Finite Group Databasebeta

← #6446192

Subgroups — #6446192

Order 512.

2032 conjugacy classes of subgroups.

The subgroup lattice has 758 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic, complemented
3#34952C211yesmaximal, characteristic
4C2×#2567C211yesmaximal, characteristic, complemented
5#25487C214yesmaximal, complemented
6#25481C214yesmaximal, complemented
7#25480C212yesmaximal, complemented
8#25477C212yesmaximal, complemented
9#25471C212yesmaximal, complemented
10#25470C212yesmaximal, complemented
11#23670C212yesmaximal, complemented
12#23617C212yesmaximal
13#21324C211yesmaximal, characteristic, complemented
14#21170C211yesmaximal, characteristic, complemented
15#21159C212yesmaximal, complemented
16#21093C212yesmaximal
17C4×#417C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3079—28no—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#476C2212yes—
25C2×#474C2212yes—
26Q8×M16—28no—
27C2×#446C2212yes—
28C2×#446C2214yes—
29C2×C8×Q8C2212yes—
30C2×#428C2212yes—
31C2×#428C2212yes—
32C2×#423C2212yes—
33C2×#423C2211yescharacteristic
34C2×#417C2211yescharacteristic
35C2×#417C2212yes—
36C2×#417C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38C2×C4×M16C2212yes—
39#2582C2211yescharacteristic
40#2582C2211yescharacteristic
41#2581C2212yes—
42#2580C2212yes—
43#2579—22no—
44#2567—22no—
45#2567—21no—
46#2567—21no—
47#2567C2212yes—
48#2567C2212yes—
49#2567C2214yes—
50#2538C2211yescharacteristic
51#2538C2211yescharacteristic
52#2487—21no—
53#2409—21no—
54#2396—22no—
55C2×#395—22no—
56C4×C22⋊Q8—21no—
57C4×C22⋊Q8—21no—
58#2297C2212yes—
59#2296C2212yes—
60C2×#391—21no—
61C2×#389C2212yes—
62C2×#389C2212yes—
63#2285C2211yescharacteristic
64C2×#384—21no—
65C2×C4×C4⋊C4C2211yescharacteristic
66#2088C2214yes—
67#2088C2214yes—
68#2087C2214yes—
69#2086C2214yes—
70#2068C2214yes—
71#2067C2214yes—
72#2066C2214yes—
73#2065C2214yes—
74#1993C2214yes—
75#1990C2212yes—
76#1990C2212yes—
77#1949C2214yes—
78#1948C2212yes—
79#1947C2212yes—
80#1944C2214yes—
81#1943C2214yes—
82#1942C2214yes—
83#1941C2214yes—
84#1936C2214yes—
85#1935C2214yes—
86#1934C2214yes—
87#1933C2214yes—
88#1763C2212yes—
89#1762C2212yes—
90C4×#103C2212yes—
91C4×#102C2212yes—
92#1736C2212yes—
93#1736C2212yes—
94#1736C2212yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8—21no—
99C23×Q8D411yescharacteristic
100C22×M16C2311yescharacteristic
101C22×M16C2312yes—
102C22×M16C2311yescharacteristic
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—24no—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C4×#41—24no—
112C4×#41—24no—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8D412yes—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2311yescharacteristic
126C2×C4×D4C2311yescharacteristic
127C2×#117—21no—
128C2×#117C2311yescharacteristic
129C2×#117C2312yes—
130C22×C4⋊C4—21no—
131C22×C4⋊C4—22no—
132C22×C4⋊C4—22no—
133C22×C4⋊C4D412yes—
134C22×C4⋊C4D411yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C42—21no—
139C22×C42—21no—
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142#477—28no—
143#476—28no—
144#475—28no—
145#474—28no—
146#446—216no—
147#446—28no—
148C8×Q8—28no—
149#428—28no—
150#428—28no—
151C2×#107C2314yes—
152C2×#107C2314yes—
153C2×#106C2314yes—
154C2×#106C2314yes—
155#423—28no—
156#423D418yes—
157C2×C4⋊C8C2314yes—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2312yes—
161#417—28no—
162#417—24no—
163#417Q818yescomplemented
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#102C2312yes—
168C2×#102C2314yes—
169C2×#102C2312yes—
170C4×M16—28no—
171C4×M16—24no—
172C2×#97C2314yes—
173C2×#97C2312yes—
174C2×C4×C8C2314yes—
175C2×C4×C8C2312yes—
176#400—22no—
177#399—22no—
178#398—22no—
179#398—22no—
180#396—22no—
181#396—22no—
182#395—22no—
183#395—24no—
184#395—24no—
185#395—24no—
186#395—24no—
187#393—22no—
188#391—22no—
189#391—24no—
190#391—22no—
191#391—21no—
192#391—21no—
193#391C2312yes—
194#391C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—21no—
198#389—21no—
199#389—22no—
200#389—22no—
201#389—22no—
202#389—22no—
203#389C2314yes—
204#389C2314yes—
205#389C2314yes—
206#389C2314yes—
207#389C2312yes—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#387—22no—
212#387—22no—
213#386—22no—
214#386—22no—
215#385—22no—
216#385—22no—
217#385—22no—
218#385—22no—
219#384—22no—
220#384—22no—
221#384—24no—
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#382C2312yes—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4C2312yes—
236C4×C4⋊C4C2312yes—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C2×#95—21no—
240C2×#95—22no—
241C2×#95—22no—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2314yes—
246#340C2314yes—
247#340C2312yes—
248#340C2314yes—
249#340C2314yes—
250#340C2312yes—
251#340C2312yes—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41C2×Q814yes—
258C22×Q8—24no—
259C22×Q8—24no—
260C22×Q8—22no—
261C22×Q8—22no—
262C22×Q8—42no—
263C22×Q8—44no—
264C22×Q8—82no—
265C22×Q8—82no—
266C22×Q8C2×D411yescharacteristic
267C22×Q8C2×D412yes—
268C22×Q8C2×D411yescharacteristic
269C22×Q8C2411yescharacteristic
270C22×D4—21no—
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4C2×D412yes—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D412yes—
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×M16—22no—
284C2×M16—24no—
285C2×M16—24no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16C2×D418yes—
291C2×M16C2×Q814yes—
292C22×C8C2412yes—
293C22×C8C2414yes—
294C22×C8C2412yes—
295C4⋊Q8—24no—
296C4⋊Q8—24no—
297C4⋊Q8—24no—
298C4⋊Q8—24no—
299C4⋊Q8—44no—
300C4⋊Q8—44no—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C4×Q8—24no—
304C4×Q8—24no—
305C4×Q8—28no—
306C4×Q8—22no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—44no—
321C4×D4—44no—
322#117—22no—
323#117—22no—
324#117—24no—
325#117—24no—
326#117—44no—
327C2×C4⋊C4—28no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—84no—
354C2×C4⋊C4—84no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×Q814yes—
363C2×C4⋊C4#4114yes—
364C2×C4⋊C4C2414yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C42—24no—
380C2×C42—24no—
381C2×C42—21no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—44no—
388C2×C42—44no—
389C2×C42C2×D412yes—
390C2×C42C2×D412yes—
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D414yes—
394C2×C42#4114yes—
395C2×C42C2412yes—
396C2×C42C2412yes—
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399#107—28no—
400#107—28no—
401#106—28no—
402#106—28no—
403C4⋊C8—28no—
404C4⋊C8—28no—
405C4⋊C8C2×D418yes—
406C4⋊C8C2×D418yes—
407#103—28no—
408#103—24no—
409#103C2×Q818yescomplemented
410#102—28no—
411#102—24no—
412#102C2×Q818yescomplemented
413#97—28no—
414#97—24no—
415C4×C8—28no—
416C4×C8—24no—
417#95—22no—
418#95—22no—
419#95—24no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433C24—21no—
434C24C22×D411yescharacteristic
435#41—28no—
436#41—28no—
437#41—44no—
438#41—44no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—42no—
452C2×Q8—416no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×Q8—82no—
458C2×Q8—82no—
459C2×Q8—816no—
460C2×Q8#13714yes—
461C2×Q8C22×Q812yes—
462C2×D4—24no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4C22×Q812yes—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—21no—
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—24no—
487C22×C4—42no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—42no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—81no—
506C22×C4—81no—
507C22×C4#12114yes—
508C22×C4#12114yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×Q812yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4112yes—
522C22×C42−1+412yes—
523C22×C42−1+412yes—
524C22×C4C2511yescharacteristic
525M16—216no—
526M16—28no—
527M16—44no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8C22×D418yes—
536C2×C8C22×Q814yes—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—24no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—42no—
565C4⋊C4—44no—
566C4⋊C4—42no—
567C4⋊C4—81no—
568C4⋊C4—81no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—84no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4—84no—
575C4⋊C4C22×Q814yes—
576C4⋊C4C2×#4114yes—
577C22⋊C4—44no—
578C22⋊C4—42no—
579C22⋊C4—44no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—44no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C42—24no—
586C42—24no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—44no—
592C42—42no—
593C42—42no—
594C42—42no—
595C42—44no—
596C42—44no—
597C42#12114yes—
598C42#12114yes—
599C42C22×D412yes—
600C42C22×D412yes—
601C42C2×#4112yes—
602C42C2×#4112yes—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—22no—
608C23—41no—
609C23—41no—
610C23—41no—
611C23—41no—
612C23—41no—
613C23—41no—
614C23C2×#12112yes—
615C23#54812yes—
616C23D4×Q812yes—
617C23C23×D411yescharacteristic
618Q8—28no—
619Q8—22no—
620Q8—22no—
621Q8—48no—
622Q8—41no—
623Q8—41no—
624Q8—44no—
625Q8—88no—
626Q8—88no—
627D4—22no—
628D4—22no—
629D4—48no—
630D4—41no—
631D4—41no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—22no—
640C2×C4—28no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—44no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—48no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—41no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—88no—
682C2×C4—84no—
683C2×C4C2×#12112yes—
684C2×C4C2×#12112yes—
685C2×C4C2×#12112yes—
686C2×C4C2×#12112yes—
687C2×C4C2×#12112yes—
688C2×C4#53314yes—
689C2×C4#54114yes—
690C2×C4#54714yes—
691C2×C4#54812yes—
692C2×C4D4×Q812yes—
693C2×C4D4×Q814yes—
694C2×C4C2×#13714yes—
695C2×C4C2×#13712yes—
696C2×C4#57512yes—
697C2×C4C23×D411yescharacteristic
698C2×C4C23×Q811yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C22×#4111yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C2×C4C2×2−1+411yescharacteristic
703C8—28no—
704C8—24no—
705C8—42no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—81no—
716C22—81no—
717C22—81no—
718C22—81no—
719C22#242914yes—
720C22C22×#12111yescharacteristic
721C22C2×#54811yescharacteristic
722C22C2×D4×Q811yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57511yescharacteristic
725C22#358511yescharacteristic
726C22#358511yescharacteristic
727C4—22no—
728C4—24no—
729C4—22no—
730C4—22no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—44no—
736C4—44no—
737C4—81no—
738C4—81no—
739C4—84no—
740C4—82no—
741C4—82no—
742C4#242914yes—
743C4#305614yes—
744C4C2×#53311yescharacteristic
745C4C2×#54111yescharacteristic
746C4C2×#54711yescharacteristic
747C4C2×D4×Q811yescharacteristic
748C2—24no—
749C2—82no—
750C2C2×#242911yescharacteristic
751C2C2×#305611yescharacteristic
752C2#6045411yescharacteristic
753C2#6131811yescharacteristic
754C2#6131911yescharacteristic
755C2#6136711yescharacteristic
756C2Q8×#13711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×Q8×M16C2semidirect
4. C2×#2567C2semidirect
5. #25487C2semidirect
6. #25481C2semidirect
7. #25480C2semidirect
8. #25477C2semidirect
9. #25471C2semidirect
10. #25470C2semidirect
11. #23670C2semidirect
13. #21324C2semidirect
14. #21170C2semidirect
15. #21159C2semidirect
163. #417Q8semidirect
409. #103C2×Q8semidirect
412. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C2×Q8×M162
#349522
C2×#25672
#254872
#254812
#254802
#254772
#254712
#254702
#236702
#236172
#213242
#211702
#211592
#210932
C4×#4172