Finite Group Databasebeta

← #355367

Subgroups — #355367

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1851C211yesmaximal, characteristic
2C2×#1753C211yesmaximal, characteristic
3C2×#1753C211yesmaximal, characteristic
4C2×#1728C211yesmaximal, characteristic
5C4×#385C211yesmaximal, characteristic
6#10876C211yesmaximal, characteristic, complemented
7#10876C211yesmaximal, characteristic, complemented
8#10875C211yesmaximal, characteristic, complemented
9#10875C211yesmaximal, characteristic, complemented
10#10835C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12#10798C211yesmaximal, characteristic, complemented
13#10797C211yesmaximal, characteristic, complemented
14#10715C211yesmaximal, characteristic
15#10712C211yesmaximal, characteristic
16C23×M16C2211yescharacteristic
17C23×M16C2211yescharacteristic
18C23×C42C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#385C412yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32#1851C418yescomplemented
33#1753—24no—
34#1753C418yescomplemented
35#1728—24no—
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
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54#1436C412yes—
55#1436C412yes—
56C22⋊#95C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2×C412yes—
60C24×C4C2311yescharacteristic
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—41no—
64C22×M16—41no—
65C22×M16—41no—
66C22×M16—41no—
67C22×M16—41no—
68C22×M16—41no—
69C22×M16—41no—
70C22×M16—41no—
71C22×M16C2×C414yes—
72C22×M16C2×C414yes—
73C22×M16D414yes—
74C22×M16Q814yes—
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C22×C4⋊C4C2×C412yes—
84C22×C4⋊C4C2×C412yes—
85C22×C22⋊C4C2×C412yes—
86C22×C22⋊C4C2×C411yescharacteristic
87C22×C22⋊C4C2×C411yescharacteristic
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—41no—
91C22×C42—41no—
92C22×C42—41no—
93C22×C42—41no—
94C22×C42C2×C412yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×C4⋊C8—41no—
102C2×C4⋊C8—41no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8C2×C414yescomplemented
106C2×C4⋊C8C2×C414yescomplemented
107C2×C22⋊C8—22no—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8C2×C414yes—
112C2×C22⋊C8C2×C414yes—
113C2×#97—22no—
114C2×#97—41no—
115C2×#97—41no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97C2×C414yescomplemented
119C2×C4×C8—22no—
120C2×C4×C8—41no—
121C2×C4×C8—41no—
122C2×C4×C8—41no—
123C2×C4×C8—41no—
124C2×C4×C8C2×C414yescomplemented
125#385—28no—
126C4×C4⋊C4—41no—
127C4×C4⋊C4—41no—
128C4×C4⋊C4—41no—
129C4×C4⋊C4—41no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4C2×C414yes—
133C4×C22⋊C4C2×C414yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C411yescharacteristic
141C2×#95C2×C411yescharacteristic
142C2×#95C2×C412yes—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—41no—
150#336—41no—
151#336—41no—
152#336—41no—
153#336—41no—
154#336—41no—
155#336—41no—
156#336—41no—
157#336C2×C414yescomplemented
158#336C2×C414yescomplemented
159#336C2×C414yescomplemented
160#336C2×C414yescomplemented
161C25C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C2×D412yes—
189C23×C4C2×Q812yes—
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4C2411yescharacteristic
193C2×M16—28no—
194C2×M16—44no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—44no—
205C2×M16—44no—
206C2×M16C4⋊C4116yescomplemented
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—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8C22×C412yes—
252C22×C8C22×C412yes—
253C22×C8C22×C412yes—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C22×C412yes—
258C22×C8C22×C412yes—
259C22×C8C2×D412yes—
260C22×C8C2×D412yes—
261C22×C8C2×Q812yes—
262C22×C8C2×Q812yes—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4C4214yes—
275C2×C22⋊C4C4214yes—
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—24no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
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—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C22×C412yes—
316C2×C42C22×C412yes—
317C2×C42C22×C412yes—
318C2×C42C22×C412yes—
319C4⋊C8—44no—
320C4⋊C8—44no—
321C22⋊C8—28no—
322C22⋊C8—28no—
323#97—44no—
324#97—44no—
325C4×C8—44no—
326C4×C8—44no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95C4214yes—
334#95C4214yes—
335#95M1618yescomplemented
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24C2×C4212yes—
345C24C4×D412yes—
346C24C4×Q812yes—
347C24C23×C411yescharacteristic
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
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—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
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—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4C2×C4212yes—
424C22×C4C2×C4212yes—
425C22×C4C2×C4212yes—
426C22×C4C2×C4212yes—
427C22×C4C2×C4212yes—
428C22×C4C2×C4⋊C414yes—
429C22×C4#11714yes—
430C22×C4C4×D414yes—
431C22×C4C4×D412yes—
432C22×C4C4×Q814yes—
433C22×C4C4×Q812yes—
434C22×C4C2×M1614yes—
435C22×C4C2×M1614yes—
436C22×C4C2×M1614yes—
437C22×C4#13114yes—
438C22×C4C23×C411yescharacteristic
439C22×C4C23×C411yescharacteristic
440C22×C4C22×D411yescharacteristic
441C22×C4C22×Q811yescharacteristic
442C22×C4C2×#4111yescharacteristic
443C22×C4C2×#4111yescharacteristic
444C22×C4H(5,𝔽2)11yescharacteristic
445C22×C4H(5,𝔽2)11yescharacteristic
446C22×C4H(5,𝔽2)11yescharacteristic
447C22×C4H(5,𝔽2)11yescharacteristic
448C22×C4H(5,𝔽2)11yescharacteristic
449C22×C4H(5,𝔽2)11yescharacteristic
450C22×C42−1+411yescharacteristic
451C22×C42−1+411yescharacteristic
452M16—416no—
453M16—416no—
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—22no—
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×C8C2×C4⋊C414yes—
509C2×C8C2×C4⋊C414yes—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C22⋊C4—28no—
513C22⋊C4—44no—
514C22⋊C4—48no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—48no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C4×C4⋊C414yes—
563C23C22×C4211yescharacteristic
564C23C2×C4×D411yescharacteristic
565C23C2×C4×Q811yescharacteristic
566C23#51811yescharacteristic
567C23#51811yescharacteristic
568C23#52011yescharacteristic
569C23#52011yescharacteristic
570C23C22×M1612yes—
571C23C2×#13112yes—
572C23#56812yes—
573C23#56812yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
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—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—432no—
611C2×C4—42no—
612C2×C4—42no—
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×C4C4×C4⋊C414yes—
637C2×C4C4×M1614yes—
638C2×C4C22×C4⋊C411yescharacteristic
639C2×C4C2×#11711yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×Q811yescharacteristic
642C2×C4#51811yescharacteristic
643C2×C4#51811yescharacteristic
644C2×C4#51811yescharacteristic
645C2×C4#51811yescharacteristic
646C2×C4#52011yescharacteristic
647C2×C4#52011yescharacteristic
648C2×C4#52011yescharacteristic
649C2×C4#52011yescharacteristic
650C2×C4#53411yescharacteristic
651C2×C4#53411yescharacteristic
652C2×C4#53511yescharacteristic
653C2×C4#53611yescharacteristic
654C2×C4#53811yescharacteristic
655C2×C4#53811yescharacteristic
656C2×C4#53811yescharacteristic
657C2×C4#53811yescharacteristic
658C2×C4#53911yescharacteristic
659C2×C4#53911yescharacteristic
660C2×C4#53911yescharacteristic
661C2×C4#53911yescharacteristic
662C2×C4#54311yescharacteristic
663C2×C4#54311yescharacteristic
664C2×C4#54411yescharacteristic
665C2×C4#54411yescharacteristic
666C8—24no—
667C8—24no—
668C8—42no—
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22C2×C4×C4⋊C411yescharacteristic
686C22#227511yescharacteristic
687C22#227511yescharacteristic
688C22#232111yescharacteristic
689C22#232211yescharacteristic
690C22#232911yescharacteristic
691C22#232911yescharacteristic
692C22C2×C4×M1611yescharacteristic
693C22#287311yescharacteristic
694C22C4×#13111yescharacteristic
695C22#287511yescharacteristic
696C22#293312yes—
697C22D4×M1612yes—
698C22#296212yes—
699C22Q8×M1612yes—
700C22#297412yes—
701C22#297512yes—
702C22#297612yes—
703C22#297812yes—
704C22#298912yes—
705C22#299012yes—
706C22#299312yes—
707C22#299512yes—
708C4—24no—
709C4—24no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—82no—
717C4#231511yescharacteristic
718C4#231711yescharacteristic
719C4#231811yescharacteristic
720C4#231811yescharacteristic
721C4#232111yescharacteristic
722C4#232211yescharacteristic
723C4#232911yescharacteristic
724C4#232911yescharacteristic
725C2—44no—
726C2#1378211yescharacteristic
727C2#2043911yescharacteristic
728C2C4⋊C4×M1611yescharacteristic
729C2#2055911yescharacteristic
730C2#2056111yescharacteristic
731C2#2056411yescharacteristic
732C2#2056811yescharacteristic
733C2#2203812yes—
734C2#2207112yes—
735C2#2229312yes—
736C2#2229712yes—
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. #10876C2semidirect
7. #10876C2semidirect
8. #10875C2semidirect
9. #10875C2semidirect
10. #10835C2semidirect
11. #10834C2semidirect
12. #10798C2semidirect
13. #10797C2semidirect
32. #1851C4semidirect
34. #1753C4semidirect
105. C2×C4⋊C8C2×C4semidirect
106. C2×C4⋊C8C2×C4semidirect
118. C2×#97C2×C4semidirect
124. C2×C4×C8C2×C4semidirect
157. #336C2×C4semidirect
158. #336C2×C4semidirect
159. #336C2×C4semidirect
160. #336C2×C4semidirect
206. C2×M16C4⋊C4semidirect
335. #95M16semidirect

Maximal subgroups

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

subgroupindex
C2×#18512
C2×#17532
C2×#17282
C4×#3852
#108762
#108752
#108352
#108342
#107982
#107972
#107152
#107122