Finite Group Databasebeta

← #7501920

Subgroups — #7501920

Order 512.

1995 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#498C211yesmaximal, characteristic, complemented
2C2×#3099C211yesmaximal, characteristic
3#35592C211yesmaximal, characteristic
4#35574C211yesmaximal, characteristic, complemented
5#30780C214yesmaximal, complemented
6#30777C214yesmaximal, complemented
7#30775C214yesmaximal, complemented
8#30774C214yesmaximal, complemented
9#23738C211yesmaximal, characteristic, complemented
10#23640C211yesmaximal, characteristic, complemented
11#23610C211yesmaximal, characteristic, complemented
12#23552C211yesmaximal, characteristic, complemented
13#23489C211yesmaximal, characteristic
14#23452C211yesmaximal, characteristic, complemented
15#21174C211yesmaximal, characteristic, complemented
16#21083C211yesmaximal, characteristic, complemented
17#21046C211yesmaximal, characteristic, complemented
18#21038C211yesmaximal, characteristic, complemented
19#20848C211yesmaximal, characteristic, complemented
20C2×#537C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#498—28no—
25C2×#498C2214yes—
26C2×#498C2214yes—
27#3099—28no—
28C2×#481C2212yes—
29C2×#485C2212yes—
30C2×#482C2212yes—
31C2×#483C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#408C2211yescharacteristic
44C2×#408C2211yescharacteristic
45C22×C4×C8C2211yescharacteristic
46#2591—22no—
47#2579—22no—
48#2492—22no—
49#2492C2214yes—
50#2491—22no—
51#2491C2214yes—
52#2452—22no—
53#2451—22no—
54#2440C2214yes—
55#2439C2214yes—
56#2436—22no—
57#2409—22no—
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2311C2211yescharacteristic
61#2311C2211yescharacteristic
62#2311C2211yescharacteristic
63#2297C2211yescharacteristic
64#2297C2211yescharacteristic
65#2296C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yes—
68#2096C2212yes—
69#2094C2212yes—
70#2092C2212yes—
71#2077C2214yes—
72#2077C2214yes—
73#2066C2214yes—
74#2065C2214yes—
75#2058C2214yes—
76#2057C2214yes—
77#2035C2214yes—
78#2034C2214yes—
79#2024C2214yes—
80#2023C2214yes—
81#1949C2214yes—
82#1948C2214yes—
83#1934C2212yes—
84#1934C2212yes—
85#1933C2212yes—
86#1933C2212yes—
87#1923C2214yes—
88#1923C2214yes—
89#1920C2214yes—
90#1919C2214yes—
91#1854C2214yes—
92#1853C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C22×#41C2311yescharacteristic
103C23×Q8—21no—
104C23×Q8—21no—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#537—24no—
110#537—42no—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8—42no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2314yes—
118C2×C4⋊Q8C2314yes—
119C2×#124—24no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—24no—
122C2×C22⋊Q8C2312yes—
123C2×#121—22no—
124C2×#121—21no—
125C2×#121C2312yes—
126#520—24no—
127#520—42no—
128C2×C4×Q8—21no—
129C2×C4×D4—22no—
130C2×C4×D4—21no—
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#498—216no—
145#498—216no—
146#485—28no—
147#483—28no—
148#482—28no—
149#481—28no—
150#428—24no—
151#428—42no—
152C2×#107C2312yes—
153C2×#107C2312yes—
154C2×#106—24no—
155C2×#106—24no—
156C2×#106—24no—
157C2×#106D418yes—
158C2×#106C2312yes—
159C2×#106C2314yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166#424—24no—
167C2×C4⋊C8C2314yes—
168#417—24no—
169#417—42no—
170#416—24no—
171#416—24no—
172C2×#103C2311yescharacteristic
173C2×#103C2312yes—
174C2×#103C2311yescharacteristic
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#102C2311yescharacteristic
178C2×#102C2312yes—
179C2×#102C2312yes—
180C2×#102C2311yescharacteristic
181C2×#102C2312yes—
182#408—24no—
183#408—24no—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2314yes—
187C2×C4×C8—24no—
188C2×C4×C8C2312yes—
189C2×C4×C8C2312yes—
190#400—24no—
191#398—24no—
192#396—24no—
193#395—24no—
194#394—24no—
195#393—24no—
196#393—24no—
197#393—24no—
198#391—21no—
199#391—21no—
200#391—24no—
201#391—24no—
202#391D412yes—
203#391C2312yes—
204#391C2312yes—
205#390—21no—
206#390—21no—
207#390D412yes—
208#390C2312yes—
209#390C2312yes—
210#389—22no—
211#389—22no—
212#389C2312yes—
213#389C2312yes—
214#389C2314yes—
215#388—24no—
216#388—24no—
217#387—24no—
218#386—22no—
219#386—22no—
220#386—21no—
221#386—21no—
222#386—22no—
223#386D412yes—
224#386D412yes—
225#385—21no—
226#385—21no—
227#385—22no—
228#385—24no—
229#384C2312yes—
230#382C2312yes—
231#380—22no—
232#380—22no—
233#380C2314yes—
234#380C2314yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4C2314yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#340C2314yes—
248#340C2312yes—
249#336C2314yes—
250#336C2312yes—
251#336C2312yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C22×Q8—24no—
265C22×Q8—24no—
266C22×Q8—44no—
267C22×Q8—44no—
268C22×Q8—84no—
269C22×Q8—84no—
270C22×Q8C2×D411yescharacteristic
271C22×Q8C2411yescharacteristic
272C22×D4—21no—
273C22×D4—22no—
274C22×D4C2×D412yes—
275C22×D4C2×D411yescharacteristic
276C22×D4C2411yescharacteristic
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—22no—
282C23×C4—21no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2412yes—
286C23×C4C2411yescharacteristic
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8—22no—
297C22×C8C2×D414yes—
298C22×C8#4114yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2411yescharacteristic
302C22×C8C2411yescharacteristic
303C22×C8C2412yes—
304C4⋊Q8—28no—
305C4⋊Q8—28no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—48no—
309C4⋊Q8—48no—
310#124—88no—
311C22⋊Q8—24no—
312C22⋊Q8—42no—
313C22⋊Q8—44no—
314C22⋊Q8—88no—
315#121—24no—
316#121—42no—
317#121—44no—
318#121—48no—
319C4×Q8—44no—
320C4×D4—44no—
321C4×D4—48no—
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—24no—
331#117—42no—
332C2×C4⋊C4—21no—
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—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—42no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—44no—
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D414yes—
378C2×C22⋊C4C2×D414yes—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—24no—
386C2×C42C2×D412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389#107—24no—
390#107—42no—
391#106—24no—
392#106—24no—
393#106—28no—
394#106—28no—
395#106—24no—
396#106—42no—
397#106C2×D418yes—
398#106C2×D418yes—
399C4⋊C8—28no—
400#103—22no—
401#103—22no—
402#103—24no—
403#103—24no—
404#103—42no—
405#102—22no—
406#102—22no—
407#102—24no—
408#102—24no—
409#102—42no—
410C22⋊C8—24no—
411C22⋊C8—24no—
412C22⋊C8—28no—
413C4×C8—24no—
414C4×C8—24no—
415#95—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—21no—
428C24C22×D412yes—
429C24C22×D411yescharacteristic
430#41—44no—
431#41—44no—
432#41—44no—
433#41—44no—
434#41—44no—
435#41—44no—
436#41—44no—
437#41—44no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—48no—
448C2×Q8—48no—
449C2×Q8—48no—
450C2×Q8—48no—
451C2×Q8—84no—
452C2×Q8—84no—
453C2×Q8—816no—
454C2×Q8—816no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—24no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—44no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—84no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—84no—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D412yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C42−1+412yes—
513C22×C42−1+412yes—
514C22×C42−1+412yes—
515C22×C4C2511yescharacteristic
516M16—44no—
517M16—44no—
518C2×C8—28no—
519C2×C8—28no—
520C2×C8—22no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8C22×D412yes—
533C2×C8C22×D412yes—
534C2×C8C2×#4114yes—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—44no—
553C4⋊C4—48no—
554C4⋊C4—48no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—84no—
561C4⋊C4—84no—
562C4⋊C4C22×D414yes—
563C4⋊C4C22×D414yes—
564C22⋊C4—24no—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—84no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C42—24no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—44no—
584C42—42no—
585C42—42no—
586C42—42no—
587C42—84no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—24no—
594C23—24no—
595C23—41no—
596C23—41no—
597C23—41no—
598C23—42no—
599C23—42no—
600C23#53612yes—
601C23D4212yes—
602C23#54712yes—
603C23C23×D411yescharacteristic
604Q8—44no—
605Q8—41no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609Q8—816no—
610Q8—816no—
611D4—48no—
612D4—44no—
613D4—41no—
614D4—41no—
615D4—41no—
616D4—41no—
617D4—84no—
618D4—164no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—84no—
657C2×C4—84no—
658C2×C4—84no—
659C2×C4—84no—
660C2×C4—84no—
661C2×C4—164no—
662C2×C4#53612yes—
663C2×C4#53614yes—
664C2×C4#54214yes—
665C2×C4D4212yes—
666C2×C4#54712yes—
667C2×C4#54814yes—
668C2×C4#56314yes—
669C2×C4#57714yes—
670C2×C4#57714yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C8—22no—
678C8—22no—
679C8—24no—
680C8—42no—
681C8—42no—
682C22—24no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—44no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—44no—
691C22—82no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—82no—
696C22#267414yes—
697C22C2×#53611yescharacteristic
698C22C2×D4211yescharacteristic
699C22C2×#54711yescharacteristic
700C22C2×#57712yes—
701C22C2×#57712yes—
702C4—24no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—42no—
708C4—82no—
709C4—81no—
710C4—82no—
711C4—82no—
712C4—81no—
713C4—81no—
714C4—82no—
715C4#267414yes—
716C4C2×#53611yescharacteristic
717C4C2×#54211yescharacteristic
718C4C2×#54811yescharacteristic
719C4C2×#56311yescharacteristic
720C2—24no—
721C2—82no—
722C2—161no—
723C2C2×#267411yescharacteristic
724C2#6088112yes—
725C2#6105011yescharacteristic
726C2#6105011yescharacteristic
727C2#6128812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
1. C22×#498C2semidirect
4. #35574C2semidirect
5. #30780C2semidirect
6. #30777C2semidirect
7. #30775C2semidirect
8. #30774C2semidirect
9. #23738C2semidirect
10. #23640C2semidirect
11. #23610C2semidirect
12. #23552C2semidirect
14. #23452C2semidirect
15. #21174C2semidirect
16. #21083C2semidirect
17. #21046C2semidirect
18. #21038C2semidirect
19. #20848C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4982
C2×#30992
#355922
#355742
#307802
#307772
#307752
#307742
#237382
#236402
#236102
#235522
#234892
#234522
#211742
#210832
#210462
#210382
#208482