Finite Group Databasebeta

← #356533

Subgroups — #356533

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1752C211yesmaximal, characteristic
2C2×#1753C211yesmaximal, characteristic
3C2×#1745C211yesmaximal, characteristic, complemented
4C2×#1729C211yesmaximal, characteristic
5#13761C211yesmaximal, characteristic
6#10798C211yesmaximal, characteristic, complemented
7#10798C211yesmaximal, characteristic, complemented
8#10798C211yesmaximal, characteristic, complemented
9#10797C211yesmaximal, characteristic, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10795C211yesmaximal, characteristic, complemented
12#10715C211yesmaximal, characteristic
13#10714C211yesmaximal, characteristic
14#10713C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic, complemented
16C23×M16C2211yescharacteristic
17C24×C8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×#376C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#1753C418yescomplemented
30#1752C418yescomplemented
31#1745—24no—
32#1729—24no—
33C2×#336C2211yescharacteristic
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
52C2×#336C2211yescharacteristic
53#1437C412yes—
54#1437C412yes—
55#1436C412yes—
56#1436C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2311yescharacteristic
60C22×M16—22no—
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—41no—
65C22×M16—41no—
66C22×M16C2×C414yes—
67C23×C8—22no—
68C23×C8—22no—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—41no—
72C23×C8—41no—
73C23×C8C2×C414yes—
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C23×C8C2311yescharacteristic
78C23×C8C2311yescharacteristic
79C23×C8C2311yescharacteristic
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—41no—
87C2×#117—41no—
88C22×C4⋊C4C2×C412yes—
89C22×C22⋊C4C2×C412yes—
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C411yescharacteristic
92C22×C22⋊C4C2×C411yescharacteristic
93C22×C42C2×C412yes—
94C22×C42C2×C412yes—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8C2×C414yes—
106C2×C22⋊C8C2×C414yes—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—41no—
112C2×#97—41no—
113C2×#97C2×C414yes—
114C2×#97C2×C414yes—
115C2×#97C2×C414yes—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8C2×C414yes—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C414yes—
122C4×C22⋊C4C2×C414yes—
123#376—41no—
124#376—41no—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C411yescharacteristic
133C2×#95C2×C411yescharacteristic
134C43—22no—
135C43—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—41no—
146#336—41no—
147#336—41no—
148#336—41no—
149#336—41no—
150#336—41no—
151#336C2×C414yes—
152#336C2×C414yes—
153#336C2×C414yes—
154#336C2×C414yes—
155#336D414yes—
156C25C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C412yes—
166C23×C4C22×C412yes—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C412yes—
172C23×C4C22×C412yes—
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4#4112yes—
177C23×C4#4112yes—
178C23×C4#4112yes—
179C23×C4#4112yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—44no—
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—44no—
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—28no—
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—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—28no—
234C22×C8—44no—
235C22×C8—44no—
236C22×C8—44no—
237C22×C8—44no—
238C22×C8—44no—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C22×C412yes—
242C22×C8C22×C412yes—
243C22×C8C22×C412yes—
244C22×C8C22×C412yes—
245C22×C8C22×C412yes—
246C22×C8C22×C412yes—
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C22×C8#4112yes—
250C22×C8#4112yes—
251#117—44no—
252#117—44no—
253#117—44no—
254#117—44no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4C4214yes—
267C2×C22⋊C4C4214yes—
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—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—21no—
284C2×C42—21no—
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—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42C22×C412yes—
305C2×C42C22×C412yes—
306C2×C42C22×C412yes—
307C2×C42C22×C412yes—
308C2×C42C2×D412yes—
309C2×C42#4112yes—
310C22⋊C8—28no—
311C22⋊C8—28no—
312#97—28no—
313#97—28no—
314#97—44no—
315C4×C8—44no—
316#95—24no—
317#95—24no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95C4214yes—
324#95C4214yes—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24C2×C4212yes—
334C24#11712yes—
335C24#11712yes—
336C24C23×C411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
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—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—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
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×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C2×C4212yes—
411C22×C4C2×C4212yes—
412C22×C4#11714yes—
413C22×C4#11714yes—
414C22×C4#11714yes—
415C22×C4#11714yes—
416C22×C4#11712yes—
417C22×C4#11712yes—
418C22×C4C4×D414yes—
419C22×C4#13114yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C42−1+411yescharacteristic
434M16—416no—
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—22no—
452C2×C8—22no—
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—416no—
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—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8C4×D414yes—
491C2×C8C4×D414yes—
492C4⋊C4—48no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—48no—
510C42—48no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23#37614yes—
546C23C22×C4211yescharacteristic
547C23C2×#11711yescharacteristic
548C23C2×#11711yescharacteristic
549C23C2×C4×D411yescharacteristic
550C23C4×#4111yescharacteristic
551C23#51811yescharacteristic
552C23#52011yescharacteristic
553C23C2×#13112yes—
554C23#56812yes—
555C23#56812yes—
556C23#56812yes—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
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×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
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#37614yes—
620C2×C4C2×#11711yescharacteristic
621C2×C4C2×#11711yescharacteristic
622C2×C4C2×#11711yescharacteristic
623C2×C4C2×#11711yescharacteristic
624C2×C4C2×C4×D411yescharacteristic
625C2×C4C2×C4×D411yescharacteristic
626C2×C4C4×#4111yescharacteristic
627C2×C4C4×#4111yescharacteristic
628C2×C4#51811yescharacteristic
629C2×C4#51811yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4#52011yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54611yescharacteristic
634C2×C4#54611yescharacteristic
635C2×C4#54811yescharacteristic
636C2×C4#55111yescharacteristic
637C2×C4#55111yescharacteristic
638C2×C4#55111yescharacteristic
639C2×C4#55111yescharacteristic
640C2×C4#55111yescharacteristic
641C2×C4#55111yescharacteristic
642C2×C4#55211yescharacteristic
643C2×C4#55211yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55311yescharacteristic
646C2×C4#55311yescharacteristic
647C2×C4#55311yescharacteristic
648C8—24no—
649C8—24no—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22C2×#37611yescharacteristic
668C22C42×D411yescharacteristic
669C22#227511yescharacteristic
670C22#234211yescharacteristic
671C22#234311yescharacteristic
672C22#235211yescharacteristic
673C22#235311yescharacteristic
674C22#287311yescharacteristic
675C22#287311yescharacteristic
676C22C4×#13111yescharacteristic
677C22#287511yescharacteristic
678C22#292312yes—
679C22#292312yes—
680C22#292412yes—
681C22#292512yes—
682C22#293012yes—
683C22#293312yes—
684C22#296712yes—
685C22#296912yes—
686C22#298012yes—
687C22#298112yes—
688C22#299212yes—
689C22#299512yes—
690C4—24no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4#234211yescharacteristic
700C4#234211yescharacteristic
701C4#234311yescharacteristic
702C4#234311yescharacteristic
703C4#235211yescharacteristic
704C4#235211yescharacteristic
705C4#235311yescharacteristic
706C4#235311yescharacteristic
707C2—44no—
708C2#1380911yescharacteristic
709C2#2036711yescharacteristic
710C2#2036811yescharacteristic
711C2#2048711yescharacteristic
712C2#2049011yescharacteristic
713C2#2056311yescharacteristic
714C2#2056611yescharacteristic
715C2#2255712yes—
716C2#2259412yes—
717C2#2275312yes—
718C2#2279012yes—
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
3. C2×#1745C2semidirect
6. #10798C2semidirect
7. #10798C2semidirect
8. #10798C2semidirect
9. #10797C2semidirect
10. #10796C2semidirect
11. #10795C2semidirect
15. #10694C2semidirect
29. #1753C4semidirect
30. #1752C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17522
C2×#17532
C2×#17452
C2×#17292
#137612
#107982
#107972
#107962
#107952
#107152
#107142
#107132
#106942