Finite Group Databasebeta

← #7502475

Subgroups — #7502475

Order 512.

1788 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#499C211yesmaximal, characteristic, complemented
2C2×#3084C211yesmaximal, characteristic
3#35640C211yesmaximal, characteristic, complemented
4#35639C211yesmaximal, characteristic
5#30960C214yesmaximal, complemented
6#30958C214yesmaximal, complemented
7#30953C214yesmaximal, complemented
8#30951C214yesmaximal, complemented
9#23738C211yesmaximal, characteristic
10#23668C211yesmaximal, characteristic, complemented
11#23660C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#23553C211yesmaximal, characteristic, complemented
14#23453C211yesmaximal, characteristic
15#21083C211yesmaximal, characteristic
16#21067C211yesmaximal, characteristic
17#21038C211yesmaximal, characteristic, complemented
18#21020C211yesmaximal, characteristic, complemented
19#20881C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#499—28no—
25C2×#499C2214yes—
26C2×#499C2214yes—
27#3084—28no—
28C2×#479C2212yes—
29C2×#475C2212yes—
30C2×#478C2212yes—
31C2×#476C2212yes—
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#416C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C22×C4×C8C2211yescharacteristic
44#2583—22no—
45#2579—22no—
46#2552—22no—
47#2552C2214yes—
48#2550—22no—
49#2550C2214yes—
50#2502—22no—
51#2487—22no—
52#2454C2214yes—
53#2453C2214yes—
54#2442—22no—
55#2441—22no—
56#2317C2211yescharacteristic
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2311C2211yescharacteristic
61#2297C2211yescharacteristic
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2274C2211yescharacteristic
65#2097C2212yes—
66#2096C2212yes—
67#2094C2212yes—
68#2092C2212yes—
69#2088C2214yes—
70#2088C2214yes—
71#2085C2214yes—
72#2083C2214yes—
73#2059C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2057C2214yes—
77#2026C2214yes—
78#2025C2214yes—
79#1934C2212yes—
80#1934C2212yes—
81#1933C2212yes—
82#1933C2212yes—
83#1928C2214yes—
84#1925C2214yes—
85#1920C2214yes—
86#1919C2214yes—
87#1916C2214yes—
88#1916C2214yes—
89#1863C2214yes—
90#1862C2214yes—
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C22×M16C2311yescharacteristic
103C22×M16C2311yescharacteristic
104C23×C8C2311yescharacteristic
105C23×C8C2311yescharacteristic
106#544—24no—
107#544—42no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8C2312yes—
110C2×#125—21no—
111C2×#125—24no—
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—22no—
115C2×#125—42no—
116C2×#125—42no—
117C2×#125C2312yes—
118C2×#125C2314yes—
119C2×#125C2314yes—
120C2×C22⋊Q8—22no—
121#520—24no—
122#520—42no—
123C2×C4×Q8C2312yes—
124C2×C4×D4—22no—
125C2×C4×D4C2312yes—
126C2×#117—21no—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—21no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#499—216no—
143#499—216no—
144#479—28no—
145#478—28no—
146#476—28no—
147#475—28no—
148#428—24no—
149#428—42no—
150C2×#107—24no—
151C2×#107—24no—
152C2×#107C2312yes—
153C2×#107C2314yes—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#106—24no—
157C2×#106D418yes—
158C2×#106C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164#424—24no—
165C2×C4⋊C8C2312yes—
166C2×C4⋊C8C2312yes—
167#417—24no—
168#417—42no—
169#416—24no—
170#416Q818yescomplemented
171C2×#103C2311yescharacteristic
172C2×#103C2312yes—
173C2×#103C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#102C2311yescharacteristic
177C2×#102C2312yes—
178C2×#102C2311yescharacteristic
179C2×#102C2312yes—
180C2×#102C2312yes—
181C2×C4×C8—24no—
182C2×C4×C8C2312yes—
183C2×C4×C8C2312yes—
184#400—24no—
185#399—24no—
186#398—24no—
187#398—24no—
188#398—24no—
189#398—24no—
190#396—24no—
191#395—24no—
192#391—24no—
193#391C2312yes—
194#391C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—22no—
198#389—22no—
199#389—24no—
200#389—24no—
201#389—24no—
202#389D412yes—
203#389D412yes—
204#389C2314yes—
205#389C2312yes—
206#389C2312yes—
207#389C2314yes—
208#389C2314yes—
209#387—24no—
210#386—21no—
211#386—21no—
212#386—22no—
213#385—21no—
214#385—21no—
215#385—24no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385D412yes—
220#385D412yes—
221#384—21no—
222#384—21no—
223#384—24no—
224#384C2312yes—
225#382—21no—
226#382—21no—
227#382C2312yes—
228#380—22no—
229#380—22no—
230#380—24no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#336C2314yes—
246#336C2314yes—
247#336C2314yes—
248#336C2312yes—
249#336C2312yes—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41C2×Q814yes—
256C22×Q8—22no—
257C22×Q8C2411yescharacteristic
258C22×D4—21no—
259C22×D4C2411yescharacteristic
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—22no—
266C23×C4C2×D411yescharacteristic
267C23×C4C2×D411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8C2×D414yes—
280C22×C8C2×Q814yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C22×C8C2412yes—
286C4⋊Q8—24no—
287C4⋊Q8—42no—
288C4⋊Q8—44no—
289#125—28no—
290#125—28no—
291#125—24no—
292#125—44no—
293#125—44no—
294#125—42no—
295#125—48no—
296#125—48no—
297#125—44no—
298C22⋊Q8—48no—
299C4×Q8—24no—
300C4×Q8—42no—
301C4×D4—24no—
302C4×D4—42no—
303C4×D4—48no—
304#117—22no—
305#117—22no—
306#117—22no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—44no—
314#117C2×Q814yes—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2×D414yes—
348C2×C4⋊C4C2×D414yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2411yescharacteristic
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4C2×D412yes—
367C2×C22⋊C4C2×D412yes—
368C2×C42—21no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—24no—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2412yes—
379#107—28no—
380#107—28no—
381#107—24no—
382#107—42no—
383#106—24no—
384#106—24no—
385#106—24no—
386#106—42no—
387#106C2×D418yes—
388#106C2×D418yes—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#103—22no—
392#103—22no—
393#103—24no—
394#103—42no—
395#103C2×Q818yes—
396#102—22no—
397#102—22no—
398#102—24no—
399#102—42no—
400#102C2×Q818yes—
401C4×C8—24no—
402C4×C8—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95C2×D414yes—
414#95C2×D414yes—
415C24—21no—
416C24C22×D411yescharacteristic
417#41—28no—
418#41—28no—
419#41—44no—
420#41—44no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—44no—
427C2×Q8—84no—
428C2×Q8C22×Q812yes—
429C2×D4—24no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4C22×Q812yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—84no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×Q812yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C22×C4C2511yescharacteristic
477M16—44no—
478M16—44no—
479C2×C8—28no—
480C2×C8—28no—
481C2×C8—22no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8C22×D412yes—
494C2×C8C22×D412yes—
495C2×C8C22×Q814yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C4⋊C4C22×Q814yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—44no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—84no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—44no—
553C42—42no—
554C42—42no—
555C42—84no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23#53612yes—
564C23D4212yes—
565C23D4×Q812yes—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—164no—
572D4—22no—
573D4—22no—
574D4—48no—
575D4—41no—
576D4—41no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—84no—
615C2×C4—84no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—164no—
619C2×C4#53612yes—
620C2×C4#53614yes—
621C2×C4#54414yes—
622C2×C4D4212yes—
623C2×C4D4×Q812yes—
624C2×C4D4×Q814yes—
625C2×C4#56314yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C23×Q811yescharacteristic
628C2×C4C2×H(5,𝔽2)11yescharacteristic
629C2×C4C2×2−1+411yescharacteristic
630C2×C4C2×2−1+411yescharacteristic
631C2×C4C2×2−1+411yescharacteristic
632C8—22no—
633C8—22no—
634C8—24no—
635C8—42no—
636C8—42no—
637C22—22no—
638C22—22no—
639C22—24no—
640C22—24no—
641C22—42no—
642C22—44no—
643C22—81no—
644C22—81no—
645C22—82no—
646C22#267714yes—
647C22C2×#53611yescharacteristic
648C22C2×D4211yescharacteristic
649C22C2×D4×Q811yescharacteristic
650C22#358412yes—
651C22#358412yes—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—81no—
662C4—82no—
663C4—82no—
664C4—81no—
665C4—82no—
666C4#267714yes—
667C4C2×#53611yescharacteristic
668C4C2×#54411yescharacteristic
669C4C2×D4×Q811yescharacteristic
670C4C2×#56311yescharacteristic
671C2—24no—
672C2—82no—
673C2C2×#267711yescharacteristic
674C2#6088212yes—
675C2#6105211yescharacteristic
676C2#6105211yescharacteristic
677C2#6137612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
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. C22×#499C2semidirect
3. #35640C2semidirect
5. #30960C2semidirect
6. #30958C2semidirect
7. #30953C2semidirect
8. #30951C2semidirect
10. #23668C2semidirect
11. #23660C2semidirect
12. #23555C2semidirect
13. #23553C2semidirect
17. #21038C2semidirect
18. #21020C2semidirect
170. #416Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4992
C2×#30842
#356402
#356392
#309602
#309582
#309532
#309512
#237382
#236682
#236602
#235552
#235532
#234532
#210832
#210672
#210382
#210202
#208812