Finite Group Databasebeta

← #6380875

Subgroups — #6380875

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×SD16C211yesmaximal, characteristic
2#34128C211yesmaximal, characteristic, complemented
3#34126C211yesmaximal, characteristic
4#22806C212yesmaximal, complemented
5#22773C212yesmaximal, complemented
6#22768C212yesmaximal, complemented
7#22683C212yesmaximal, complemented
8#22609C212yesmaximal, complemented
9#22575C212yesmaximal, complemented
10#22570C212yesmaximal
11C22⋊C4×SD16C212yesmaximal
12#21274C212yesmaximal, complemented
13#21120C212yesmaximal, complemented
14#21114C212yesmaximal, complemented
15#20818C211yesmaximal, characteristic, complemented
16#20735C211yesmaximal, characteristic, complemented
17#20733C211yesmaximal, characteristic
18C42×SD16C212yesmaximal, complemented
19C8×#117C211yesmaximal, characteristic, complemented
20C23×SD16C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C2×C4×SD16C4116yescomplemented
24C2×C4×SD16C2214yes—
25C2×C4×SD16C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×C4×SD16C2211yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×C4×SD16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×C4×SD16C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×C4×C8C2211yescharacteristic
38#2357C2212yes—
39#2356—22no—
40#2356C2212yescomplemented
41#2353—22no—
42#2351C2212yescomplemented
43#2349C2212yes—
44#2345C2212yes—
45#2344—22no—
46#2344C2212yes—
47#2342—22no—
48Q8×C22⋊C4C2212yes—
49D4×C22⋊C4C2212yes—
50#2295—22no—
51#2295C2212yescomplemented
52#2293C2212yes—
53#2290—22no—
54#2283—21no—
55#2283C2211yescharacteristic
56#2283C2211yescharacteristic
57#2280—21no—
58C42×Q8C2212yescomplemented
59C42×D4C2212yes—
60C4×#117—21no—
61C4×#117C2211yescharacteristic
62C2×C4×C22⋊C4—21no—
63#1968C2214yes—
64#1941C2214yescomplemented
65#1934C2214yes—
66#1933C2214yescomplemented
67#1928C2214yescomplemented
68#1925C2214yes—
69#1920C2214yescomplemented
70#1919C2214yescomplemented
71#1880C2214yes—
72#1877C2214yes—
73#1863C2214yescomplemented
74#1862C2214yes—
75#1846C2212yescomplemented
76#1835C2212yescomplemented
77#1824C2212yes—
78#1823C2212yes—
79#1808C2212yes—
80#1807C2212yes—
81#1805C2212yescomplemented
82#1805C2212yescomplemented
83#1804C2212yes—
84#1804C2212yes—
85#1787C2212yes—
86#1786C2212yes—
87C4×#107C2212yescomplemented
88C4×#106C2214yescomplemented
89C4×#106C2212yescomplemented
90C8×C4⋊C4C2212yescomplemented
91C8×C4⋊C4C2212yescomplemented
92C4×#103C2214yescomplemented
93C4×#103C2212yescomplemented
94C4×#103C2212yescomplemented
95C4×#102C2214yes—
96C4×#102C2212yes—
97C4×#102C2212yes—
98C8×C22⋊C4C2212yes—
99C8×C22⋊C4C2212yes—
100#1744C2212yescomplemented
101C42×C8C2212yescomplemented
102C23×Q8C2311yescharacteristic
103C23×D4C2311yescharacteristic
104C24×C4—21no—
105C22×SD16—28no—
106C22×SD16C2×C418yes—
107C22×SD16C2311yescharacteristic
108C22×SD16C2311yescharacteristic
109C22×SD16C2311yescharacteristic
110C22×SD16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—24no—
116C2×C4×Q8C2×C418yescomplemented
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4C2×C418yes—
126C2×C4×D4C2314yes—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×C4×D4C2311yescharacteristic
130C2×C4×D4C2311yescharacteristic
131C2×#117—22no—
132C2×#117—22no—
133C2×#117D412yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4C2311yescharacteristic
136C22×C4⋊C4C2311yescharacteristic
137C22×C22⋊C4—21no—
138C22×C22⋊C4—22no—
139C22×C42—21no—
140C22×C42—22no—
141C22×C42C2311yescharacteristic
142C4×SD16—216no—
143C4×SD16C2×C418yes—
144C4×SD16C2×C418yes—
145C4×SD16C2×C418yes—
146C4×SD16C2×C418yes—
147C4×SD16C2×C418yescomplemented
148C4×SD16C2×C418yescomplemented
149C4×SD16C2×C418yescomplemented
150C4×SD16C2×C418yescomplemented
151#431—24no—
152C2×#107C2314yes—
153C2×#106C2×C418yescomplemented
154C2×#106C2314yes—
155C2×#106C2311yescharacteristic
156C2×#106C2311yescharacteristic
157C2×#106C2311yescharacteristic
158C2×#106C2311yescharacteristic
159C2×C4⋊C8C2312yes—
160C2×#103C2×C418yescomplemented
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#103C2311yescharacteristic
164C2×#103C2311yescharacteristic
165C2×#103C2311yescharacteristic
166C2×#103C2311yescharacteristic
167C2×#102C2×C418yes—
168C2×#102C2314yes—
169C2×#102C2314yes—
170C2×#102C2311yescharacteristic
171C2×#102C2311yescharacteristic
172C2×#102C2311yescharacteristic
173C2×#102C2311yescharacteristic
174C2×C22⋊C8C2312yes—
175C2×C4×C8C2×C418yescomplemented
176C2×C4×C8C2312yes—
177C2×C4×C8C2314yes—
178C2×C4×C8C2314yes—
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183#391—24no—
184#391C2314yes—
185#390C2314yes—
186#389—24no—
187#389C2314yes—
188#389C2314yes—
189#388—24no—
190#388—24no—
191#387—24no—
192#387—24no—
193#387—24no—
194#387—24no—
195#386—24no—
196#385—24no—
197#383—21no—
198#383—21no—
199#383—22no—
200#382—22no—
201#382C2312yes—
202#382C2312yes—
203#381—21no—
204#381—21no—
205#381—22no—
206#381—22no—
207#380—22no—
208#380—22no—
209#380—22no—
210#380C2312yes—
211#380C2312yes—
212#379—22no—
213C4×C4⋊C4—24no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4D412yes—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C4⋊C4C2312yes—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—24no—
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D411yescharacteristic
232C4×C22⋊C4D411yescharacteristic
233C4×C22⋊C4C2312yes—
234#376—22no—
235#376—21no—
236#376—21no—
237#376—22no—
238#376D412yes—
239C2×#95—22no—
240C43—22no—
241C43C2312yes—
242#340C2314yes—
243#340C2314yes—
244#340C2314yes—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#336C2314yes—
249#336C2314yes—
250C25—21no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—24no—
254C22×Q8—42no—
255C22×Q8C22×C414yes—
256C22×Q8C2411yescharacteristic
257C22×Q8C2411yescharacteristic
258C22×D4—24no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—42no—
262C22×D4C22×C414yes—
263C22×D4C2411yescharacteristic
264C22×D4C2411yescharacteristic
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—28no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—24no—
274C23×C4—84no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—232no—
281C2×SD16—24no—
282C2×SD16C22×C414yes—
283C2×SD16C22×C414yes—
284C2×SD16C22×C414yes—
285C2×SD16C22×C414yes—
286C2×SD16#4118yes—
287C22×C8—24no—
288C22×C8—42no—
289C22×C8C22×C414yes—
290C22×C8C22×C414yes—
291C22×C8C2414yes—
292C22×C8C2411yescharacteristic
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C22×C8C2411yescharacteristic
296C4×Q8—28no—
297C4×Q8—28no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—22no—
301C4×Q8—22no—
302C4×Q8—28no—
303C4×Q8—44no—
304C4×Q8C22×C414yes—
305C4×Q8C22×C414yes—
306C4×Q8C22×C414yes—
307C4×Q8C22×C414yes—
308C4×D4—28no—
309C4×D4—28no—
310C4×D4—28no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—44no—
316C4×D4C22×C414yes—
317C4×D4C22×C414yes—
318C4×D4C22×C414yes—
319C4×D4C22×C414yes—
320#117—28no—
321#117—48no—
322#117—84no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C22×C414yes—
336C2×C4⋊C4C22×C414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2414yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—28no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—48no—
355C2×C22⋊C4C2×D412yes—
356C2×C22⋊C4C2×D412yes—
357C2×C42—28no—
358C2×C42—21no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—84no—
368C2×C42C22×C414yes—
369C2×C42C2×D414yes—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42#4114yes—
375C2×C42C2414yes—
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—44no—
379#106—28no—
380#106C22×C414yes—
381#106C22×C414yes—
382#106C22×C414yes—
383#106C22×C414yes—
384C4⋊C8—24no—
385#103—28no—
386#103—44no—
387#103C22×C414yes—
388#103C22×C414yes—
389#103C22×C414yes—
390#103C22×C414yes—
391#102—28no—
392#102—44no—
393#102C22×C414yes—
394#102C22×C414yes—
395#102C22×C414yes—
396#102C22×C414yes—
397C22⋊C8—24no—
398C4×C8—28no—
399C4×C8—24no—
400C4×C8—44no—
401C4×C8C22×C414yes—
402C4×C8C22×C414yes—
403C4×C8C22×C414yes—
404C4×C8C22×C414yes—
405#95—22no—
406#95—24no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—22no—
414#95—22no—
415#95—24no—
416#95—22no—
417#95—24no—
418#95—22no—
419#95C2×D412yes—
420#95C2×D412yes—
421#95C2×D414yes—
422C24—21no—
423C24—21no—
424C24—24no—
425C24—44no—
426C24—44no—
427C24—82no—
428C24C22×D411yescharacteristic
429C2×Q8—24no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—48no—
439C2×Q8C23×C412yes—
440C2×Q8C23×C412yes—
441C2×Q8C2×#4114yes—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—44no—
450C2×D4—48no—
451C2×D4—44no—
452C2×D4C23×C412yes—
453C2×D4C23×C412yes—
454C2×D4C2×#4114yes—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—48no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—816no—
481C22×C4—84no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4C4×D414yes—
486C22×C4C4×D414yes—
487C22×C4C23×C412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D414yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C2×#4112yes—
496C22×C4C2511yescharacteristic
497SD16—28no—
498SD16—28no—
499SD16—28no—
500SD16#117116yescomplemented
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—28no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—44no—
509C2×C8C23×C412yes—
510C2×C8C23×C412yes—
511C2×C8C23×C412yes—
512C2×C8C23×C412yes—
513C2×C8C2×#4114yes—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—28no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—84no—
532C4⋊C4C23×C412yes—
533C4⋊C4C23×C412yes—
534C4⋊C4C23×C412yes—
535C4⋊C4C23×C412yes—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4#13518yes—
547C42—24no—
548C42—24no—
549C42—24no—
550C42—24no—
551C42—28no—
552C42—24no—
553C42—24no—
554C42—24no—
555C42—24no—
556C42—48no—
557C42—48no—
558C42—44no—
559C42—44no—
560C42—84no—
561C42C4×D414yes—
562C42C4×D414yes—
563C42C23×C412yes—
564C42C23×C412yes—
565C42C2×#4112yes—
566C42C2×#4112yes—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—22no—
572C23—48no—
573C23—42no—
574C23—42no—
575C23—48no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—84no—
580C23—816no—
581C23C2×C4×D412yes—
582C23C2×#13514yes—
583C23C23×D411yescharacteristic
584Q8—22no—
585Q8—24no—
586Q8—22no—
587Q8—22no—
588Q8—48no—
589Q8—42no—
590Q8C2×#11714yes—
591D4—22no—
592D4—22no—
593D4—22no—
594D4—24no—
595D4—48no—
596D4—42no—
597D4C2×#11714yes—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—816no—
637C2×C4—84no—
638C2×C4C2×C4×D412yes—
639C2×C4C2×C4×D412yes—
640C2×C4C2×C4×D412yes—
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D412yes—
643C2×C4C2×#13518yes—
644C2×C4#57714yes—
645C2×C4C24×C411yescharacteristic
646C2×C4C22×#4112yes—
647C2×C4C22×#4111yescharacteristic
648C2×C4#58512yes—
649C8—22no—
650C8—22no—
651C8—22no—
652C8—44no—
653C8C2×#11714yes—
654C22—22no—
655C22—22no—
656C22—28no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—48no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—816no—
665C22—84no—
666C22C4×#13514yes—
667C22C22×C4×D411yescharacteristic
668C22D4×#4112yes—
669C22C22×#13512yes—
670C22C2×#57712yes—
671C4—22no—
672C4—22no—
673C4—24no—
674C4—22no—
675C4—22no—
676C4—28no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—84no—
684C4—84no—
685C4#294914yes—
686C4C22×#11711yescharacteristic
687C4#342811yescharacteristic
688C4#347512yes—
689C2—24no—
690C2—44no—
691C2—82no—
692C2D4×#11711yescharacteristic
693C2C2×C4×#13511yescharacteristic
694C2C2×#294911yescharacteristic
695C2#6143214yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34128C2semidirect
4. #22806C2semidirect
5. #22773C2semidirect
6. #22768C2semidirect
7. #22683C2semidirect
8. #22609C2semidirect
9. #22575C2semidirect
12. #21274C2semidirect
13. #21120C2semidirect
14. #21114C2semidirect
15. #20818C2semidirect
16. #20735C2semidirect
18. C42×SD16C2semidirect
19. C8×#117C2semidirect
23. C2×C4×SD16C4semidirect
40. #2356C22semidirect
42. #2351C22semidirect
51. #2295C22semidirect
58. C42×Q8C22semidirect
64. #1941C22semidirect
66. #1933C22semidirect
67. #1928C22semidirect
69. #1920C22semidirect
70. #1919C22semidirect
73. #1863C22semidirect
75. #1846C22semidirect
76. #1835C22semidirect
81. #1805C22semidirect
82. #1805C22semidirect
87. C4×#107C22semidirect
88. C4×#106C22semidirect
89. C4×#106C22semidirect
90. C8×C4⋊C4C22semidirect
91. C8×C4⋊C4C22semidirect
92. C4×#103C22semidirect
93. C4×#103C22semidirect
94. C4×#103C22semidirect
100. #1744C22semidirect
101. C42×C8C22semidirect
116. C2×C4×Q8C2×C4semidirect
147. C4×SD16C2×C4semidirect
148. C4×SD16C2×C4semidirect
149. C4×SD16C2×C4semidirect
150. C4×SD16C2×C4semidirect
153. C2×#106C2×C4semidirect
160. C2×#103C2×C4semidirect
175. C2×C4×C8C2×C4semidirect
500. SD16#117semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SD162
#341282
#341262
#228062
#227732
#227682
#226832
#226092
#225752
#225702
C22⋊C4×SD162
#212742
#211202
#211142
#208182
#207352
#207332
C42×SD162
C8×#1172