Finite Group Databasebeta

← #360469

Subgroups — #360469

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1845C211yesmaximal, characteristic, complemented
2C2×#1808C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×C4×#107C211yesmaximal, characteristic, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic
6#13876C212yesmaximal
7#11267C212yesmaximal, complemented
8#11096C212yesmaximal
9#10907C212yesmaximal, complemented
10#10862C211yesmaximal, characteristic, complemented
11#10862C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#388—22no—
22C2×#384C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1845—24no—
28#1808—24no—
29#1808—24no—
30C4×#107—24no—
31C8×C22⋊C4C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1442—22no—
40#1439—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—22no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—24no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#107D414yes—
83C2×#107D414yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8C2×C414yes—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8C2×C414yes—
92C2×C4×C8C2×C414yes—
93#388—48no—
94#384—24no—
95#384—42no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—42no—
102C4×C22⋊C4C2×C414yes—
103C4×C22⋊C4Q814yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#336C2×C414yes—
119#336C2×C414yes—
120C25C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×Q812yes—
144C23×C4C2411yescharacteristic
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—28no—
163C22×C8—28no—
164C22×C8—44no—
165C22×C8—44no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C22×C8C22×C412yes—
172C22×C8C2×D412yes—
173C22×C8C2×D412yes—
174C22×C8#4112yes—
175C22×C8#4112yes—
176C22×C8#4112yes—
177C22×C8#4112yes—
178C22×C8#4112yes—
179C22×C8#4112yes—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
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—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—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D414yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4C4⋊C414yes—
219C2×C22⋊C4C4⋊C414yes—
220C2×C22⋊C4D814yes—
221C2×C22⋊C4Q1614yes—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—82no—
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C2×Q812yes—
236C2×C42C2×Q812yes—
237#107—28no—
238#107—28no—
239#107—48no—
240#107—44no—
241#107—44no—
242C22⋊C8—28no—
243C4×C8—28no—
244C4×C8—28no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—82no—
252#95C4⋊C414yes—
253#95C4⋊C414yes—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24C2×C4⋊C412yes—
267C24C2×D812yes—
268C24C2×Q1612yes—
269C24C22×D411yescharacteristic
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C2×C4⋊C412yes—
351C22×C4C2×C4⋊C412yes—
352C22×C4C2×C4⋊C412yes—
353C22×C4C2×C4⋊C412yes—
354C22×C4C2×C4⋊C412yes—
355C22×C4C2×D812yes—
356C22×C4C2×D812yes—
357C22×C4C2×Q1612yes—
358C22×C4C2×Q1612yes—
359C22×C4#13512yes—
360C22×C4#13512yes—
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×Q811yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—416no—
396C2×C8#11714yes—
397C2×C8#11714yes—
398C2×C8C4×D414yes—
399C2×C8C4×D414yes—
400C2×C8#12114yes—
401C2×C8#12314yes—
402C2×C8#12414yes—
403C2×C8#12614yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
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—88no—
421C4⋊C4—88no—
422C22⋊C4—28no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C22⋊C4#107116yescomplemented
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—88no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C2×#10714yes—
470C23C22×C4⋊C411yescharacteristic
471C23#53211yescharacteristic
472C23#53211yescharacteristic
473C23D4211yescharacteristic
474C23D4211yescharacteristic
475C23#54711yescharacteristic
476C23#54711yescharacteristic
477C23C22×D811yescharacteristic
478C23C22×Q1611yescharacteristic
479C23C2×#13511yescharacteristic
480C23C2×#13511yescharacteristic
481C23#57611yescharacteristic
482C23#57611yescharacteristic
483C23#57811yescharacteristic
484C23#57811yescharacteristic
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
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—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4C2×#10714yes—
554C2×C4C2×#10714yes—
555C2×C4#42714yes—
556C2×C4C2×#11711yescharacteristic
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4#52011yescharacteristic
561C2×C4#52011yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12311yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4C2×#12611yescharacteristic
566C2×C4#53311yescharacteristic
567C2×C4#53311yescharacteristic
568C2×C4#54011yescharacteristic
569C2×C4#54111yescharacteristic
570C2×C4#54211yescharacteristic
571C2×C4#54211yescharacteristic
572C2×C4#54611yescharacteristic
573C2×C4#54711yescharacteristic
574C2×C4#54811yescharacteristic
575C2×C4D4×Q811yescharacteristic
576C2×C4D4×Q811yescharacteristic
577C2×C4#55011yescharacteristic
578C2×C4#55211yescharacteristic
579C2×C4#55311yescharacteristic
580C2×C4#55411yescharacteristic
581C2×C4#55411yescharacteristic
582C2×C4#55511yescharacteristic
583C2×C4#55611yescharacteristic
584C8—24no—
585C8—42no—
586C8—42no—
587C8#38818yescomplemented
588C22—24no—
589C22—24no—
590C22—24no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22#228511yescharacteristic
604C22D4×C4⋊C411yescharacteristic
605C22D4×C4⋊C411yescharacteristic
606C22#242211yescharacteristic
607C22#242811yescharacteristic
608C22#249111yescharacteristic
609C22#249611yescharacteristic
610C22C22×#10711yescharacteristic
611C22C2×#42711yescharacteristic
612C22#291311yescharacteristic
613C22#291311yescharacteristic
614C22#316911yescharacteristic
615C22#316911yescharacteristic
616C22#317011yescharacteristic
617C22#317011yescharacteristic
618C22#317511yescharacteristic
619C22#317511yescharacteristic
620C22#317611yescharacteristic
621C22#317711yescharacteristic
622C22D4×D811yescharacteristic
623C22D4×D811yescharacteristic
624C22#328011yescharacteristic
625C22#328311yescharacteristic
626C22#328511yescharacteristic
627C22D4×Q1611yescharacteristic
628C22D4×Q1611yescharacteristic
629C22#328711yescharacteristic
630C22#333411yescharacteristic
631C22#333411yescharacteristic
632C22#333611yescharacteristic
633C22#333811yescharacteristic
634C22#333811yescharacteristic
635C22#333911yescharacteristic
636C22#334111yescharacteristic
637C22#334211yescharacteristic
638C4—24no—
639C4—24no—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—84no—
644C4—84no—
645C4—162no—
646C4—162no—
647C4C2×#38811yescharacteristic
648C4#231411yescharacteristic
649C4#235211yescharacteristic
650C4#235411yescharacteristic
651C4#242311yescharacteristic
652C4#242911yescharacteristic
653C4#249211yescharacteristic
654C4#249711yescharacteristic
655C2—44no—
656C2#1387611yescharacteristic
657C2#2081711yescharacteristic
658C2#2082111yescharacteristic
659C2D4×#10711yescharacteristic
660C2D4×#10711yescharacteristic
661C2#2265611yescharacteristic
662C2#2265811yescharacteristic
663C2#2503411yescharacteristic
664C2#2503511yescharacteristic
665C2#2541411yescharacteristic
666C2#2541511yescharacteristic
667C2#2779111yescharacteristic
668C2#2779211yescharacteristic
669C2#2783911yescharacteristic
670C2#2784011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1845C2semidirect
4. C2×C4×#107C2semidirect
7. #11267C2semidirect
9. #10907C2semidirect
10. #10862C2semidirect
11. #10862C2semidirect
31. C8×C22⋊C4C4semidirect
425. C22⋊C4#107semidirect
587. C8#388semidirect

Maximal subgroups

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

subgroupindex
C2×#18452
C2×#18082
C2×C4×#1072
C2×C8×C22⋊C42
#138762
#112672
#110962
#109072
#108622