302#117—22no—
303#117—24no—
304#117—28no—
305#117—22no—
306#117—22no—
307#117—84no—
308#117—84no—
309#117C22×C414yes—
310#117#4114yes—
311C2×C4⋊C4—28no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—82no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4C22×C418yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D412yes—
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4C2×D414yes—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—28no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42C22×C414yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42C2414yes—
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373#107—28no—
374#107C22×C4116yes—
375#106—28no—
376#106C22×C4116yes—
377C4⋊C8C22×C418yes—
378C4⋊C8C22×C418yes—
379#103—22no—
380#103—22no—
381#103—22no—
382#103—22no—
383#103C22×C4116yes—
384#102—22no—
385#102—22no—
386#102—22no—
387#102—22no—
388#102C22×C4116yes—
389#97—28no—
390C4×C8—28no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—22no—
394C4×C8—22no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—24no—
403#95—24no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95C2×D414yes—
408#95C2×D414yes—
409C24—21no—
410C24C22×D411yescharacteristic
411#41—28no—
412#41—28no—
413#41—44no—
414#41—44no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—82no—
425C2×Q8—82no—
426C2×Q8—816no—
427C2×Q8C23×C412yes—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C23×C412yes—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4C4×D414yes—
466C22×C4C4×D414yes—
467C22×C4C23×C412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C42−1+412yes—
479C22×C42−1+412yes—
480C22×C4C2511yescharacteristic
481M16—216no—
482C2×C8—28no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—44no—
496C2×C8C23×C418yes—
497C2×C8C2×#4112yes—
498C2×C8C2×#4112yes—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4C23×C418yes—
529C4⋊C4C23×C412yes—
530C4⋊C4C23×C412yes—
531C4⋊C4C2×#4112yes—
532C4⋊C4C2×#4112yes—
533C22⋊C4—24no—
534C22⋊C4—28no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—42no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—24no—
545C42—24no—
546C42—24no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—84no—
553C42—84no—
554C42—84no—
555C42—84no—
556C42C4×D414yes—
557C42C4×D414yes—
558C42C23×C412yes—
559C42C23×C412yes—
560C42C2×#4112yes—
561C42C2×#4112yes—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—41no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23C2×C4×D412yes—
574C23#54712yes—
575C23#54812yes—
576C23C23×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—416no—
580Q8—41no—
581Q8—41no—
582Q8—88no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4—41no—
587D4—41no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—88no—
631C2×C4—164no—
632C2×C4C2×C4×D412yes—
633C2×C4C2×C4×D412yes—
634C2×C4C2×C4×D412yes—
635C2×C4C2×C4×D412yes—
636C2×C4C2×C4×D412yes—
637C2×C4C4×#4114yes—
638C2×C4#51914yes—
639C2×C4#54712yes—
640C2×C4#54812yes—
641C2×C4#55214yes—
642C2×C4#55614yes—
643C2×C4#57612yes—
644C2×C4#57712yes—
645C2×C4#57712yes—
646C2×C4#57812yes—
647C2×C4C24×C411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—28no—
654C8—22no—
655C8—22no—
656C8—41no—
657C8—41no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22#236214yes—
672C22C22×C4×D411yescharacteristic
673C22C2×#54711yescharacteristic
674C22C2×#54811yescharacteristic
675C22C2×#57611yescharacteristic
676C22C2×#57711yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57811yescharacteristic
679C4—22no—
680C4—22no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—48no—
689C4—81no—
690C4—81no—
691C4—84no—
692C4#236214yes—
693C4#294614yes—
694C4#294714yes—
695C4C2×C4×#4111yescharacteristic
696C4C2×#51911yescharacteristic
697C4C2×#55211yescharacteristic
698C4C2×#55611yescharacteristic
699C2—24no—
700C2—82no—
701C2C2×#236211yescharacteristic
702C2C2×#294611yescharacteristic
703C2C2×#294711yescharacteristic
704C2#6128711yescharacteristic
705C2#6128811yescharacteristic
706C2#6132911yescharacteristic
707C2#6133011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
4. #23069C2semidirect
5. #23068C2semidirect
6. #23067C2semidirect
7. #23066C2semidirect
8. #23063C2semidirect
9. #23062C2semidirect
10. #23061C2semidirect
11. #23060C2semidirect
12. C2×#1990C2semidirect
13. #21171C2semidirect
16. #21036C2semidirect
17. #20453C2semidirect
18. #20437C2semidirect
22. #3079C4semidirect
137. #477C2×C4semidirect
138. #476C2×C4semidirect
139. #475C2×C4semidirect
140. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#342162
#341882
#230692
#230682
#230672
#230662
#230632
#230622
#230612
#230602
C2×#19902
#211712
#210922
#210622
#210362
#204532
#204372
C4×#4162