Finite Group Databasebeta

← #7503477

Subgroups — #7503477

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3096C211yesmaximal, characteristic
2C22×#476C211yesmaximal, characteristic, complemented
3#35574C211yesmaximal, characteristic, complemented
4#35152C211yesmaximal, characteristic
5#31108C214yesmaximal, complemented
6#31104C214yesmaximal, complemented
7#31090C214yesmaximal, complemented
8#31086C214yesmaximal, complemented
9#23666C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23640C211yesmaximal, characteristic, complemented
12#23632C211yesmaximal, characteristic, complemented
13#23552C211yesmaximal, characteristic, complemented
14#23483C211yesmaximal, characteristic
15#21089C211yesmaximal, characteristic, complemented
16#21052C211yesmaximal, characteristic, complemented
17#20873C211yesmaximal, characteristic, complemented
18#20848C211yesmaximal, characteristic, complemented
19#20802C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3096—28no—
25C2×#481C2212yes—
26C2×#482C2212yes—
27C2×#482C2212yes—
28C2×#480C2212yes—
29C2×#476—28no—
30C2×#476C2214yes—
31C2×#476C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2849—22no—
47#2622—22no—
48#2579—22no—
49#2452—22no—
50#2451—22no—
51#2447—22no—
52#2445—22no—
53#2440C2214yescomplemented
54#2439C2214yescomplemented
55#2422C2214yes—
56#2420C2214yes—
57#2409—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2309C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2296C2211yescharacteristic
67#2088C2212yescomplemented
68#2088C2212yescomplemented
69#2087C2212yescomplemented
70#2086C2212yescomplemented
71#2085C2214yescomplemented
72#2084C2214yescomplemented
73#2077C2214yescomplemented
74#2077C2214yescomplemented
75#2073C2214yes—
76#2071C2214yes—
77#2058C2214yescomplemented
78#2057C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#1936C2214yes—
82#1934C2214yes—
83#1924C2214yescomplemented
84#1923C2214yescomplemented
85#1864C2214yes—
86#1862C2214yes—
87#1854C2214yescomplemented
88#1853C2214yescomplemented
89#1847C2212yescomplemented
90#1847C2212yescomplemented
91#1846C2212yescomplemented
92#1844C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×D4C2311yescharacteristic
104C24×C4—21no—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#535—24no—
110#535—42no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—22no—
113C2×C4⋊Q8—42no—
114C2×C4⋊Q8C2314yes—
115C2×#127—21no—
116C2×#124—21no—
117C2×#123—24no—
118C2×C22⋊Q8—24no—
119C2×#121C2312yes—
120C2×#121C2312yes—
121C2×#120—22no—
122#520—24no—
123#520—42no—
124C2×C4×Q8—21no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—24no—
128C2×C4×D4—22no—
129C2×C4×D4—42no—
130C2×C4×D4C2314yes—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—21no—
143C22×C42C2311yescharacteristic
144#482—28no—
145#482—28no—
146#481—28no—
147#480—28no—
148#476—216no—
149#476—216no—
150#428—24no—
151#428—42no—
152#428—42no—
153#427—24no—
154C2×#107C2312yes—
155C2×#107C2311yescharacteristic
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#106—24no—
160C2×#106—24no—
161C2×#106C2312yes—
162C2×#106C2311yescharacteristic
163C2×#106C2311yescharacteristic
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2314yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169C2×C4⋊C8—24no—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172#418—24no—
173#418—24no—
174C2×#102—24no—
175C2×#102D418yes—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2314yes—
179C2×#102C2314yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182#408—24no—
183#408—24no—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2314yes—
187#405—24no—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#398—24no—
191#397—24no—
192#397—24no—
193#396—24no—
194#395—24no—
195#394—24no—
196#394—24no—
197#393—24no—
198#391—21no—
199#391—21no—
200#391—24no—
201#391D412yes—
202#390—21no—
203#390—21no—
204#390—22no—
205#390D412yes—
206#390C2314yes—
207#389—22no—
208#389—22no—
209#389—22no—
210#389C2312yes—
211#389C2312yes—
212#389C2314yes—
213#389C2312yes—
214#389C2312yes—
215#387—24no—
216#386—22no—
217#386—22no—
218#386—24no—
219#386D412yes—
220#386D412yes—
221#385—21no—
222#385—21no—
223#385—21no—
224#385—21no—
225#385—22no—
226#385—22no—
227#385—24no—
228#385—24no—
229#384C2312yes—
230#384C2312yes—
231#382C2312yes—
232#382C2312yes—
233#381—24no—
234#380—22no—
235#380C2314yes—
236#380C2314yes—
237#379—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2312yes—
250#336C2314yes—
251#336C2314yes—
252C25—21no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C22×Q8—24no—
260C22×Q8—44no—
261C22×Q8—84no—
262C22×Q8C2×D411yescharacteristic
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—22no—
271C22×D4C2×D412yes—
272C22×D4C2×D411yescharacteristic
273C22×D4C2×D414yes—
274C22×D4C2412yes—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4—24no—
280C23×C4—44no—
281C23×C4—84no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2412yes—
285C23×C4C2411yescharacteristic
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16#4114yes—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8C2×D414yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C22×C8C2412yes—
303C4⋊Q8—28no—
304C4⋊Q8—44no—
305C4⋊Q8—48no—
306#127—44no—
307#124—44no—
308#123—88no—
309C22⋊Q8—88no—
310#121—24no—
311#121—24no—
312#121—42no—
313#121—42no—
314#120—48no—
315C4×Q8—44no—
316C4×D4—28no—
317C4×D4—44no—
318C4×D4—48no—
319C4×D4—44no—
320C4×D4—48no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—24no—
332#117—42no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4C2×D412yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D414yes—
379C2×C22⋊C4C2×D414yes—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C2×D412yes—
388C2×C42C2412yes—
389C2×C42C2412yes—
390#107—22no—
391#107—22no—
392#107—24no—
393#107—42no—
394#107—42no—
395#106—24no—
396#106—24no—
397#106—22no—
398#106—22no—
399#106—28no—
400#106—24no—
401#106—42no—
402#106—42no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#102—28no—
406#102—24no—
407#102—24no—
408#102—28no—
409#102C2×D418yes—
410#102C2×D418yes—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—28no—
414#97—24no—
415C4×C8—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—22no—
428C24—24no—
429C24—42no—
430C24—42no—
431C24—82no—
432C24—82no—
433C24C22×D412yes—
434C24C22×D411yescharacteristic
435#41—44no—
436#41—44no—
437#41—44no—
438#41—44no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—48no—
443C2×Q8—48no—
444C2×Q8—84no—
445C2×Q8—816no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—28no—
450C2×D4—22no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—28no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—44no—
464C2×D4—42no—
465C2×D4—44no—
466C2×D4—84no—
467C2×D4—84no—
468C2×D4C22×D414yes—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—48no—
490C22×C4—44no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—48no—
494C22×C4—84no—
495C22×C4—816no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—84no—
500C22×C4C22×D412yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+412yes—
509C22×C42−1+412yes—
510C22×C42−1+412yes—
511C22×C4C2511yescharacteristic
512M16—28no—
513M16—44no—
514C2×C8—28no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—22no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—44no—
528C2×C8C22×D412yes—
529C2×C8C22×D412yes—
530C2×C8C2×#4114yes—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—44no—
551C4⋊C4—48no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—82no—
558C4⋊C4—84no—
559C4⋊C4—84no—
560C4⋊C4—84no—
561C4⋊C4C22×D414yes—
562C22⋊C4—24no—
563C22⋊C4—28no—
564C22⋊C4—44no—
565C22⋊C4—48no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—84no—
574C22⋊C4—84no—
575C22⋊C4—84no—
576C22⋊C4—84no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—44no—
583C42—42no—
584C42—42no—
585C42—42no—
586C42—42no—
587C23—22no—
588C23—22no—
589C23—24no—
590C23—22no—
591C23—24no—
592C23—22no—
593C23—22no—
594C23—24no—
595C23—44no—
596C23—42no—
597C23—48no—
598C23—42no—
599C23—44no—
600C23—84no—
601C23—84no—
602C23—88no—
603C23—88no—
604C23#53612yes—
605C23D4212yes—
606C23#54712yes—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—816no—
610D4—24no—
611D4—24no—
612D4—48no—
613D4—48no—
614D4—44no—
615D4—42no—
616D4—42no—
617D4—164no—
618D4—164no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—816no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—164no—
661C2×C4#53614yes—
662C2×C4#53612yes—
663C2×C4#53914yes—
664C2×C4D4212yes—
665C2×C4#54712yes—
666C2×C4#54714yes—
667C2×C4#56314yes—
668C2×C4#57714yes—
669C2×C4C23×D411yescharacteristic
670C2×C4C22×#4111yescharacteristic
671C2×C4C2×H(5,𝔽2)11yescharacteristic
672C2×C4C2×2−1+411yescharacteristic
673C2×C4C2×2−1+411yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C8—24no—
676C8—22no—
677C8—22no—
678C8—42no—
679C8—42no—
680C22—24no—
681C22—22no—
682C22—22no—
683C22—24no—
684C22—44no—
685C22—44no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—42no—
690C22—44no—
691C22—44no—
692C22—816no—
693C22—84no—
694C22—84no—
695C22#268014yes—
696C22C2×#53611yescharacteristic
697C22C2×D4211yescharacteristic
698C22C2×#54711yescharacteristic
699C22C2×#57712yes—
700C22#358512yes—
701C4—24no—
702C4—44no—
703C4—42no—
704C4—44no—
705C4—42no—
706C4—82no—
707C4—81no—
708C4—82no—
709C4—81no—
710C4—82no—
711C4—81no—
712C4—81no—
713C4—82no—
714C4#268014yes—
715C4C2×#53611yescharacteristic
716C4C2×#53911yescharacteristic
717C4C2×#54711yescharacteristic
718C4C2×#56311yescharacteristic
719C2—24no—
720C2—82no—
721C2—84no—
722C2C2×#268011yescharacteristic
723C2#6089212yes—
724C2#6105011yescharacteristic
725C2#6105811yescharacteristic
726C2#6127912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#476C2semidirect
3. #35574C2semidirect
5. #31108C2semidirect
6. #31104C2semidirect
7. #31090C2semidirect
8. #31086C2semidirect
9. #23666C2semidirect
10. #23661C2semidirect
11. #23640C2semidirect
12. #23632C2semidirect
13. #23552C2semidirect
15. #21089C2semidirect
16. #21052C2semidirect
17. #20873C2semidirect
18. #20848C2semidirect
19. #20802C2semidirect
53. #2440C22semidirect
54. #2439C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2087C22semidirect
70. #2086C22semidirect
71. #2085C22semidirect
72. #2084C22semidirect
73. #2077C22semidirect
74. #2077C22semidirect
77. #2058C22semidirect
78. #2057C22semidirect
83. #1924C22semidirect
84. #1923C22semidirect
87. #1854C22semidirect
88. #1853C22semidirect
89. #1847C22semidirect
90. #1847C22semidirect
91. #1846C22semidirect
92. #1844C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30962
C22×#4762
#355742
#351522
#311082
#311042
#310902
#310862
#236662
#236612
#236402
#236322
#235522
#234832
#210892
#210522
#208732
#208482
#208022