Finite Group Databasebeta

← #7646455

Subgroups — #7646455

Order 512.

7336 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#559C211yesmaximal, characteristic, complemented
2C2×#3461C211yesmaximal, characteristic, complemented
3#42881C218yesmaximal, complemented
4#42877C218yesmaximal, complemented
5#39726C218yesmaximal, complemented
6#39137C218yesmaximal, complemented
7#37546C214yesmaximal, complemented
8#36722C218yesmaximal, complemented
9#35677C214yesmaximal, complemented
10#35272C212yesmaximal, complemented
11#35236C214yesmaximal, complemented
12#35174C212yesmaximal, complemented
13#35168C212yesmaximal, complemented
14#34952C212yesmaximal, complemented
15#34182C211yesmaximal, characteristic, complemented
16C2×#559—216no—
17C2×#559C2218yes—
18C2×#559C2218yes—
19#3461—216no—
20C2×#544C2211yescharacteristic
21C2×#542C2214yescomplemented
22C2×#541C2214yescomplemented
23C2×#539C2214yes—
24C2×#537C2211yescharacteristic
25C2×#536C2212yes—
26C2×#535C2214yes—
27C2×#533C2214yescomplemented
28C22×#127C2211yescharacteristic
29C2×#529C2214yes—
30C2×#529C2214yescomplemented
31C22×#124C2212yes—
32C22×#124C2212yes—
33C22×#123C2214yes—
34C2×#519C2212yes—
35C2×C4×#41C2212yes—
36C22×C4×Q8C2211yescharacteristic
37C22×C4×Q8C2211yescharacteristic
38C22×C4×D4C2212yes—
39C22×C4×D4C2212yes—
40#2856C2218yes—
41#2848C2218yes—
42#2847C2218yes—
43#2820C2218yescomplemented
44#2819C2218yescomplemented
45#2707C22132yescomplemented
46#2705C22132yescomplemented
47#2682C22132yescomplemented
48#2681C22132yescomplemented
49#2662C22116yescomplemented
50#2636C22116yescomplemented
51#2626C2218yes—
52#2620C22116yescomplemented
53#2619C2218yes—
54#2617C2218yes—
55#2592C2218yescomplemented
56#2582C2214yescomplemented
57#2581C2214yescomplemented
58#2580C2214yescomplemented
59#2579C2218yes—
60#2562C2218yescomplemented
61#2548C2214yes—
62#2548C2214yes—
63#2548C22116yescomplemented
64#2545C22116yescomplemented
65#2541C2214yes—
66#2541C2214yes—
67#2540C2218yescomplemented
68#2540C2218yescomplemented
69#2538C2218yescomplemented
70#2538C2214yescomplemented
71#2537C2218yescomplemented
72#2487C2214yes—
73#2454C22116yescomplemented
74#2452C22116yescomplemented
75#2428C22116yescomplemented
76#2427C22116yescomplemented
77#2426C22116yescomplemented
78#2426C22116yescomplemented
79#2411C22116yes—
80#2411C22116yes—
81#2409C2214yes—
82#2408C2218yes—
83#2408C2218yes—
84#2396C2218yescomplemented
85#2351C2218yescomplemented
86Q8×C4⋊C4C2218yescomplemented
87#2324C22116yescomplemented
88#2316C2212yes—
89#2316C2214yes—
90#2309C2214yes—
91C4×C22⋊Q8C2218yescomplemented
92#2297C2214yes—
93#2296C2214yes—
94#2285C2212yes—
95#2285C2214yes—
96C22×#41C2312yes—
97C23×Q8C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C23×D4C2312yes—
100C23×D4C2312yes—
101C23×D4C2312yes—
102C24×C4C2314yes—
103#559—264no—
104#559—264no—
105#544—28no—
106#542—232no—
107#541—232no—
108#539—232no—
109#537—28no—
110#536—216no—
111#535—416no—
112#533—232no—
113C2×C4⋊Q8C2314yes—
114C2×C4⋊Q8C2314yes—
115C2×#127—28no—
116C2×#127C2314yes—
117C2×#127C2314yes—
118C2×#127C2318yes—
119#529—232no—
120#529—232no—
121C2×#126C2318yes—
122C2×#126C2318yes—
123C2×#125C2314yes—
124C2×#125C2314yes—
125C2×#125C2318yes—
126C2×#124—216no—
127C2×#124—216no—
128C2×#124C2314yes—
129C2×#124C2318yes—
130C2×#124C2318yes—
131C2×#124C23116yes—
132C2×#124C2314yes—
133C2×#124C2318yes—
134C2×#123—232no—
135C2×#123C23132yes—
136C2×#123C2318yes—
137C2×#123C2318yes—
138C2×C22⋊Q8C2314yes—
139C2×C22⋊Q8C2314yes—
140C2×C22⋊Q8C2314yes—
141C2×C22⋊Q8C2314yes—
142C2×C22⋊Q8C2318yes—
143C2×C22⋊Q8C2318yes—
144C2×C22⋊Q8C23116yes—
145C2×#121C2314yes—
146C2×#121C2314yes—
147C2×#121C23116yes—
148C2×#121C23116yes—
149C2×#121C23116yes—
150C2×#121C23116yes—
151C2×#120C23116yes—
152#519—48no—
153C4×#41—216no—
154C2×C4×Q8—28no—
155C2×C4×Q8D4116yescomplemented
156C2×C4×Q8C2314yes—
157C2×C4×Q8C2312yes—
158C2×C4×Q8C2312yes—
159C2×C4×Q8C2314yes—
160C2×C4×Q8C2314yes—
161C2×C4×Q8C23116yes—
162C2×C4×Q8C2314yes—
163C2×C4×Q8C2314yes—
164C2×C4×D4—216no—
165C2×C4×D4—216no—
166C2×C4×D4C2318yes—
167C2×C4×D4C2314yes—
168C2×C4×D4C2318yes—
169C2×C4×D4C2312yes—
170C2×C4×D4C23116yes—
171C2×C4×D4C2314yes—
172C2×C4×D4C2312yes—
173C2×C4×D4C2314yes—
174C2×#117C2312yes—
175C2×#117C2314yes—
176C2×#117C2318yes—
177C2×#117C2314yes—
178C2×#117C2318yes—
179C22×C4⋊C4C2312yes—
180C22×C4⋊C4C2311yescharacteristic
181C22×C4⋊C4C2311yescharacteristic
182C22×C4⋊C4C2312yes—
183C22×C22⋊C4C2314yes—
184C22×C22⋊C4C2314yes—
185C22×C22⋊C4C2314yes—
186C22×C42C2314yes—
187C22×C42C2312yes—
188C22×C42C2312yes—
189C22×C42C2311yescharacteristic
190#400C2318yes—
191#399C23116yes—
192#399C2318yes—
193#398C23116yes—
194#397C23116yes—
195#396C23132yes—
196#396C23116yes—
197#396C23132yes—
198#396C23132yes—
199#395C2318yes—
200#394C23116yes—
201#394C23132yes—
202#394C23132yes—
203#394C23116yes—
204#394C23132yes—
205#393C2318yes—
206#393C23116yes—
207#392C23116yes—
208#392C23116yes—
209#392C23116yes—
210#391C2318yes—
211#391C2318yes—
212#390C23116yes—
213#390C2318yes—
214#390C2318yes—
215#390C23116yes—
216#390C23132yes—
217#390C2318yes—
218#390C23116yes—
219#390C23132yes—
220#389C23132yes—
221#389C23116yes—
222#389C2318yes—
223#389C2318yes—
224#389C2318yes—
225#388C23132yes—
226#388C23132yes—
227#387C23116yes—
228#386C2318yes—
229#386C2318yes—
230#386C23132yes—
231#385C23116yes—
232#385C2318yes—
233#385C2318yes—
234#385C23132yes—
235#384C2314yes—
236#384C2318yes—
237#382C2318yes—
238#382C2314yes—
239#380C23116yes—
240#380C23116yes—
241#380C2314yes—
242#380C2318yes—
243#380C2318yes—
244#380C2314yes—
245#379C23116yes—
246C4×C4⋊C4C2318yes—
247C4×C4⋊C4C2318yes—
248C4×C4⋊C4C2314yes—
249C4×C4⋊C4C2314yes—
250C4×C4⋊C4C23116yes—
251C4×C22⋊C4C23116yes—
252C25C2414yes—
253C2×#41—28no—
254C2×#41—28no—
255C2×#41—28no—
256C22×Q8—24no—
257C22×Q8—24no—
258C22×Q8—24no—
259C22×Q8—44no—
260C22×Q8C2×D418yes—
261C22×Q8C2412yes—
262C22×Q8C2412yes—
263C22×Q8C2414yes—
264C22×Q8C2412yes—
265C22×Q8C2414yes—
266C22×D4—28no—
267C22×D4—28no—
268C22×D4—28no—
269C22×D4—28no—
270C22×D4—28no—
271C22×D4—48no—
272C22×D4—84no—
273C22×D4C2414yes—
274C22×D4C2414yes—
275C22×D4C2418yes—
276C22×D4C2412yes—
277C22×D4C2414yes—
278C22×D4C2414yes—
279C22×D4C2418yes—
280C22×D4C24132yes—
281C23×C4—216no—
282C23×C4—88no—
283C23×C4C2414yes—
284C23×C4C2414yes—
285C23×C4C2412yes—
286C23×C4C2411yescharacteristic
287C23×C4C2412yes—
288C23×C4C2412yes—
289C23×C4C24116yes—
290C23×C4C2414yes—
291C23×C4C2412yes—
292C23×C4C2411yescharacteristic
293C23×C4C2411yescharacteristic
294C4⋊Q8—216no—
295C4⋊Q8—216no—
296#127—216no—
297#127—216no—
298#127—416no—
299#126—232no—
300#126—232no—
301#125—216no—
302#125—232no—
303#125—216no—
304#124—264no—
305#124—216no—
306#124—232no—
307#124—232no—
308#124—216no—
309#124—416no—
310#123—232no—
311#123—2128no—
312#123—232no—
313C22⋊Q8—216no—
314C22⋊Q8—216no—
315C22⋊Q8—216no—
316C22⋊Q8—216no—
317C22⋊Q8—232no—
318C22⋊Q8—416no—
319C22⋊Q8—816no—
320#121—216no—
321#121—216no—
322#121—432no—
323#121—432no—
324#121—816no—
325#121—816no—
326#120—432no—
327C4×Q8—216no—
328C4×Q8—216no—
329C4×Q8—28no—
330C4×Q8—28no—
331C4×Q8—216no—
332C4×Q8—432no—
333C4×Q8C2×D4132yes—
334C4×Q8C2×D4132yes—
335C4×D4—216no—
336C4×D4—216no—
337C4×D4—264no—
338C4×D4—232no—
339C4×D4—232no—
340C4×D4—28no—
341C4×D4—28no—
342C4×D4—216no—
343#117—28no—
344#117—216no—
345#117—216no—
346#117—232no—
347#117—416no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—28no—
350C2×C4⋊C4C2×D418yes—
351C2×C4⋊C4C2×D4116yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2418yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2418yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C2418yes—
362C2×C4⋊C4C2414yes—
363C2×C4⋊C4C2414yes—
364C2×C4⋊C4C2418yes—
365C2×C4⋊C4C2418yes—
366C2×C4⋊C4C24116yes—
367C2×C4⋊C4C2418yes—
368C2×C4⋊C4C24116yes—
369C2×C4⋊C4C24116yes—
370C2×C4⋊C4C24116yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2418yes—
373C2×C22⋊C4—216no—
374C2×C22⋊C4—216no—
375C2×C22⋊C4—216no—
376C2×C22⋊C4C24116yes—
377C2×C22⋊C4C24116yes—
378C2×C22⋊C4C24116yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2414yes—
381C2×C22⋊C4C2414yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C2418yes—
384C2×C22⋊C4C2418yes—
385C2×C22⋊C4C2418yes—
386C2×C22⋊C4C24132yes—
387C2×C22⋊C4C24116yes—
388C2×C22⋊C4C24132yes—
389C2×C22⋊C4C24132yes—
390C2×C22⋊C4C24116yes—
391C2×C42—28no—
392C2×C42—216no—
393C2×C42C2×D4116yes—
394C2×C42C2×D418yes—
395C2×C42C2414yes—
396C2×C42C2414yes—
397C2×C42C2418yes—
398C2×C42C2418yes—
399C2×C42C2414yes—
400C2×C42C24116yes—
401C2×C42C2414yes—
402C2×C42C2412yes—
403C2×C42C2414yes—
404C2×C42C2412yes—
405#95C24116yes—
406#95C24116yes—
407#95C2418yes—
408#95C24116yes—
409#95C24116yes—
410#95C2418yes—
411#95C24116yes—
412C24—28no—
413C24—28no—
414C24—28no—
415C24—48no—
416C24—48no—
417C24C2518yes—
418C24C2514yes—
419C24C2511yescharacteristic
420#41—432no—
421C2×Q8—232no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—28no—
426C2×Q8—28no—
427C2×Q8—24no—
428C2×Q8—28no—
429C2×Q8—24no—
430C2×Q8—28no—
431C2×Q8—44no—
432C2×Q8—48no—
433C2×Q8—416no—
434C2×Q8C22×D4116yes—
435C2×D4—28no—
436C2×D4—28no—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—216no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—216no—
444C2×D4—216no—
445C2×D4—24no—
446C2×D4—216no—
447C2×D4—24no—
448C2×D4—216no—
449C2×D4—48no—
450C2×D4—432no—
451C2×D4—48no—
452C2×D4—416no—
453C2×D4—432no—
454C2×D4—432no—
455C2×D4—432no—
456C2×D4—432no—
457C2×D4—816no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—216no—
468C22×C4—28no—
469C22×C4—232no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—832no—
478C22×C4C22×D414yes—
479C22×C4C22×D418yes—
480C22×C4C22×D418yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C42−1+414yes—
485C22×C4C25116yes—
486C22×C4C2514yes—
487C22×C4C2514yes—
488C22×C4C2514yes—
489C22×C4C2514yes—
490C22×C4C2514yes—
491C22×C4C2511yescharacteristic
492C22×C4C2512yes—
493C22×C4C2514yes—
494C22×C4C2514yes—
495C22×C4C2512yes—
496C22×C4C2511yescharacteristic
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—216no—
500C4⋊C4—24no—
501C4⋊C4—28no—
502C4⋊C4—24no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—416no—
512C4⋊C4—48no—
513C4⋊C4—416no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C4⋊C4C22×D4116yes—
517C4⋊C4C22×D4132yes—
518C22⋊C4—28no—
519C22⋊C4—216no—
520C22⋊C4—232no—
521C22⋊C4—232no—
522C22⋊C4—232no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—432no—
529C22⋊C4—432no—
530C22⋊C4—416no—
531C22⋊C4—432no—
532C22⋊C4—416no—
533C42—28no—
534C42—28no—
535C42—216no—
536C42—28no—
537C42—216no—
538C42—416no—
539C42C22×D418yes—
540C42C22×D4116yes—
541C42C22×D418yes—
542C42C22×D4116yes—
543C23—22no—
544C23—22no—
545C23—24no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—216no—
553C23—216no—
554C23—28no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—816no—
559C23—816no—
560C23C23×D412yes—
561C23C2×H(5,𝔽2)12yes—
562C23C2×2−1+412yes—
563C23C2611yescharacteristic
564Q8—232no—
565Q8—48no—
566Q8—48no—
567Q8—48no—
568Q8—88no—
569D4—416no—
570D4—416no—
571D4—432no—
572D4—48no—
573D4—416no—
574D4—88no—
575D4—816no—
576D4—1632no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—216no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—216no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—216no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—216no—
598C2×C4—24no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—88no—
606C2×C4—832no—
607C2×C4#537116yes—
608C2×C4C23×D414yes—
609C2×C4C23×D412yes—
610C2×C4C23×D418yes—
611C2×C4C23×D418yes—
612C2×C4C23×D414yes—
613C2×C4C23×D412yes—
614C2×C4C2×H(5,𝔽2)14yes—
615C2×C4C2×2−1+414yes—
616C2×C4#58514yes—
617C22—22no—
618C22—22no—
619C22—24no—
620C22—24no—
621C22—44no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—48no—
627C22—48no—
628C22—416no—
629C22—48no—
630C22—416no—
631C22—832no—
632C22C2×#53714yes—
633C22C24×D411yescharacteristic
634C22C22×H(5,𝔽2)11yescharacteristic
635C22C22×2−1+411yescharacteristic
636C22C2×#58512yes—
637C22H(7,𝔽2)12yes—
638C4—22no—
639C4—28no—
640C4—24no—
641C4—28no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—416no—
648C4C2×#53718yes—
649C4#348714yes—
650C2—24no—
651C2—84no—
652C2—88no—
653C2C22×#53711yescharacteristic
654C2C2×#348711yescharacteristic
655C2#6285211yescharacteristic
656C2#6296112yes—
657C2#6296212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#559C2semidirect
2. C2×#3461C2semidirect
3. #42881C2semidirect
4. #42877C2semidirect
5. #39726C2semidirect
6. #39137C2semidirect
7. #37546C2semidirect
8. #36722C2semidirect
9. #35677C2semidirect
10. #35272C2semidirect
11. #35236C2semidirect
12. #35174C2semidirect
13. #35168C2semidirect
14. #34952C2semidirect
15. #34182C2semidirect
21. C2×#542C22semidirect
22. C2×#541C22semidirect
27. C2×#533C22semidirect
30. C2×#529C22semidirect
43. #2820C22semidirect
44. #2819C22semidirect
45. #2707C22semidirect
46. #2705C22semidirect
47. #2682C22semidirect
48. #2681C22semidirect
49. #2662C22semidirect
50. #2636C22semidirect
52. #2620C22semidirect
55. #2592C22semidirect
56. #2582C22semidirect
57. #2581C22semidirect
58. #2580C22semidirect
60. #2562C22semidirect
63. #2548C22semidirect
64. #2545C22semidirect
67. #2540C22semidirect
68. #2540C22semidirect
69. #2538C22semidirect
70. #2538C22semidirect
71. #2537C22semidirect
73. #2454C22semidirect
74. #2452C22semidirect
75. #2428C22semidirect
76. #2427C22semidirect
77. #2426C22semidirect
78. #2426C22semidirect
84. #2396C22semidirect
85. #2351C22semidirect
86. Q8×C4⋊C4C22semidirect
87. #2324C22semidirect
91. C4×C22⋊Q8C22semidirect
155. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5592
C2×#34612
#428812
#428772
#397262
#391372
#375462
#367222
#356772
#352722
#352362
#351742
#351682
#349522
#341822