Finite Group Databasebeta

← #4307610

Subgroups — #4307610

Order 512.

3666 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic, complemented
2C2×#2351C211yesmaximal, characteristic, complemented
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5C2×Q8×C22⋊C4C211yesmaximal, characteristic
6C2×Q8×C22⋊C4C211yesmaximal, characteristic
7#15237C213yesmaximal, complemented
8#15237C213yesmaximal, complemented
9#15220C213yesmaximal
10#15220C213yesmaximal
11#15189C213yesmaximal, complemented
12#15184C213yesmaximal, complemented
13#15131C213yesmaximal
14C4×#396C213yesmaximal
15Q8×#95C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23C2×#396C2213yes—
24#2351—28no—
25#2351—28no—
26Q8×C4⋊C4—28no—
27Q8×C22⋊C4—28no—
28Q8×C22⋊C4—28no—
29Q8×C22⋊C4—28no—
30C2×#391C2213yes—
31C2×#391C2213yes—
32C2×#391C2213yes—
33C2×#391C2213yes—
34C2×#391C2213yes—
35C2×#391C2213yes—
36C2×#389C2213yes—
37C2×#389C2213yes—
38C2×#389C2213yes—
39C2×#389C2213yes—
40C2×#389C2213yes—
41C2×#389C2213yes—
42C2×#384C2213yes—
43C2×#382C2213yes—
44C2×#382C2213yes—
45C2×#380C2213yes—
46C2×#380C2213yes—
47C2×#380C2213yes—
48C2×C4×C4⋊C4C2213yes—
49C2×C4×C4⋊C4C2213yes—
50C2×C4×C4⋊C4C2213yes—
51C2×C4×C22⋊C4C2213yes—
52C2×C4×C22⋊C4C2213yes—
53C2×C4×C22⋊C4C2213yes—
54#1443C2213yes—
55#1443C2216yes—
56#1443C2216yes—
57#1441C2216yes—
58#1441C2216yes—
59#1441C2213yes—
60#1441C2213yes—
61#1441C2213yes—
62#1441C2216yes—
63#1441C2216yes—
64#1439C2216yes—
65#1439C2211yescharacteristic
66#1439C2213yes—
67#1439C2213yes—
68C4⋊#95C2213yes—
69C4⋊#95C2213yes—
70C4⋊#95C2213yes—
71C4×#95C2213yes—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—24no—
77C23×Q8—24no—
78C23×Q8Q818yes—
79C23×Q8C2313yes—
80C23×Q8C2311yescharacteristic
81C24×C4C2313yes—
82C2×C4×Q8—24no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—24no—
87C2×C4×Q8—24no—
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×Q8D418yes—
98C2×C4×Q8D418yes—
99C2×C4×Q8D418yes—
100C22×C4⋊C4C2313yes—
101C22×C4⋊C4C2313yes—
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C4⋊C4C2313yes—
108C22×C4⋊C4C2313yes—
109C22×C4⋊C4C2313yes—
110C22×C4⋊C4C2313yes—
111C22×C4⋊C4C2313yes—
112C22×C4⋊C4C2316yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2313yes—
116C22×C22⋊C4C2313yes—
117C22×C22⋊C4C2313yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2313yes—
120C22×C42C2313yes—
121C22×C42C2313yes—
122C22×C42C2313yes—
123C22×C42C2313yes—
124C22×C42C2313yes—
125#396—212no—
126#391—212no—
127#391—212no—
128#391—212no—
129#391—212no—
130#391—212no—
131#391—212no—
132#389—212no—
133#389—212no—
134#389—212no—
135#389—212no—
136#389—212no—
137#389—212no—
138#384—212no—
139#382—212no—
140#382—212no—
141#380—212no—
142#380—212no—
143#380—212no—
144C4×C4⋊C4—212no—
145C4×C4⋊C4—212no—
146C4×C4⋊C4—212no—
147C4×C22⋊C4—212no—
148C4×C22⋊C4—212no—
149C4×C22⋊C4—212no—
150C2×#95C2313yes—
151C2×#95C2311yescharacteristic
152C2×#95C2313yes—
153C2×#95C2311yescharacteristic
154C2×#95C2313yes—
155C2×#95C2316yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2316yes—
159C2×#95C2311yescharacteristic
160C2×#95C2313yes—
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2313yes—
164C2×#95C2313yes—
165C2×#95C2316yes—
166C2×#95C2316yes—
167C2×#95C2313yes—
168C2×#95C2316yes—
169C2×#95C2313yes—
170C2×#95C2313yes—
171C2×#95C2316yes—
172C25C2411yescharacteristic
173C22×Q8—232no—
174C22×Q8—232no—
175C22×Q8—232no—
176C22×Q8—26no—
177C22×Q8—26no—
178C22×Q8—26no—
179C22×Q8—26no—
180C22×Q8—26no—
181C22×Q8—26no—
182C22×Q8—416no—
183C22×Q8—416no—
184C22×Q8—416no—
185C22×Q8—416no—
186C22×Q8C2×D414yes—
187C22×Q8C2×D414yes—
188C22×Q8C2×D414yes—
189C22×Q8C2×Q8112yes—
190C22×Q8C2×Q814yes—
191C22×Q8#4114yes—
192C22×Q8#4114yes—
193C22×Q8#4114yes—
194C23×C4—26no—
195C23×C4—26no—
196C23×C4—26no—
197C23×C4—26no—
198C23×C4—26no—
199C23×C4—26no—
200C23×C4C2×Q8112yes—
201C23×C4C2413yes—
202C23×C4C2413yes—
203C23×C4C2413yes—
204C23×C4C2413yes—
205C23×C4C2413yes—
206C23×C4C2413yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2413yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2413yes—
215C4×Q8—232no—
216C4×Q8—232no—
217C4×Q8—232no—
218C4×Q8—416no—
219C4×Q8—416no—
220C4×Q8—416no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—26no—
226C2×C4⋊C4—26no—
227C2×C4⋊C4—26no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
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—26no—
243C2×C4⋊C4—26no—
244C2×C4⋊C4—26no—
245C2×C4⋊C4—26no—
246C2×C4⋊C4—26no—
247C2×C4⋊C4—26no—
248C2×C4⋊C4—26no—
249C2×C4⋊C4—212no—
250C2×C4⋊C4—43no—
251C2×C4⋊C4—43no—
252C2×C4⋊C4—43no—
253C2×C4⋊C4—43no—
254C2×C4⋊C4—43no—
255C2×C4⋊C4—46no—
256C2×C4⋊C4—46no—
257C2×C4⋊C4C2×D4112yes—
258C2×C4⋊C4C2×D4112yes—
259C2×C4⋊C4C2×D4112yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—26no—
269C2×C22⋊C4—26no—
270C2×C22⋊C4—26no—
271C2×C22⋊C4—26no—
272C2×C22⋊C4—26no—
273C2×C22⋊C4—26no—
274C2×C22⋊C4—43no—
275C2×C22⋊C4—43no—
276C2×C22⋊C4—43no—
277C2×C22⋊C4#4114yes—
278C2×C42—26no—
279C2×C42—26no—
280C2×C42—26no—
281C2×C42—26no—
282C2×C42—26no—
283C2×C42—26no—
284C2×C42—26no—
285C2×C42—26no—
286C2×C42—26no—
287C2×C42—26no—
288C2×C42—26no—
289C2×C42—26no—
290C2×C42—26no—
291C2×C42—26no—
292C2×C42—26no—
293C2×C42C2×D4112yes—
294C2×C42C2×D4112yes—
295C2×C42C2×D4112yes—
296#95—26no—
297#95—26no—
298#95—26no—
299#95—26no—
300#95—26no—
301#95—26no—
302#95—22no—
303#95—22no—
304#95—26no—
305#95—26no—
306#95—26no—
307#95—26no—
308#95—212no—
309#95—212no—
310#95—212no—
311#95—212no—
312#95—212no—
313#95—26no—
314#95—22no—
315#95—26no—
316#95—43no—
317#95—46no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24C22×Q812yes—
331C24C2×#4112yes—
332C242−1+412yes—
333C24C2511yescharacteristic
334C2×Q8—224no—
335C2×Q8—224no—
336C2×Q8—224no—
337C2×Q8—4128no—
338C2×Q8—412no—
339C2×Q8—412no—
340C2×Q8—412no—
341C2×Q8—412no—
342C2×Q8#121116yes—
343C2×Q8#121116yes—
344C2×Q8#121116yes—
345C2×Q8C22⋊Q8116yes—
346C2×Q8C22⋊Q8116yes—
347C2×Q8C22⋊Q8116yes—
348C2×Q8#126116yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—23no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—23no—
363C22×C4—23no—
364C22×C4—23no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—23no—
373C22×C4—23no—
374C22×C4—23no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—23no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—21no—
397C22×C4—23no—
398C22×C4—23no—
399C22×C4—23no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—224no—
408C22×C4—224no—
409C22×C4—224no—
410C22×C4—23no—
411C22×C4—23no—
412C22×C4—23no—
413C22×C4—23no—
414C22×C4—23no—
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—23no—
426C22×C4—23no—
427C22×C4—23no—
428C22×C4—23no—
429C22×C4—23no—
430C22×C4—23no—
431C22×C4—412no—
432C22×C4—412no—
433C22×C4—412no—
434C22×C4—412no—
435C22×C4C22×D412yes—
436C22×C4C22×D416yes—
437C22×C4C22×D416yes—
438C22×C4C22×D412yes—
439C22×C4C22×D416yes—
440C22×C4C22×D412yes—
441C22×C4C22×D416yes—
442C22×C4C22×D416yes—
443C22×C4C22×D416yes—
444C22×C4C22×Q816yes—
445C22×C4C22×Q816yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4116yes—
449C22×C4C2×#4116yes—
450C22×C4C2×#4116yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C4⋊C4—28no—
455C4⋊C4—224no—
456C4⋊C4—224no—
457C4⋊C4—224no—
458C4⋊C4—412no—
459C4⋊C4—412no—
460C4⋊C4—412no—
461C4⋊C4—412no—
462C4⋊C4—412no—
463C4⋊C4—412no—
464C4⋊C4—412no—
465C4⋊C4—412no—
466C4⋊C4—812no—
467C22⋊C4—28no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—412no—
471C22⋊C4—412no—
472C22⋊C4—412no—
473C42—224no—
474C42—224no—
475C42—224no—
476C42—412no—
477C42—412no—
478C42—412no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#55714yes—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4111yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+411yescharacteristic
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23#58511yescharacteristic
527C23#58511yescharacteristic
528C23#58511yescharacteristic
529Q8—448no—
530Q8#396164yescomplemented
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—26no—
540C2×C4—26no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—26no—
545C2×C4—22no—
546C2×C4—26no—
547C2×C4—26no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—26no—
558C2×C4—22no—
559C2×C4—26no—
560C2×C4—26no—
561C2×C4—26no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—43no—
568C2×C4—43no—
569C2×C4—43no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—43no—
578C2×C4—43no—
579C2×C4—43no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—43no—
588C2×C4—41no—
589C2×C4—43no—
590C2×C4—41no—
591C2×C4—43no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—43no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—43no—
600C2×C4—448no—
601C2×C4—43no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—43no—
605C2×C4—43no—
606C2×C4—43no—
607C2×C4—43no—
608C2×C4—43no—
609C2×C4—43no—
610C2×C4—43no—
611C2×C4—43no—
612C2×C4—43no—
613C2×C4—43no—
614C2×C4—43no—
615C2×C4—43no—
616C2×C4—43no—
617C2×C4—43no—
618C2×C4—43no—
619C2×C4—43no—
620C2×C4—43no—
621C2×C4—43no—
622C2×C4C2×#121112yes—
623C2×C4C2×#121112yes—
624C2×C4C2×#121112yes—
625C2×C4C2×C22⋊Q8112yes—
626C2×C4C2×C22⋊Q8112yes—
627C2×C4C2×C22⋊Q8112yes—
628C2×C4C2×#126112yes—
629C2×C4#54814yes—
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22C22×#12111yescharacteristic
646C22C22×#12111yescharacteristic
647C22C22×#12111yescharacteristic
648C22C22×C22⋊Q811yescharacteristic
649C22C22×C22⋊Q811yescharacteristic
650C22C22×C22⋊Q811yescharacteristic
651C22C22×#12611yescharacteristic
652C22C2×#54811yescharacteristic
653C22D4×#4111yescharacteristic
654C22D4×#4111yescharacteristic
655C22C2×#55711yescharacteristic
656C22#348011yescharacteristic
657C22#348211yescharacteristic
658C22#348211yescharacteristic
659C22#348611yescharacteristic
660C22#348611yescharacteristic
661C22#348611yescharacteristic
662C22#348711yescharacteristic
663C22#348911yescharacteristic
664C22#348911yescharacteristic
665C22#349711yescharacteristic
666C22#349911yescharacteristic
667C22#350011yescharacteristic
668C22#350211yescharacteristic
669C22#350211yescharacteristic
670C22#350211yescharacteristic
671C22#351611yescharacteristic
672C22#351611yescharacteristic
673C22#351611yescharacteristic
674C22#351611yescharacteristic
675C22#351811yescharacteristic
676C22#351811yescharacteristic
677C22#352311yescharacteristic
678C22#352311yescharacteristic
679C22#352411yescharacteristic
680C4—24no—
681C4—46no—
682C4—46no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—46no—
687C4—42no—
688C4—42no—
689C4—46no—
690C4—83no—
691C4—83no—
692C4—83no—
693C4C2×#396124yes—
694C2—44no—
695C2C22×#39611yescharacteristic
696C2#3570011yescharacteristic
697C2#3571111yescharacteristic
698C2#3573711yescharacteristic
699C2#3577111yescharacteristic
700C2#3578611yescharacteristic
701C2#3584511yescharacteristic
702C2#3588311yescharacteristic
703C2#3643011yescharacteristic
704C2#3643411yescharacteristic
705C2#3643511yescharacteristic
706C2#3643511yescharacteristic
707C2#3644411yescharacteristic
708C2#3644811yescharacteristic
709C2#3657711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2351C2semidirect
2. C2×#2351C2semidirect
3. C2×Q8×C4⋊C4C2semidirect
7. #15237C2semidirect
8. #15237C2semidirect
11. #15189C2semidirect
12. #15184C2semidirect
15. Q8×#95C2semidirect
530. Q8#396semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×Q8×C4⋊C42
C2×Q8×C22⋊C42
#152372
#152202
#151892
#151842
#151312
C4×#3962
Q8×#952