Finite Group Databasebeta

← #356309

Subgroups — #356309

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1798C211yesmaximal, characteristic
2C2×#1752C211yesmaximal, characteristic
3C2×#1753C211yesmaximal, characteristic
4C2×#1729C211yesmaximal, characteristic
5#13761C211yesmaximal, characteristic
6#10877C211yesmaximal, characteristic, complemented
7#10877C211yesmaximal, characteristic, complemented
8#10877C211yesmaximal, characteristic, complemented
9#10876C211yesmaximal, 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×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×#376C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1798C418yescomplemented
32#1753—24no—
33#1752—24no—
34#1729C418yescomplemented
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—41no—
83C2×#117—41no—
84C2×#117—41no—
85C2×#117—41no—
86C2×#117C2×C414yes—
87C2×#117C2×C414yes—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C22⋊C4C2×C412yes—
91C22×C22⋊C4C2×C411yescharacteristic
92C22×C22⋊C4C2×C411yescharacteristic
93C22×C22⋊C4C2×C412yes—
94C22×C42C2×C412yes—
95C22×C42C2×C412yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C2×C4⋊C8—41no—
103C2×C4⋊C8—41no—
104C2×C4⋊C8—41no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8C2×C414yes—
107C2×C4⋊C8C2×C414yes—
108C2×C22⋊C8—22no—
109C2×C22⋊C8—22no—
110C2×C22⋊C8—22no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8C2×C414yes—
113C2×C22⋊C8C2×C414yes—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—22no—
120C2×#97—41no—
121C2×#97—41no—
122C2×#97—41no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—41no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4D414yes—
129C4×C22⋊C4Q814yes—
130#376—41no—
131#376—41no—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C43—41no—
141C43—41no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—41no—
151#336—41no—
152#336—41no—
153#336—41no—
154#336C2×C414yes—
155#336C2×C414yes—
156#336C2×C414yes—
157#336C2×C414yes—
158#336C2×C414yes—
159#336C2×C414yes—
160C25C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4C22×C412yes—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C412yes—
179C23×C4C22×C412yes—
180C23×C4C2×D412yes—
181C23×C4C2×Q812yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—44no—
188C2×M16—44no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—44no—
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—28no—
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—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—28no—
238C22×C8—44no—
239C22×C8—44no—
240C22×C8—44no—
241C22×C8—44no—
242C22×C8—44no—
243C22×C8C22×C412yes—
244C22×C8C22×C412yes—
245C22×C8C22×C412yes—
246C22×C8C22×C412yes—
247C22×C8C22×C412yes—
248C22×C8C22×C412yes—
249C22×C8C22×C412yes—
250C22×C8C22×C412yes—
251C22×C8#4112yes—
252C22×C8#4112yes—
253C22×C8#4112yes—
254C22×C8#4112yes—
255#117—44no—
256#117—44no—
257#117—44no—
258#117—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4C4214yes—
264C2×C4⋊C4C4214yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4C4214yes—
275C2×C22⋊C4C4214yes—
276C2×C22⋊C4C4⋊C418yes—
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—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—24no—
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—42no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C22×C412yes—
312C2×C42C22×C412yes—
313C2×C42C22×C412yes—
314C2×C42C22×C412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×Q812yes—
318C2×C42C2×Q812yes—
319C4⋊C8—44no—
320C4⋊C8—44no—
321C22⋊C8—28no—
322C22⋊C8—28no—
323#97—44no—
324#97—44no—
325#97—44no—
326C4×C8—44no—
327#95—24no—
328#95—24no—
329#95—42no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95C4⋊C418yes—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24C2×C4212yes—
344C24C2×C4⋊C412yes—
345C24#11712yes—
346C24C23×C411yescharacteristic
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
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—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—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
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×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×C4C2×C4212yes—
417C22×C4C2×C4212yes—
418C22×C4C2×C4212yes—
419C22×C4C2×C4212yes—
420C22×C4C2×C4212yes—
421C22×C4C2×C4⋊C414yes—
422C22×C4C2×C4⋊C414yes—
423C22×C4C2×C4⋊C412yes—
424C22×C4#11712yes—
425C22×C4C4×D414yes—
426C22×C4C4×D414yes—
427C22×C4C4×Q814yes—
428C22×C4C4×Q814yes—
429C22×C4#13114yes—
430C22×C4C23×C411yescharacteristic
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×Q811yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C4H(5,𝔽2)11yescharacteristic
444M16—416no—
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—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—416no—
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×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#11714yes—
501C2×C8#11714yes—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—48no—
525C42—48no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—22no—
547C23—22no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23C4×C4⋊C414yes—
557C23C22×C4211yescharacteristic
558C23C22×C4⋊C411yescharacteristic
559C23C2×#11711yescharacteristic
560C23C4×#4111yescharacteristic
561C23C4×#4111yescharacteristic
562C23#51811yescharacteristic
563C23#51811yescharacteristic
564C23C2×#13112yes—
565C23#56812yes—
566C23#56812yes—
567C23#56812yes—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
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—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×C4C4×C4⋊C414yes—
631C2×C4C2×#11711yescharacteristic
632C2×C4C2×#11711yescharacteristic
633C2×C4C2×C4×D411yescharacteristic
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×Q811yescharacteristic
636C2×C4C2×C4×Q811yescharacteristic
637C2×C4C4×#4111yescharacteristic
638C2×C4C4×#4111yescharacteristic
639C2×C4#51811yescharacteristic
640C2×C4#51811yescharacteristic
641C2×C4#51811yescharacteristic
642C2×C4#51811yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54611yescharacteristic
647C2×C4#55111yescharacteristic
648C2×C4#55111yescharacteristic
649C2×C4#55111yescharacteristic
650C2×C4#55111yescharacteristic
651C2×C4#55311yescharacteristic
652C2×C4#55311yescharacteristic
653C2×C4#55311yescharacteristic
654C2×C4#55311yescharacteristic
655C2×C4#55411yescharacteristic
656C2×C4#55411yescharacteristic
657C2×C4#55411yescharacteristic
658C2×C4#55411yescharacteristic
659C8—24no—
660C8—24no—
661C8—42no—
662C8—42no—
663C8—42no—
664C8—42no—
665C8—42no—
666C8—42no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—42no—
678C22C2×C4×C4⋊C411yescharacteristic
679C22C4×#11711yescharacteristic
680C22#227311yescharacteristic
681C22#234911yescharacteristic
682C22#234911yescharacteristic
683C22#235311yescharacteristic
684C22#235311yescharacteristic
685C22C4×#13111yescharacteristic
686C22#287511yescharacteristic
687C22#287511yescharacteristic
688C22#287511yescharacteristic
689C22#290512yes—
690C22#290612yes—
691C22#292312yes—
692C22#292412yes—
693C22#296712yes—
694C22#296712yes—
695C22#296812yes—
696C22#296912yes—
697C22#297612yes—
698C22#297812yes—
699C22#298012yes—
700C22#298112yes—
701C4—24no—
702C4—44no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—44no—
709C4—82no—
710C4#234211yescharacteristic
711C4#234211yescharacteristic
712C4#234511yescharacteristic
713C4#234511yescharacteristic
714C4#236411yescharacteristic
715C4#236411yescharacteristic
716C4#236811yescharacteristic
717C4#236811yescharacteristic
718C2—44no—
719C2#1380211yescharacteristic
720C2#2042711yescharacteristic
721C2#2043011yescharacteristic
722C2#2047511yescharacteristic
723C2#2047811yescharacteristic
724C2#2053111yescharacteristic
725C2#2053211yescharacteristic
726C2#2269912yes—
727C2#2270212yes—
728C2#2279012yes—
729C2#2279412yes—
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. #10877C2semidirect
7. #10877C2semidirect
8. #10877C2semidirect
9. #10876C2semidirect
10. #10796C2semidirect
11. #10795C2semidirect
14. #10694C2semidirect
15. #10693C2semidirect
31. #1798C4semidirect
34. #1729C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17982
C2×#17522
C2×#17532
C2×#17292
#137612
#108772
#108762
#107962
#107952
#107152
#107142
#106942
#106932