Finite Group Databasebeta

← #6453003

Subgroups — #6453003

Order 512.

3454 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#3095C211yesmaximal, characteristic, complemented
3C22×#428C211yesmaximal, characteristic, complemented
4#34809C211yesmaximal, characteristic
5#24109C218yesmaximal, complemented
6#24098C214yesmaximal, complemented
7#24097C214yesmaximal, complemented
8#23552C212yesmaximal, complemented
9#23481C212yesmaximal, complemented
10#20972C211yesmaximal, characteristic, complemented
11#20898C211yesmaximal, characteristic
12#20874C212yesmaximal, complemented
13#20844C212yesmaximal, complemented
14#20700C211yesmaximal, characteristic, complemented
15C23×M16C2211yescharacteristic
16C2×#554—28no—
17C2×#554—22no—
18C2×#554—22no—
19C2×#554—28no—
20C2×#554C2214yes—
21C2×#554C2214yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125—21no—
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8—21no—
26C22×C22⋊Q8C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4×D4—21no—
30C22×C4×D4C2211yescharacteristic
31C22×#117C2211yescharacteristic
32C23×C4⋊C4C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#3095—28no—
35C2×#484C2212yescomplemented
36C2×#485C2214yescomplemented
37C2×#483C2212yescomplemented
38C2×#428—28no—
39C2×#428C2212yes—
40C2×#428C2212yes—
41C2×#428C2212yes—
42C2×#428C2214yescomplemented
43C22×#107C2212yescomplemented
44C22×#106C2212yescomplemented
45C2×#419C2212yes—
46C2×#419C2212yes—
47C2×#407C2211yescharacteristic
48C2×#407C2212yes—
49#2487C2212yes—
50#2486C2212yes—
51#2485C2214yes—
52#2484—21no—
53#2483—21no—
54#2482—21no—
55#2405—21no—
56#2395—22no—
57C4×#123—22no—
58#2296C2212yes—
59#2280C2212yes—
60#2280C2211yescharacteristic
61#2058C2218yescomplemented
62#2057C2218yescomplemented
63#2036C2218yescomplemented
64#2035C2218yescomplemented
65#1894C2214yescomplemented
66#1892C2214yescomplemented
67#1870C2214yes—
68#1869C2214yes—
69#1865C2218yescomplemented
70#1863C2218yescomplemented
71#1853C2218yescomplemented
72#1852C2218yescomplemented
73#1824C2214yescomplemented
74#1823C2214yescomplemented
75#1736C2212yes—
76#1736C2212yes—
77#1736C2212yes—
78#1728C2212yes—
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C23×D4D411yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C22×M16—24no—
87C22×M16—28no—
88C22×M16C2312yes—
89C22×M16C2312yes—
90C22×M16C2312yes—
91C23×C8C2312yes—
92#554—232no—
93#554—232no—
94#554—216no—
95#554—216no—
96#554—216no—
97#554—216no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8C2314yes—
102C2×#125—216no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#123—22no—
106C2×C22⋊Q8—216no—
107C2×C22⋊Q8—216no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×C22⋊Q8—24no—
118C2×C22⋊Q8D418yes—
119C2×C22⋊Q8C2314yes—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C4×Q8—24no—
123C2×C4×Q8—24no—
124C2×C4×Q8—22no—
125C2×C4×Q8C2314yes—
126C2×C4×D4—216no—
127C2×C4×D4—24no—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4D412yes—
132C2×C4×D4D412yes—
133C2×C4×D4C2312yes—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117—24no—
138C2×#117—24no—
139C2×#117C2312yes—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—22no—
145C22×C4⋊C4—24no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—21no—
149C22×C4⋊C4—21no—
150C22×C4⋊C4—21no—
151C22×C4⋊C4—21no—
152C22×C4⋊C4—24no—
153C22×C4⋊C4—24no—
154C22×C4⋊C4D411yescharacteristic
155C22×C4⋊C4D418yes—
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2314yes—
159C22×C4⋊C4C2312yes—
160C22×C4⋊C4C2312yes—
161C22×C4⋊C4C2311yescharacteristic
162C22×C4⋊C4C2311yescharacteristic
163C22×C4⋊C4C2311yescharacteristic
164C22×C22⋊C4—21no—
165C22×C22⋊C4—21no—
166C22×C22⋊C4—21no—
167C22×C22⋊C4D411yescharacteristic
168C22×C22⋊C4D411yescharacteristic
169C22×C42—21no—
170C22×C42—21no—
171C22×C42C2311yescharacteristic
172#485—216no—
173#484—28no—
174#483—28no—
175#428—28no—
176#428—28no—
177#428—28no—
178#428—216no—
179C2×#107—28no—
180C2×#107C2314yes—
181C2×#107C2314yes—
182C2×#107C2314yes—
183C2×#106—28no—
184C2×#106C2314yes—
185C2×#106C2314yes—
186C2×#106C2314yes—
187#419—28no—
188#419—28no—
189C2×#103C2314yes—
190C2×#103C2314yes—
191C2×#103C2318yes—
192#407—24no—
193#407—28no—
194C2×C22⋊C8C2312yes—
195C2×C22⋊C8C2312yes—
196C2×C22⋊C8C2318yes—
197#400—22no—
198#399—22no—
199#398—22no—
200#397—22no—
201#397—24no—
202#394—24no—
203#389C2314yes—
204#389C2314yes—
205#388—24no—
206#387—24no—
207#386—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#381—21no—
212#381—22no—
213#381—22no—
214#380C2312yes—
215#380C2312yes—
216#380C2314yes—
217#380C2314yes—
218#379—22no—
219#379—21no—
220C4×C4⋊C4—24no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4—24no—
223C4×C22⋊C4—21no—
224#340C2318yes—
225#340C2318yes—
226#340C2318yes—
227#336C2318yes—
228C25C2×D411yescharacteristic
229C25C2×D411yescharacteristic
230C22×Q8—28no—
231C22×Q8—22no—
232C22×Q8—21no—
233C22×Q8—24no—
234C22×Q8—24no—
235C22×Q8—24no—
236C22×Q8—84no—
237C22×Q8C2×D414yes—
238C22×Q8C2412yes—
239C22×D4—24no—
240C22×D4—28no—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—28no—
250C23×C4—28no—
251C23×C4—24no—
252C23×C4—22no—
253C23×C4—22no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—84no—
261C23×C4—84no—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D412yes—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D414yes—
269C23×C4C2×D414yes—
270C23×C4C2×D411yescharacteristic
271C23×C4C2412yes—
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2411yescharacteristic
275C2×M16—232no—
276C2×M16—24no—
277C2×M16—24no—
278C2×M16—24no—
279C2×M16—24no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—28no—
283C2×M16#4118yes—
284C22×C8—24no—
285C22×C8—28no—
286C22×C8C2414yes—
287C22×C8C2414yes—
288C4⋊Q8—216no—
289C4⋊Q8—28no—
290C4⋊Q8—28no—
291#125—232no—
292#125—232no—
293#123—48no—
294C22⋊Q8—216no—
295C22⋊Q8—232no—
296C22⋊Q8—232no—
297C22⋊Q8—232no—
298C22⋊Q8—232no—
299C22⋊Q8—28no—
300C22⋊Q8—28no—
301C22⋊Q8—28no—
302C22⋊Q8—24no—
303C22⋊Q8—24no—
304C22⋊Q8—28no—
305C22⋊Q8C2×D418yes—
306C22⋊Q8C2×D418yes—
307C4×Q8—216no—
308C4×Q8—28no—
309C4×Q8—28no—
310C4×D4—232no—
311C4×D4—232no—
312C4×D4—28no—
313C4×D4—28no—
314C4×D4—24no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—24no—
318#117—24no—
319#117—24no—
320#117—24no—
321#117—24no—
322#117—24no—
323#117—24no—
324#117—44no—
325#117—44no—
326C2×C4⋊C4—28no—
327C2×C4⋊C4—216no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—28no—
330C2×C4⋊C4—28no—
331C2×C4⋊C4—28no—
332C2×C4⋊C4—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—28no—
353C2×C4⋊C4—28no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D412yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2×D414yes—
366C2×C4⋊C4C2×D414yes—
367C2×C4⋊C4C2×D418yes—
368C2×C4⋊C4C2×D418yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2414yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—28no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—22no—
389C2×C22⋊C4—44no—
390C2×C22⋊C4C2×D412yes—
391C2×C22⋊C4C2×D412yes—
392C2×C22⋊C4C2×D412yes—
393C2×C22⋊C4C2×D412yes—
394C2×C42—28no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—24no—
398C2×C42—22no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—44no—
402C2×C42C2×D412yes—
403C2×C42C2412yes—
404#107—28no—
405#107—28no—
406#107—28no—
407#106—28no—
408#106—28no—
409#106—28no—
410#103—28no—
411#103—28no—
412#103—216no—
413C22⋊C8—24no—
414C22⋊C8—24no—
415C22⋊C8—216no—
416#95—24no—
417#95—24no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422C24—22no—
423C24—24no—
424C24—24no—
425C24—24no—
426C24—24no—
427C24—42no—
428C24—42no—
429C24#12014yes—
430C24C22×D411yescharacteristic
431C24C22×D412yes—
432C24C22×D411yescharacteristic
433C24C22×D411yescharacteristic
434C2×Q8—216no—
435C2×Q8—24no—
436C2×Q8—22no—
437C2×Q8—216no—
438C2×Q8—24no—
439C2×Q8—24no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—44no—
443C2×Q8—88no—
444C2×Q8—816no—
445C2×Q8C22×D414yes—
446C2×D4—232no—
447C2×D4—24no—
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—24no—
451C2×D4—24no—
452C2×D4#13718yes—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—216no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—216no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—48no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—82no—
500C22×C4—84no—
501C22×C4—84no—
502C22×C4—88no—
503C22×C4—816no—
504C22×C4—816no—
505C22×C4#12014yes—
506C22×C4#12014yes—
507C22×C4#12014yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×D412yes—
520C22×C4C2×#4114yes—
521C22×C42−1+414yes—
522C22×C4C2511yescharacteristic
523M16—216no—
524M16—216no—
525M16—48no—
526C2×C8—216no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8C2×#4118yes—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—216no—
536C4⋊C4—216no—
537C4⋊C4—24no—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—28no—
543C4⋊C4—28no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—48no—
558C4⋊C4—84no—
559C4⋊C4C22×D414yes—
560C4⋊C4C22×D414yes—
561C4⋊C4C22×D414yes—
562C22⋊C4—216no—
563C22⋊C4—28no—
564C22⋊C4—28no—
565C22⋊C4—24no—
566C22⋊C4—24no—
567C22⋊C4—24no—
568C22⋊C4—24no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—48no—
574C22⋊C4—82no—
575C22⋊C4—84no—
576C42—216no—
577C42—24no—
578C42—22no—
579C42—22no—
580C42—44no—
581C42—44no—
582C42—44no—
583C23—22no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—28no—
588C23—24no—
589C23—22no—
590C23—28no—
591C23—22no—
592C23—24no—
593C23—22no—
594C23—28no—
595C23—22no—
596C23—22no—
597C23—44no—
598C23—416no—
599C23—416no—
600C23—44no—
601C23—44no—
602C23C2×#12012yes—
603C23C2×#12012yes—
604C23C2×#12012yes—
605C23#54714yes—
606C23C2×#13714yes—
607C23C2×#13714yes—
608C23C23×D411yescharacteristic
609Q8—28no—
610Q8—48no—
611Q8—42no—
612Q8—42no—
613Q8—816no—
614D4—216no—
615D4—216no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—28no—
623C2×C4—28no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—48no—
639C2×C4—48no—
640C2×C4—48no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—48no—
647C2×C4—44no—
648C2×C4—48no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4—816no—
662C2×C4—84no—
663C2×C4—84no—
664C2×C4C2×#12012yes—
665C2×C4C2×#12012yes—
666C2×C4C2×#12012yes—
667C2×C4#52514yes—
668C2×C4#53614yes—
669C2×C4#54714yes—
670C2×C4#54718yes—
671C2×C4C2×#13714yes—
672C2×C4C23×D411yescharacteristic
673C2×C4C23×D411yescharacteristic
674C2×C4C22×#4112yes—
675C2×C4C2×2−1+412yes—
676C8—28no—
677C8—44no—
678C22—28no—
679C22—28no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—28no—
685C22—22no—
686C22—22no—
687C22—24no—
688C22—22no—
689C22—416no—
690C22—416no—
691C22—44no—
692C22—44no—
693C22—42no—
694C22—42no—
695C22#240614yes—
696C22#300414yes—
697C22C22×#12011yescharacteristic
698C22C2×#54712yes—
699C22C22×#13712yes—
700C22#358512yes—
701C4—22no—
702C4—22no—
703C4—24no—
704C4—42no—
705C4—44no—
706C4—42no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4#240614yes—
714C4C2×#52511yescharacteristic
715C4C2×#53611yescharacteristic
716C4C2×#54712yes—
717C2—24no—
718C2—24no—
719C2—44no—
720C2—42no—
721C2C2×#240611yescharacteristic
722C2C2×#300411yescharacteristic
723C2#6036711yescharacteristic
724C2#6125414yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3095C2semidirect
3. C22×#428C2semidirect
5. #24109C2semidirect
6. #24098C2semidirect
7. #24097C2semidirect
8. #23552C2semidirect
9. #23481C2semidirect
10. #20972C2semidirect
12. #20874C2semidirect
13. #20844C2semidirect
14. #20700C2semidirect
35. C2×#484C22semidirect
36. C2×#485C22semidirect
37. C2×#483C22semidirect
42. C2×#428C22semidirect
43. C22×#107C22semidirect
44. C22×#106C22semidirect
61. #2058C22semidirect
62. #2057C22semidirect
63. #2036C22semidirect
64. #2035C22semidirect
65. #1894C22semidirect
66. #1892C22semidirect
69. #1865C22semidirect
70. #1863C22semidirect
71. #1853C22semidirect
72. #1852C22semidirect
73. #1824C22semidirect
74. #1823C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#30952
C22×#4282
#348092
#241092
#240982
#240972
#235522
#234812
#209722
#208982
#208742
#208442
#207002