Finite Group Databasebeta

← #7630524

Subgroups — #7630524

Order 512.

11182 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3449C212yesmaximal, complemented
2C22×#537C211yesmaximal, characteristic, complemented
3C22×#520C211yesmaximal, characteristic, complemented
4#35633C218yesmaximal, complemented
5#35564C2116yesmaximal, complemented
6#35563C218yesmaximal, complemented
7#35223C214yesmaximal, complemented
8#35168C214yesmaximal, complemented
9#35153C214yesmaximal, complemented
10#35150C214yesmaximal, complemented
11#35000C218yesmaximal, complemented
12C2×#2316C211yesmaximal, characteristic, complemented
13#34044C212yesmaximal, complemented
14C23×#41C2212yes—
15C23×#41C2211yescharacteristic
16#3449—232no—
17C2×#537—216no—
18C2×#537C2214yes—
19C2×#537C2214yes—
20C2×#537C2218yescomplemented
21C2×#537C2218yescomplemented
22C2×#536C2214yes—
23C2×#536C2214yescomplemented
24C2×#535C2214yes—
25C2×#535C2214yescomplemented
26C2×#534C2218yes—
27C2×#532C2218yescomplemented
28C2×#525C2218yes—
29C2×#525C2218yescomplemented
30C22×C22⋊Q8C2214yescomplemented
31C22×#121C2214yes—
32C22×#121C2214yescomplemented
33C2×#520—216no—
34C2×#520C2218yes—
35C2×#520C2218yescomplemented
36C22×C4×Q8C2212yescomplemented
37C22×C4×D4C2214yes—
38C22×C4×D4C2212yescomplemented
39C22×#117C2212yes—
40C22×#117C2211yescharacteristic
41C22×#117C2212yescomplemented
42C23×C4⋊C4C2211yescharacteristic
43C23×C4⋊C4C2212yescomplemented
44#2854C22116yescomplemented
45#2847C22116yescomplemented
46#2618C22116yescomplemented
47#2617C22116yescomplemented
48#2543C22132yescomplemented
49#2534C22132yescomplemented
50#2486C22116yescomplemented
51#2429C22132yescomplemented
52#2428C22132yescomplemented
53#2427C22132yescomplemented
54#2426C22132yescomplemented
55#2417C22132yescomplemented
56#2416C22132yescomplemented
57#2416C22132yescomplemented
58#2415C22132yescomplemented
59#2414C22132yescomplemented
60#2408C22116yescomplemented
61#2316C2218yescomplemented
62#2316C2214yescomplemented
63#2316C2214yescomplemented
64#2311C2214yescomplemented
65#2311C2214yescomplemented
66#2311C2218yescomplemented
67#2310C2214yescomplemented
68#2309C2214yescomplemented
69#2308C2218yes—
70#2297C2218yescomplemented
71#2297C2218yescomplemented
72C2×#389C2218yescomplemented
73#2280C2218yescomplemented
74C2×#380C2214yes—
75C22×#41—28no—
76C22×#41—216no—
77C22×#41—216no—
78C22×#41—28no—
79C22×#41—216no—
80C22×#41D4116yescomplemented
81C22×#41C2314yes—
82C22×#41C2314yes—
83C22×#41C2314yes—
84C22×#41C2314yes—
85C22×#41C2314yes—
86C22×#41C2314yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×D4C2314yes—
90C23×D4C2312yes—
91C23×D4C2312yes—
92C23×D4C2311yescharacteristic
93C23×D4C2314yes—
94C24×C4C2314yes—
95C24×C4C2312yes—
96C24×C4C2314yes—
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99#537—264no—
100#537—232no—
101#537—232no—
102#537—264no—
103#536—232no—
104#536—232no—
105#535—232no—
106#535—232no—
107#534—264no—
108#532—264no—
109C2×C4⋊Q8C2318yes—
110C2×#127C2318yes—
111C2×#124C2318yes—
112C2×#124C2318yes—
113#525—264no—
114#525—264no—
115C2×#123C23116yes—
116C2×#123C23116yes—
117C2×C22⋊Q8—232no—
118C2×C22⋊Q8C23116yes—
119C2×C22⋊Q8C23116yes—
120C2×C22⋊Q8C23116yes—
121C2×C22⋊Q8C23116yes—
122C2×#121—232no—
123C2×#121—232no—
124C2×#121C23116yes—
125C2×#121C23116yes—
126C2×#121C23116yes—
127C2×#121C23116yes—
128C2×#121C23116yes—
129C2×#121C23116yes—
130C2×#121C23132yes—
131C2×#121C23132yes—
132C2×#120C23116yes—
133C2×#120C23116yes—
134#520—264no—
135#520—264no—
136C2×C4×Q8—216no—
137C2×C4×Q8C23116yes—
138C2×C4×D4—232no—
139C2×C4×D4—216no—
140C2×C4×D4C23116yes—
141C2×C4×D4C23116yes—
142C2×C4×D4C23116yes—
143C2×C4×D4C23132yes—
144C2×#117—28no—
145C2×#117—216no—
146C2×#117—216no—
147C2×#117C2314yes—
148C2×#117C2314yes—
149C2×#117C2314yes—
150C2×#117C2314yes—
151C2×#117C2314yes—
152C2×#117C2314yes—
153C2×#117C23116yes—
154C22×C4⋊C4—216no—
155C22×C4⋊C4D4116yescomplemented
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2314yes—
160C22×C4⋊C4C23116yes—
161C22×C4⋊C4C2314yes—
162C22×C4⋊C4C2318yes—
163C22×C4⋊C4C2312yes—
164C22×C4⋊C4C2312yes—
165C22×C4⋊C4C2314yes—
166C22×C4⋊C4C2314yes—
167C22×C22⋊C4C2312yes—
168C22×C22⋊C4C2314yes—
169C22×C22⋊C4C2314yes—
170C22×C22⋊C4C2318yes—
171C22×C42C2314yes—
172C22×C42C2312yes—
173C22×C42C2314yes—
174#398C23132yes—
175#397C23132yes—
176#396C23164yes—
177#394C23164yes—
178#394C23132yes—
179#392C23132yes—
180#391C2318yes—
181#391C2318yes—
182#391C23116yes—
183#390C2318yes—
184#390C2318yes—
185#390C23116yes—
186#390C23132yes—
187#389C23132yes—
188#389C23132yes—
189#389C23132yes—
190#388C23164yes—
191#386C23116yes—
192#386C23116yes—
193#386C23132yes—
194#386C23164yes—
195#381C23116yes—
196#380C23116yes—
197#380C23116yes—
198#380C23116yes—
199#380C23116yes—
200#380C23132yes—
201#379C23116yes—
202C4×C22⋊C4C23132yes—
203C2×#95C2318yes—
204C25C2414yes—
205C25C2412yes—
206C25C2411yescharacteristic
207C2×#41—2128no—
208C2×#41—216no—
209C2×#41—216no—
210C2×#41—216no—
211C2×#41—216no—
212C2×#41—216no—
213C2×#41—216no—
214C2×#41—216no—
215C2×#41—216no—
216C2×#41—216no—
217C2×#41—216no—
218C2×#41—216no—
219C2×#41—216no—
220C2×#41—216no—
221C2×#41—216no—
222C2×#41—216no—
223C2×#41—216no—
224C2×#41—4128no—
225C2×#41C2×D4132yes—
226C2×#41C2×D4132yescomplemented
227C22×Q8—24no—
228C22×Q8—28no—
229C22×Q8—28no—
230C22×Q8—24no—
231C22×Q8—28no—
232C22×Q8C2×D418yescomplemented
233C22×Q8C2414yes—
234C22×Q8C2414yes—
235C22×Q8C2414yes—
236C22×D4—28no—
237C22×D4—28no—
238C22×D4—216no—
239C22×D4—216no—
240C22×D4—216no—
241C22×D4—24no—
242C22×D4—28no—
243C22×D4—28no—
244C22×D4—216no—
245C22×D4—24no—
246C22×D4—28no—
247C22×D4—88no—
248C22×D4C2×D4116yes—
249C22×D4C2×D418yescomplemented
250C22×D4C2418yes—
251C22×D4C2418yes—
252C22×D4C2414yes—
253C22×D4C2414yes—
254C22×D4C2414yes—
255C22×D4C2414yes—
256C22×D4C2414yes—
257C22×D4C24132yes—
258C22×D4C24116yes—
259C23×C4—28no—
260C23×C4—28no—
261C23×C4—216no—
262C23×C4—216no—
263C23×C4—216no—
264C23×C4—216no—
265C23×C4—84no—
266C23×C4—88no—
267C23×C4C2×D4116yes—
268C23×C4C2×D418yes—
269C23×C4C2×D418yes—
270C23×C4C2×D4116yescomplemented
271C23×C4#4118yes—
272C23×C4C2414yes—
273C23×C4C2418yes—
274C23×C4C2418yes—
275C23×C4C2418yes—
276C23×C4C2412yes—
277C23×C4C2412yes—
278C23×C4C2418yes—
279C23×C4C2412yes—
280C23×C4C24116yes—
281C23×C4C2414yes—
282C23×C4C2414yes—
283C23×C4C2414yes—
284C23×C4C2411yescharacteristic
285C23×C4C2412yes—
286C23×C4C2412yes—
287C23×C4C2411yescharacteristic
288C4⋊Q8—232no—
289#127—232no—
290#124—232no—
291#124—232no—
292#123—264no—
293#123—264no—
294C22⋊Q8—264no—
295C22⋊Q8—264no—
296C22⋊Q8—264no—
297C22⋊Q8—264no—
298#121—264no—
299#121—2128no—
300#121—264no—
301#121—264no—
302#121—264no—
303#121—264no—
304#121—264no—
305#121—832no—
306#120—264no—
307#120—264no—
308C4×Q8—264no—
309C4×D4—264no—
310C4×D4—264no—
311C4×D4—2128no—
312C4×D4—264no—
313#117—216no—
314#117—216no—
315#117—216no—
316#117—216no—
317#117—216no—
318#117—216no—
319#117—264no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—216no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—216no—
324C2×C4⋊C4—216no—
325C2×C4⋊C4—264no—
326C2×C4⋊C4—88no—
327C2×C4⋊C4C2×D4116yes—
328C2×C4⋊C4C2×D4116yes—
329C2×C4⋊C4C2×D4132yes—
330C2×C4⋊C4C2×D4132yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C24116yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2418yes—
335C2×C4⋊C4C2418yes—
336C2×C4⋊C4C24116yes—
337C2×C4⋊C4C24132yes—
338C2×C4⋊C4C24132yes—
339C2×C4⋊C4C2418yes—
340C2×C4⋊C4C2418yes—
341C2×C22⋊C4—28no—
342C2×C22⋊C4—216no—
343C2×C22⋊C4—216no—
344C2×C22⋊C4—232no—
345C2×C22⋊C4C2414yes—
346C2×C22⋊C4C2414yes—
347C2×C22⋊C4C24116yes—
348C2×C22⋊C4C2418yes—
349C2×C22⋊C4C2418yes—
350C2×C22⋊C4C24132yes—
351C2×C22⋊C4C24132yes—
352C2×C22⋊C4C24132yes—
353C2×C22⋊C4C24132yes—
354C2×C22⋊C4C24132yes—
355C2×C42—216no—
356C2×C42—28no—
357C2×C42—216no—
358C2×C42C2418yes—
359C2×C42C2418yes—
360C2×C42C24116yes—
361C2×C42C2418yes—
362#95C24132yes—
363#95C24132yes—
364#95C24132yes—
365C24—24no—
366C24—24no—
367C24—28no—
368C24—28no—
369C24—28no—
370C24—42no—
371C24—44no—
372C24—42no—
373C24—44no—
374C24—48no—
375C24—48no—
376C24C22×D418yes—
377C24C22×D414yes—
378C24C22×D414yes—
379C24C2×#4114yes—
380C24C2518yes—
381C24C2514yes—
382C24C2511yescharacteristic
383C24C2511yescharacteristic
384C24C2511yescharacteristic
385#41—2128no—
386#41—2128no—
387#41—464no—
388#41—464no—
389#41—464no—
390#41—464no—
391C2×Q8—232no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—28no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—28no—
400C2×Q8—432no—
401C2×Q8C22×D4116yes—
402C2×D4—264no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—28no—
406C2×D4—28no—
407C2×D4—216no—
408C2×D4—216no—
409C2×D4—216no—
410C2×D4—216no—
411C2×D4—216no—
412C2×D4—216no—
413C2×D4—264no—
414C2×D4—232no—
415C2×D4—28no—
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—232no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—432no—
425C2×D4—432no—
426C2×D4—432no—
427C2×D4—464no—
428C2×D4—432no—
429C2×D4—832no—
430C2×D4C22×D4116yes—
431C2×D4C22×D4116yes—
432C2×D4C22×D4116yes—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—264no—
436C22×C4—216no—
437C22×C4—216no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—216no—
444C22×C4—216no—
445C22×C4—216no—
446C22×C4—216no—
447C22×C4—216no—
448C22×C4—216no—
449C22×C4—216no—
450C22×C4—232no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—432no—
458C22×C4—48no—
459C22×C4—464no—
460C22×C4—816no—
461C22×C4—832no—
462C22×C4#121164yescomplemented
463C22×C4C22×D4132yes—
464C22×C4C22×D418yes—
465C22×C4C22×D4116yes—
466C22×C4C22×D4116yes—
467C22×C4C22×D4116yes—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D418yes—
472C22×C4C22×D418yes—
473C22×C4C22×D418yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C2×#4118yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4118yes—
480C22×C4C2514yes—
481C22×C4C25116yes—
482C22×C4C2514yes—
483C22×C4C2518yes—
484C22×C4C2518yes—
485C22×C4C2514yes—
486C22×C4C2512yes—
487C22×C4C2512yes—
488C4⋊C4—28no—
489C4⋊C4—232no—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—216no—
493C4⋊C4—232no—
494C4⋊C4—264no—
495C4⋊C4—432no—
496C4⋊C4C22×D4132yes—
497C4⋊C4C22×D4132yes—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—264no—
503C22⋊C4—232no—
504C22⋊C4—264no—
505C22⋊C4—264no—
506C22⋊C4—432no—
507C22⋊C4—432no—
508C42—216no—
509C42—232no—
510C42—216no—
511C42—216no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—216no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—216no—
527C23—216no—
528C23—44no—
529C23—44no—
530C23—48no—
531C23—416no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—416no—
536C23—416no—
537C23—416no—
538C23—416no—
539C23—816no—
540C23—816no—
541C23C2×#121116yes—
542C23C23×D418yes—
543C23C23×D412yes—
544C23C23×D412yes—
545C23C23×D412yes—
546C23C23×D412yes—
547C23C23×D412yes—
548C23C23×D412yes—
549C23C22×#4112yes—
550C23C22×#4112yes—
551C23C22×#4112yes—
552C23C2611yescharacteristic
553Q8—232no—
554Q8—416no—
555Q8—416no—
556D4—232no—
557D4—232no—
558D4—232no—
559D4—432no—
560D4—432no—
561D4—416no—
562D4—416no—
563D4—832no—
564D4—864no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—216no—
568C2×C4—216no—
569C2×C4—264no—
570C2×C4—24no—
571C2×C4—216no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—216no—
578C2×C4—28no—
579C2×C4—216no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—416no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—416no—
586C2×C4—432no—
587C2×C4—432no—
588C2×C4—832no—
589C2×C4—816no—
590C2×C4C2×#121116yes—
591C2×C4C2×#121132yes—
592C2×C4C2×#121116yes—
593C2×C4C2×#121132yes—
594C2×C4C23×D4116yes—
595C2×C4C23×D418yes—
596C2×C4C23×D414yes—
597C2×C4C23×D414yes—
598C2×C4C23×D414yes—
599C2×C4C23×D414yes—
600C2×C4C22×#4114yes—
601C2×C4C22×#4114yes—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—22no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—216no—
615C22—28no—
616C22—416no—
617C22—432no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—416no—
624C22—416no—
625C22—48no—
626C22—832no—
627C22C22×#12114yes—
628C22C22×#12114yes—
629C22C22×#12114yes—
630C22C24×D411yescharacteristic
631C22C24×D411yescharacteristic
632C22C23×#4111yescharacteristic
633C22H(7,𝔽2)12yes—
634C222−1+612yes—
635C4—216no—
636C4—28no—
637C4—44no—
638C4—44no—
639C4—416no—
640C4—48no—
641C4C22×#12118yes—
642C4C22×#12118yes—
643C2—24no—
644C2—24no—
645C2—24no—
646C2—48no—
647C2—88no—
648C2C23×#12111yescharacteristic
649C2#6284211yescharacteristic
650C2#6284811yescharacteristic
651C2#6285312yes—
652C2#6285612yes—
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. C2×#3449C2semidirect
2. C22×#537C2semidirect
3. C22×#520C2semidirect
4. #35633C2semidirect
5. #35564C2semidirect
6. #35563C2semidirect
7. #35223C2semidirect
8. #35168C2semidirect
9. #35153C2semidirect
10. #35150C2semidirect
11. #35000C2semidirect
12. C2×#2316C2semidirect
13. #34044C2semidirect
20. C2×#537C22semidirect
21. C2×#537C22semidirect
23. C2×#536C22semidirect
25. C2×#535C22semidirect
27. C2×#532C22semidirect
29. C2×#525C22semidirect
30. C22×C22⋊Q8C22semidirect
32. C22×#121C22semidirect
35. C2×#520C22semidirect
36. C22×C4×Q8C22semidirect
38. C22×C4×D4C22semidirect
41. C22×#117C22semidirect
43. C23×C4⋊C4C22semidirect
44. #2854C22semidirect
45. #2847C22semidirect
46. #2618C22semidirect
47. #2617C22semidirect
48. #2543C22semidirect
49. #2534C22semidirect
50. #2486C22semidirect
51. #2429C22semidirect
52. #2428C22semidirect
53. #2427C22semidirect
54. #2426C22semidirect
55. #2417C22semidirect
56. #2416C22semidirect
57. #2416C22semidirect
58. #2415C22semidirect
59. #2414C22semidirect
60. #2408C22semidirect
61. #2316C22semidirect
62. #2316C22semidirect
63. #2316C22semidirect
64. #2311C22semidirect
65. #2311C22semidirect
66. #2311C22semidirect
67. #2310C22semidirect
68. #2309C22semidirect
70. #2297C22semidirect
71. #2297C22semidirect
72. C2×#389C22semidirect
73. #2280C22semidirect
80. C22×#41D4semidirect
155. C22×C4⋊C4D4semidirect
226. C2×#41C2×D4semidirect
232. C22×Q8C2×D4semidirect
249. C22×D4C2×D4semidirect
270. C23×C4C2×D4semidirect
462. C22×C4#121semidirect

Maximal subgroups

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

subgroupindex
C2×#34492
C22×#5372
C22×#5202
#356332
#355642
#355632
#352232
#351682
#351532
#351502
#350002
C2×#23162
#340442