Finite Group Databasebeta

← #355658

Subgroups — #355658

Order 512.

1374 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3C4×#388C211yesmaximal, characteristic
4#10921C211yesmaximal, characteristic
5#10921C211yesmaximal, characteristic
6#10876C211yesmaximal, characteristic, complemented
7#10874C211yesmaximal, characteristic, complemented
8#10834C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10797C211yesmaximal, characteristic, complemented
11#10795C211yesmaximal, characteristic, complemented
12#10714C211yesmaximal, characteristic
13#10714C211yesmaximal, characteristic
14C8×#95C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C2×#388C412yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#1923C418yescomplemented
33C8×C22⋊C4C418yescomplemented
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1437C412yes—
53#1436C412yes—
54#1434C412yes—
55C4×#95C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C24×C4C2311yescharacteristic
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—41no—
61C23×C8—41no—
62C23×C8—41no—
63C23×C8—41no—
64C23×C8C2×C414yes—
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C22×C4⋊C4C2×C412yes—
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C42C2×C412yes—
79C22×C42C2×C412yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8C2×C414yes—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—41no—
92C2×C22⋊C8—41no—
93C2×C22⋊C8—41no—
94C2×C22⋊C8—41no—
95C2×C22⋊C8C2×C414yes—
96C2×C22⋊C8C2×C414yes—
97C2×C22⋊C8C2×C414yes—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—41no—
101C2×C4×C8—41no—
102C2×C4×C8—41no—
103C2×C4×C8—41no—
104C2×C4×C8—41no—
105C2×C4×C8—41no—
106C2×C4×C8—41no—
107C2×C4×C8—41no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8C2×C414yes—
111C2×C4×C8C2×C414yes—
112C2×C4×C8C2×C414yes—
113#388C8116yescomplemented
114C4×C4⋊C4—41no—
115C4×C4⋊C4—41no—
116C4×C22⋊C4—41no—
117C4×C22⋊C4—41no—
118C4×C22⋊C4—41no—
119C4×C22⋊C4—41no—
120C4×C22⋊C4C2×C414yes—
121C4×C22⋊C4C2×C414yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C411yescharacteristic
128C2×#95C2×C411yescharacteristic
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C43—41no—
132C43—41no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—41no—
140#336—41no—
141#336—41no—
142#336—41no—
143#336—41no—
144#336—41no—
145#336—41no—
146#336—41no—
147#336C2×C414yes—
148#336C2×C414yes—
149#336C2×C414yes—
150#336C2×C414yes—
151C25C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C412yes—
167C23×C4C22×C412yes—
168C23×C4C22×C412yes—
169C23×C4C22×C412yes—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—28no—
221C22×C8—44no—
222C22×C8—44no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8C22×C412yes—
228C22×C8C22×C412yes—
229C22×C8C22×C412yes—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8#4112yes—
236C22×C8#4112yes—
237C22×C8#4112yes—
238C22×C8#4112yes—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C2×C818yescomplemented
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C4214yes—
248C2×C22⋊C4C4214yes—
249C2×C22⋊C4C2×C818yes—
250C2×C22⋊C4C2×C818yes—
251C2×C22⋊C4C2×C818yes—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—24no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42C2×C818yescomplemented
290C2×C42C22×C412yes—
291C2×C42C22×C412yes—
292C2×C42C22×C412yes—
293C2×C42C22×C412yes—
294C4⋊C8—44no—
295C22⋊C8—28no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298C4×C8—44no—
299C4×C8—44no—
300C4×C8—44no—
301C4×C8—44no—
302C4×C8—44no—
303#95—24no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95C4214yes—
309#95C4214yes—
310#95C2×C818yescomplemented
311#95C2×C818yescomplemented
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C2×C4212yes—
322C24C22×C814yes—
323C24C23×C411yescharacteristic
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
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—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4C2×C4212yes—
396C22×C4C2×C4212yes—
397C22×C4C2×C4212yes—
398C22×C4C2×C4212yes—
399C22×C4C2×C4212yes—
400C22×C4C22×C814yes—
401C22×C4C22×C814yes—
402C22×C4C22×C814yes—
403C22×C4C22×C814yes—
404C22×C4C22×C814yes—
405C22×C4C22×C814yes—
406C22×C4#13114yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C23×C411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4C2×#4111yescharacteristic
411C22×C4C2×#4111yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4H(5,𝔽2)11yescharacteristic
414C22×C4H(5,𝔽2)11yescharacteristic
415C22×C4H(5,𝔽2)11yescharacteristic
416C22×C4H(5,𝔽2)11yescharacteristic
417C22×C4H(5,𝔽2)11yescharacteristic
418C22×C4H(5,𝔽2)11yescharacteristic
419C22×C42−1+411yescharacteristic
420C22×C42−1+411yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—416no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8#11714yes—
477C2×C8#11714yes—
478C4⋊C4—48no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4C4×C8116yescomplemented
483C42—22no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—48no—
492C42—48no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23C2×C4×C814yes—
532C23C22×C4211yescharacteristic
533C23C4×#4111yescharacteristic
534C23C4×#4111yescharacteristic
535C23#51811yescharacteristic
536C23#51811yescharacteristic
537C23#52011yescharacteristic
538C23#52011yescharacteristic
539C23C23×C812yes—
540C23C2×#13112yes—
541C23#56812yes—
542C23#56812yes—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
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—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4C2×C4×C814yes—
604C2×C4C2×C4×C814yes—
605C2×C4#40514yes—
606C2×C4C2×#11711yescharacteristic
607C2×C4C2×#11711yescharacteristic
608C2×C4C4×#4111yescharacteristic
609C2×C4C4×#4111yescharacteristic
610C2×C4#51811yescharacteristic
611C2×C4#51811yescharacteristic
612C2×C4#51811yescharacteristic
613C2×C4#51811yescharacteristic
614C2×C4#52011yescharacteristic
615C2×C4#52011yescharacteristic
616C2×C4#52011yescharacteristic
617C2×C4#52011yescharacteristic
618C2×C4#53811yescharacteristic
619C2×C4#53811yescharacteristic
620C2×C4#53811yescharacteristic
621C2×C4#53811yescharacteristic
622C2×C4#53811yescharacteristic
623C2×C4#53811yescharacteristic
624C2×C4#53911yescharacteristic
625C2×C4#53911yescharacteristic
626C2×C4#54011yescharacteristic
627C2×C4#54011yescharacteristic
628C2×C4#54011yescharacteristic
629C2×C4#54111yescharacteristic
630C2×C4#54211yescharacteristic
631C2×C4#54211yescharacteristic
632C2×C4#54211yescharacteristic
633C2×C4#54211yescharacteristic
634C8—24no—
635C8—24no—
636C8—42no—
637C8—42no—
638C8—42no—
639C8—42no—
640C8—42no—
641C8—42no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—44no—
646C22—44no—
647C22—42no—
648C22—42no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—42no—
653C22C4×#11711yescharacteristic
654C22#227511yescharacteristic
655C22#227511yescharacteristic
656C22#233311yescharacteristic
657C22#233411yescharacteristic
658C22#233511yescharacteristic
659C22#233711yescharacteristic
660C22C22×C4×C811yescharacteristic
661C22C2×#40511yescharacteristic
662C22#287511yescharacteristic
663C22#287511yescharacteristic
664C22C8×#4112yes—
665C22C8×#4112yes—
666C22#296712yes—
667C22#296712yes—
668C22#297212yes—
669C22#297212yes—
670C22#297612yes—
671C22#298112yes—
672C22#298812yes—
673C22#298812yes—
674C22#299212yes—
675C22#299312yes—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—44no—
680C4—44no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—42no—
685C4#231211yescharacteristic
686C4#231411yescharacteristic
687C4#231811yescharacteristic
688C4#231911yescharacteristic
689C4#233311yescharacteristic
690C4#233411yescharacteristic
691C4#233511yescharacteristic
692C4#233711yescharacteristic
693C2—44no—
694C2#1378811yescharacteristic
695C2C8×#11711yescharacteristic
696C2#2047511yescharacteristic
697C2#2055511yescharacteristic
698C2#2055511yescharacteristic
699C2#2056011yescharacteristic
700C2#2056511yescharacteristic
701C2#2232912yes—
702C2#2234112yes—
703C2#2235712yes—
704C2#2238512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC22C22×C4×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10876C2semidirect
7. #10874C2semidirect
8. #10834C2semidirect
9. #10797C2semidirect
10. #10797C2semidirect
11. #10795C2semidirect
14. C8×#95C2semidirect
15. C4×#336C2semidirect
32. #1923C4semidirect
33. C8×C22⋊C4C4semidirect
113. #388C8semidirect
241. C2×C4⋊C4C2×C8semidirect
289. C2×C42C2×C8semidirect
310. #95C2×C8semidirect
311. #95C2×C8semidirect
482. C22⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×C8×C22⋊C42
C4×#3882
#109212
#108762
#108742
#108342
#107972
#107952
#107142
C8×#952
C4×#3362