Finite Group Databasebeta

← #364854

Subgroups — #364854

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic, complemented
4C2×#1800C211yesmaximal, characteristic
5C2×C4×#100C211yesmaximal, characteristic
6#11204C212yesmaximal, complemented
7#11044C212yesmaximal
8#10962C212yesmaximal
9#10913C212yesmaximal, complemented
10#10857C211yesmaximal, characteristic, complemented
11#10857C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C412yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15Q8×M16—24no—
16Q8×M16—24no—
17C2×#446—22no—
18C2×#446—21no—
19C2×C8×Q8—21no—
20C2×#423C2212yes—
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×#100C2211yescharacteristic
24C22×#100C2212yes—
25C22×#100C2211yescharacteristic
26C2×C4×M16C2212yes—
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2538—21no—
30#2538—21no—
31#2538—22no—
32#2538—21no—
33#2538—21no—
34#2538—22no—
35C2×#396C414yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#384C2211yescharacteristic
39C2×#380C412yes—
40#1985—22no—
41#1984—21no—
42#1984—21no—
43#1800—24no—
44C4×#100—24no—
45C2×#343C2212yes—
46C2×#343C2212yes—
47C2×#343C2212yes—
48C2×#343C2211yescharacteristic
49C2×#343C2211yescharacteristic
50C2×#343C2212yes—
51C2×#343C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C23×Q8C2×C411yescharacteristic
55C23×D4C2311yescharacteristic
56C24×C4C2×C412yes—
57C22×M16C2312yes—
58C22×M16C2312yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—24no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—48no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D412yes—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×Q8D412yes—
77C2×C4×D4—22no—
78C2×C4×D4—22no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C22×C4⋊C4C2×C412yes—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C414yes—
88C22×C4⋊C4C2×C412yes—
89C22×C4⋊C4C2×C412yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2×C412yes—
93C22×C22⋊C4C2×C414yes—
94C22×C22⋊C4C2×C414yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97#446—216no—
98#446—28no—
99C8×Q8—28no—
100#423—24no—
101#423—24no—
102#423—22no—
103#423—22no—
104#423—22no—
105#423D414yes—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×#100—22no—
113C2×#100—22no—
114C2×#100—22no—
115C2×#100—22no—
116C2×#100—22no—
117C2×#100—24no—
118C2×#100—24no—
119C2×#100—42no—
120C2×#100Q814yescomplemented
121C4×M16—24no—
122C4×M16—24no—
123C4×M16—22no—
124C4×M16—22no—
125C4×M16—22no—
126C4×M16D414yes—
127C2×#97—22no—
128C2×#97—21no—
129C2×#97—21no—
130C2×C4×C8—22no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133#396—24no—
134#396—24no—
135#396—24no—
136#396—24no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#384—21no—
146#384—21no—
147#384—21no—
148#384—21no—
149#380—24no—
150#380—22no—
151#380—22no—
152C2×#95C2×C414yes—
153#343—24no—
154#343—22no—
155#343—22no—
156#343—24no—
157#343—24no—
158#343—42no—
159#343—42no—
160#336—22no—
161#336—22no—
162#336—22no—
163#336—22no—
164C25C22×C412yes—
165C22×Q8—24no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—42no—
169C22×Q8C22⋊C411yescharacteristic
170C22×Q8C22⋊C412yes—
171C22×Q8C22⋊C411yescharacteristic
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C2×Q812yes—
185C22×D4#4112yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C22×C412yes—
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C414yes—
195C23×C4C22×C412yes—
196C23×C4C22×C412yes—
197C23×C4C22×C411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—22no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—22no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—22no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—22no—
219C2×M16—21no—
220C2×M16—22no—
221C2×M16—21no—
222C2×M16—22no—
223C2×M16—22no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16C2×D414yes—
229C2×M16C2×D412yes—
230C2×M16C2×D412yes—
231C2×M16C2×Q812yes—
232C2×M16C2×Q812yes—
233C2×M16#4112yes—
234C2×M16#4112yes—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—22no—
239C22×C8—21no—
240C22×C8—22no—
241C22⋊Q8—48no—
242C22⋊Q8—416no—
243C22⋊Q8—48no—
244C4×Q8—22no—
245C4×Q8—24no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—24no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—24no—
253C4×Q8—22no—
254C4×Q8—24no—
255C4×Q8—22no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—42no—
259C4×D4—42no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—82no—
289C2×C4⋊C4C22⋊C412yes—
290C2×C4⋊C4C22⋊C411yescharacteristic
291C2×C4⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22⋊C412yes—
293C2×C4⋊C4C22⋊C412yes—
294C2×C4⋊C4C22⋊C412yes—
295C2×C4⋊C4C22⋊C412yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—84no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42#4112yes—
321C2×C42#4112yes—
322C4⋊C8—28no—
323C4⋊C8—28no—
324C4⋊C8—24no—
325C4⋊C8—24no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328#100—28no—
329#100—28no—
330#100—48no—
331#97—28no—
332#97—24no—
333#97—42no—
334C4×C8—28no—
335C4×C8—24no—
336C4×C8—42no—
337#95—24no—
338#95—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—42no—
346C24—42no—
347C24—42no—
348C24—42no—
349C24C4×Q814yes—
350C24C23×C411yescharacteristic
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8#10014yescomplemented
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C4×D412yes—
429C22×C4C4×D412yes—
430C22×C4C4×D412yes—
431C22×C4C4×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×Q811yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C22×C4C2×#4111yescharacteristic
440C22×C4C2×#4111yescharacteristic
441C22×C4C2×#4111yescharacteristic
442C22×C4H(5,𝔽2)11yescharacteristic
443C22×C42−1+411yescharacteristic
444C22×C42−1+411yescharacteristic
445C22×C42−1+411yescharacteristic
446M16—22no—
447M16—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—22no—
452M16—22no—
453M16—24no—
454M16—24no—
455M16—22no—
456M16—22no—
457M16—42no—
458M16—41no—
459M16—42no—
460M16—41no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—42no—
466M16—42no—
467M16—42no—
468M16—42no—
469M16C22⋊Q814yescomplemented
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—42no—
481C2×C8—41no—
482C2×C8—42no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—42no—
486C2×C8—42no—
487C2×C8—42no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—82no—
508C4⋊C4—82no—
509C22⋊C4—48no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—48no—
513C22⋊C4—84no—
514C22⋊C4—84no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42#12012yes—
524C42#12012yes—
525C42#12112yes—
526C42#12112yes—
527C42#12112yes—
528C42#12112yes—
529C42#12312yes—
530C42#12312yes—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—82no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23—82no—
557C23—82no—
558C23C22×C22⋊C411yescharacteristic
559C23C2×C4×D411yescharacteristic
560C23C2×C4×D411yescharacteristic
561C23C2×C4×Q811yescharacteristic
562C23C4×#4111yescharacteristic
563C23#51911yescharacteristic
564C23#52011yescharacteristic
565Q8—28no—
566Q8—48no—
567Q8—44no—
568D4—42no—
569D4—42no—
570D4—42no—
571D4—42no—
572D4—42no—
573D4—42no—
574D4—42no—
575D4—42no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—88no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4—88no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4#38612yes—
614C2×C4#38612yes—
615C2×C4#38612yes—
616C2×C4#38612yes—
617C2×C4C2×#10014yes—
618C2×C4C2×#10012yes—
619C2×C4#41312yes—
620C2×C4C2×#12011yescharacteristic
621C2×C4C2×#12111yescharacteristic
622C2×C4C2×#12111yescharacteristic
623C2×C4C2×C22⋊Q811yescharacteristic
624C2×C4C2×#12311yescharacteristic
625C2×C4#52511yescharacteristic
626C2×C4#52911yescharacteristic
627C2×C4#53311yescharacteristic
628C2×C4#53611yescharacteristic
629C2×C4#53711yescharacteristic
630C2×C4#54111yescharacteristic
631C2×C4#54211yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54711yescharacteristic
634C2×C4#54811yescharacteristic
635C2×C4#54811yescharacteristic
636C2×C4#54811yescharacteristic
637C2×C4#54811yescharacteristic
638C2×C4#54811yescharacteristic
639C2×C4D4×Q811yescharacteristic
640C2×C4D4×Q811yescharacteristic
641C2×C4D4×Q811yescharacteristic
642C2×C4#55011yescharacteristic
643C2×C4#55011yescharacteristic
644C2×C4#55211yescharacteristic
645C2×C4#55211yescharacteristic
646C2×C4#55411yescharacteristic
647C2×C4#55611yescharacteristic
648C8—24no—
649C8—42no—
650C8—44no—
651C8—42no—
652C8—42no—
653C8—82no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22C2×#38611yescharacteristic
678C22C4×C22⋊Q811yescharacteristic
679C22#232311yescharacteristic
680C22Q8×C22⋊C411yescharacteristic
681C22#234111yescharacteristic
682C22#236011yescharacteristic
683C22#236211yescharacteristic
684C22C22×#10011yescharacteristic
685C22C2×#41311yescharacteristic
686C22#288811yescharacteristic
687C22#288911yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—84no—
696C4—82no—
697C4#190014yes—
698C4#240711yescharacteristic
699C4#240911yescharacteristic
700C4#242111yescharacteristic
701C4#242311yescharacteristic
702C4#242611yescharacteristic
703C4#242911yescharacteristic
704C4#253611yescharacteristic
705C4#253811yescharacteristic
706C2—21no—
707C2—21no—
708C2—21no—
709C2—21no—
710C2—84no—
711C2#1396911yescharacteristic
712C2C2×#190011yescharacteristic
713C2#2099211yescharacteristic
714C2Q8×#10011yescharacteristic
715C2#2251511yescharacteristic
716C2#2253611yescharacteristic
717C2#2253711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×Q8×M16C2semidirect
3. #21303C2semidirect
6. #11204C2semidirect
9. #10913C2semidirect
10. #10857C2semidirect
11. #10857C2semidirect
120. C2×#100Q8semidirect
361. C2×Q8#100semidirect
469. M16C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C2×#25382
#213032
C2×#18002
C2×C4×#1002
#112042
#110442
#109622
#109132
#108572