Finite Group Databasebeta

← #518324

Subgroups — #518324

Order 512.

3916 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2341C211yesmaximal, characteristic
3C2×#2341C211yesmaximal, characteristic
4C2×#2341C211yesmaximal, characteristic
5C2×#2276C211yesmaximal, characteristic, complemented
6#14260C213yesmaximal, complemented
7#14141C213yesmaximal, complemented
8#14022C213yesmaximal, complemented
9#14001C213yesmaximal, complemented
10#13993C213yesmaximal, complemented
11#13974C213yesmaximal, complemented
12#13876C213yesmaximal, complemented
13#13842C213yesmaximal, complemented
14Q8×#95C211yesmaximal, characteristic, complemented
15Q8×#95C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23Q8×C4⋊C4—28no—
24#2341—28no—
25#2341—28no—
26#2341C4116yescomplemented
27C2×#390C2213yes—
28C2×#390C2213yes—
29C2×#390C2213yes—
30C2×#390C2213yes—
31C2×#390C2213yes—
32C2×#390C2213yes—
33C2×#389C2213yes—
34C2×#389C2213yes—
35C2×#388C2216yes—
36C2×#388C2216yes—
37C2×#384C2213yes—
38C2×#380C2213yes—
39C2×#380C2213yes—
40C2×#380C2213yes—
41#2276—28no—
42C2×C4×C4⋊C4C2213yes—
43C2×C4×C4⋊C4C2213yes—
44C2×C4×C4⋊C4C2213yes—
45C2×C4×C4⋊C4C2213yes—
46C2×C4×C22⋊C4C2213yes—
47C2×C4×C22⋊C4C2213yes—
48C2×C4×C22⋊C4C2213yes—
49C2×#376C2213yes—
50#1443C2213yes—
51#1443C2213yes—
52#1442C2211yescharacteristic
53#1442C2213yes—
54#1440C2213yes—
55#1440C2213yes—
56#1440C2216yes—
57#1439C2213yes—
58#1439C2213yes—
59#1439C2216yes—
60#1439C2216yes—
61#1437C2216yes—
62#1436C2216yes—
63C4⋊#95C2213yes—
64C4⋊#95C2213yes—
65C4⋊#95C2213yes—
66C4⋊#95C2213yes—
67C4⋊#95C2213yes—
68C4⋊#95C2213yes—
69C4×#95C2213yes—
70C4×#95C2213yes—
71C22×#41—24no—
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—24no—
75C22×#41—24no—
76C22×#41—24no—
77C22×#41C2×C418yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2313yes—
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⋊C4C2311yescharacteristic
102C22×C4⋊C4C2313yes—
103C22×C4⋊C4C2313yes—
104C22×C4⋊C4C2313yes—
105C22×C4⋊C4C2313yes—
106C22×C4⋊C4C2313yes—
107C22×C4⋊C4C2316yes—
108C22×C22⋊C4C2313yes—
109C22×C22⋊C4C2313yes—
110C22×C22⋊C4C2313yes—
111C22×C22⋊C4C2316yes—
112C22×C22⋊C4C2313yes—
113C22×C22⋊C4C2313yes—
114C22×C22⋊C4C2316yes—
115C22×C22⋊C4C2316yes—
116C22×C22⋊C4C2313yes—
117C22×C42C2313yes—
118C22×C42C2313yes—
119C22×C42C2313yes—
120C22×C42C2313yes—
121C22×C42C2313yes—
122C22×C42C2313yes—
123C22×C42C2316yes—
124#390—212no—
125#390—212no—
126#390—212no—
127#390—212no—
128#390C2×C4124yes—
129#390C2×C4124yes—
130#389—212no—
131#389—212no—
132#388—412no—
133#388—412no—
134#384—212no—
135#380—212no—
136#380—212no—
137#380C2×C4124yes—
138C4×C4⋊C4—212no—
139C4×C4⋊C4—212no—
140C4×C4⋊C4—212no—
141C4×C4⋊C4—212no—
142C4×C22⋊C4—212no—
143C4×C22⋊C4—212no—
144C4×C22⋊C4C2×C4124yes—
145#376—212no—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2313yes—
149C2×#95C2313yes—
150C2×#95C2313yes—
151C2×#95C2313yes—
152C2×#95C2313yes—
153C2×#95C2313yes—
154C2×#95C2313yes—
155C2×#95C2313yes—
156C2×#95C2313yes—
157C2×#95C2311yescharacteristic
158C2×#95C2313yes—
159C2×#95C2311yescharacteristic
160C2×#95C2311yescharacteristic
161C2×#95C2313yes—
162C2×#95C2313yes—
163C2×#95C2313yes—
164C2×#95C2316yes—
165C2×#95C2316yes—
166C2×#95C2313yes—
167C2×#95C2311yescharacteristic
168C25C2413yes—
169C2×#41—232no—
170C2×#41—232no—
171C2×#41—232no—
172C2×#41—416no—
173C2×#41—416no—
174C2×#41—416no—
175C2×#41—416no—
176C22×Q8C22×C414yes—
177C22×Q8C2×D414yes—
178C22×Q8C2×D414yes—
179C22×Q8#4114yes—
180C22×Q8#4114yes—
181C22×Q8#4114yes—
182C22×Q8#4114yes—
183C22×D4—26no—
184C22×D4—26no—
185C22×D4—26no—
186C22×D4—26no—
187C22×D4—26no—
188C22×D4—26no—
189C22×D4C22×C4112yes—
190C23×C4—26no—
191C23×C4—26no—
192C23×C4—26no—
193C23×C4—26no—
194C23×C4—26no—
195C23×C4—26no—
196C23×C4C22×C4112yes—
197C23×C4C2413yes—
198C23×C4C2413yes—
199C23×C4C2413yes—
200C23×C4C2413yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2413yes—
204C23×C4C2413yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2413yes—
211C4×Q8—232no—
212C4×Q8—232no—
213C4×Q8—232no—
214C4×Q8—232no—
215C4×Q8—416no—
216C4×Q8—416no—
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—26no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—26no—
222C2×C4⋊C4—26no—
223C2×C4⋊C4—26no—
224C2×C4⋊C4—26no—
225C2×C4⋊C4—26no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—26no—
229C2×C4⋊C4—26no—
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—212no—
237C2×C4⋊C4—43no—
238C2×C4⋊C4—46no—
239C2×C4⋊C4—46no—
240C2×C4⋊C4C22×C4112yes—
241C2×C4⋊C4C22×C4112yes—
242C2×C4⋊C4C2×D4112yes—
243C2×C4⋊C4C2×D4112yes—
244C2×C22⋊C4—26no—
245C2×C22⋊C4—26no—
246C2×C22⋊C4—26no—
247C2×C22⋊C4—26no—
248C2×C22⋊C4—26no—
249C2×C22⋊C4—26no—
250C2×C22⋊C4—26no—
251C2×C22⋊C4—26no—
252C2×C22⋊C4—212no—
253C2×C22⋊C4—212no—
254C2×C22⋊C4—43no—
255C2×C22⋊C4—46no—
256C2×C22⋊C4—43no—
257C2×C22⋊C4—43no—
258C2×C22⋊C4—43no—
259C2×C22⋊C4—43no—
260C2×C22⋊C4—43no—
261C2×C22⋊C4—43no—
262C2×C22⋊C4—43no—
263C2×C22⋊C4—46no—
264C2×C22⋊C4—46no—
265C2×C22⋊C4—46no—
266C2×C22⋊C4—46no—
267C2×C22⋊C4—46no—
268C2×C22⋊C4C22×C4112yes—
269C2×C22⋊C4C22×C4112yes—
270C2×C22⋊C4C22×C4124yes—
271C2×C42—26no—
272C2×C42—26no—
273C2×C42—26no—
274C2×C42—26no—
275C2×C42—26no—
276C2×C42—26no—
277C2×C42—26no—
278C2×C42—26no—
279C2×C42—26no—
280C2×C42—26no—
281C2×C42—26no—
282C2×C42—26no—
283C2×C42—26no—
284C2×C42—26no—
285C2×C42—212no—
286C2×C42—46no—
287C2×C42—46no—
288C2×C42C22×C4112yes—
289C2×C42C22×C4112yes—
290C2×C42C2×D4112yes—
291C2×C42C2×D4112yes—
292#95—22no—
293#95—26no—
294#95—26no—
295#95—26no—
296#95—22no—
297#95—22no—
298#95—26no—
299#95—26no—
300#95—26no—
301#95—26no—
302#95—26no—
303#95—26no—
304#95—26no—
305#95—22no—
306#95—22no—
307#95—43no—
308#95—43no—
309#95—43no—
310#95—46no—
311#95—46no—
312#95C22×C414yes—
313#95C22×C4112yes—
314C24—23no—
315C24—23no—
316C24—23no—
317C24—23no—
318C24—23no—
319C24—23no—
320C24—23no—
321C24—23no—
322C24—23no—
323C24—23no—
324C24—23no—
325C24—23no—
326C24—23no—
327C24—23no—
328C24C23×C416yes—
329C24C2511yescharacteristic
330#41—4128no—
331C2×Q8#117116yes—
332C2×Q8C4×D4116yes—
333C2×Q8C4×D4116yes—
334C2×Q8#121116yes—
335C2×Q8#123116yes—
336C2×Q8#124116yes—
337C2×Q8#126116yes—
338C2×D4—224no—
339C2×D4—224no—
340C2×D4—224no—
341C2×D4—412no—
342C2×D4—412no—
343C2×D4—412no—
344C2×D4—412no—
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—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—23no—
367C22×C4—23no—
368C22×C4—23no—
369C22×C4—23no—
370C22×C4—23no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—23no—
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—23no—
387C22×C4—23no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—23no—
395C22×C4—23no—
396C22×C4—23no—
397C22×C4—23no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—23no—
407C22×C4—23no—
408C22×C4—23no—
409C22×C4—224no—
410C22×C4—224no—
411C22×C4—224no—
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—412no—
427C22×C4—412no—
428C22×C4—412no—
429C22×C4—412no—
430C22×C4C23×C412yes—
431C22×C4C23×C412yes—
432C22×C4C23×C416yes—
433C22×C4C23×C416yes—
434C22×C4C23×C412yes—
435C22×C4C23×C416yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D416yes—
439C22×C4C22×D416yes—
440C22×C4C22×D416yes—
441C22×C4C22×D416yes—
442C22×C4C2×#4116yes—
443C22×C4C2×#4116yes—
444C22×C4C2×#4116yes—
445C22×C4C2×#4116yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+412yes—
449C22×C42−1+412yes—
450C4⋊C4—28no—
451C4⋊C4—224no—
452C4⋊C4—224no—
453C4⋊C4—224no—
454C4⋊C4—224no—
455C4⋊C4—412no—
456C4⋊C4—412no—
457C4⋊C4—412no—
458C4⋊C4—812no—
459C22⋊C4—224no—
460C22⋊C4—412no—
461C22⋊C4—424no—
462C22⋊C4—412no—
463C22⋊C4—412no—
464C22⋊C4—412no—
465C22⋊C4—412no—
466C22⋊C4—812no—
467C22⋊C4—812no—
468C42—224no—
469C42—224no—
470C42—224no—
471C42—224no—
472C42—412no—
473C42—412no—
474C42—812no—
475C23—26no—
476C23—26no—
477C23—26no—
478C23—26no—
479C23—26no—
480C23—26no—
481C23—26no—
482C23—43no—
483C23—43no—
484C23—43no—
485C23—43no—
486C23—43no—
487C23—43no—
488C23—43no—
489C23—43no—
490C23—43no—
491C23—43no—
492C23—43no—
493C23—43no—
494C23—43no—
495C23—43no—
496C23—43no—
497C23—43no—
498C23—43no—
499C23—43no—
500C23—43no—
501C23—43no—
502C23—43no—
503C23—43no—
504C23—43no—
505C23—43no—
506C23—43no—
507C23—43no—
508C23—43no—
509C23—43no—
510C23C24×C411yescharacteristic
511C23C23×D411yescharacteristic
512C23C23×D411yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23#58511yescharacteristic
522C23#58511yescharacteristic
523C23#58511yescharacteristic
524C23#58511yescharacteristic
525Q8#388164yescomplemented
526D4—448no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—26no—
536C2×C4—26no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—26no—
540C2×C4—22no—
541C2×C4—26no—
542C2×C4—26no—
543C2×C4—26no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—26no—
547C2×C4—22no—
548C2×C4—26no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—26no—
555C2×C4—26no—
556C2×C4—22no—
557C2×C4—26no—
558C2×C4—22no—
559C2×C4—26no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—43no—
572C2×C4—41no—
573C2×C4—43no—
574C2×C4—43no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—43no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—43no—
581C2×C4—43no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—43no—
585C2×C4—43no—
586C2×C4—43no—
587C2×C4—43no—
588C2×C4—43no—
589C2×C4—43no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—43no—
597C2×C4—43no—
598C2×C4—43no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—43no—
603C2×C4—448no—
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×C4C2×#117112yes—
619C2×C4C2×C4×D4112yes—
620C2×C4C2×C4×D4112yes—
621C2×C4#52014yes—
622C2×C4C2×#121112yes—
623C2×C4C2×#123112yes—
624C2×C4C2×#124112yes—
625C2×C4C2×#126112yes—
626C22—46no—
627C22—46no—
628C22—46no—
629C22—46no—
630C22—46no—
631C22—46no—
632C22—46no—
633C22—83no—
634C22—83no—
635C22—83no—
636C22—83no—
637C22—83no—
638C22—83no—
639C22—83no—
640C22—83no—
641C22C22×#11711yescharacteristic
642C22C22×C4×D411yescharacteristic
643C22C22×C4×D411yescharacteristic
644C22C2×#52011yescharacteristic
645C22#342811yescharacteristic
646C22C4×H(5,𝔽2)11yescharacteristic
647C22C4×2−1+411yescharacteristic
648C22C22×#12111yescharacteristic
649C22C22×#12311yescharacteristic
650C22C22×#12411yescharacteristic
651C22C22×#12611yescharacteristic
652C22#348611yescharacteristic
653C22#348711yescharacteristic
654C22#348711yescharacteristic
655C22#348811yescharacteristic
656C22#348811yescharacteristic
657C22#348911yescharacteristic
658C22#349711yescharacteristic
659C22#349811yescharacteristic
660C22#351611yescharacteristic
661C22#351611yescharacteristic
662C22#351711yescharacteristic
663C22#351711yescharacteristic
664C22#351711yescharacteristic
665C22#351711yescharacteristic
666C22#351711yescharacteristic
667C22#351811yescharacteristic
668C22#351811yescharacteristic
669C22#351811yescharacteristic
670C22#352211yescharacteristic
671C22#352211yescharacteristic
672C22#352211yescharacteristic
673C22#352411yescharacteristic
674C22#352411yescharacteristic
675C22#352411yescharacteristic
676C4—24no—
677C4—46no—
678C4—42no—
679C4—46no—
680C4—42no—
681C4—42no—
682C4—46no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—46no—
687C4—83no—
688C4—83no—
689C4C2×#388124yes—
690C2—86no—
691C2C22×#38811yescharacteristic
692C2#3419311yescharacteristic
693C2#3426311yescharacteristic
694C2#3429311yescharacteristic
695C2#3429411yescharacteristic
696C2#3429511yescharacteristic
697C2#3431311yescharacteristic
698C2#3577111yescharacteristic
699C2#3580111yescharacteristic
700C2#3589811yescharacteristic
701C2#3589811yescharacteristic
702C2#3594811yescharacteristic
703C2#3600511yescharacteristic
704C2#3602911yescharacteristic
705C2#3604911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
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×Q8×C4⋊C4C2semidirect
5. C2×#2276C2semidirect
6. #14260C2semidirect
7. #14141C2semidirect
8. #14022C2semidirect
9. #14001C2semidirect
10. #13993C2semidirect
11. #13974C2semidirect
12. #13876C2semidirect
13. #13842C2semidirect
14. Q8×#95C2semidirect
15. Q8×#95C2semidirect
26. #2341C4semidirect
525. Q8#388semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#23412
C2×#22762
#142602
#141412
#140222
#140012
#139932
#139742
#138762
#138422
Q8×#952