Finite Group Databasebeta

← #357104

Subgroups — #357104

Order 512.

1422 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×#1770C211yesmaximal, characteristic, complemented
3C2×#1752C211yesmaximal, characteristic
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5C4×#388C211yesmaximal, characteristic
6#10922C211yesmaximal, characteristic
7#10922C211yesmaximal, characteristic
8#10877C211yesmaximal, characteristic, complemented
9#10875C211yesmaximal, characteristic, complemented
10#10797C211yesmaximal, characteristic, complemented
11#10797C211yesmaximal, characteristic, complemented
12#10714C211yesmaximal, characteristic
13#10696C211yesmaximal, characteristic, complemented
14#10693C211yesmaximal, characteristic, complemented
15C4×#336C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#388C412yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1923C418yescomplemented
35#1770—24no—
36#1752—24no—
37C8×C22⋊C4C418yescomplemented
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#1437C412yes—
55#1436C412yes—
56#1434C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8—22no—
64C23×C8—41no—
65C23×C8—41no—
66C23×C8C2×C414yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42C2×C412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—41no—
96C2×C22⋊C8—41no—
97C2×C22⋊C8C2×C414yes—
98C2×C22⋊C8C2×C414yes—
99C2×C22⋊C8C2×C414yes—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97—22no—
107C2×#97—22no—
108C2×#97—41no—
109C2×#97—41no—
110C2×#97—41no—
111C2×#97—41no—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—41no—
115C2×C4×C8—41no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8C2×C414yes—
119C2×C4×C8C2×C414yes—
120C2×C4×C8C2×C414yes—
121#388—28no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—41no—
127C4×C22⋊C4—41no—
128C4×C22⋊C4C2×C414yes—
129C4×C22⋊C4C2×C414yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139C43—41no—
140C43—41no—
141#336—22no—
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#336C2×C414yes—
153#336C2×C414yes—
154#336C2×C414yes—
155#336C2×C414yes—
156#336D414yes—
157C25C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
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×C4—21no—
170C23×C4C22×C412yes—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2411yescharacteristic
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—21no—
221C22×C8—21no—
222C22×C8—28no—
223C22×C8—28no—
224C22×C8—44no—
225C22×C8—44no—
226C22×C8—44no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8C22×C412yes—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8#4112yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4C4214yes—
256C2×C22⋊C4C4214yes—
257C2×C22⋊C4M1618yes—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
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—24no—
271C2×C42—24no—
272C2×C42—24no—
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—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C2×D412yes—
299C2×C42#4112yes—
300C4⋊C8—28no—
301C22⋊C8—28no—
302C22⋊C8—28no—
303C22⋊C8—44no—
304#97—44no—
305#97—44no—
306#97—44no—
307#97—44no—
308C4×C8—28no—
309C4×C8—44no—
310C4×C8—44no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—42no—
318#95C4214yes—
319#95C4214yes—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C2×C4212yes—
330C24C2×M1614yes—
331C24C23×C411yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
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—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4C2×C4212yes—
406C22×C4C2×C4212yes—
407C22×C4C2×C4212yes—
408C22×C4C2×C4212yes—
409C22×C4C2×C4212yes—
410C22×C4C4×D414yes—
411C22×C4C2×M1614yes—
412C22×C4C2×M1614yes—
413C22×C4#13114yes—
414C22×C4#13114yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C4H(5,𝔽2)11yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C42−1+411yescharacteristic
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—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#11714yes—
483C2×C8#11714yes—
484C2×C8C4×D414yes—
485C2×C8C4×D414yes—
486C4⋊C4—48no—
487C22⋊C4—28no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4#97116yescomplemented
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—48no—
504C42—48no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C2×#9714yes—
540C23C22×C4211yescharacteristic
541C23C2×C4×D411yescharacteristic
542C23C4×#4111yescharacteristic
543C23C4×#4111yescharacteristic
544C23C4×#4111yescharacteristic
545C23#51811yescharacteristic
546C23#52011yescharacteristic
547C23C22×M1612yes—
548C23C2×#13112yes—
549C23C2×#13112yes—
550C23#56812yes—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
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×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4C2×#9714yes—
612C2×C4C2×#9714yes—
613C2×C4#40514yes—
614C2×C4C2×#11711yescharacteristic
615C2×C4C2×#11711yescharacteristic
616C2×C4C2×C4×D411yescharacteristic
617C2×C4C2×C4×D411yescharacteristic
618C2×C4C4×#4111yescharacteristic
619C2×C4C4×#4111yescharacteristic
620C2×C4C4×#4111yescharacteristic
621C2×C4C4×#4111yescharacteristic
622C2×C4#51811yescharacteristic
623C2×C4#51811yescharacteristic
624C2×C4#52011yescharacteristic
625C2×C4#52011yescharacteristic
626C2×C4#54611yescharacteristic
627C2×C4#54611yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54811yescharacteristic
630C2×C4#55111yescharacteristic
631C2×C4#55111yescharacteristic
632C2×C4#55211yescharacteristic
633C2×C4#55311yescharacteristic
634C2×C4#55311yescharacteristic
635C2×C4#55311yescharacteristic
636C2×C4#55311yescharacteristic
637C2×C4#55311yescharacteristic
638C2×C4#55611yescharacteristic
639C2×C4#55611yescharacteristic
640C2×C4#55911yescharacteristic
641C2×C4#55911yescharacteristic
642C8—24no—
643C8—24no—
644C8—24no—
645C8—24no—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—44no—
654C22—42no—
655C22—44no—
656C22—42no—
657C22—44no—
658C22—44no—
659C22—42no—
660C22—42no—
661C22C4×#11711yescharacteristic
662C22C42×D411yescharacteristic
663C22#227511yescharacteristic
664C22#236411yescharacteristic
665C22#236711yescharacteristic
666C22#237311yescharacteristic
667C22#237411yescharacteristic
668C22C22×#9711yescharacteristic
669C22C2×#40511yescharacteristic
670C22C4×#13111yescharacteristic
671C22#287511yescharacteristic
672C22#293112yes—
673C22D4×M1612yes—
674C22#296512yes—
675C22#296512yes—
676C22#296612yes—
677C22#296812yes—
678C22#296912yes—
679C22#297012yes—
680C22#297412yes—
681C22#297612yes—
682C22#299012yes—
683C22#299212yes—
684C4—24no—
685C4—24no—
686C4—24no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—82no—
692C4—82no—
693C4#234211yescharacteristic
694C4#234711yescharacteristic
695C4#235211yescharacteristic
696C4#235711yescharacteristic
697C4#236411yescharacteristic
698C4#236711yescharacteristic
699C4#237311yescharacteristic
700C4#237411yescharacteristic
701C2—44no—
702C2#1382211yescharacteristic
703C2#2047311yescharacteristic
704C2#2047811yescharacteristic
705C2#2048511yescharacteristic
706C2D4×#9711yescharacteristic
707C2#2055811yescharacteristic
708C2#2056011yescharacteristic
709C2#2311812yes—
710C2#2320612yes—
711C2#2328812yes—
712C2#2330012yes—
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
2. C2×#1770C2semidirect
8. #10877C2semidirect
9. #10875C2semidirect
10. #10797C2semidirect
11. #10797C2semidirect
13. #10696C2semidirect
14. #10693C2semidirect
15. C4×#336C2semidirect
34. #1923C4semidirect
37. C8×C22⋊C4C4semidirect
490. C22⋊C4#97semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#17702
C2×#17522
C2×C8×C22⋊C42
C4×#3882
#109222
#108772
#108752
#107972
#107142
#106962
#106932
C4×#3362