Finite Group Databasebeta

← #356124

Subgroups — #356124

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2298C211yesmaximal, characteristic
2C2×#1818C211yesmaximal, characteristic
3C2×#1752C211yesmaximal, characteristic
4C2×#1753C211yesmaximal, characteristic
5C2×#1729C211yesmaximal, characteristic
6#10875C211yesmaximal, characteristic, complemented
7#10875C211yesmaximal, characteristic, complemented
8#10875C211yesmaximal, characteristic, complemented
9#10874C211yesmaximal, characteristic, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10795C211yesmaximal, characteristic, complemented
12#10715C211yesmaximal, characteristic
13#10714C211yesmaximal, characteristic
14#10694C211yesmaximal, characteristic, complemented
15#10693C211yesmaximal, characteristic, complemented
16C23×M16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2298—24no—
26#2298—24no—
27C2×#388C412yes—
28C2×#388C412yes—
29C2×#387C412yes—
30C2×#387C412yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×#376C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#1818C418yescomplemented
38#1753—24no—
39#1752—24no—
40#1729C418yescomplemented
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59C2×#336C2211yescharacteristic
60C2×#336C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—22no—
65C22×M16—22no—
66C22×M16—41no—
67C22×M16—41no—
68C22×M16C2×C414yes—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—41no—
74C23×C8—41no—
75C23×C8C2×C414yes—
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—22no—
87C2×#117—22no—
88C2×#117C2×C414yes—
89C2×#117C2×C414yes—
90C22×C4⋊C4C2×C412yes—
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C4⋊C4C2×C411yescharacteristic
93C22×C4⋊C4C2×C412yes—
94C22×C22⋊C4C2×C412yes—
95C22×C22⋊C4C2×C411yescharacteristic
96C22×C22⋊C4C2×C411yescharacteristic
97C22×C22⋊C4C2×C412yes—
98C22×C42C2×C412yes—
99C22×C42C2×C412yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8C2×C414yes—
111C2×C22⋊C8C2×C414yes—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—41no—
119C2×#97—41no—
120C2×#97—41no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—41no—
124#388—28no—
125#388—28no—
126#387—28no—
127#387—28no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4D414yes—
135C4×C22⋊C4D414yes—
136#376—22no—
137#376—22no—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C43—22no—
145C43—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—41no—
155#336—41no—
156#336—41no—
157#336—41no—
158#336C2×C414yes—
159#336C2×C414yes—
160#336C2×C414yes—
161#336C2×C414yes—
162#336C2×C414yes—
163#336C2×C414yes—
164#336C2×C414yes—
165#336C2×C414yes—
166C25C22×C411yescharacteristic
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C412yes—
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C412yes—
185C23×C4C22×C412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—44no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
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—28no—
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—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—28no—
244C22×C8—44no—
245C22×C8—44no—
246C22×C8—44no—
247C22×C8—44no—
248C22×C8—44no—
249C22×C8C22×C412yes—
250C22×C8C22×C412yes—
251C22×C8C22×C412yes—
252C22×C8C22×C412yes—
253C22×C8C22×C412yes—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8#4112yes—
258C22×C8#4112yes—
259C22×C8#4112yes—
260C22×C8#4112yes—
261#117—28no—
262#117—28no—
263#117—28no—
264#117—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4C4214yes—
276C2×C4⋊C4C4214yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4C4214yes—
287C2×C22⋊C4C4214yes—
288C2×C22⋊C4C22⋊C418yes—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42C22×C412yes—
324C2×C42C22×C412yes—
325C2×C42C22×C412yes—
326C2×C42C22×C412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C22⋊C8—28no—
332C22⋊C8—28no—
333#97—44no—
334#97—44no—
335#97—44no—
336C4×C8—44no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95C22⋊C418yes—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24C2×C4212yes—
352C24C2×C22⋊C412yes—
353C24#11712yes—
354C24C23×C411yescharacteristic
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
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—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4C2×C4212yes—
425C22×C4C2×C4212yes—
426C22×C4C2×C4212yes—
427C22×C4C2×C4212yes—
428C22×C4C2×C4212yes—
429C22×C4C2×C22⋊C414yes—
430C22×C4C2×C22⋊C414yes—
431C22×C4C2×C22⋊C412yes—
432C22×C4#11712yes—
433C22×C4C4×D414yes—
434C22×C4C4×D414yes—
435C22×C4C4×D414yes—
436C22×C4C4×D414yes—
437C22×C4#13114yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4C2×#4111yescharacteristic
445C22×C4C2×#4111yescharacteristic
446C22×C4C2×#4111yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C42−1+411yescharacteristic
451C22×C42−1+411yescharacteristic
452M16—416no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—416no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8#11714yes—
509C2×C8#11714yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—22no—
534C42—48no—
535C42—48no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23C4×C22⋊C414yes—
567C23C22×C4211yescharacteristic
568C23C22×C22⋊C411yescharacteristic
569C23C2×#11711yescharacteristic
570C23C4×#4111yescharacteristic
571C23C4×#4111yescharacteristic
572C23#52011yescharacteristic
573C23#52011yescharacteristic
574C23C2×#13112yes—
575C23#56812yes—
576C23#56812yes—
577C23#56812yes—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4C4×C22⋊C414yes—
641C2×C4C2×#11711yescharacteristic
642C2×C4C2×#11711yescharacteristic
643C2×C4C2×C4×D411yescharacteristic
644C2×C4C2×C4×D411yescharacteristic
645C2×C4C2×C4×D411yescharacteristic
646C2×C4C2×C4×D411yescharacteristic
647C2×C4C4×#4111yescharacteristic
648C2×C4C4×#4111yescharacteristic
649C2×C4#51811yescharacteristic
650C2×C4#51911yescharacteristic
651C2×C4#52011yescharacteristic
652C2×C4#52011yescharacteristic
653C2×C4#54611yescharacteristic
654C2×C4#54611yescharacteristic
655C2×C4#54611yescharacteristic
656C2×C4#54611yescharacteristic
657C2×C4#54711yescharacteristic
658C2×C4#54711yescharacteristic
659C2×C4#54711yescharacteristic
660C2×C4#54711yescharacteristic
661C2×C4#55111yescharacteristic
662C2×C4#55111yescharacteristic
663C2×C4#55111yescharacteristic
664C2×C4#55111yescharacteristic
665C2×C4#55211yescharacteristic
666C2×C4#55211yescharacteristic
667C2×C4#55211yescharacteristic
668C2×C4#55211yescharacteristic
669C8—24no—
670C8—24no—
671C8—42no—
672C8—42no—
673C8—42no—
674C8—42no—
675C8—42no—
676C8—42no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—44no—
681C22—44no—
682C22—42no—
683C22—42no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22C2×C4×C22⋊C411yescharacteristic
689C22C4×#11711yescharacteristic
690C22#227411yescharacteristic
691C22#233911yescharacteristic
692C22#233911yescharacteristic
693C22#234311yescharacteristic
694C22#234311yescharacteristic
695C22C4×#13111yescharacteristic
696C22#287511yescharacteristic
697C22#287511yescharacteristic
698C22#287511yescharacteristic
699C22#287912yes—
700C22#288012yes—
701C22#292312yes—
702C22#292512yes—
703C22#296712yes—
704C22#296712yes—
705C22#296812yes—
706C22#296912yes—
707C22#299212yes—
708C22#299312yes—
709C22#299512yes—
710C22#299512yes—
711C4—24no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—82no—
720C4#234211yescharacteristic
721C4#234211yescharacteristic
722C4#234411yescharacteristic
723C4#234411yescharacteristic
724C4#235911yescharacteristic
725C4#235911yescharacteristic
726C4#235911yescharacteristic
727C4#235911yescharacteristic
728C2—44no—
729C2#1379911yescharacteristic
730C2#2037511yescharacteristic
731C2#2037811yescharacteristic
732C2#2047511yescharacteristic
733C2#2047811yescharacteristic
734C2#2055311yescharacteristic
735C2#2055411yescharacteristic
736C2#2247212yes—
737C2#2247512yes—
738C2#2259412yes—
739C2#2259812yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
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
6. #10875C2semidirect
7. #10875C2semidirect
8. #10875C2semidirect
9. #10874C2semidirect
10. #10796C2semidirect
11. #10795C2semidirect
14. #10694C2semidirect
15. #10693C2semidirect
37. #1818C4semidirect
40. #1729C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22982
C2×#18182
C2×#17522
C2×#17532
C2×#17292
#108752
#108742
#107962
#107952
#107152
#107142
#106942
#106932