Finite Group Databasebeta

← #7657431

Subgroups — #7657431

Order 512.

6432 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3502C211yesmaximal, characteristic, complemented
2C2×Q8×#41C211yesmaximal, characteristic
3#42814C216yesmaximal, complemented
4#42811C216yesmaximal, complemented
5#42436C2112yesmaximal, complemented
6#39724C216yesmaximal, complemented
7#39597C216yesmaximal, complemented
8#37641C213yesmaximal, complemented
9#37635C213yesmaximal, complemented
10#36577C212yesmaximal, complemented
11Q8×C22⋊Q8C212yesmaximal, complemented
12#35937C212yesmaximal, complemented
13#35910C212yesmaximal, complemented
14#35723C214yesmaximal, complemented
15#35297C213yesmaximal
16#34627C213yesmaximal
17#34196C211yesmaximal, characteristic, complemented
18#3502—216no—
19Q8×#41—216no—
20C2×Q82C2213yes—
21C2×Q82C2211yescharacteristic
22C2×Q82C2211yescharacteristic
23C2×#557C2213yes—
24C2×#557C2213yes—
25C2×#557C2216yes—
26C2×#557C2212yes—
27C2×#557C2211yescharacteristic
28C2×#554C2213yes—
29C2×#554C2216yes—
30C2×D4×Q8C2211yescharacteristic
31C2×D4×Q8C2212yes—
32C2×#544C2216yes—
33C2×#533C2213yes—
34C2×#533C2213yes—
35C2×#533C2213yes—
36C2×#533C2216yes—
37C2×#519C2212yes—
38C2×C4×#41C2213yes—
39C22×C4×Q8C2212yes—
40C22×C4×Q8C2211yescharacteristic
41#2857C2216yes—
42#2857C2216yes—
43#2822C22112yes—
44#2817C22112yes—
45#2719C22124yes—
46#2713C22124yes—
47#2690C22124yes—
48#2677C22124yes—
49#2660C2216yes—
50#2642C2216yes—
51#2638C22112yes—
52#2634C22112yes—
53#2634C2216yes—
54#2633C2216yes—
55#2628C2216yes—
56#2625C2213yes—
57#2625C2216yes—
58#2625C2213yes—
59#2625C22112yes—
60#2621C22112yes—
61#2621C2216yes—
62#2616C2211yescharacteristic
63#2616C2213yes—
64#2584C2216yes—
65#2582C2216yes—
66#2582C2216yes—
67#2581C22112yes—
68#2581C2216yes—
69#2568C2216yes—
70#2568C2216yes—
71#2568C2216yes—
72#2567C2216yes—
73#2562C2216yes—
74#2551C22112yes—
75#2550C22112yes—
76#2546C2216yes—
77#2546C2216yes—
78#2536C2216yes—
79#2499C22112yes—
80#2492C22112yes—
81#2489C22112yes—
82#2485C22112yes—
83#2453C22112yes—
84#2451C22112yes—
85#2446C22112yes—
86#2446C22112yes—
87#2442C22112yes—
88#2440C22112yes—
89#2423C22112yes—
90#2421C22112yes—
91#2415C22112yes—
92#2399C2216yes—
93#2398C2216yes—
94#2395C22112yes—
95#2355C2216yes—
96#2351C2214yes—
97Q8×C4⋊C4C2214yes—
98Q8×C4⋊C4C2212yes—
99#2344C2216yes—
100Q8×C22⋊C4C2214yes—
101Q8×C22⋊C4C2214yes—
102Q8×C22⋊C4C2212yes—
103#2323C22112yes—
104#2320C2216yes—
105C4×C22⋊Q8C2216yes—
106C4×C22⋊Q8C2216yes—
107#2285C2213yes—
108#2283C2213yes—
109#2276C2212yes—
110C22×#41C2311yescharacteristic
111C23×Q8C2312yes—
112C23×Q8C2311yescharacteristic
113Q82—28no—
114Q82—28no—
115Q82—224no—
116#557—224no—
117#557—28no—
118#557—224no—
119#557—216no—
120#557—248no—
121#554—224no—
122#554—248no—
123D4×Q8—216no—
124D4×Q8—28no—
125#544—248no—
126#533—248no—
127#533—224no—
128#533—224no—
129#533—224no—
130C2×C4⋊Q8C2313yes—
131C2×C4⋊Q8C2313yes—
132C2×C4⋊Q8C2313yes—
133C2×C4⋊Q8C2313yes—
134C2×C4⋊Q8C2313yes—
135C2×C4⋊Q8C2313yes—
136C2×C4⋊Q8C2316yes—
137C2×C4⋊Q8C2316yes—
138C2×C4⋊Q8C23112yes—
139C2×C4⋊Q8C2316yes—
140C2×C4⋊Q8C2316yes—
141C2×C4⋊Q8C23112yes—
142C2×#125C2316yes—
143C2×#125C2316yes—
144C2×#125C2316yes—
145C2×#125C23112yes—
146C2×#125C23112yes—
147C2×#125C2316yes—
148C2×#125C2316yes—
149C2×C22⋊Q8C2316yes—
150C2×C22⋊Q8C2316yes—
151C2×C22⋊Q8C2316yes—
152C2×C22⋊Q8C2312yes—
153C2×C22⋊Q8C2311yescharacteristic
154C2×C22⋊Q8C2313yes—
155C2×C22⋊Q8C2316yes—
156C2×C22⋊Q8C2316yes—
157C2×C22⋊Q8C2316yes—
158C2×C22⋊Q8C23112yes—
159C2×C22⋊Q8C2316yes—
160C2×C22⋊Q8C23112yes—
161#519—216no—
162C4×#41—224no—
163C2×C4×Q8—216no—
164C2×C4×Q8D4116yes—
165C2×C4×Q8C2316yes—
166C2×C4×Q8C2316yes—
167C2×C4×Q8C2316yes—
168C2×C4×Q8C2316yes—
169C2×C4×Q8C2313yes—
170C2×C4×Q8C2316yes—
171C2×C4×Q8C2313yes—
172C2×C4×Q8C2313yes—
173C2×C4×Q8C2313yes—
174C2×C4×Q8C2314yes—
175C2×C4×Q8C2312yes—
176C2×C4×Q8C2312yes—
177C2×C4×Q8C2312yes—
178C2×C4×Q8C2316yes—
179C2×C4×Q8C2316yes—
180C2×C4×Q8C2316yes—
181C2×C4×Q8C2313yes—
182C2×C4×Q8C2311yescharacteristic
183C2×C4×Q8C2313yes—
184C2×C4×Q8C2311yescharacteristic
185C2×C4×D4C2313yes—
186C2×C4×D4C2313yes—
187C2×C4×D4C2316yes—
188C2×C4×D4C2316yes—
189C2×#117C2316yes—
190C2×#117C2313yes—
191C2×#117C2316yes—
192C2×#117C2316yes—
193C2×#117C2316yes—
194C22×C4⋊C4C2316yes—
195C22×C4⋊C4C2313yes—
196C22×C42C2316yes—
197C22×C42C2313yes—
198#400C23124yes—
199#400C23124yes—
200#400C23112yes—
201#400C2316yes—
202#400C23124yes—
203#400C2316yes—
204#400C23112yes—
205#399C23112yes—
206#398C2316yes—
207#398C23124yes—
208#398C23124yes—
209#398C23112yes—
210#398C23124yes—
211#398C23112yes—
212#398C23112yes—
213#398C2316yes—
214#397C23112yes—
215#397C23124yes—
216#397C23112yes—
217#397C23112yes—
218#397C23112yes—
219#395C23124yes—
220#395C23124yes—
221#395C2316yes—
222#395C2312yes—
223#395C23112yes—
224#394C2316yes—
225#394C2312yes—
226#394C2314yes—
227#394C23112yes—
228#393C23112yes—
229#391C23112yes—
230#391C23112yes—
231#391C2316yes—
232#391C2316yes—
233#391C23112yes—
234#391C23112yes—
235#391C23112yes—
236#391C23112yes—
237#391C23112yes—
238#391C23112yes—
239#391C23112yes—
240#389C2316yes—
241#389C23112yes—
242#389C23112yes—
243#389C23124yes—
244#389C23112yes—
245#389C2316yes—
246#389C23112yes—
247#389C23112yes—
248#389C23112yes—
249#389C23112yes—
250#388C23124yes—
251#388C23124yes—
252#387C23124yes—
253#387C23124yes—
254#387C23124yes—
255#387C23124yes—
256#387C23112yes—
257#387C23112yes—
258#385C23112yes—
259#385C23112yes—
260#384C23112yes—
261#384C2316yes—
262#383C2316yes—
263#382C2316yes—
264#382C23112yes—
265#382C23112yes—
266#382C2316yes—
267#381C2316yes—
268#380C23112yes—
269#380C23112yes—
270#380C2316yes—
271#380C2316yes—
272#380C23112yes—
273#380C2311yescharacteristic
274#380C2313yes—
275#380C2312yes—
276C4×C4⋊C4C2316yes—
277C4×C4⋊C4C2316yes—
278C4×C4⋊C4C2316yes—
279C4×C4⋊C4C23112yes—
280C4×C4⋊C4C23112yes—
281C4×C4⋊C4C23112yes—
282C4×C22⋊C4C2316yes—
283C4×C22⋊C4C23112yes—
284C4×C22⋊C4C23112yes—
285#376C2316yes—
286C2×#41—24no—
287C2×#41—28no—
288C22×Q8—28no—
289C22×Q8—28no—
290C22×Q8—28no—
291C22×Q8—28no—
292C22×Q8C2×D418yes—
293C22×Q8C2411yescharacteristic
294C22×Q8C2413yes—
295C22×Q8C2411yescharacteristic
296C22×Q8C2416yes—
297C22×Q8C2413yes—
298C22×Q8C2416yes—
299C22×Q8C2416yes—
300C22×Q8C2416yes—
301C22×Q8C2411yescharacteristic
302C22×D4C2411yescharacteristic
303C22×D4C2412yes—
304C23×C4C2416yes—
305C23×C4C2413yes—
306C23×C4C2411yescharacteristic
307C23×C4C2412yes—
308C23×C4C2416yes—
309C23×C4C2413yes—
310C4⋊Q8—224no—
311C4⋊Q8—224no—
312C4⋊Q8—224no—
313C4⋊Q8—248no—
314C4⋊Q8—224no—
315C4⋊Q8—212no—
316C4⋊Q8—212no—
317C4⋊Q8—212no—
318C4⋊Q8—212no—
319C4⋊Q8—212no—
320C4⋊Q8—212no—
321C4⋊Q8—812no—
322#125—248no—
323#125—224no—
324#125—248no—
325#125—224no—
326#125—224no—
327#125—224no—
328#125—224no—
329C22⋊Q8—224no—
330C22⋊Q8—24no—
331C22⋊Q8—212no—
332C22⋊Q8—224no—
333C22⋊Q8—224no—
334C22⋊Q8—224no—
335C22⋊Q8—224no—
336C22⋊Q8—224no—
337C22⋊Q8—28no—
338C22⋊Q8—224no—
339C22⋊Q8—248no—
340C22⋊Q8—812no—
341C4×Q8—28no—
342C4×Q8—224no—
343C4×Q8—28no—
344C4×Q8—224no—
345C4×Q8—28no—
346C4×Q8—216no—
347C4×Q8—224no—
348C4×Q8—224no—
349C4×Q8—212no—
350C4×Q8—212no—
351C4×Q8—224no—
352C4×Q8—224no—
353C4×Q8—224no—
354C4×Q8—212no—
355C4×Q8—212no—
356C4×Q8—224no—
357C4×Q8C2×D418yes—
358C4×Q8C2×D4124yes—
359C4×Q8C2×D418yes—
360C4×Q8C2×D4124yes—
361C4×D4—224no—
362C4×D4—224no—
363C4×D4—212no—
364C4×D4—212no—
365#117—212no—
366#117—224no—
367#117—224no—
368#117—224no—
369#117—86no—
370C2×C4⋊C4—224no—
371C2×C4⋊C4C2×D4124yes—
372C2×C4⋊C4C24112yes—
373C2×C4⋊C4C2416yes—
374C2×C4⋊C4C2416yes—
375C2×C4⋊C4C2416yes—
376C2×C4⋊C4C2413yes—
377C2×C4⋊C4C2413yes—
378C2×C4⋊C4C2416yes—
379C2×C4⋊C4C2416yes—
380C2×C4⋊C4C2416yes—
381C2×C4⋊C4C2416yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2416yes—
384C2×C4⋊C4C2413yes—
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2416yes—
387C2×C4⋊C4C24112yes—
388C2×C4⋊C4C2416yes—
389C2×C4⋊C4C2416yes—
390C2×C4⋊C4C2416yes—
391C2×C4⋊C4C2416yes—
392C2×C4⋊C4C24112yes—
393C2×C4⋊C4C24112yes—
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2416yes—
396C2×C4⋊C4C2414yes—
397C2×C4⋊C4C24112yes—
398C2×C4⋊C4C24112yes—
399C2×C4⋊C4C24124yes—
400C2×C4⋊C4C24112yes—
401C2×C4⋊C4C24112yes—
402C2×C4⋊C4C2413yes—
403C2×C4⋊C4C2416yes—
404C2×C4⋊C4C2413yes—
405C2×C22⋊C4C2412yes—
406C2×C22⋊C4C2416yes—
407C2×C22⋊C4C2416yes—
408C2×C22⋊C4C2416yes—
409C2×C22⋊C4C2416yes—
410C2×C22⋊C4C2414yes—
411C2×C22⋊C4C24112yes—
412C2×C22⋊C4C2416yes—
413C2×C22⋊C4C24112yes—
414C2×C22⋊C4C2412yes—
415C2×C22⋊C4C2414yes—
416C2×C42—224no—
417C2×C42C2×D4124yes—
418C2×C42C2416yes—
419C2×C42C2416yes—
420C2×C42C2416yes—
421C2×C42C24112yes—
422C2×C42C2416yes—
423C2×C42C2416yes—
424C2×C42C24112yes—
425C2×C42C24112yes—
426C2×C42C2413yes—
427C2×C42C2413yes—
428C2×C42C2413yes—
429C2×C42C2413yes—
430#95C2412yes—
431#95C2416yes—
432#95C24112yes—
433#95C24112yes—
434#95C24112yes—
435#95C24112yes—
436#95C24112yes—
437#95C24112yes—
438#95C2414yes—
439#95C24112yes—
440#95C24112yes—
441#95C24112yes—
442#95C24112yes—
443#95C24112yes—
444#95C24124yes—
445#95C24124yes—
446#95C24124yes—
447C24C2511yescharacteristic
448C24C2512yes—
449#41—232no—
450C2×Q8—232no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—26no—
454C2×Q8—212no—
455C2×Q8—212no—
456C2×Q8—212no—
457C2×Q8—212no—
458C2×Q8—212no—
459C2×Q8—24no—
460C2×Q8—212no—
461C2×Q8—24no—
462C2×Q8—212no—
463C2×Q8—212no—
464C2×Q8—46no—
465C2×Q8—46no—
466C2×Q8—46no—
467C2×Q8—46no—
468C2×Q8—46no—
469C2×Q8—46no—
470C2×Q8—432no—
471C2×Q8C22×D4112yes—
472C2×Q8C22×D414yes—
473C2×Q82−1+418yes—
474C2×D4—24no—
475C2×D4—24no—
476C2×D4—24no—
477C2×D4—22no—
478C2×D4—24no—
479C22×C4—24no—
480C22×C4—212no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—212no—
484C22×C4—212no—
485C22×C4—212no—
486C22×C4—212no—
487C22×C4—43no—
488C22×C4—43no—
489C22×C4—46no—
490C22×C4—46no—
491C22×C4C22×D414yes—
492C22×C4C22×D4112yes—
493C22×C4C22×D4112yes—
494C22×C42−1+418yes—
495C22×C4C2514yes—
496C22×C4C25112yes—
497C22×C4C2516yes—
498C22×C4C2512yes—
499C22×C4C2516yes—
500C22×C4C2516yes—
501C22×C4C2512yes—
502C22×C4C2516yes—
503C22×C4C2513yes—
504C22×C4C2511yescharacteristic
505C22×C4C2513yes—
506C22×C4C2513yes—
507C22×C4C2512yes—
508C22×C4C2513yes—
509C22×C4C2511yescharacteristic
510C4⋊C4—224no—
511C4⋊C4—212no—
512C4⋊C4—212no—
513C4⋊C4—212no—
514C4⋊C4—26no—
515C4⋊C4—26no—
516C4⋊C4—212no—
517C4⋊C4—212no—
518C4⋊C4—212no—
519C4⋊C4—212no—
520C4⋊C4—24no—
521C4⋊C4—212no—
522C4⋊C4—26no—
523C4⋊C4—22no—
524C4⋊C4—224no—
525C4⋊C4—212no—
526C4⋊C4—212no—
527C4⋊C4—212no—
528C4⋊C4—212no—
529C4⋊C4—212no—
530C4⋊C4—224no—
531C4⋊C4—224no—
532C4⋊C4—24no—
533C4⋊C4—212no—
534C4⋊C4—28no—
535C4⋊C4—224no—
536C4⋊C4—412no—
537C4⋊C4—412no—
538C4⋊C4—424no—
539C4⋊C4—412no—
540C4⋊C4C22×D4112yes—
541C4⋊C4C22×D4124yes—
542C4⋊C4C22×D4112yes—
543C22⋊C4—24no—
544C22⋊C4—212no—
545C22⋊C4—212no—
546C22⋊C4—28no—
547C22⋊C4—224no—
548C22⋊C4—212no—
549C22⋊C4—212no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—46no—
553C22⋊C4—412no—
554C42—212no—
555C42—212no—
556C42—212no—
557C42—224no—
558C42—212no—
559C42—212no—
560C42—224no—
561C42—412no—
562C42C22×D4112yes—
563C42C22×D4112yes—
564C42C22×D4112yes—
565C42C22×D4112yes—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—21no—
573C23—22no—
574C23—21no—
575C23—22no—
576C23C23×D412yes—
577C23C2×2−1+414yes—
578C23C2611yescharacteristic
579Q8—28no—
580Q8—224no—
581Q8—424no—
582Q8—44no—
583Q8—412no—
584Q8—812no—
585Q8—812no—
586Q8—812no—
587Q8#536116yes—
588D4—28no—
589D4—216no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—26no—
594C2×C4—23no—
595C2×C4—23no—
596C2×C4—23no—
597C2×C4—23no—
598C2×C4—224no—
599C2×C4—28no—
600C2×C4—216no—
601C2×C4—26no—
602C2×C4—22no—
603C2×C4—26no—
604C2×C4—22no—
605C2×C4—26no—
606C2×C4—26no—
607C2×C4—24no—
608C2×C4—212no—
609C2×C4—212no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—212no—
613C2×C4—26no—
614C2×C4—26no—
615C2×C4—212no—
616C2×C4—26no—
617C2×C4—26no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—26no—
622C2×C4—26no—
623C2×C4—412no—
624C2×C4—424no—
625C2×C4—812no—
626C2×C4C23×D416yes—
627C2×C4C23×D416yes—
628C2×C4C23×D412yes—
629C2×C4C23×D416yes—
630C2×C4C23×D416yes—
631C2×C4C23×D412yes—
632C2×C4C2×2−1+414yes—
633C2×C4C2×2−1+4112yes—
634C2×C4C2×2−1+414yes—
635C22—22no—
636C22—24no—
637C22—22no—
638C22—22no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22C24×D411yescharacteristic
648C22C22×2−1+412yes—
649C22C22×2−1+412yes—
650C22H(7,𝔽2)11yescharacteristic
651C222−1+611yescharacteristic
652C4—22no—
653C4—24no—
654C4—26no—
655C4—26no—
656C4—26no—
657C4—46no—
658C4—42no—
659C4—46no—
660C4—46no—
661C4—42no—
662C4—46no—
663C4—412no—
664C4—44no—
665C4C2×#536112yes—
666C4#348614yes—
667C2—42no—
668C2—44no—
669C2C22×#53611yescharacteristic
670C2C2×#348611yescharacteristic
671C2#6285311yescharacteristic
672C2#6294312yes—
673C2#6294712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radicalC22×#41C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3502C2semidirect
3. #42814C2semidirect
4. #42811C2semidirect
5. #42436C2semidirect
6. #39724C2semidirect
7. #39597C2semidirect
8. #37641C2semidirect
9. #37635C2semidirect
10. #36577C2semidirect
11. Q8×C22⋊Q8C2semidirect
12. #35937C2semidirect
13. #35910C2semidirect
14. #35723C2semidirect
17. #34196C2semidirect

Maximal subgroups

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

subgroupindex
C2×#35022
C2×Q8×#412
#428142
#428112
#424362
#397242
#395972
#376412
#376352
#365772
Q8×C22⋊Q82
#359372
#359102
#357232
#352972
#346272
#341962