Finite Group Databasebeta

← #518273

Subgroups — #518273

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic, complemented
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3C2×Q8×C22⋊C4C211yesmaximal, characteristic
4C2×Q8×C22⋊C4C211yesmaximal, characteristic
5C2×#2276C211yesmaximal, characteristic, complemented
6#14261C213yesmaximal, complemented
7#14142C213yesmaximal, complemented
8#14021C213yesmaximal, complemented
9#14003C213yesmaximal, complemented
10#13993C213yesmaximal
11#13973C213yesmaximal
12#13875C213yesmaximal
13#13842C213yesmaximal
14Q8×#95C211yesmaximal, characteristic, complemented
15Q8×#95C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#2351—28no—
24Q8×C22⋊C4—28no—
25Q8×C22⋊C4—28no—
26Q8×C22⋊C4C4116yescomplemented
27C2×#391C2213yes—
28C2×#391C2213yes—
29C2×#391C2213yes—
30C2×#391C2213yes—
31C2×#391C2213yes—
32C2×#391C2213yes—
33C2×#389C2213yes—
34C2×#389C2213yes—
35C2×#382C2213yes—
36C2×#380C2213yes—
37C2×#380C2213yes—
38C2×#380C2213yes—
39#2276—28no—
40C2×C4×C4⋊C4C2213yes—
41C2×C4×C4⋊C4C2213yes—
42C2×C4×C4⋊C4C2213yes—
43C2×C4×C4⋊C4C2213yes—
44C2×C4×C22⋊C4C2213yes—
45C2×C4×C22⋊C4C2213yes—
46C2×C4×C22⋊C4C2213yes—
47C2×#376C2213yes—
48#1443C2213yes—
49#1443C2213yes—
50#1443C2216yes—
51#1443C2216yes—
52#1442C2213yes—
53#1442C2213yes—
54#1442C2216yes—
55#1440C2211yescharacteristic
56#1440C2213yes—
57#1439C2213yes—
58#1439C2213yes—
59#1437C2216yes—
60#1436C2216yes—
61C4⋊#95C2213yes—
62C4⋊#95C2213yes—
63C4⋊#95C2213yes—
64C4⋊#95C2213yes—
65C4⋊#95C2213yes—
66C4⋊#95C2213yes—
67#1434C2216yes—
68#1434C2216yes—
69C4×#95C2213yes—
70C4×#95C2213yes—
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—24no—
77C23×Q8C2×C418yes—
78C23×Q8C2313yes—
79C23×Q8C2311yescharacteristic
80C24×C4C2313yes—
81C2×C4×Q8—24no—
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×Q8C2×C418yes—
96C2×C4×Q8C2×C418yes—
97C2×C4×Q8D418yes—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2313yes—
100C22×C4⋊C4C2313yes—
101C22×C4⋊C4C2313yes—
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2313yes—
113C22×C42C2313yes—
114C22×C42C2313yes—
115C22×C42C2313yes—
116C22×C42C2313yes—
117C22×C42C2313yes—
118C22×C42C2313yes—
119C22×C42C2316yes—
120#391—212no—
121#391—212no—
122#391—212no—
123#391—212no—
124#391C2×C4124yes—
125#391C2×C4124yes—
126#389—212no—
127#389—212no—
128#382—212no—
129#380—212no—
130#380—212no—
131#380C2×C4124yes—
132C4×C4⋊C4—212no—
133C4×C4⋊C4—212no—
134C4×C4⋊C4—212no—
135C4×C4⋊C4—212no—
136C4×C22⋊C4—212no—
137C4×C22⋊C4—212no—
138C4×C22⋊C4C2×C4124yes—
139#376—212no—
140C2×#95C2311yescharacteristic
141C2×#95C2311yescharacteristic
142C2×#95C2313yes—
143C2×#95C2313yes—
144C2×#95C2313yes—
145C2×#95C2316yes—
146C2×#95C2316yes—
147C2×#95C2316yes—
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2311yescharacteristic
152C2×#95C2313yes—
153C2×#95C2311yescharacteristic
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2313yes—
158C2×#95C2313yes—
159C2×#95C2313yes—
160C2×#95C2313yes—
161C2×#95C2316yes—
162C2×#95C2316yes—
163C2×#95C2313yes—
164C2×#95C2316yes—
165C2×#95C2313yes—
166C25C2411yescharacteristic
167C22×Q8—232no—
168C22×Q8—232no—
169C22×Q8—232no—
170C22×Q8—26no—
171C22×Q8—26no—
172C22×Q8—26no—
173C22×Q8—26no—
174C22×Q8—26no—
175C22×Q8—26no—
176C22×Q8—416no—
177C22×Q8—416no—
178C22×Q8—416no—
179C22×Q8—416no—
180C22×Q8C22×C4112yes—
181C22×Q8C22×C414yes—
182C22×Q8C2×D414yes—
183C22×Q8C2×D414yes—
184C22×Q8#4114yes—
185C22×Q8#4114yes—
186C22×Q8#4114yes—
187C22×Q8#4114yes—
188C23×C4—26no—
189C23×C4—26no—
190C23×C4—26no—
191C23×C4—26no—
192C23×C4—26no—
193C23×C4—26no—
194C23×C4C22×C4112yes—
195C23×C4C2413yes—
196C23×C4C2413yes—
197C23×C4C2413yes—
198C23×C4C2413yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2413yes—
202C23×C4C2411yescharacteristic
203C23×C4C2413yes—
204C23×C4C2413yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2413yes—
209C4×Q8—232no—
210C4×Q8—232no—
211C4×Q8—232no—
212C4×Q8—232no—
213C4×Q8—416no—
214C4×Q8—416no—
215C2×C4⋊C4—26no—
216C2×C4⋊C4—26no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—26no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—26no—
226C2×C4⋊C4—26no—
227C2×C4⋊C4—26no—
228C2×C4⋊C4—26no—
229C2×C4⋊C4—26no—
230C2×C4⋊C4—26no—
231C2×C4⋊C4—26no—
232C2×C4⋊C4—43no—
233C2×C4⋊C4—43no—
234C2×C4⋊C4—43no—
235C2×C4⋊C4C22×C4112yes—
236C2×C4⋊C4C22×C4112yes—
237C2×C4⋊C4C2×D4112yes—
238C2×C4⋊C4C2×D4112yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—26no—
251C2×C22⋊C4—43no—
252C2×C22⋊C4—43no—
253C2×C22⋊C4—43no—
254C2×C22⋊C4—43no—
255C2×C22⋊C4C22×C414yes—
256C2×C22⋊C4C22×C4112yes—
257C2×C42—26no—
258C2×C42—26no—
259C2×C42—26no—
260C2×C42—26no—
261C2×C42—26no—
262C2×C42—26no—
263C2×C42—26no—
264C2×C42—26no—
265C2×C42—26no—
266C2×C42—26no—
267C2×C42—26no—
268C2×C42—26no—
269C2×C42—26no—
270C2×C42—26no—
271C2×C42—212no—
272C2×C42—46no—
273C2×C42—46no—
274C2×C42C22×C4112yes—
275C2×C42C22×C4112yes—
276C2×C42C2×D4112yes—
277C2×C42C2×D4112yes—
278#95—22no—
279#95—26no—
280#95—26no—
281#95—26no—
282#95—26no—
283#95—26no—
284#95—26no—
285#95—26no—
286#95—22no—
287#95—26no—
288#95—212no—
289#95—212no—
290#95—212no—
291#95—22no—
292#95—22no—
293#95—26no—
294#95—43no—
295#95—43no—
296#95—43no—
297#95—43no—
298#95—43no—
299#95—46no—
300#95—46no—
301#95C22×C4112yes—
302#95C22×C4112yes—
303#95C22×C4124yes—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24C23×C412yes—
317C242−1+412yes—
318C242−1+412yes—
319C24C2511yescharacteristic
320C2×Q8—224no—
321C2×Q8—224no—
322C2×Q8—224no—
323C2×Q8—4128no—
324C2×Q8—412no—
325C2×Q8—412no—
326C2×Q8—412no—
327C2×Q8—412no—
328C2×Q8#117116yes—
329C2×Q8C4×D4116yes—
330C2×Q8C4×D4116yes—
331C2×Q8#121116yes—
332C2×Q8#123116yes—
333C2×Q8#124116yes—
334C2×Q8#126116yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—23no—
339C22×C4—23no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—23no—
343C22×C4—23no—
344C22×C4—23no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—23no—
355C22×C4—23no—
356C22×C4—23no—
357C22×C4—23no—
358C22×C4—23no—
359C22×C4—23no—
360C22×C4—23no—
361C22×C4—23no—
362C22×C4—23no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—23no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—23no—
379C22×C4—23no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—23no—
388C22×C4—23no—
389C22×C4—23no—
390C22×C4—23no—
391C22×C4—23no—
392C22×C4—23no—
393C22×C4—23no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—224no—
402C22×C4—224no—
403C22×C4—224no—
404C22×C4—23no—
405C22×C4—23no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—23no—
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—412no—
419C22×C4—412no—
420C22×C4—412no—
421C22×C4—412no—
422C22×C4C23×C416yes—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C23×C416yes—
426C22×C4C23×C416yes—
427C22×C4C23×C416yes—
428C22×C4C22×D412yes—
429C22×C4C22×D416yes—
430C22×C4C22×D416yes—
431C22×C4C22×D412yes—
432C22×C4C22×D416yes—
433C22×C4C22×D416yes—
434C22×C4C2×#4116yes—
435C22×C4C2×#4116yes—
436C22×C4C2×#4116yes—
437C22×C4C2×#4116yes—
438C22×C42−1+412yes—
439C22×C42−1+412yes—
440C4⋊C4—224no—
441C4⋊C4—224no—
442C4⋊C4—224no—
443C4⋊C4—224no—
444C4⋊C4—412no—
445C4⋊C4—412no—
446C4⋊C4—412no—
447C4⋊C4—412no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—412no—
452C22⋊C4—412no—
453C22⋊C4—412no—
454C42—224no—
455C42—224no—
456C42—224no—
457C42—224no—
458C42—412no—
459C42—412no—
460C42—812no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23#51914yes—
496C23C24×C411yescharacteristic
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C22×#4111yescharacteristic
500C23C22×#4111yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4111yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23#58511yescharacteristic
508C23#58511yescharacteristic
509C23#58511yescharacteristic
510C23#58511yescharacteristic
511Q8—448no—
512Q8#388164yes—
513C2×C4—26no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—26no—
517C2×C4—26no—
518C2×C4—26no—
519C2×C4—26no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—26no—
523C2×C4—26no—
524C2×C4—26no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—26no—
528C2×C4—22no—
529C2×C4—26no—
530C2×C4—22no—
531C2×C4—26no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—26no—
538C2×C4—26no—
539C2×C4—22no—
540C2×C4—26no—
541C2×C4—22no—
542C2×C4—26no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—43no—
548C2×C4—43no—
549C2×C4—41no—
550C2×C4—43no—
551C2×C4—43no—
552C2×C4—43no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—43no—
556C2×C4—43no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—43no—
563C2×C4—41no—
564C2×C4—43no—
565C2×C4—43no—
566C2×C4—41no—
567C2×C4—43no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—43no—
571C2×C4—43no—
572C2×C4—41no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—43no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—448no—
591C2×C4—43no—
592C2×C4—43no—
593C2×C4—43no—
594C2×C4—43no—
595C2×C4—43no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4—43no—
599C2×C4—43no—
600C2×C4—43no—
601C2×C4—43no—
602C2×C4—43no—
603C2×C4—43no—
604C2×C4—43no—
605C2×C4C2×#117112yes—
606C2×C4C2×C4×D4112yes—
607C2×C4C2×C4×D4112yes—
608C2×C4C2×#121112yes—
609C2×C4C2×#123112yes—
610C2×C4C2×#124112yes—
611C2×C4C2×#126112yes—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C22×#11711yescharacteristic
628C22C22×C4×D411yescharacteristic
629C22C22×C4×D411yescharacteristic
630C22C2×#51911yescharacteristic
631C22#342811yescharacteristic
632C22C4×2−1+411yescharacteristic
633C22C4×2−1+411yescharacteristic
634C22C22×#12111yescharacteristic
635C22C22×#12311yescharacteristic
636C22C22×#12411yescharacteristic
637C22C22×#12611yescharacteristic
638C22#348611yescharacteristic
639C22#348611yescharacteristic
640C22#348711yescharacteristic
641C22#348911yescharacteristic
642C22#348911yescharacteristic
643C22#348911yescharacteristic
644C22#349711yescharacteristic
645C22#350011yescharacteristic
646C22#351611yescharacteristic
647C22#351611yescharacteristic
648C22#351611yescharacteristic
649C22#351611yescharacteristic
650C22#351611yescharacteristic
651C22#351611yescharacteristic
652C22#351611yescharacteristic
653C22#351611yescharacteristic
654C22#351811yescharacteristic
655C22#351811yescharacteristic
656C22#352311yescharacteristic
657C22#352311yescharacteristic
658C22#352311yescharacteristic
659C22#352411yescharacteristic
660C22#352411yescharacteristic
661C22#352411yescharacteristic
662C4—46no—
663C4—42no—
664C4—42no—
665C4—46no—
666C4—46no—
667C4—42no—
668C4—42no—
669C4—46no—
670C4—42no—
671C4—42no—
672C4—46no—
673C4—83no—
674C4—83no—
675C4C2×#388124yes—
676C2—44no—
677C2C22×#38811yescharacteristic
678C2#3419211yescharacteristic
679C2#3426211yescharacteristic
680C2#3429111yescharacteristic
681C2#3429111yescharacteristic
682C2#3429311yescharacteristic
683C2#3431211yescharacteristic
684C2#3572311yescharacteristic
685C2#3573711yescharacteristic
686C2#3577111yescharacteristic
687C2#3585411yescharacteristic
688C2#3588311yescharacteristic
689C2#3588311yescharacteristic
690C2#3591011yescharacteristic
691C2#3598311yescharacteristic
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
1. C2×#2351C2semidirect
5. C2×#2276C2semidirect
6. #14261C2semidirect
7. #14142C2semidirect
8. #14021C2semidirect
9. #14003C2semidirect
14. Q8×#95C2semidirect
15. Q8×#95C2semidirect
26. Q8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×Q8×C22⋊C42
C2×#22762
#142612
#141422
#140212
#140032
#139932
#139732
#138752
#138422
Q8×#952