Finite Group Databasebeta

← #7503936

Subgroups — #7503936

Order 512.

1920 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic, complemented
3#35639C211yesmaximal, characteristic
4#35154C211yesmaximal, characteristic, complemented
5#31249C214yesmaximal, complemented
6#31248C214yesmaximal, complemented
7#31243C214yesmaximal, complemented
8#31242C214yesmaximal, complemented
9#23615C211yesmaximal, characteristic, complemented
10#23613C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic
12#23609C211yesmaximal, characteristic
13#23555C211yesmaximal, characteristic, complemented
14#23553C211yesmaximal, characteristic, complemented
15#21234C211yesmaximal, characteristic
16#21157C211yesmaximal, characteristic, complemented
17#21147C211yesmaximal, characteristic, complemented
18#20931C211yesmaximal, characteristic
19#20926C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#3084—28no—
26C2×#479—28no—
27C2×#479C2214yes—
28C2×#479C2214yes—
29C2×#479C2212yes—
30C2×#475C2212yes—
31C2×#478C2212yes—
32C2×#476C2212yes—
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C22×C4⋊C8C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C2×#405C2211yescharacteristic
48#2855—22no—
49#2627—22no—
50#2579—22no—
51#2487—22no—
52#2454C2214yes—
53#2453—22no—
54#2453C2214yes—
55#2453C2214yes—
56#2451—22no—
57#2451C2214yes—
58#2442—22no—
59#2441—22no—
60#2317C2211yescharacteristic
61#2316C2211yescharacteristic
62#2316C2211yescharacteristic
63#2311C2211yescharacteristic
64#2310C2211yescharacteristic
65#2310C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68#2296C2211yescharacteristic
69#2068C2214yes—
70#2068C2214yes—
71#2068C2214yes—
72#2068C2212yes—
73#2067C2214yes—
74#2067C2212yes—
75#2066C2212yes—
76#2066C2214yes—
77#2066C2214yes—
78#2065C2212yes—
79#2059C2214yes—
80#2059C2214yes—
81#2059C2214yes—
82#2057C2214yes—
83#1973C2212yes—
84#1971C2212yes—
85#1969C2212yes—
86#1968C2212yes—
87#1944C2214yes—
88#1941C2214yes—
89#1940C2214yes—
90#1939C2214yes—
91#1880C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1877C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C2×#137—42no—
104C2×#137—42no—
105C2×#137Q818yescomplemented
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112#544—24no—
113#544—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8C2312yes—
116C2×#125—21no—
117C2×#125—24no—
118C2×#125—22no—
119C2×#125—42no—
120C2×#125C2312yes—
121C2×#125C2314yes—
122C2×C22⋊Q8—22no—
123#520—24no—
124#520—42no—
125C2×C4×Q8—22no—
126C2×C4×Q8—24no—
127C2×C4×Q8—42no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2314yes—
130C2×C4×D4—22no—
131C2×C4×D4C2312yes—
132C2×#117—21no—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147#479—216no—
148#479—216no—
149#479—28no—
150#478—28no—
151#476—28no—
152#475—28no—
153#428—24no—
154#428—42no—
155#427—42no—
156C2×#107—24no—
157C2×#107C2312yes—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106D418yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167#424—24no—
168#424—42no—
169C2×C4⋊C8—24no—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2314yes—
175#419—42no—
176#417—24no—
177#416—24no—
178#416—42no—
179C2×#103—24no—
180C2×#103—24no—
181C2×#103C2311yescharacteristic
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2311yescharacteristic
187C2×#103C2314yes—
188C2×#103C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2312yes—
191C2×#102C2311yescharacteristic
192C2×#102C2312yes—
193#405—42no—
194C2×#97C2312yes—
195C2×C4×C8C2312yes—
196#400—24no—
197#400—24no—
198#399—24no—
199#398—24no—
200#398—24no—
201#398—24no—
202#396—24no—
203#395—24no—
204#391—21no—
205#391—22no—
206#391—21no—
207#391—24no—
208#391C2312yes—
209#391C2314yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#390C2312yes—
214#390C2312yes—
215#389—22no—
216#389—22no—
217#389—24no—
218#389—24no—
219#389D412yes—
220#389D412yes—
221#389C2314yes—
222#389C2312yes—
223#389C2312yes—
224#389C2314yes—
225#389C2314yes—
226#387—24no—
227#387—24no—
228#387—24no—
229#386—21no—
230#386—21no—
231#386—22no—
232#385—21no—
233#385—21no—
234#385—24no—
235#385—22no—
236#385D412yes—
237#385D412yes—
238#384—21no—
239#382—21no—
240#381—22no—
241#381—22no—
242#380—22no—
243#380—22no—
244#380—22no—
245#380—24no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—22no—
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41C2×Q814yes—
262C22×Q8—21no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8C2×Q814yes—
270C22×Q8C2412yes—
271C22×Q8C2411yescharacteristic
272C22×D4—21no—
273C22×D4C2411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—22no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283#137—416no—
284C2×Q16—42no—
285C2×Q16—42no—
286C2×Q16C2×Q818yes—
287C2×SD16—42no—
288C2×SD16—42no—
289C2×SD16C2×Q818yes—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16C2×Q814yes—
296C22×C8—22no—
297C22×C8—22no—
298C22×C8—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8C2×D414yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2411yescharacteristic
305C22×C8C2411yescharacteristic
306C22×C8C2412yes—
307C4⋊Q8—24no—
308C4⋊Q8—42no—
309C4⋊Q8—44no—
310#125—28no—
311#125—24no—
312#125—44no—
313#125—42no—
314#125—48no—
315#125—44no—
316C22⋊Q8—48no—
317C4×Q8—28no—
318C4×Q8—24no—
319C4×Q8—44no—
320C4×Q8—42no—
321C4×Q8—48no—
322C4×D4—24no—
323C4×D4—42no—
324C4×D4—48no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—44no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4—44no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—24no—
389C2×C42—44no—
390C2×C42C2×D412yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2412yes—
394#107—28no—
395#107—24no—
396#107—42no—
397#107—42no—
398#106—24no—
399#106—42no—
400#106—42no—
401#106C2×D418yes—
402#106C2×D418yes—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—44no—
408#103—24no—
409#103—24no—
410#103—22no—
411#103—22no—
412#103—28no—
413#103—24no—
414#103—42no—
415#103—44no—
416#102—22no—
417#102—22no—
418#102—24no—
419#102—42no—
420#97—42no—
421C4×C8—42no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95C2×D414yes—
435#95C2×D414yes—
436C24—21no—
437C24C22×D411yescharacteristic
438#41—28no—
439#41—28no—
440#41—44no—
441#41—44no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—21no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—22no—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—22no—
454C2×Q8—21no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—44no—
458C2×Q8—42no—
459C2×Q8—44no—
460C2×Q8—84no—
461C2×Q8C22×Q814yes—
462C2×Q8C22×Q812yes—
463C2×D4—24no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4C22×Q812yes—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—84no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D412yes—
506C22×C4C22×Q812yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C4H(5,𝔽2)12yes—
509C22×C42−1+412yes—
510C22×C42−1+412yes—
511C22×C4C2511yescharacteristic
512Q16—48no—
513SD16—48no—
514M16—28no—
515M16—44no—
516C2×C8—28no—
517C2×C8—21no—
518C2×C8—22no—
519C2×C8—21no—
520C2×C8—22no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—44no—
530C2×C8C22×D412yes—
531C2×C8C22×D412yes—
532C2×C8C22×Q814yes—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—48no—
554C4⋊C4—48no—
555C4⋊C4—48no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—84no—
562C4⋊C4—82no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4C22×D414yes—
566C4⋊C4C22×D414yes—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—84no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C42—24no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—48no—
584C42—44no—
585C42—42no—
586C42—42no—
587C42—42no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—42no—
592C23—42no—
593C23—41no—
594C23—41no—
595C23#53712yes—
596C23D4212yes—
597C23D4×Q812yes—
598C23C23×D411yescharacteristic
599Q8—24no—
600Q8—22no—
601Q8—24no—
602Q8—22no—
603Q8—42no—
604Q8—41no—
605Q8—48no—
606Q8—42no—
607Q8—41no—
608Q8—164no—
609D4—22no—
610D4—22no—
611D4—48no—
612D4—41no—
613D4—41no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—48no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—48no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—41no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4—164no—
657C2×C4#53714yes—
658C2×C4#53712yes—
659C2×C4#54314yes—
660C2×C4D4212yes—
661C2×C4D4×Q812yes—
662C2×C4D4×Q814yes—
663C2×C4#56314yes—
664C2×C4C23×D411yescharacteristic
665C2×C4C23×Q811yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×H(5,𝔽2)11yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4C2×2−1+411yescharacteristic
670C8—22no—
671C8—22no—
672C8—24no—
673C8—42no—
674C8—42no—
675C22—24no—
676C22—22no—
677C22—22no—
678C22—24no—
679C22—42no—
680C22—44no—
681C22—82no—
682C22—81no—
683C22—81no—
684C22#268314yes—
685C22C2×#53711yescharacteristic
686C22C2×D4211yescharacteristic
687C22C2×D4×Q811yescharacteristic
688C22#358412yes—
689C22#358612yes—
690C4—44no—
691C4—42no—
692C4—42no—
693C4—44no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—42no—
698C4—44no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—81no—
704C4#268314yes—
705C4C2×#53711yescharacteristic
706C4C2×#54311yescharacteristic
707C4C2×D4×Q811yescharacteristic
708C4C2×#56311yescharacteristic
709C2—24no—
710C2—82no—
711C2C2×#268311yescharacteristic
712C2#6093212yes—
713C2#6105211yescharacteristic
714C2#6106011yescharacteristic
715C2#6138612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#479C2semidirect
4. #35154C2semidirect
5. #31249C2semidirect
6. #31248C2semidirect
7. #31243C2semidirect
8. #31242C2semidirect
9. #23615C2semidirect
10. #23613C2semidirect
13. #23555C2semidirect
14. #23553C2semidirect
16. #21157C2semidirect
17. #21147C2semidirect
105. C2×#137Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C22×#4792
#356392
#351542
#312492
#312482
#312432
#312422
#236152
#236132
#236112
#236092
#235552
#235532
#212342
#211572
#211472
#209312
#209262