Finite Group Databasebeta

← #367049

Subgroups — #367049

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#467C211yesmaximal, characteristic
2C2×#1969C211yesmaximal, characteristic, complemented
3C2×#1969C211yesmaximal, characteristic, complemented
4C2×#1942C211yesmaximal, characteristic, complemented
5C2×#1808C211yesmaximal, characteristic
6C2×C4×#107C211yesmaximal, characteristic, complemented
7C2×C8×C22⋊C4C211yesmaximal, characteristic
8#13997C212yesmaximal
9#11067C212yesmaximal, complemented
10#10970C212yesmaximal
11#10932C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C2×#467—24no—
16C2×#467—24no—
17C2×#467C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33#1969—24no—
34#1969—24no—
35#1942—24no—
36#1808—24no—
37C4×#107—24no—
38C8×C22⋊C4—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44#1445—22no—
45#1439—22no—
46#1437—22no—
47C4⋊#95—22no—
48C23×Q8C2312yes—
49C24×C4C2311yescharacteristic
50C22×Q16—22no—
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—22no—
54C22×Q16—22no—
55C22×Q16—22no—
56C22×Q16C2×C414yes—
57C23×C8—22no—
58C23×C8—22no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—22no—
63C23×C8C2×C414yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8—24no—
71C2×C22⋊Q8—42no—
72C2×C22⋊Q8—42no—
73C2×C22⋊Q8C2×C418yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—22no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#467—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107C2×C414yes—
93C2×#107D414yes—
94C2×#107D414yes—
95C2×#107D414yes—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103C2×C418yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yes—
114#391—24no—
115#391—24no—
116#391—24no—
117#391—42no—
118#389—24no—
119#389—42no—
120#380—24no—
121#380—42no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135#340—24no—
136#340—24no—
137#340—24no—
138#336—22no—
139#336—22no—
140C25C2×D411yescharacteristic
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8C22×C414yes—
155C23×C4—22no—
156C23×C4—21no—
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×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4C22×C412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4112yes—
178C23×C4C2411yescharacteristic
179C2×Q16—28no—
180C2×Q16—28no—
181C2×Q16—28no—
182C2×Q16—44no—
183C2×Q16—44no—
184C2×Q16—44no—
185C2×Q16—44no—
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—28no—
205C22×C8—28no—
206C22×C8—28no—
207C22×C8—44no—
208C22×C8—44no—
209C22×C8—44no—
210C22×C8—44no—
211C22×C8C22×C412yes—
212C22×C8C22×C412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C22×C8#4112yes—
219C22×C8#4112yes—
220C22×C8#4112yes—
221C22⋊Q8—216no—
222C22⋊Q8—48no—
223C22⋊Q8—48no—
224C22⋊Q8—48no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4C22×C414yes—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—82no—
260C2×C22⋊C4Q1614yescomplemented
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—21no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—82no—
274C2×C42C2×D412yes—
275C2×C42#4112yes—
276#107—28no—
277#107—44no—
278#107C22⋊C4116yescomplemented
279#103—216no—
280#103—48no—
281#103—48no—
282C22⋊C8—28no—
283C4×C8—28no—
284C4×C8—44no—
285#95—22no—
286#95—22no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24C4×D412yes—
307C24C2×Q1612yes—
308C24#13512yes—
309C24C22×D411yescharacteristic
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—42no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
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—24no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
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—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4C4×D412yes—
423C22×C4#12112yes—
424C22×C4#12112yes—
425C22×C4C2×Q1612yes—
426C22×C4C2×Q1612yes—
427C22×C4#13512yes—
428C22×C4#13512yes—
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4H(5,𝔽2)11yescharacteristic
440C22×C4H(5,𝔽2)11yescharacteristic
441C22×C4H(5,𝔽2)11yescharacteristic
442C22×C42−1+411yescharacteristic
443Q16—416no—
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—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—416no—
466C2×C8C2×C22⋊C414yes—
467C2×C8C4×D414yes—
468C2×C8C4×D414yes—
469C2×C8#12114yes—
470C2×C8#12114yes—
471C2×C8#12414yes—
472C2×C8#12714yes—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—416no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—88no—
486C4⋊C4C2×C22⋊C418yes—
487C22⋊C4—28no—
488C22⋊C4—28no—
489C22⋊C4—416no—
490C22⋊C4—88no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—88no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23C4×Q1614yes—
535C23C2×C4×D411yescharacteristic
536C23C2×#12111yescharacteristic
537C23#53211yescharacteristic
538C23D4211yescharacteristic
539C23D4211yescharacteristic
540C23D4211yescharacteristic
541C23#54711yescharacteristic
542C23C22×Q1611yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×#13511yescharacteristic
545C23C2×#13511yescharacteristic
546C23#57611yescharacteristic
547C23#57811yescharacteristic
548C23#57811yescharacteristic
549C23#57811yescharacteristic
550Q8—44no—
551Q8—44no—
552Q8—44no—
553Q8—44no—
554Q8—82no—
555Q8—82no—
556Q8—82no—
557Q8—82no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
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—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4#46714yescomplemented
629C2×C4C22×C22⋊C411yescharacteristic
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C4×#4111yescharacteristic
633C2×C4#51811yescharacteristic
634C2×C4#52011yescharacteristic
635C2×C4C2×#12111yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4#52511yescharacteristic
638C2×C4C2×#12411yescharacteristic
639C2×C4#52911yescharacteristic
640C2×C4C2×#12711yescharacteristic
641C2×C4#53511yescharacteristic
642C2×C4#53711yescharacteristic
643C2×C4#54011yescharacteristic
644C2×C4#54211yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54611yescharacteristic
647C2×C4#54611yescharacteristic
648C2×C4#54811yescharacteristic
649C2×C4#54811yescharacteristic
650C2×C4#54811yescharacteristic
651C2×C4#55011yescharacteristic
652C2×C4#55011yescharacteristic
653C2×C4#55011yescharacteristic
654C2×C4#55311yescharacteristic
655C2×C4#55311yescharacteristic
656C2×C4#55511yescharacteristic
657C8—42no—
658C8—42no—
659C8—42no—
660C8#39018yes—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22C4×#12111yescharacteristic
677C22D4×C22⋊C411yescharacteristic
678C22#236111yescharacteristic
679C22#241611yescharacteristic
680C22#242811yescharacteristic
681C22#249111yescharacteristic
682C22#254111yescharacteristic
683C22C2×C4×Q1611yescharacteristic
684C22C4×#13511yescharacteristic
685C22#295011yescharacteristic
686C22#295111yescharacteristic
687C22C2×#46711yescharacteristic
688C22#305011yescharacteristic
689C22#305911yescharacteristic
690C22#306111yescharacteristic
691C22#316811yescharacteristic
692C22#317011yescharacteristic
693C22#317511yescharacteristic
694C22#317711yescharacteristic
695C22#328011yescharacteristic
696C22#328011yescharacteristic
697C22#328011yescharacteristic
698C22#328311yescharacteristic
699C22#328311yescharacteristic
700C22#328311yescharacteristic
701C22#328511yescharacteristic
702C22#328511yescharacteristic
703C22#328511yescharacteristic
704C22D4×Q1611yescharacteristic
705C22D4×Q1611yescharacteristic
706C22D4×Q1611yescharacteristic
707C22#333611yescharacteristic
708C22#333611yescharacteristic
709C22#333811yescharacteristic
710C22#334211yescharacteristic
711C4—24no—
712C4—42no—
713C4—42no—
714C4—42no—
715C4—48no—
716C4—42no—
717C4—84no—
718C4—84no—
719C4—162no—
720C4C2×#39011yescharacteristic
721C4#232411yescharacteristic
722C4#234111yescharacteristic
723C4#236311yescharacteristic
724C4#241411yescharacteristic
725C4#242611yescharacteristic
726C4#249011yescharacteristic
727C4#254011yescharacteristic
728C2—44no—
729C2#1399811yescharacteristic
730C2C4×#46711yescharacteristic
731C2#2152511yescharacteristic
732C2#2244811yescharacteristic
733C2C22⋊C4×Q1611yescharacteristic
734C2#2302411yescharacteristic
735C2#2302611yescharacteristic
736C2#2477911yescharacteristic
737C2#2478111yescharacteristic
738C2#2541311yescharacteristic
739C2#2541511yescharacteristic
740C2#2778911yescharacteristic
741C2#2779211yescharacteristic
742C2#2848411yescharacteristic
743C2#2848611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#1969C2semidirect
3. C2×#1969C2semidirect
4. C2×#1942C2semidirect
6. C2×C4×#107C2semidirect
9. #11067C2semidirect
11. #10932C2semidirect
17. C2×#467C4semidirect
260. C2×C22⋊C4Q16semidirect
278. #107C22⋊C4semidirect
628. C2×C4#467semidirect

Maximal subgroups

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

subgroupindex
C22×#4672
C2×#19692
C2×#19422
C2×#18082
C2×C4×#1072
C2×C8×C22⋊C42
#139972
#110672
#109702
#109322