Finite Group Databasebeta

← #518082

Subgroups — #518082

Order 512.

4998 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#519C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5C2×Q8×C22⋊C4C211yesmaximal, characteristic
6#14264C213yesmaximal
7#14129C213yesmaximal
8#14017C213yesmaximal, complemented
9#14017C213yesmaximal, complemented
10#13978C213yesmaximal, complemented
11#13976C213yesmaximal, complemented
12#13879C213yesmaximal
13#13844C213yesmaximal
14Q8×#95C211yesmaximal, characteristic, complemented
15Q8×#95C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C2×#519—28no—
18C2×#519—28no—
19C2×#519—28no—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2213yes—
23C22×C4×Q8C2213yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×#117C2213yes—
29C22×#117C2213yes—
30#2351—28no—
31Q8×C4⋊C4—28no—
32Q8×C22⋊C4—28no—
33Q8×C22⋊C4C4116yescomplemented
34C2×#391C2213yes—
35C2×#391C2213yes—
36C2×#391C2213yes—
37C2×#391C2213yes—
38C2×#389C2213yes—
39C2×#389C2213yes—
40C2×#389C2213yes—
41C2×#389C2213yes—
42C2×#384C2213yes—
43C2×#382C2213yes—
44C2×#380C2213yes—
45C2×#380C2213yes—
46C2×C4×C4⋊C4C2213yes—
47C2×C4×C4⋊C4C2213yes—
48C2×C4×C22⋊C4C2213yes—
49C2×C4×C22⋊C4C2213yes—
50#1444C2213yes—
51#1444C2213yes—
52#1444C2216yes—
53#1442C2213yes—
54#1442C2213yes—
55#1442C2216yes—
56#1439C2211yescharacteristic
57#1439C2213yes—
58#1439C2213yes—
59#1439C2213yes—
60#1439C2216yes—
61#1436C2216yes—
62#1436C2216yes—
63#1436C2216yes—
64#1436C2216yes—
65C4⋊#95C2213yes—
66C4⋊#95C2213yes—
67C4⋊#95C2213yes—
68C4⋊#95C2213yes—
69C4⋊#95C2213yes—
70C4⋊#95C2213yes—
71C4×#95C2213yes—
72C4×#95C2213yes—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—24no—
77C23×Q8C2×C418yes—
78C23×Q8D418yes—
79C23×Q8Q818yes—
80C23×Q8C2313yes—
81C23×Q8C2311yescharacteristic
82C24×C4C2313yes—
83#519—2128no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—212no—
87C2×C4×Q8—212no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—212no—
101C2×C4×Q8—212no—
102C2×C4×Q8—212no—
103C2×C4×Q8—212no—
104C2×C4×Q8C2×C418yes—
105C2×C4×Q8C2×C418yes—
106C2×C4×Q8D418yes—
107C2×C4×Q8D418yes—
108C2×#117—212no—
109C2×#117—212no—
110C2×#117—212no—
111C2×#117—212no—
112C2×#117—212no—
113C2×#117—212no—
114C22×C4⋊C4C2313yes—
115C22×C4⋊C4C2313yes—
116C22×C4⋊C4C2313yes—
117C22×C4⋊C4C2313yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2313yes—
120C22×C4⋊C4C2313yes—
121C22×C4⋊C4C2313yes—
122C22×C4⋊C4C2313yes—
123C22×C4⋊C4C2313yes—
124C22×C4⋊C4C2313yes—
125C22×C4⋊C4C2316yes—
126C22×C4⋊C4C2313yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2313yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2313yes—
132C22×C22⋊C4C2313yes—
133C22×C42C2316yes—
134C22×C42C2313yes—
135C22×C42C2313yes—
136C22×C42C2313yes—
137C22×C42C2313yes—
138C22×C42C2313yes—
139C22×C42C2313yes—
140#391—212no—
141#391—212no—
142#391C2×C4124yes—
143#391C2×C4124yes—
144#389—212no—
145#389—212no—
146#389—212no—
147#389—212no—
148#384—212no—
149#382—212no—
150#380—212no—
151#380C2×C4124yes—
152C4×C4⋊C4—212no—
153C4×C4⋊C4—212no—
154C4×C22⋊C4—212no—
155C4×C22⋊C4C2×C4124yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2316yes—
163C2×#95C2313yes—
164C2×#95C2313yes—
165C2×#95C2316yes—
166C2×#95C2311yescharacteristic
167C2×#95C2313yes—
168C2×#95C2313yes—
169C2×#95C2313yes—
170C2×#95C2313yes—
171C2×#95C2313yes—
172C2×#95C2316yes—
173C2×#95C2316yes—
174C2×#95C2313yes—
175C2×#95C2316yes—
176C2×#95C2313yes—
177C25C2411yescharacteristic
178C22×Q8—232no—
179C22×Q8—232no—
180C22×Q8—232no—
181C22×Q8—232no—
182C22×Q8—232no—
183C22×Q8—232no—
184C22×Q8—26no—
185C22×Q8—26no—
186C22×Q8—26no—
187C22×Q8—26no—
188C22×Q8C4⋊C4164yescomplemented
189C22×Q8C22×C4112yes—
190C22×Q8C22×C414yes—
191C22×Q8C2×D4112yes—
192C22×Q8C2×D414yes—
193C22×Q8C2×D414yes—
194C22×Q8C2×D414yes—
195C22×Q8C2×Q8112yes—
196C22×Q8C2×Q814yes—
197C22×Q8#4114yes—
198C22×Q8#4114yes—
199C23×C4—26no—
200C23×C4—26no—
201C23×C4—26no—
202C23×C4—26no—
203C23×C4C22×C4112yes—
204C23×C4C2×D4112yes—
205C23×C4C2×Q8112yes—
206C23×C4C2413yes—
207C23×C4C2413yes—
208C23×C4C2413yes—
209C23×C4C2413yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2413yes—
213C23×C4C2413yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2413yes—
218C23×C4C2411yescharacteristic
219C23×C4C2413yes—
220C4×Q8—296no—
221C4×Q8—296no—
222C4×Q8—232no—
223C4×Q8—232no—
224C4×Q8—232no—
225C4×Q8—232no—
226C4×Q8—416no—
227C4×Q8—416no—
228#117—296no—
229#117—296no—
230C2×C4⋊C4—26no—
231C2×C4⋊C4—26no—
232C2×C4⋊C4—26no—
233C2×C4⋊C4—26no—
234C2×C4⋊C4—26no—
235C2×C4⋊C4—26no—
236C2×C4⋊C4—26no—
237C2×C4⋊C4—26no—
238C2×C4⋊C4—26no—
239C2×C4⋊C4—26no—
240C2×C4⋊C4—26no—
241C2×C4⋊C4—26no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—26no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—26no—
250C2×C4⋊C4—26no—
251C2×C4⋊C4—26no—
252C2×C4⋊C4—26no—
253C2×C4⋊C4—26no—
254C2×C4⋊C4—26no—
255C2×C4⋊C4—26no—
256C2×C4⋊C4—26no—
257C2×C4⋊C4—26no—
258C2×C4⋊C4—212no—
259C2×C4⋊C4—43no—
260C2×C4⋊C4—43no—
261C2×C4⋊C4—43no—
262C2×C4⋊C4—43no—
263C2×C4⋊C4—46no—
264C2×C4⋊C4—46no—
265C2×C4⋊C4C22×C4112yes—
266C2×C4⋊C4C22×C4112yes—
267C2×C4⋊C4C2×D4112yes—
268C2×C4⋊C4C2×D4112yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—26no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—26no—
281C2×C22⋊C4—43no—
282C2×C22⋊C4—43no—
283C2×C22⋊C4—43no—
284C2×C22⋊C4—43no—
285C2×C22⋊C4C22×C414yes—
286C2×C22⋊C4C22×C4112yes—
287C2×C42—212no—
288C2×C42—212no—
289C2×C42—212no—
290C2×C42—26no—
291C2×C42—26no—
292C2×C42—26no—
293C2×C42—26no—
294C2×C42—26no—
295C2×C42—26no—
296C2×C42—26no—
297C2×C42—26no—
298C2×C42—26no—
299C2×C42—26no—
300C2×C42—26no—
301C2×C42—26no—
302C2×C42—26no—
303C2×C42—26no—
304C2×C42C22×C4112yes—
305C2×C42C22×C4112yes—
306C2×C42C2×D4112yes—
307C2×C42C2×D4112yes—
308#95—26no—
309#95—26no—
310#95—22no—
311#95—26no—
312#95—26no—
313#95—26no—
314#95—212no—
315#95—212no—
316#95—22no—
317#95—26no—
318#95—22no—
319#95—26no—
320#95—43no—
321#95—43no—
322#95—43no—
323#95—43no—
324#95—46no—
325#95—46no—
326#95C22×C4112yes—
327#95C22×C4112yes—
328#95C22×C4124yes—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24C23×C412yes—
342C24C22×D412yes—
343C24C22×Q812yes—
344C24C2511yescharacteristic
345C2×Q8—2256no—
346C2×Q8—224no—
347C2×Q8—224no—
348C2×Q8—224no—
349C2×Q8—224no—
350C2×Q8—224no—
351C2×Q8—224no—
352C2×Q8C2×C4⋊C4148yes—
353C2×Q8C2×C4⋊C4116yes—
354C2×Q8C4×D4116yes—
355C2×Q8C4×D4116yes—
356C2×Q8#120116yes—
357C2×Q8C22⋊Q8116yes—
358C2×Q8C22⋊Q8116yes—
359C2×Q8#123116yes—
360C22×C4—21no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—21no—
364C22×C4—23no—
365C22×C4—23no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—224no—
369C22×C4—224no—
370C22×C4—224no—
371C22×C4—224no—
372C22×C4—224no—
373C22×C4—224no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—21no—
381C22×C4—23no—
382C22×C4—21no—
383C22×C4—23no—
384C22×C4—23no—
385C22×C4—21no—
386C22×C4—23no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—23no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—23no—
410C22×C4—23no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—23no—
416C22×C4—23no—
417C22×C4—23no—
418C22×C4—23no—
419C22×C4—23no—
420C22×C4—23no—
421C22×C4—23no—
422C22×C4—23no—
423C22×C4—23no—
424C22×C4—23no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—23no—
429C22×C4—23no—
430C22×C4—23no—
431C22×C4—23no—
432C22×C4—23no—
433C22×C4—23no—
434C22×C4—23no—
435C22×C4—23no—
436C22×C4—23no—
437C22×C4—23no—
438C22×C4—23no—
439C22×C4—23no—
440C22×C4—23no—
441C22×C4—23no—
442C22×C4C2×C4⋊C4148yes—
443C22×C4C23×C412yes—
444C22×C4C23×C416yes—
445C22×C4C23×C416yes—
446C22×C4C23×C416yes—
447C22×C4C23×C412yes—
448C22×C4C23×C416yes—
449C22×C4C22×D416yes—
450C22×C4C22×D412yes—
451C22×C4C22×D416yes—
452C22×C4C22×D412yes—
453C22×C4C22×D416yes—
454C22×C4C22×D416yes—
455C22×C4C22×D416yes—
456C22×C4C22×D416yes—
457C22×C4C22×Q816yes—
458C22×C4C22×Q816yes—
459C22×C4C2×#4116yes—
460C22×C4C2×#4116yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C42−1+412yes—
465C4⋊C4—28no—
466C4⋊C4—224no—
467C4⋊C4—224no—
468C4⋊C4—224no—
469C4⋊C4—224no—
470C4⋊C4—224no—
471C4⋊C4—224no—
472C4⋊C4—412no—
473C4⋊C4—412no—
474C4⋊C4—412no—
475C4⋊C4—412no—
476C4⋊C4—412no—
477C4⋊C4—812no—
478C22⋊C4—28no—
479C22⋊C4—224no—
480C22⋊C4—28no—
481C22⋊C4—28no—
482C22⋊C4—412no—
483C22⋊C4—412no—
484C42—248no—
485C42—224no—
486C42—224no—
487C42—224no—
488C42—224no—
489C42—412no—
490C42—412no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C4⋊C414yes—
526C23C24×C411yescharacteristic
527C23C23×D411yescharacteristic
528C23C23×D411yescharacteristic
529C23C23×D411yescharacteristic
530C23C23×Q811yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23C2×2−1+411yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23#58511yescharacteristic
538C23#58511yescharacteristic
539C23#58511yescharacteristic
540C23#58511yescharacteristic
541Q8—296no—
542Q8#385164yes—
543C2×C4—22no—
544C2×C4—26no—
545C2×C4—26no—
546C2×C4—22no—
547C2×C4—26no—
548C2×C4—26no—
549C2×C4—26no—
550C2×C4—26no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—296no—
556C2×C4—22no—
557C2×C4—26no—
558C2×C4—26no—
559C2×C4—22no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—22no—
563C2×C4—26no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—22no—
567C2×C4—26no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—26no—
576C2×C4—26no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—26no—
580C2×C4—26no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—43no—
586C2×C4—41no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—41no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—43no—
597C2×C4—41no—
598C2×C4—43no—
599C2×C4—41no—
600C2×C4—43no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—43no—
604C2×C4—41no—
605C2×C4—43no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—43no—
610C2×C4—41no—
611C2×C4—43no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—43no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—43no—
618C2×C4—43no—
619C2×C4—43no—
620C2×C4—43no—
621C2×C4—43no—
622C2×C4—43no—
623C2×C4—43no—
624C2×C4—43no—
625C2×C4—43no—
626C2×C4—43no—
627C2×C4—43no—
628C2×C4—43no—
629C2×C4—43no—
630C2×C4—43no—
631C2×C4—43no—
632C2×C4—43no—
633C2×C4C22×C4⋊C4112yes—
634C2×C4C22×C4⋊C4112yes—
635C2×C4C2×C4×D4112yes—
636C2×C4C2×C4×D4112yes—
637C2×C4#51914yescomplemented
638C2×C4C2×#120112yes—
639C2×C4C2×C22⋊Q8112yes—
640C2×C4C2×C22⋊Q8112yes—
641C2×C4C2×#123112yes—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22C23×C4⋊C411yescharacteristic
658C22C22×C4×D411yescharacteristic
659C22C22×C4×D411yescharacteristic
660C22C2×#51911yescharacteristic
661C22#342811yescharacteristic
662C22C4×2−1+411yescharacteristic
663C22C4×2−1+411yescharacteristic
664C22C22×#12011yescharacteristic
665C22C22×C22⋊Q811yescharacteristic
666C22C22×C22⋊Q811yescharacteristic
667C22C22×#12311yescharacteristic
668C22#348611yescharacteristic
669C22#348611yescharacteristic
670C22#348711yescharacteristic
671C22#348711yescharacteristic
672C22#348911yescharacteristic
673C22#348911yescharacteristic
674C22#349711yescharacteristic
675C22#349711yescharacteristic
676C22#349711yescharacteristic
677C22#349711yescharacteristic
678C22#349911yescharacteristic
679C22#350011yescharacteristic
680C22#350211yescharacteristic
681C22#350211yescharacteristic
682C22#350211yescharacteristic
683C22#350211yescharacteristic
684C22#351611yescharacteristic
685C22#351611yescharacteristic
686C22#351811yescharacteristic
687C22#351811yescharacteristic
688C22#352311yescharacteristic
689C22#352311yescharacteristic
690C22#352411yescharacteristic
691C22#352411yescharacteristic
692C4—212no—
693C4—24no—
694C4—42no—
695C4—46no—
696C4—42no—
697C4—46no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—46no—
702C4—46no—
703C4—83no—
704C4—83no—
705C4C2×#385124yes—
706C2—44no—
707C2C22×#38511yescharacteristic
708C2#3418211yescharacteristic
709C2#3424911yescharacteristic
710C2#3429111yescharacteristic
711C2#3429311yescharacteristic
712C2#3431111yescharacteristic
713C2#3431211yescharacteristic
714C2#3568711yescharacteristic
715C2#3571111yescharacteristic
716C2#3572311yescharacteristic
717C2#3575711yescharacteristic
718C2#3578611yescharacteristic
719C2#3578611yescharacteristic
720C2#3580111yescharacteristic
721C2#3584511yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2351C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
8. #14017C2semidirect
9. #14017C2semidirect
10. #13978C2semidirect
11. #13976C2semidirect
14. Q8×#95C2semidirect
15. Q8×#95C2semidirect
33. Q8×C22⋊C4C4semidirect
188. C22×Q8C4⋊C4semidirect
637. C2×C4#519semidirect

Maximal subgroups

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

subgroupindex
C22×#5192
C2×#23512
C2×Q8×C4⋊C42
C2×Q8×C22⋊C42
#142642
#141292
#140172
#139782
#139762
#138792
#138442
Q8×#952