Finite Group Databasebeta

← #6457218

Subgroups — #6457218

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3095C211yesmaximal, characteristic
2C22×#485C211yesmaximal, characteristic
3#34203C211yesmaximal, characteristic
4#34052C211yesmaximal, characteristic
5#27657C214yesmaximal, complemented
6#27656C214yesmaximal, complemented
7#27652C214yesmaximal, complemented
8#27649C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23633C211yesmaximal, characteristic, complemented
12#23633C211yesmaximal, characteristic, complemented
13#21174C211yesmaximal, characteristic, complemented
14#21173C211yesmaximal, characteristic, complemented
15#20701C211yesmaximal, characteristic
16#20699C211yesmaximal, characteristic
17#20651C211yesmaximal, characteristic, complemented
18#20651C211yesmaximal, characteristic, complemented
19#20628C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C2×#519C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25#3095—28no—
26C2×#484C2212yes—
27C2×#485—28no—
28C2×#485C2212yes—
29C2×#485C2214yes—
30C2×#485C2212yes—
31C2×#485C2214yes—
32C2×#483C2212yes—
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C22×#103C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45#2362—22no—
46#2360—22no—
47#2344—22no—
48#2342—22no—
49#2323C2214yes—
50#2317C2212yes—
51#2317C2212yes—
52#2317C2211yescharacteristic
53#2317C2211yescharacteristic
54#2317C2211yescharacteristic
55#2317C2211yescharacteristic
56#2316—22no—
57#2316C2211yescharacteristic
58#2315—22no—
59#2310C2211yescharacteristic
60C4×C22⋊Q8C2214yes—
61#2296—22no—
62#2296—22no—
63#2285C2212yes—
64#2285C2212yes—
65#2280C2211yescharacteristic
66#2274C2211yescharacteristic
67#2085C2214yescomplemented
68#2085C2214yescomplemented
69#2084C2214yescomplemented
70#2083C2214yescomplemented
71#2074C2214yescomplemented
72#2074C2214yescomplemented
73#2072C2214yescomplemented
74#2072C2214yescomplemented
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1825C2212yes—
80#1825C2214yes—
81#1825C2212yes—
82#1825C2214yes—
83#1824C2212yes—
84#1823C2212yes—
85#1805C2214yescomplemented
86#1805C2214yescomplemented
87#1803C2214yescomplemented
88#1803C2214yescomplemented
89#1798C2214yes—
90#1797C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×Q8C2311yescharacteristic
102C24×C4C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8D418yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113#519—24no—
114#519—24no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2314yes—
117C2×#117—24no—
118C2×#117—24no—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117—21no—
122C2×#117—21no—
123C2×#117—22no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117—24no—
127C2×#117—24no—
128C2×#117D412yes—
129C2×#117D412yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C2×#117C2312yes—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4D412yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C22⋊C4—21no—
149C22×C22⋊C4—21no—
150C22×C22⋊C4—21no—
151C22×C22⋊C4D412yes—
152C22×C42—21no—
153C22×C42—21no—
154C22×C42C2312yes—
155#485—216no—
156#485—216no—
157#485—28no—
158#485—28no—
159#484—28no—
160#483—28no—
161#428—24no—
162#428—24no—
163C2×#107—24no—
164C2×#107C2314yes—
165C2×#107C2312yes—
166C2×#107C2312yes—
167C2×#106—24no—
168C2×#106C2312yes—
169C2×#106C2312yes—
170C2×#106C2314yes—
171#424—24no—
172#424—24no—
173C2×C4⋊C8C2314yes—
174C2×C4⋊C8C2314yes—
175#419—24no—
176#419—24no—
177C2×#103—24no—
178C2×#103—24no—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2314yes—
182C2×#103C2314yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185#407—24no—
186C2×C22⋊C8—24no—
187C2×C22⋊C8C2312yes—
188C2×C22⋊C8C2312yes—
189C2×C22⋊C8C2312yes—
190C2×C22⋊C8C2312yes—
191#391—22no—
192#391C2314yes—
193#389—24no—
194#389—24no—
195#389—22no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—24no—
201#389C2314yes—
202#388—24no—
203#388—24no—
204#387—24no—
205#387—24no—
206#385—24no—
207#385—22no—
208#385—22no—
209#385—24no—
210#385—24no—
211#385—22no—
212#385—22no—
213#384C2312yes—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382C2312yes—
218#382C2312yes—
219#382C2312yes—
220#382C2312yes—
221#381—21no—
222#381—22no—
223#380—24no—
224#380—24no—
225#380—22no—
226#380C2312yes—
227#380C2312yes—
228#379—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—24no—
231C4×C4⋊C4—24no—
232C4×C4⋊C4—24no—
233C4×C4⋊C4C2314yes—
234C4×C22⋊C4—24no—
235C4×C22⋊C4—21no—
236C4×C22⋊C4—21no—
237C4×C22⋊C4—22no—
238#340C2312yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2314yes—
247#340C2314yes—
248#340C2312yes—
249#340C2312yes—
250#336C2314yes—
251#336C2314yes—
252C25C2×D411yescharacteristic
253C22×Q8—21no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—22no—
259C22×Q8C2×D414yes—
260C22×Q8C2412yes—
261C23×C4—22no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D414yes—
275C23×C4C2×D414yes—
276C23×C4#4114yes—
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C22×C8—22no—
287C22×C8—22no—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22⋊Q8—28no—
297C22⋊Q8—28no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8C2×D418yes—
301C22⋊Q8C2×D418yes—
302C4×Q8—28no—
303C4×Q8—28no—
304C4×Q8—48no—
305#117—22no—
306#117—22no—
307#117—28no—
308#117—28no—
309#117—24no—
310#117—24no—
311#117—22no—
312#117—22no—
313#117—24no—
314#117—24no—
315#117—22no—
316#117—22no—
317#117—24no—
318#117—24no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—44no—
328#117—44no—
329#117—44no—
330#117—44no—
331#117—48no—
332#117#4114yes—
333#117#4114yes—
334C2×C4⋊C4—28no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—28no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—28no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—28no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—44no—
386C2×C22⋊C4—44no—
387C2×C22⋊C4C2×D412yes—
388C2×C22⋊C4C2×D414yes—
389C2×C22⋊C4C2×D412yes—
390C2×C42—24no—
391C2×C42—24no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—24no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—24no—
402C2×C42—44no—
403C2×C42—44no—
404C2×C42C2×D414yes—
405C2×C42C2412yes—
406C2×C42C2412yes—
407#107—28no—
408#107—24no—
409#107—24no—
410#106—28no—
411#106—24no—
412#106—24no—
413C4⋊C8—28no—
414C4⋊C8—28no—
415#103—28no—
416#103—28no—
417#103—24no—
418#103—24no—
419#103—24no—
420#103—24no—
421C22⋊C8—24no—
422C22⋊C8—24no—
423C22⋊C8—24no—
424C22⋊C8—24no—
425#95—22no—
426#95—24no—
427#95—24no—
428#95—22no—
429#95—24no—
430#95—24no—
431#95—24no—
432#95—24no—
433#95—24no—
434#95—24no—
435C24—22no—
436C24—22no—
437C24—42no—
438C24—42no—
439C24#12114yes—
440C24C22×D411yescharacteristic
441C24C22×D411yescharacteristic
442C24C22×D411yescharacteristic
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—28no—
446C2×Q8—28no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—44no—
454C2×Q8—44no—
455C2×Q8C22×D414yes—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—28no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—82no—
498C22×C4—82no—
499C22×C4—82no—
500C22×C4#12114yes—
501C22×C4#12114yes—
502C22×C4#12114yes—
503C22×C4C22×D414yes—
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D414yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D412yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C42−1+412yes—
517C22×C42−1+412yes—
518C22×C4C2511yescharacteristic
519M16—44no—
520M16—44no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
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—44no—
534C2×C8—44no—
535C4⋊C4—28no—
536C4⋊C4—28no—
537C4⋊C4—24no—
538C4⋊C4—28no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—48no—
562C4⋊C4—84no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4—84no—
566C4⋊C4—84no—
567C4⋊C4#12318yes—
568C4⋊C4C22×D414yes—
569C22⋊C4—28no—
570C22⋊C4—28no—
571C22⋊C4—24no—
572C22⋊C4—28no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—44no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—82no—
583C22⋊C4#12318yes—
584C42—28no—
585C42—22no—
586C42—22no—
587C42—44no—
588C42—42no—
589C42—42no—
590C42—44no—
591C42—42no—
592C42—42no—
593C42—42no—
594C42—48no—
595C42—42no—
596C42—84no—
597C42C2×#4114yes—
598C42C2×#4114yes—
599C23—22no—
600C23—22no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—28no—
610C23—44no—
611C23—44no—
612C23—84no—
613C23—84no—
614C23C2×#12112yes—
615C23C2×#12112yes—
616C23C2×#12112yes—
617C23C2×#12312yes—
618C23#53612yes—
619C23C23×D411yescharacteristic
620Q8—24no—
621Q8—24no—
622Q8—42no—
623Q8—42no—
624Q8—48no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—28no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—48no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—88no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12318yes—
679C2×C4C2×#12312yes—
680C2×C4#53612yes—
681C2×C4#54714yes—
682C2×C4#54714yes—
683C2×C4#55214yes—
684C2×C4#55214yes—
685C2×C4C23×D411yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C2×2−1+411yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C8—42no—
692C8—42no—
693C8—42no—
694C8—42no—
695C22—22no—
696C22—22no—
697C22—24no—
698C22—24no—
699C22—22no—
700C22—22no—
701C22—22no—
702C22—22no—
703C22—22no—
704C22—22no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—42no—
709C22—88no—
710C22#248614yes—
711C22C22×#12111yescharacteristic
712C22C22×#12311yescharacteristic
713C22C2×#53611yescharacteristic
714C22#358412yes—
715C22#358612yes—
716C4—28no—
717C4—22no—
718C4—22no—
719C4—44no—
720C4—44no—
721C4—44no—
722C4—82no—
723C4—81no—
724C4—81no—
725C4—82no—
726C4—81no—
727C4—81no—
728C4—82no—
729C4—82no—
730C4#248614yes—
731C4C2×#54711yescharacteristic
732C4C2×#54711yescharacteristic
733C4C2×#55211yescharacteristic
734C4C2×#55211yescharacteristic
735C2—24no—
736C2—42no—
737C2—42no—
738C2C2×#248611yescharacteristic
739C2#6042211yescharacteristic
740C2#6042411yescharacteristic
741C2#6052412yes—
742C2#6087612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #27657C2semidirect
6. #27656C2semidirect
7. #27652C2semidirect
8. #27649C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
11. #23633C2semidirect
12. #23633C2semidirect
13. #21174C2semidirect
14. #21173C2semidirect
17. #20651C2semidirect
18. #20651C2semidirect
67. #2085C22semidirect
68. #2085C22semidirect
69. #2084C22semidirect
70. #2083C22semidirect
71. #2074C22semidirect
72. #2074C22semidirect
73. #2072C22semidirect
74. #2072C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
85. #1805C22semidirect
86. #1805C22semidirect
87. #1803C22semidirect
88. #1803C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30952
C22×#4852
#342032
#340522
#276572
#276562
#276522
#276492
#236612
#236602
#236332
#211742
#211732
#207012
#206992
#206512
#206282