Finite Group Databasebeta

← #366409

Subgroups — #366409

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1939C211yesmaximal, characteristic, complemented
2C2×#1797C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5C4×#380C211yesmaximal, characteristic
6#11298C212yesmaximal, complemented
7#11042C212yesmaximal
8#10921C212yesmaximal
9#10876C212yesmaximal, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2212yes—
22C2×#380C411yescharacteristic
23C2×#380C411yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2212yes—
27#1939—24no—
28#1797—24no—
29C8×C4⋊C4—24no—
30C4×C22⋊C8—24no—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C4⋊#95C411yescharacteristic
41C4⋊#95C411yescharacteristic
42#1434C412yes—
43C22⋊#95C412yes—
44C4×#95C2212yes—
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2×C412yes—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2312yes—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8D414yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8Q814yes—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—24no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—42no—
97C2×C4×C8—42no—
98C2×C4×C8D414yes—
99#380—24no—
100#380C818yescomplemented
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C22⋊C4—42no—
106C4×C22⋊C4—42no—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115#336—24no—
116#336—24no—
117#336—22no—
118#336—22no—
119#336—24no—
120#336—24no—
121#336—42no—
122#336—42no—
123#336—42no—
124C25C22×C411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4C2×C814yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×Q812yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C2×D412yes—
199C22×C8C2×D414yes—
200C22×C8C2×D412yes—
201C22×C8C2×Q812yes—
202C22×C8C2×Q812yes—
203C22×C8#4112yes—
204C22×C8#4112yes—
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—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×C814yes—
215C2×C4⋊C4C2×C814yes—
216C2×C4⋊C4M1614yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4C2×C818yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—28no—
258C4⋊C8—48no—
259C4⋊C8—44no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262C22⋊C8—48no—
263C4×C8—28no—
264C4×C8—48no—
265C4×C8—44no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95C2×C818yes—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—21no—
286C24C4×Q812yes—
287C24C22×C812yes—
288C24#13112yes—
289C24C23×C411yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—28no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—28no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4#11712yes—
380C22×C4#11712yes—
381C22×C4C4×D412yes—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C4×Q812yes—
386C22×C4C22×C812yes—
387C22×C4C22×C814yes—
388C22×C4C22×C814yes—
389C22×C4C22×C812yes—
390C22×C4C2×M1614yes—
391C22×C4C2×M1612yes—
392C22×C4#13112yes—
393C22×C4#13112yes—
394C22×C4#13112yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4H(5,𝔽2)11yescharacteristic
406C22×C4H(5,𝔽2)11yescharacteristic
407C22×C4H(5,𝔽2)11yescharacteristic
408C22×C42−1+411yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8C22⋊Q814yes—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42#12114yes—
472C42#12314yes—
473C42#12414yes—
474C42#12614yes—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C8×Q814yes—
510C23C2×#11711yescharacteristic
511C23C2×C4×D411yescharacteristic
512C23C2×C4×D411yescharacteristic
513C23C2×C4×Q811yescharacteristic
514C23C4×#4111yescharacteristic
515C23#51811yescharacteristic
516C23#52011yescharacteristic
517C23C23×C811yescharacteristic
518C23C22×M1611yescharacteristic
519C23C2×#13111yescharacteristic
520C23C2×#13111yescharacteristic
521C23C2×#13111yescharacteristic
522C23C2×#13111yescharacteristic
523C23#56811yescharacteristic
524C23#56811yescharacteristic
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—416no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4#38814yes—
591C2×C4#38814yes—
592C2×C4#43114yes—
593C2×C4#43314yes—
594C2×C4C8×D414yes—
595C2×C4C8×D414yes—
596C2×C4#43514yes—
597C2×C4#43614yes—
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×#12311yescharacteristic
601C2×C4#52511yescharacteristic
602C2×C4C2×#12411yescharacteristic
603C2×C4C2×#12611yescharacteristic
604C2×C4#52911yescharacteristic
605C2×C4#53311yescharacteristic
606C2×C4#53511yescharacteristic
607C2×C4#53711yescharacteristic
608C2×C4#54011yescharacteristic
609C2×C4#54211yescharacteristic
610C2×C4#54611yescharacteristic
611C2×C4#54611yescharacteristic
612C2×C4D4×Q811yescharacteristic
613C2×C4#55011yescharacteristic
614C2×C4#55211yescharacteristic
615C2×C4#55311yescharacteristic
616C2×C4#55311yescharacteristic
617C2×C4#55311yescharacteristic
618C2×C4#55311yescharacteristic
619C2×C4#55311yescharacteristic
620C2×C4#55411yescharacteristic
621C2×C4#55411yescharacteristic
622C2×C4#55411yescharacteristic
623C2×C4#55511yescharacteristic
624C2×C4#55611yescharacteristic
625C2×C4#55611yescharacteristic
626C8—24no—
627C8—42no—
628C8—44no—
629C8—42no—
630C8—42no—
631C8—82no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C2×#38811yescharacteristic
648C22C4×C22⋊Q811yescharacteristic
649C22#232411yescharacteristic
650C22#234511yescharacteristic
651C22#234711yescharacteristic
652C22#236011yescharacteristic
653C22#236311yescharacteristic
654C22C2×#43111yescharacteristic
655C22C2×#43311yescharacteristic
656C22#292411yescharacteristic
657C22#292411yescharacteristic
658C22C2×C8×D411yescharacteristic
659C22C2×C8×D411yescharacteristic
660C22C2×#43511yescharacteristic
661C22C2×#43611yescharacteristic
662C22#292911yescharacteristic
663C22#293011yescharacteristic
664C22#293311yescharacteristic
665C22#293311yescharacteristic
666C22C2×C8×Q811yescharacteristic
667C22#296011yescharacteristic
668C22#296211yescharacteristic
669C22Q8×M1611yescharacteristic
670C22C8×#4111yescharacteristic
671C22#296611yescharacteristic
672C22#296911yescharacteristic
673C22#297111yescharacteristic
674C22#297211yescharacteristic
675C22#297411yescharacteristic
676C22#297911yescharacteristic
677C22#298111yescharacteristic
678C22#298811yescharacteristic
679C22#299011yescharacteristic
680C22#299311yescharacteristic
681C22#299411yescharacteristic
682C4—24no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4—42no—
689C4—82no—
690C4#192318yes—
691C4#248511yescharacteristic
692C4#248711yescharacteristic
693C4#249011yescharacteristic
694C4#249211yescharacteristic
695C4#249411yescharacteristic
696C4#249711yescharacteristic
697C4#253511yescharacteristic
698C4#253811yescharacteristic
699C2—44no—
700C2#1399311yescharacteristic
701C2C2×#192311yescharacteristic
702C2#2105011yescharacteristic
703C2C8×C22⋊Q811yescharacteristic
704C2#2159611yescharacteristic
705C2#2213311yescharacteristic
706C2#2213511yescharacteristic
707C2#2262511yescharacteristic
708C2#2262611yescharacteristic
709C2#2263911yescharacteristic
710C2#2264011yescharacteristic
711C2#2295211yescharacteristic
712C2#2295711yescharacteristic
713C2#2308311yescharacteristic
714C2#2308811yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1939C2semidirect
3. C2×C8×C4⋊C4C2semidirect
6. #11298C2semidirect
9. #10876C2semidirect
10. #10834C2semidirect
11. #10834C2semidirect
100. #380C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19392
C2×#17972
C2×C8×C4⋊C42
C2×C4×C22⋊C82
C4×#3802
#112982
#110422
#109212
#108762
#108342