Finite Group Databasebeta

← #6445579

Subgroups — #6445579

Order 512.

2118 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic, complemented
2C2×C4×#137C211yesmaximal, characteristic
3#35333C211yesmaximal, characteristic
4#34182C211yesmaximal, characteristic, complemented
5#24826C214yesmaximal, complemented
6#24824C214yesmaximal, complemented
7#24821C214yesmaximal, complemented
8#24819C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic
12#23610C211yesmaximal, characteristic
13#21325C211yesmaximal, characteristic, complemented
14#21145C211yesmaximal, characteristic, complemented
15#20930C211yesmaximal, characteristic
16#20930C211yesmaximal, characteristic
17#20651C211yesmaximal, characteristic, complemented
18#20651C211yesmaximal, characteristic, complemented
19#20645C211yesmaximal, characteristic
20C22×#137C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#519C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C2×#479—28no—
27C2×#479C2214yes—
28C2×#479C2214yes—
29C4×#137—28no—
30C2×#441C2212yes—
31C2×#440C2212yes—
32C2×C4×Q16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#428C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C2×C4×M16C2211yescharacteristic
47#2851—22no—
48#2629—22no—
49#2579—22no—
50#2579—22no—
51#2566C2212yes—
52#2566C2212yes—
53#2565C2212yes—
54#2565C2212yes—
55#2351—22no—
56#2351C2214yes—
57Q8×C4⋊C4—22no—
58Q8×C4⋊C4C2214yes—
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2316—22no—
62#2316C2211yescharacteristic
63#2316C2211yescharacteristic
64#2311C2211yescharacteristic
65#2311C2211yescharacteristic
66#2285—22no—
67#2285C2211yescharacteristic
68#2285C2211yescharacteristic
69#2085C2214yes—
70#2085C2214yes—
71#2084C2214yes—
72#2083C2214yes—
73#2068C2214yes—
74#2067C2214yes—
75#2066C2214yes—
76#2065C2214yes—
77#1992C2214yes—
78#1991C2214yes—
79#1940C2214yes—
80#1939C2214yes—
81#1880C2214yes—
82#1880C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1805C2212yes—
86#1805C2214yes—
87#1805C2214yes—
88#1804C2212yes—
89#1803C2212yes—
90#1803C2214yes—
91#1803C2214yes—
92#1802C2212yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C2×#137—24no—
104C2×#137—24no—
105C2×#137Q818yes—
106C22×Q16C2312yes—
107C22×SD16C2312yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#125—24no—
115C2×#125—24no—
116C2×#125C2312yes—
117C2×#125C2312yes—
118#519—24no—
119#519—24no—
120#519—42no—
121#519—42no—
122C4×#41—24no—
123C4×#41—24no—
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—24no—
129C2×C4×Q8D418yes—
130C2×C4×Q8C2311yescharacteristic
131C2×C4×Q8C2311yescharacteristic
132C2×C4×Q8C2314yes—
133C2×C4×Q8C2314yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×D4—21no—
137C2×C4×D4—21no—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×#117—22no—
141C2×#117—22no—
142C2×#117—21no—
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C2×#117C2311yescharacteristic
146C2×#117C2311yescharacteristic
147C2×#117C2311yescharacteristic
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C4⋊C4C2311yescharacteristic
154C22×C42—21no—
155C22×C42—21no—
156C22×C42C2311yescharacteristic
157#479—216no—
158#479—216no—
159#441—28no—
160#440—28no—
161C4×Q16—28no—
162C4×SD16—28no—
163#428—24no—
164C2×#107—24no—
165C2×#107C2314yes—
166C2×#107C2312yes—
167C2×#106—24no—
168C2×#106C2314yes—
169C2×#106C2312yes—
170#424—42no—
171#424—42no—
172C2×C4⋊C8—24no—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×C4⋊C8C2314yes—
176C2×C4⋊C8C2314yes—
177#419—24no—
178#417—24no—
179#416—42no—
180#416—42no—
181C2×#103—24no—
182C2×#103—24no—
183C2×#103C2314yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2312yes—
189C2×#103C2312yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C4×M16—24no—
194C2×#97C2312yes—
195C2×C4×C8C2312yes—
196#398—24no—
197#398—24no—
198#396—24no—
199#396—24no—
200#391C2312yes—
201#391C2312yes—
202#390C2312yes—
203#390C2312yes—
204#389—24no—
205#389—24no—
206#389—22no—
207#389—22no—
208#389—22no—
209#389—22no—
210#389—24no—
211#389—24no—
212#389C2314yes—
213#389C2314yes—
214#387—24no—
215#387—24no—
216#386—22no—
217#386—22no—
218#385—22no—
219#385—22no—
220#385—24no—
221#385—24no—
222#384—24no—
223#384D412yes—
224#384C2312yes—
225#384C2312yes—
226#384C2312yes—
227#382—24no—
228#382D412yes—
229#382C2312yes—
230#382C2312yes—
231#382C2312yes—
232#380—21no—
233#380—21no—
234#380—22no—
235#380—22no—
236#380—22no—
237#380—22no—
238#380D412yes—
239#380D412yes—
240C4×C4⋊C4—24no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—24no—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
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×Q8C2×D414yes—
267C22×Q8C2×Q814yes—
268C22×Q8#4114yes—
269C22×Q8C2411yescharacteristic
270C22×Q8C2412yes—
271C22×D4—21no—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283#137—232no—
284C2×Q16—24no—
285C2×Q16—24no—
286C2×Q16C2×Q818yes—
287C2×SD16—24no—
288C2×SD16—24no—
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×C8—22no—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2412yes—
306#125—24no—
307#125—24no—
308#125—24no—
309#125—24no—
310#125—44no—
311#125—44no—
312C4×Q8—28no—
313C4×Q8—28no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—28no—
317C4×Q8—28no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—48no—
325C4×Q8—48no—
326C4×Q8C2×D418yes—
327C4×Q8C2×D418yes—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—44no—
333C4×D4—44no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—48no—
345#117—48no—
346#117—44no—
347C2×C4⋊C4—28no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—44no—
373C2×C4⋊C4—44no—
374C2×C4⋊C4C2×D412yes—
375C2×C4⋊C4C2×D412yes—
376C2×C4⋊C4C2×D414yes—
377C2×C4⋊C4C2×D414yes—
378C2×C4⋊C4C2×D414yes—
379C2×C4⋊C4C2412yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2411yescharacteristic
382C2×C4⋊C4C2411yescharacteristic
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2412yes—
385C2×C4⋊C4C2412yes—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D412yes—
395C2×C42—28no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—24no—
400C2×C42—24no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—44no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D414yes—
409C2×C42#4114yes—
410C2×C42C2412yes—
411C2×C42C2412yes—
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414#107—28no—
415#107—24no—
416#106—28no—
417#106—24no—
418C4⋊C8—24no—
419C4⋊C8—24no—
420C4⋊C8—44no—
421C4⋊C8—44no—
422#103—28no—
423#103—28no—
424#103—24no—
425#103—24no—
426#103—24no—
427#103—42no—
428#103—42no—
429#102—24no—
430#102—42no—
431#102—42no—
432#97—24no—
433C4×C8—24no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95—22no—
438#95—24no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95C2×D414yes—
443#95C2×D414yes—
444C24—21no—
445C24C22×D411yescharacteristic
446#41—28no—
447#41—28no—
448#41—44no—
449#41—44no—
450C2×Q8—24no—
451C2×Q8—28no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—28no—
455C2×Q8—28no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C22⋊Q8116yes—
465C2×Q8C22×D414yes—
466C2×Q8C22×Q812yes—
467C2×Q8C22×Q814yes—
468C2×Q8C2×#4114yes—
469C2×D4—24no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4C22×Q812yes—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4#12114yes—
506C22×C4#12114yes—
507C22×C4C22×D414yes—
508C22×C4C22×D414yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×Q812yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C42−1+412yes—
520C22×C4C2511yescharacteristic
521Q16—216no—
522SD16—216no—
523M16—28no—
524M16—44no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—44no—
537C2×C8—44no—
538C2×C8C22×Q814yes—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—48no—
560C4⋊C4—48no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—44no—
567C4⋊C4—44no—
568C4⋊C4—42no—
569C4⋊C4—42no—
570C4⋊C4—82no—
571C4⋊C4—84no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C4⋊C4C22×D414yes—
576C22⋊C4—24no—
577C22⋊C4—24no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C22⋊C4—84no—
584C22⋊C4—84no—
585C42—28no—
586C42—28no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—48no—
592C42—48no—
593C42—42no—
594C42—42no—
595C42—42no—
596C42—42no—
597C42—42no—
598C42#12114yes—
599C42#12114yes—
600C42C22×D412yes—
601C42C22×D412yes—
602C42C2×#4112yes—
603C42C2×#4112yes—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—42no—
608C23—42no—
609C23—41no—
610C23—41no—
611C23C2×#12112yes—
612C23#54612yes—
613C23D4×Q812yes—
614C23C23×D411yescharacteristic
615Q8—216no—
616Q8—24no—
617Q8—24no—
618Q8—24no—
619Q8—22no—
620Q8—22no—
621Q8—41no—
622Q8—41no—
623Q8C2×C22⋊Q818yes—
624D4—22no—
625D4—22no—
626D4—48no—
627D4—41no—
628D4—41no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—216no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—48no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—84no—
666C2×C4—84no—
667C2×C4—84no—
668C2×C4—84no—
669C2×C4—84no—
670C2×C4C2×#12112yes—
671C2×C4C2×#12112yes—
672C2×C4C2×#12112yes—
673C2×C4C2×#12112yes—
674C2×C4C2×#12112yes—
675C2×C4C2×C22⋊Q814yes—
676C2×C4#53714yes—
677C2×C4#54612yes—
678C2×C4D4×Q812yes—
679C2×C4#55214yes—
680C2×C4#55414yes—
681C2×C4C23×D411yescharacteristic
682C2×C4C23×Q811yescharacteristic
683C2×C4C22×#4111yescharacteristic
684C2×C4C22×#4111yescharacteristic
685C2×C4C2×H(5,𝔽2)11yescharacteristic
686C2×C4C2×2−1+411yescharacteristic
687C8—24no—
688C8—42no—
689C8—42no—
690C8—42no—
691C22—22no—
692C22—22no—
693C22—24no—
694C22—24no—
695C22—42no—
696C22—44no—
697C22—81no—
698C22—81no—
699C22—82no—
700C22#241714yes—
701C22C22×#12111yescharacteristic
702C22C2×#54611yescharacteristic
703C22C2×D4×Q811yescharacteristic
704C22#358412yes—
705C22#358612yes—
706C4—22no—
707C4—22no—
708C4—22no—
709C4—22no—
710C4—28no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—42no—
715C4—82no—
716C4—81no—
717C4—82no—
718C4—81no—
719C4—82no—
720C4—82no—
721C4#241714yes—
722C4C22×C22⋊Q811yescharacteristic
723C4C2×#53711yescharacteristic
724C4C2×#55211yescharacteristic
725C4C2×#55411yescharacteristic
726C2—24no—
727C2—82no—
728C2C2×#241711yescharacteristic
729C2#6042211yescharacteristic
730C2#6042411yescharacteristic
731C2#6120412yes—
732C2#6138612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#479C2semidirect
4. #34182C2semidirect
5. #24826C2semidirect
6. #24824C2semidirect
7. #24821C2semidirect
8. #24819C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
13. #21325C2semidirect
14. #21145C2semidirect
17. #20651C2semidirect
18. #20651C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×C4×#1372
#353332
#341822
#248262
#248242
#248212
#248192
#236612
#236602
#236112
#236102
#213252
#211452
#209302
#206512
#206452