Finite Group Databasebeta

← #7503489

Subgroups — #7503489

Order 512.

2088 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#481C211yesmaximal, characteristic
2C2×#3081C211yesmaximal, characteristic, complemented
3#35540C211yesmaximal, characteristic
4#35224C211yesmaximal, characteristic, complemented
5#31096C214yesmaximal, complemented
6#31094C214yesmaximal, complemented
7#31088C214yesmaximal, complemented
8#31086C214yesmaximal, complemented
9#23668C211yesmaximal, characteristic, complemented
10#23661C211yesmaximal, characteristic, complemented
11#23638C211yesmaximal, characteristic, complemented
12#23632C211yesmaximal, characteristic, complemented
13#23556C211yesmaximal, characteristic, complemented
14#23480C211yesmaximal, characteristic
15#21089C211yesmaximal, characteristic, complemented
16#21053C211yesmaximal, characteristic, complemented
17#20876C211yesmaximal, characteristic, complemented
18#20844C211yesmaximal, characteristic, complemented
19#20802C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C2×#481—28no—
25C2×#481C2214yes—
26C2×#481C2214yes—
27#3081—28no—
28C2×#478C2212yescomplemented
29C2×#478C2212yescomplemented
30C2×#476C2212yescomplemented
31C2×#474C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#407C2211yescharacteristic
44C22×C22⋊C8C2211yescharacteristic
45C2×#405C2211yescharacteristic
46#2848—22no—
47#2627—22no—
48#2483—22no—
49#2442—22no—
50#2442C2214yescomplemented
51#2440—22no—
52#2440C2214yescomplemented
53#2422—22no—
54#2422C2214yes—
55#2420—22no—
56#2420C2214yes—
57#2405—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2316C2211yescharacteristic
63#2309C2211yescharacteristic
64#2280C2211yescharacteristic
65#2280C2211yescharacteristic
66#2280C2211yescharacteristic
67#2088C2214yescomplemented
68#2088C2214yescomplemented
69#2085C2214yescomplemented
70#2084C2214yescomplemented
71#2077C2212yescomplemented
72#2077C2212yescomplemented
73#2076C2212yescomplemented
74#2075C2212yescomplemented
75#2073C2214yescomplemented
76#2071C2214yescomplemented
77#2059C2214yescomplemented
78#2058C2214yescomplemented
79#2034C2214yes—
80#2033C2214yes—
81#1936C2214yescomplemented
82#1934C2214yescomplemented
83#1924C2214yescomplemented
84#1923C2214yescomplemented
85#1864C2214yescomplemented
86#1862C2214yescomplemented
87#1853C2214yescomplemented
88#1852C2214yescomplemented
89#1847C2212yescomplemented
90#1847C2212yescomplemented
91#1846C2212yescomplemented
92#1844C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×D4—21no—
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C24×C4C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#544—24no—
110#544—42no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2312yes—
113C2×#125—21no—
114C2×#125C2312yes—
115C2×#123—24no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—24no—
118C2×#121—22no—
119C2×#121—24no—
120C2×#121—42no—
121C2×#121C2314yes—
122#518—24no—
123#518—42no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C4⋊C4—24no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—22no—
137C22×C4⋊C4—42no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2314yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—21no—
143C22×C22⋊C4—22no—
144#481—216no—
145#481—216no—
146#478—28no—
147#478—28no—
148#476—28no—
149#474—28no—
150#428—24no—
151#428—42no—
152#428—42no—
153#427—24no—
154C2×#107C2311yescharacteristic
155C2×#107C2312yes—
156C2×#107C2311yescharacteristic
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#106—24no—
160C2×#106—24no—
161C2×#106C2311yescharacteristic
162C2×#106C2312yes—
163C2×#106C2314yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2311yescharacteristic
167C2×#106C2312yes—
168C2×#106C2312yes—
169#424—24no—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172#418—24no—
173#418—24no—
174C2×#102—24no—
175C2×#102D418yes—
176C2×#102C2312yes—
177C2×#102C2314yes—
178C2×#102C2312yes—
179C2×#102C2314yes—
180C2×#102C2312yes—
181C2×#102C2312yes—
182#407—24no—
183C2×C22⋊C8—24no—
184C2×C22⋊C8C2312yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2314yes—
187#405—24no—
188C2×#97C2312yes—
189C2×C4×C8C2312yes—
190#400—24no—
191#398—24no—
192#398—24no—
193#397—24no—
194#397—24no—
195#395—24no—
196#394—24no—
197#394—24no—
198#390—22no—
199#390C2314yes—
200#389—22no—
201#389—22no—
202#389—22no—
203#389C2314yes—
204#389C2314yes—
205#389C2314yes—
206#386—24no—
207#385—22no—
208#385—22no—
209#385—22no—
210#385—22no—
211#385—24no—
212#385—24no—
213#385—24no—
214#384—21no—
215#384—21no—
216#384C2312yes—
217#384C2312yes—
218#382—21no—
219#382—21no—
220#382C2312yes—
221#382C2312yes—
222#381—21no—
223#381—21no—
224#381—24no—
225#381D412yes—
226#380—22no—
227#380—24no—
228#380D412yes—
229#380D412yes—
230#380C2312yes—
231#380C2312yes—
232#380C2312yes—
233#380C2312yes—
234#379—21no—
235#379—21no—
236#379—22no—
237#379D412yes—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4C2314yes—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#336C2314yes—
251#336C2314yes—
252C25—21no—
253C25C2×D411yescharacteristic
254C22×Q8—22no—
255C22×D4—24no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—44no—
263C22×D4—84no—
264C22×D4C2×D414yes—
265C22×D4C2412yes—
266C23×C4—21no—
267C23×C4—22no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—22no—
273C23×C4—24no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—44no—
279C23×C4—84no—
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D412yes—
282C23×C4C2412yes—
283C23×C4C2411yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16#4114yes—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8C2×D414yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2412yes—
301C4⋊Q8—24no—
302C4⋊Q8—42no—
303C4⋊Q8—44no—
304#125—24no—
305#125—42no—
306#125—44no—
307#123—88no—
308C22⋊Q8—48no—
309C22⋊Q8—88no—
310#121—28no—
311#121—44no—
312#121—48no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—42no—
316C4×D4—42no—
317C4×D4—44no—
318C4×D4—44no—
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—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—28no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
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—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—48no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2411yescharacteristic
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—48no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D414yes—
383C2×C22⋊C4C2×D414yes—
384C2×C42—21no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42C2412yes—
389C2×C42C2412yes—
390#107—22no—
391#107—22no—
392#107—24no—
393#107—42no—
394#107—42no—
395#106—24no—
396#106—24no—
397#106—22no—
398#106—22no—
399#106—28no—
400#106—24no—
401#106—42no—
402#106—42no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#102—28no—
406#102—24no—
407#102—24no—
408#102—28no—
409#102C2×D418yes—
410#102C2×D418yes—
411C22⋊C8—24no—
412C22⋊C8—24no—
413C22⋊C8—28no—
414#97—24no—
415C4×C8—24no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—22no—
428C24—24no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—24no—
433C24—42no—
434C24—42no—
435C24—42no—
436C24—82no—
437C24—82no—
438C24C22×D412yes—
439C24C22×D411yescharacteristic
440C2×Q8—44no—
441C2×Q8—84no—
442C2×D4—28no—
443C2×D4—24no—
444C2×D4—28no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—44no—
456C2×D4—84no—
457C2×D4—816no—
458C2×D4C22×D414yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—48no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—82no—
494C22×C4—84no—
495C22×C4—82no—
496C22×C4—82no—
497C22×C4—816no—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C2×#4112yes—
505C22×C4H(5,𝔽2)12yes—
506C22×C42−1+412yes—
507C22×C42−1+412yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510M16—28no—
511M16—44no—
512C2×C8—28no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—22no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—44no—
526C2×C8C22×D412yes—
527C2×C8C22×D412yes—
528C2×C8C2×#4114yes—
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—48no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—84no—
557C4⋊C4—84no—
558C4⋊C4—84no—
559C4⋊C4—84no—
560C4⋊C4—84no—
561C4⋊C4C22×D414yes—
562C22⋊C4—24no—
563C22⋊C4—24no—
564C22⋊C4—28no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—48no—
569C22⋊C4—44no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C22⋊C4—82no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—42no—
584C42—42no—
585C42—42no—
586C42—42no—
587C23—22no—
588C23—22no—
589C23—28no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23—22no—
595C23—22no—
596C23—24no—
597C23—48no—
598C23—48no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—42no—
603C23—42no—
604C23—44no—
605C23—48no—
606C23—84no—
607C23—84no—
608C23—164no—
609C23—164no—
610C23#53612yes—
611C23D4212yes—
612C23#54712yes—
613C23#57714yes—
614C23C23×D411yescharacteristic
615Q8—164no—
616D4—24no—
617D4—24no—
618D4—48no—
619D4—42no—
620D4—42no—
621D4—816no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—48no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—84no—
659C2×C4—816no—
660C2×C4—84no—
661C2×C4—84no—
662C2×C4—164no—
663C2×C4—164no—
664C2×C4#53614yes—
665C2×C4#53612yes—
666C2×C4#53914yes—
667C2×C4D4212yes—
668C2×C4#54712yes—
669C2×C4#54714yes—
670C2×C4#56314yes—
671C2×C4C23×D411yescharacteristic
672C2×C4C22×#4111yescharacteristic
673C2×C4C2×H(5,𝔽2)11yescharacteristic
674C2×C4C2×2−1+411yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C8—24no—
678C8—22no—
679C8—22no—
680C8—42no—
681C8—42no—
682C22—24no—
683C22—22no—
684C22—22no—
685C22—24no—
686C22—24no—
687C22—24no—
688C22—24no—
689C22—416no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—84no—
698C22—84no—
699C22—168no—
700C22#268014yes—
701C22C2×#53611yescharacteristic
702C22C2×D4211yescharacteristic
703C22C2×#54711yescharacteristic
704C22C2×#57712yes—
705C22#358512yes—
706C4—44no—
707C4—42no—
708C4—42no—
709C4—44no—
710C4—42no—
711C4—82no—
712C4—81no—
713C4—82no—
714C4—81no—
715C4—82no—
716C4—81no—
717C4—81no—
718C4—82no—
719C4#268014yes—
720C4C2×#53611yescharacteristic
721C4C2×#53911yescharacteristic
722C4C2×#54711yescharacteristic
723C4C2×#56311yescharacteristic
724C2—24no—
725C2—44no—
726C2—84no—
727C2C2×#268011yescharacteristic
728C2#6087312yes—
729C2#6105011yescharacteristic
730C2#6105811yescharacteristic
731C2#6127212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3081C2semidirect
4. #35224C2semidirect
5. #31096C2semidirect
6. #31094C2semidirect
7. #31088C2semidirect
8. #31086C2semidirect
9. #23668C2semidirect
10. #23661C2semidirect
11. #23638C2semidirect
12. #23632C2semidirect
13. #23556C2semidirect
15. #21089C2semidirect
16. #21053C2semidirect
17. #20876C2semidirect
18. #20844C2semidirect
19. #20802C2semidirect
28. C2×#478C22semidirect
29. C2×#478C22semidirect
30. C2×#476C22semidirect
31. C2×#474C22semidirect
50. #2442C22semidirect
52. #2440C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2085C22semidirect
70. #2084C22semidirect
71. #2077C22semidirect
72. #2077C22semidirect
73. #2076C22semidirect
74. #2075C22semidirect
75. #2073C22semidirect
76. #2071C22semidirect
77. #2059C22semidirect
78. #2058C22semidirect
81. #1936C22semidirect
82. #1934C22semidirect
83. #1924C22semidirect
84. #1923C22semidirect
85. #1864C22semidirect
86. #1862C22semidirect
87. #1853C22semidirect
88. #1852C22semidirect
89. #1847C22semidirect
90. #1847C22semidirect
91. #1846C22semidirect
92. #1844C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4812
C2×#30812
#355402
#352242
#310962
#310942
#310882
#310862
#236682
#236612
#236382
#236322
#235562
#234802
#210892
#210532
#208762
#208442
#208022