Finite Group Databasebeta

← #6449927

Subgroups — #6449927

Order 512.

2018 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic, complemented
2C2×#2925C211yesmaximal, characteristic
3#35280C211yesmaximal, characteristic
4C4×#536C211yesmaximal, characteristic
5#24719C214yesmaximal, complemented
6#24715C214yesmaximal, complemented
7#24711C212yesmaximal, complemented
8#24710C212yesmaximal, complemented
9#24703C212yesmaximal, complemented
10#24702C212yesmaximal, complemented
11#23643C212yesmaximal
12#23484C212yesmaximal
13#21311C211yesmaximal, characteristic, complemented
14#21171C211yesmaximal, characteristic, complemented
15#21160C212yesmaximal, complemented
16#21088C212yesmaximal, complemented
17C2×#1803C211yesmaximal, characteristic, complemented
18C2×#536C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#3078—28no—
23C2×#479C2214yes—
24C2×#477C2212yes—
25C2×#475C2212yes—
26#2925—28no—
27C2×#433C2214yes—
28C2×#432C2212yes—
29C2×#431C2212yes—
30C2×#428C2212yes—
31C2×#428C2212yes—
32C2×#423C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#419C2212yes—
35C22×#103C2212yes—
36C2×#408C2212yes—
37C2×#405C2212yes—
38#2857C2212yes—
39#2857C2212yes—
40#2629C2214yes—
41#2582—21no—
42#2582—21no—
43#2538—21no—
44#2538—21no—
45#2417—22no—
46#2417—22no—
47#2327—21no—
48#2326—21no—
49#2323C2212yes—
50#2323C2212yes—
51#2322—22no—
52#2317C2211yescharacteristic
53#2316C2212yes—
54#2311C2211yescharacteristic
55#2311C2212yes—
56#2310C2211yescharacteristic
57C4×C4⋊Q8—21no—
58C4×#124—21no—
59C4×C22⋊Q8C2212yes—
60C4×C22⋊Q8C2212yes—
61C4×#123—22no—
62C2×#384C2211yescharacteristic
63C4×#117C2211yescharacteristic
64#2077C2214yes—
65#2077C2214yes—
66#2076C2214yes—
67#2075C2214yes—
68#2036C2214yes—
69#2036C2214yes—
70#2035C2214yes—
71#2035C2214yes—
72#1989C2212yes—
73#1989C2212yes—
74#1988C2211yescharacteristic
75#1988C2211yescharacteristic
76#1987C2211yescharacteristic
77#1987C2211yescharacteristic
78#1948C2212yes—
79#1948C2212yes—
80#1947C2212yes—
81#1947C2212yes—
82#1944C2214yes—
83#1943C2214yes—
84#1942C2214yes—
85#1941C2214yes—
86#1935C2214yes—
87#1935C2214yes—
88#1933C2214yes—
89#1933C2214yes—
90#1803C2212yes—
91#1803C2212yes—
92#1803C2214yes—
93#1736C2212yes—
94#1735C2212yes—
95#1735C2212yes—
96C2×#340C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2312yes—
100C23×C8C2312yes—
101#536—24no—
102#536—24no—
103#533—24no—
104#533—24no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#125C2312yes—
109C2×#124—21no—
110C2×#123—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—22no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C4×#41—24no—
118C4×#41—24no—
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×C4×Q8D411yescharacteristic
126C2×C4×Q8D411yescharacteristic
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4D412yes—
130C2×C4×D4D411yescharacteristic
131C2×C4×D4D412yes—
132C2×C4×D4D411yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2312yes—
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2312yes—
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—21no—
141C22×C4⋊C4C2312yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C42—21no—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#479—216no—
147#477—28no—
148#475—28no—
149#433—216no—
150#432—28no—
151#431—28no—
152#428—28no—
153#428—28no—
154C2×#107C2314yes—
155C2×#107C2314yes—
156C2×#106C2314yes—
157C2×#106C2314yes—
158#423D418yes—
159C2×C4⋊C8—24no—
160C2×C4⋊C8C2311yescharacteristic
161C2×C4⋊C8C2311yescharacteristic
162C2×C4⋊C8C2312yes—
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165#419—28no—
166C2×#103—28no—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#103C2314yes—
171#408—28no—
172C2×C22⋊C8C2314yes—
173C2×C22⋊C8C2314yes—
174#405—28no—
175C2×#97C2314yes—
176C2×C4×C8C2314yes—
177#398—24no—
178#398—24no—
179#396—24no—
180#396—24no—
181#391—21no—
182#391—22no—
183#391—21no—
184#391—21no—
185#391—22no—
186#391—22no—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#390—22no—
191#389—22no—
192#389—22no—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389C2314yes—
198#389C2314yes—
199#389C2312yes—
200#389C2312yes—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—24no—
211#385—22no—
212#385—22no—
213#385—22no—
214#384—21no—
215#384—21no—
216#384—21no—
217#384C2312yes—
218#384C2312yes—
219#382—21no—
220#381—22no—
221#380—22no—
222#380—22no—
223#380—24no—
224#380C2312yes—
225#380C2312yes—
226#380C2314yes—
227#380C2314yes—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4C2311yescharacteristic
231C4×C4⋊C4C2311yescharacteristic
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—21no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238#376—21no—
239C43—21no—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2312yes—
250#340C2312yes—
251#340C2312yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C22×Q8—21no—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—22no—
264C22×Q8—22no—
265C22×Q8—22no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8#4114yes—
268C22×Q8C2412yes—
269C22×D4C2×D412yes—
270C22×D4C2×D411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—21no—
274C23×C4C2×D412yes—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C2×M16—24no—
281C2×M16—24no—
282C2×M16C2×D418yes—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—28no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294#125—24no—
295#125—24no—
296#124—28no—
297#123—216no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—44no—
303C22⋊Q8—44no—
304C22⋊Q8—84no—
305C22⋊Q8—84no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×D4—28no—
317C4×D4—28no—
318C4×D4—24no—
319C4×D4—24no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—24no—
324#117—24no—
325#117—24no—
326#117—24no—
327#117—22no—
328#117—22no—
329#117#4114yes—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—82no—
354C2×C4⋊C4—82no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2414yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—24no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4C2×D412yes—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D412yes—
379C2×C42—21no—
380C2×C42—24no—
381C2×C42—24no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—44no—
391C2×C42C2×D412yes—
392C2×C42C2×D412yes—
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D414yes—
396C2×C42#4114yes—
397C2×C42C2412yes—
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—28no—
401#107—28no—
402#106—28no—
403#106—28no—
404C4⋊C8—22no—
405C4⋊C8—22no—
406C4⋊C8—24no—
407C4⋊C8C2×D418yes—
408C4⋊C8C2×D418yes—
409#103—28no—
410#103—28no—
411#103—28no—
412#103—28no—
413C22⋊C8—28no—
414C22⋊C8—28no—
415#97—28no—
416C4×C8—28no—
417#95—21no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433C24C22×D412yes—
434C24C22×D411yescharacteristic
435#41—44no—
436#41—44no—
437#41—44no—
438#41—44no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—28no—
445C2×Q8—28no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—44no—
457C2×Q8—44no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8C2×#4114yes—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—81no—
498C22×C4—81no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4#12114yes—
502C22×C4#12114yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C42−1+412yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518M16—216no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—48no—
528C2×C8C22×D418yes—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4C2×#4112yes—
566C4⋊C4C2×#4112yes—
567C22⋊C4—28no—
568C22⋊C4—28no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—24no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C42—24no—
580C42—24no—
581C42—24no—
582C42—22no—
583C42—22no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42—42no—
588C42—42no—
589C42—42no—
590C42—81no—
591C42—81no—
592C42#12114yes—
593C42#12114yes—
594C42C22×D412yes—
595C42C22×D412yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C23—21no—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23C2×#12112yes—
610C23#52512yes—
611C23#53612yes—
612C23C23×D411yescharacteristic
613Q8—24no—
614Q8—24no—
615Q8—48no—
616Q8—44no—
617Q8—42no—
618Q8—42no—
619Q8—84no—
620Q8—84no—
621D4—48no—
622D4—44no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—28no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—48no—
647C2×C4—42no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—48no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—164no—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4#52512yes—
682C2×C4#53612yes—
683C2×C4#54714yes—
684C2×C4#54814yes—
685C2×C4#55214yes—
686C2×C4#55614yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4C23×D411yescharacteristic
690C2×C4C22×#4111yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C2×C4C2×2−1+411yescharacteristic
694C2×C4C2×2−1+411yescharacteristic
695C8—28no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—42no—
710C22—42no—
711C22#241514yes—
712C22C22×#12111yescharacteristic
713C22C2×#52511yescharacteristic
714C22C2×#53611yescharacteristic
715C22C2×#57711yescharacteristic
716C22C2×#57811yescharacteristic
717C22#358511yescharacteristic
718C22#358611yescharacteristic
719C4—22no—
720C4—22no—
721C4—22no—
722C4—42no—
723C4—42no—
724C4—42no—
725C4—44no—
726C4—42no—
727C4—42no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—84no—
732C4—81no—
733C4—81no—
734C4#241514yes—
735C4#306314yes—
736C4C2×#54711yescharacteristic
737C4C2×#54811yescharacteristic
738C4C2×#55211yescharacteristic
739C4C2×#55611yescharacteristic
740C2—24no—
741C2—82no—
742C2C2×#241511yescharacteristic
743C2C2×#306311yescharacteristic
744C2#6045611yescharacteristic
745C2#6059311yescharacteristic
746C2#6059411yescharacteristic
747C2#6088911yescharacteristic
748C2#6089311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3078C2semidirect
5. #24719C2semidirect
6. #24715C2semidirect
7. #24711C2semidirect
8. #24710C2semidirect
9. #24703C2semidirect
10. #24702C2semidirect
13. #21311C2semidirect
14. #21171C2semidirect
15. #21160C2semidirect
16. #21088C2semidirect
17. C2×#1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29252
#352802
C4×#5362
#247192
#247152
#247112
#247102
#247032
#247022
#236432
#234842
#213112
#211712
#211602
#210882
C2×#18032