Finite Group Databasebeta

← #368605

Subgroups — #368605

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1954C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic
3C2×#1923C212yesmaximal, complemented
4C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
6C4×#390C211yesmaximal, characteristic
7#11147C212yesmaximal, complemented
8#11042C212yesmaximal, complemented
9#10921C212yesmaximal, complemented
10#10834C211yesmaximal, characteristic, complemented
11#10832C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24C2×#390C411yescharacteristic
25C2×#390C411yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2212yes—
29C2×C43C2211yescharacteristic
30#1954—24no—
31#1954—24no—
32#1923—28no—
33C8×C4⋊C4—24no—
34C4×C4⋊C8—24no—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1437C412yes—
43#1437C412yes—
44C4⋊#95C411yescharacteristic
45C4⋊#95C411yescharacteristic
46C23×D4C2×C411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yes—
87C2×C4⋊C8D414yescomplemented
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—42no—
97C2×C22⋊C8—42no—
98C2×C22⋊C8—42no—
99C2×C22⋊C8—42no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—24no—
103C2×C4×C8—24no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—42no—
108C2×C4×C8D414yes—
109#390—24no—
110#390C818yescomplemented
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—24no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—42no—
116C4×C22⋊C4—42no—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C43—22no—
123C43—22no—
124#336—24no—
125#336—22no—
126#336—22no—
127#336—24no—
128#336—24no—
129#336—42no—
130#336—42no—
131C25C22×C412yes—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4C2×C814yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2411yescharacteristic
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
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×D412yes—
202C22×C8C2×D412yes—
203C22×C8#4112yes—
204C22×C8#4112yes—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4C2×C814yes—
217C2×C4⋊C4M1614yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4C2×C818yes—
232C2×C22⋊C4C2×C818yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—21no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42C2×C814yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42#4112yes—
267C2×C42#4112yes—
268C2×C42#4112yes—
269C4⋊C8—28no—
270C4⋊C8—28no—
271C4⋊C8—28no—
272C4⋊C8—44no—
273C22⋊C8—48no—
274C22⋊C8—48no—
275C22⋊C8—48no—
276C22⋊C8—48no—
277C4×C8—28no—
278C4×C8—48no—
279C4×C8—44no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—42no—
284#95—42no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C22×C814yes—
300C24C23×C411yescharacteristic
301C2×D4—28no—
302C2×D4—28no—
303C2×D4—28no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
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—42no—
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×C4—42no—
381C22×C4#11712yes—
382C22×C4#11712yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C4×D412yes—
388C22×C4C4×D412yes—
389C22×C4C22×C812yes—
390C22×C4C22×C814yes—
391C22×C4C22×C812yes—
392C22×C4C22×C812yes—
393C22×C4C2×M1614yes—
394C22×C4C2×M1612yes—
395C22×C4#13112yes—
396C22×C4#13112yes—
397C22×C4#13112yes—
398C22×C4#13112yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4H(5,𝔽2)11yescharacteristic
410C22×C4H(5,𝔽2)11yescharacteristic
411C22×C4H(5,𝔽2)11yescharacteristic
412C22×C42−1+411yescharacteristic
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—42no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—42no—
447C2×C8—42no—
448C2×C8—42no—
449C2×C8—42no—
450C2×C8—42no—
451C2×C8—42no—
452C2×C8—42no—
453C2×C8—42no—
454C2×C8#12114yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—44no—
458C22⋊C4—216no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—28no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—28no—
469C42—44no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—42no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—44no—
479C42#12114yes—
480C42#12314yes—
481C42#12414yes—
482C42#12614yes—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23C2×#11711yescharacteristic
519C23C2×C4×D411yescharacteristic
520C23C2×C4×D411yescharacteristic
521C23C2×C4×D411yescharacteristic
522C23C4×#4111yescharacteristic
523C23#51811yescharacteristic
524C23#52011yescharacteristic
525C23C23×C811yescharacteristic
526C23C22×M1611yescharacteristic
527C23C2×#13111yescharacteristic
528C23C2×#13111yescharacteristic
529C23C2×#13111yescharacteristic
530C23C2×#13111yescharacteristic
531C23#56811yescharacteristic
532C23#56811yescharacteristic
533D4—416no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4#38814yes—
597C2×C4#38814yes—
598C2×C4#43114yes—
599C2×C4#43314yes—
600C2×C4C8×D414yes—
601C2×C4C8×D414yes—
602C2×C4C8×D414yes—
603C2×C4#43514yes—
604C2×C4#43614yes—
605C2×C4C2×#12111yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12311yescharacteristic
608C2×C4#52511yescharacteristic
609C2×C4C2×#12411yescharacteristic
610C2×C4C2×#12611yescharacteristic
611C2×C4#52911yescharacteristic
612C2×C4#53211yescharacteristic
613C2×C4#53511yescharacteristic
614C2×C4#53711yescharacteristic
615C2×C4#54011yescharacteristic
616C2×C4#54211yescharacteristic
617C2×C4D4211yescharacteristic
618C2×C4#54611yescharacteristic
619C2×C4#54711yescharacteristic
620C2×C4#54811yescharacteristic
621C2×C4#55011yescharacteristic
622C2×C4#55011yescharacteristic
623C2×C4#55011yescharacteristic
624C2×C4#55311yescharacteristic
625C2×C4#55311yescharacteristic
626C2×C4#55511yescharacteristic
627C2×C4#55611yescharacteristic
628C2×C4#55611yescharacteristic
629C2×C4#55911yescharacteristic
630C2×C4#55911yescharacteristic
631C2×C4#55911yescharacteristic
632C2×C4#55911yescharacteristic
633C8—24no—
634C8—42no—
635C8—42no—
636C8—42no—
637C8—44no—
638C8—82no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22C2×#38811yescharacteristic
655C22C4×#12111yescharacteristic
656C22#232411yescharacteristic
657C22#235411yescharacteristic
658C22#235711yescharacteristic
659C22#236611yescharacteristic
660C22#236711yescharacteristic
661C22C2×#43111yescharacteristic
662C22C2×#43311yescharacteristic
663C22#292511yescharacteristic
664C22#292511yescharacteristic
665C22C2×C8×D411yescharacteristic
666C22C2×C8×D411yescharacteristic
667C22C2×C8×D411yescharacteristic
668C22C2×#43511yescharacteristic
669C22C2×#43611yescharacteristic
670C22#292911yescharacteristic
671C22#293011yescharacteristic
672C22#293111yescharacteristic
673C22D4×M1611yescharacteristic
674C22#293511yescharacteristic
675C22#293511yescharacteristic
676C22#293511yescharacteristic
677C22C8×#4111yescharacteristic
678C22#296611yescharacteristic
679C22#296911yescharacteristic
680C22#297111yescharacteristic
681C22#297211yescharacteristic
682C22#297411yescharacteristic
683C22#297911yescharacteristic
684C22#298111yescharacteristic
685C22#298811yescharacteristic
686C22#299111yescharacteristic
687C22#299311yescharacteristic
688C22#299411yescharacteristic
689C4—24no—
690C4—24no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—44no—
695C4—42no—
696C4—82no—
697C4#192318yescomplemented
698C4#241411yescharacteristic
699C4#241611yescharacteristic
700C4#244511yescharacteristic
701C4#244811yescharacteristic
702C4#251211yescharacteristic
703C4#251311yescharacteristic
704C4#254811yescharacteristic
705C4#259211yescharacteristic
706C2—84no—
707C2#1402211yescharacteristic
708C2C2×#192311yescharacteristic
709C2#2105211yescharacteristic
710C2C8×#12111yescharacteristic
711C2#2148811yescharacteristic
712C2#2213311yescharacteristic
713C2#2213711yescharacteristic
714C2#2279511yescharacteristic
715C2#2279611yescharacteristic
716C2#2283511yescharacteristic
717C2#2283611yescharacteristic
718C2#2317311yescharacteristic
719C2#2317411yescharacteristic
720C2#2320411yescharacteristic
721C2#2320511yescharacteristic
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C23×C8
socleC24C23×C4
prime radicalC23×D4C2×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
3. C2×#1923C2semidirect
4. C2×C8×C4⋊C4C2semidirect
5. C2×C4×C4⋊C8C2semidirect
7. #11147C2semidirect
8. #11042C2semidirect
9. #10921C2semidirect
10. #10834C2semidirect
11. #10832C2semidirect
87. C2×C4⋊C8D4semidirect
110. #390C8semidirect
697. C4#1923semidirect

Maximal subgroups

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

subgroupindex
C2×#19542
C2×#19232
C2×C8×C4⋊C42
C2×C4×C4⋊C82
C4×#3902
#111472
#110422
#109212
#108342
#108322