Finite Group Databasebeta

← #7502452

Subgroups — #7502452

Order 512.

1980 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic, complemented
2C22×#474C211yesmaximal, characteristic
3#35576C211yesmaximal, characteristic, complemented
4#35174C211yesmaximal, characteristic
5#30966C214yesmaximal, complemented
6#30962C214yesmaximal, complemented
7#30950C214yesmaximal, complemented
8#30946C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic
10#23666C211yesmaximal, characteristic, complemented
11#23660C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#23552C211yesmaximal, characteristic, complemented
14#23456C211yesmaximal, characteristic
15#21091C211yesmaximal, characteristic
16#21063C211yesmaximal, characteristic
17#21037C211yesmaximal, characteristic, complemented
18#21024C211yesmaximal, characteristic, complemented
19#20873C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3164—28no—
25C2×#501C2212yescomplemented
26C2×#501C2212yescomplemented
27C2×#500C2212yescomplemented
28C2×#499C2212yescomplemented
29C2×#474—28no—
30C2×#474C2214yes—
31C2×#474C2214yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#427C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39C2×#418C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C22×#102C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C2×#405C2211yescharacteristic
44#2848—22no—
45#2626—22no—
46#2583—22no—
47#2551—22no—
48#2551C2214yescomplemented
49#2550—22no—
50#2550C2214yescomplemented
51#2454—22no—
52#2454C2214yes—
53#2452—22no—
54#2452C2214yes—
55#2434—22no—
56#2317C2211yescharacteristic
57#2317C2211yescharacteristic
58#2316C2211yescharacteristic
59#2309C2211yescharacteristic
60#2309C2211yescharacteristic
61#2296C2211yescharacteristic
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2274C2211yescharacteristic
65#2094C2212yes—
66#2093C2212yes—
67#2092C2212yes—
68#2092C2212yes—
69#2088C2212yescomplemented
70#2088C2212yescomplemented
71#2087C2212yescomplemented
72#2086C2212yescomplemented
73#2085C2214yescomplemented
74#2083C2214yescomplemented
75#2059C2214yescomplemented
76#2058C2214yescomplemented
77#2057C2214yescomplemented
78#2057C2214yescomplemented
79#2025C2214yes—
80#2023C2214yes—
81#1936C2214yes—
82#1934C2214yes—
83#1926C2214yes—
84#1925C2214yes—
85#1921C2214yescomplemented
86#1918C2214yescomplemented
87#1917C2214yescomplemented
88#1916C2214yescomplemented
89#1864C2214yes—
90#1862C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1735C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×Q8—21no—
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—24no—
110#544—42no—
111#544—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8C2312yes—
119C2×C4⋊Q8C2314yes—
120C2×C4⋊Q8C2312yes—
121C2×#125—21no—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×C22⋊Q8—24no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—22no—
128C2×C4×D4—24no—
129C2×C4×D4—22no—
130C2×C4×D4—42no—
131C2×C4×D4C2314yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—21no—
143C22×C42C2311yescharacteristic
144#501—28no—
145#501—28no—
146#500—28no—
147#499—28no—
148#474—216no—
149#474—216no—
150#428—24no—
151#428—24no—
152#428—42no—
153#427—24no—
154C2×#107—24no—
155C2×#107D418yes—
156C2×#107C2312yes—
157C2×#107C2314yes—
158C2×#107C2312yes—
159C2×#107C2311yescharacteristic
160C2×#107C2311yescharacteristic
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2311yescharacteristic
167C2×#106C2311yescharacteristic
168C2×#106C2312yes—
169C2×C4⋊C8—24no—
170C2×C4⋊C8C2312yes—
171C2×C4⋊C8C2312yes—
172#418—42no—
173#418Q818yescomplemented
174C2×#102—24no—
175C2×#102—24no—
176C2×#102C2312yes—
177C2×#102C2312yes—
178C2×#102C2314yes—
179C2×#102C2312yes—
180C2×#102C2312yes—
181C2×#102C2314yes—
182#405—24no—
183C2×#97C2312yes—
184C2×C4×C8C2312yes—
185#399—24no—
186#398—24no—
187#396—24no—
188#396—24no—
189#395—24no—
190#395—24no—
191#395—24no—
192#393—24no—
193#391—24no—
194#390—21no—
195#390—22no—
196#390—21no—
197#390C2312yes—
198#390C2312yes—
199#390C2314yes—
200#389—22no—
201#389—22no—
202#389—24no—
203#389—24no—
204#389D412yes—
205#389D412yes—
206#389C2312yes—
207#389C2314yes—
208#389C2312yes—
209#389C2314yes—
210#389C2312yes—
211#389C2312yes—
212#385—21no—
213#385—21no—
214#385—21no—
215#385—21no—
216#385—24no—
217#385—22no—
218#385—22no—
219#385—24no—
220#385D412yes—
221#385D412yes—
222#384—21no—
223#384C2312yes—
224#382—21no—
225#382C2312yes—
226#380—22no—
227#380—24no—
228#379—22no—
229#379—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236#340C2312yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2312yes—
241#340C2312yes—
242#340C2312yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249#336C2312yes—
250C25—21no—
251C22×Q8—24no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—44no—
255C22×Q8—84no—
256C22×D4—22no—
257C22×D4—21no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4C2×Q814yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—24no—
268C23×C4—44no—
269C23×C4—84no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C2×M16—22no—
277C2×M16—22no—
278C2×M16C2×Q814yes—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8C2×D414yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C22×C8C2411yescharacteristic
288C22×C8C2411yescharacteristic
289C22×C8C2412yes—
290C4⋊Q8—28no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—44no—
294C4⋊Q8—42no—
295C4⋊Q8—42no—
296C4⋊Q8—48no—
297C4⋊Q8—44no—
298C4⋊Q8—44no—
299C4⋊Q8—88no—
300#125—24no—
301#125—24no—
302#125—42no—
303#125—42no—
304#125—44no—
305#125—44no—
306C22⋊Q8—48no—
307C22⋊Q8—48no—
308C22⋊Q8—88no—
309C4×D4—28no—
310C4×D4—44no—
311C4×D4—48no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117C2×Q814yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—24no—
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—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—44no—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—24no—
367C2×C22⋊C4—24no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—44no—
372C2×C22⋊C4C2×D412yes—
373C2×C22⋊C4C2×D412yes—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—24no—
381C2×C42C2×D412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2412yes—
385#107—22no—
386#107—22no—
387#107—28no—
388#107—24no—
389#107—24no—
390#107—42no—
391#107C2×D418yes—
392#107C2×D418yes—
393#106—22no—
394#106—22no—
395#106—24no—
396#106—24no—
397#106—42no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400#102—24no—
401#102—24no—
402#102—28no—
403#102—42no—
404#102—42no—
405#102C2×Q8116yescomplemented
406#97—24no—
407C4×C8—24no—
408#95—24no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95C2×D414yes—
417#95C2×D414yes—
418C24—22no—
419C24—24no—
420C24—42no—
421C24—42no—
422C24—82no—
423C24—82no—
424C24C22×D411yescharacteristic
425C2×Q8—48no—
426C2×Q8—44no—
427C2×Q8—44no—
428C2×Q8—48no—
429C2×Q8—84no—
430C2×Q8—84no—
431C2×Q8—84no—
432C2×Q8—816no—
433C2×D4—28no—
434C2×D4—28no—
435C2×D4—24no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—42no—
442C2×D4—44no—
443C2×D4—42no—
444C2×D4—44no—
445C2×D4C22×Q814yes—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—48no—
462C22×C4—42no—
463C22×C4—48no—
464C22×C4—84no—
465C22×C4—816no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D412yes—
476C22×C4C22×Q812yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C22×C42−1+412yes—
480C22×C42−1+412yes—
481C22×C4C2511yescharacteristic
482M16—28no—
483M16—44no—
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—44no—
498C2×C8C22×D412yes—
499C2×C8C22×D412yes—
500C2×C8C22×Q814yes—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—82no—
535C4⋊C4—84no—
536C4⋊C4—84no—
537C4⋊C4—84no—
538C4⋊C4C22×D414yes—
539C4⋊C4C22×D414yes—
540C4⋊C4C22×Q814yes—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—48no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—42no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—84no—
552C42—24no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—84no—
564C23—22no—
565C23—22no—
566C23—24no—
567C23—44no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—88no—
573C23—84no—
574C23—84no—
575C23—88no—
576C23#53612yes—
577C23D4212yes—
578C23D4×Q812yes—
579C23C23×D411yescharacteristic
580Q8—816no—
581Q8—164no—
582Q8—164no—
583D4—24no—
584D4—24no—
585D4—48no—
586D4—42no—
587D4—42no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—816no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#53612yes—
627C2×C4#53614yes—
628C2×C4#54414yes—
629C2×C4D4212yes—
630C2×C4D4×Q812yes—
631C2×C4D4×Q814yes—
632C2×C4#56314yes—
633C2×C4#57614yes—
634C2×C4C23×D411yescharacteristic
635C2×C4C23×Q811yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×2−1+411yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—22no—
641C8—22no—
642C8—24no—
643C8—42no—
644C8—42no—
645C22—22no—
646C22—22no—
647C22—24no—
648C22—24no—
649C22—44no—
650C22—48no—
651C22—816no—
652C22—82no—
653C22—82no—
654C22—84no—
655C22#267714yes—
656C22C2×#53611yescharacteristic
657C22C2×D4211yescharacteristic
658C22C2×D4×Q811yescharacteristic
659C22C2×#57612yes—
660C22#358612yes—
661C4—24no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—82no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—81no—
674C4—82no—
675C4#267714yes—
676C4C2×#53611yescharacteristic
677C4C2×#54411yescharacteristic
678C4C2×D4×Q811yescharacteristic
679C4C2×#56311yescharacteristic
680C2—24no—
681C2—84no—
682C2C2×#267711yescharacteristic
683C2#6088812yes—
684C2#6104911yescharacteristic
685C2#6106111yescharacteristic
686C2#6138112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3164C2semidirect
3. #35576C2semidirect
5. #30966C2semidirect
6. #30962C2semidirect
7. #30950C2semidirect
8. #30946C2semidirect
10. #23666C2semidirect
11. #23660C2semidirect
12. #23555C2semidirect
13. #23552C2semidirect
17. #21037C2semidirect
18. #21024C2semidirect
25. C2×#501C22semidirect
26. C2×#501C22semidirect
27. C2×#500C22semidirect
28. C2×#499C22semidirect
48. #2551C22semidirect
50. #2550C22semidirect
69. #2088C22semidirect
70. #2088C22semidirect
71. #2087C22semidirect
72. #2086C22semidirect
73. #2085C22semidirect
74. #2083C22semidirect
75. #2059C22semidirect
76. #2058C22semidirect
77. #2057C22semidirect
78. #2057C22semidirect
85. #1921C22semidirect
86. #1918C22semidirect
87. #1917C22semidirect
88. #1916C22semidirect
173. #418Q8semidirect
405. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
C22×#4742
#355762
#351742
#309662
#309622
#309502
#309462
#237352
#236662
#236602
#235552
#235522
#234562
#210912
#210632
#210372
#210242
#208732