Finite Group Databasebeta

← #7501441

Subgroups — #7501441

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#488C211yesmaximal, characteristic, complemented
2C2×#3097C211yesmaximal, characteristic, complemented
3#35677C211yesmaximal, characteristic
4#35639C211yesmaximal, characteristic, complemented
5#30644C214yesmaximal, complemented
6#30643C214yesmaximal, complemented
7#30642C214yesmaximal, complemented
8#30641C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic, complemented
11#23553C211yesmaximal, characteristic, complemented
12#23506C211yesmaximal, characteristic, complemented
13#23486C211yesmaximal, characteristic
14#23484C211yesmaximal, characteristic, complemented
15#21151C211yesmaximal, characteristic, complemented
16#21046C211yesmaximal, characteristic, complemented
17#20914C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#488—28no—
27C2×#488C2214yes—
28C2×#488C2214yes—
29#3097—28no—
30C2×#484C2212yescomplemented
31C2×#485C2212yescomplemented
32C2×#485C2212yescomplemented
33C2×#483C2212yescomplemented
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C2×#427C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C2×#408C2211yescharacteristic
48C2×#408C2211yescharacteristic
49C22×C4×C8C2211yescharacteristic
50#2579—22no—
51#2548—22no—
52#2545—22no—
53#2487—22no—
54#2454—22no—
55#2453—22no—
56#2442C2214yescomplemented
57#2441C2214yescomplemented
58#2426C2214yes—
59#2426C2214yes—
60#2411—22no—
61#2411—22no—
62#2317C2211yescharacteristic
63#2316C2211yescharacteristic
64#2311C2211yescharacteristic
65#2309C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68#2296C2211yescharacteristic
69#2094C2212yes—
70#2093C2212yes—
71#2092C2212yes—
72#2092C2212yes—
73#2067C2214yes—
74#2065C2214yes—
75#2059C2214yescomplemented
76#2059C2214yescomplemented
77#2042C2214yes—
78#2042C2214yes—
79#2036C2214yescomplemented
80#2035C2214yescomplemented
81#2034C2214yes—
82#2033C2214yes—
83#1942C2212yescomplemented
84#1942C2212yescomplemented
85#1941C2212yescomplemented
86#1941C2212yescomplemented
87#1923C2214yescomplemented
88#1923C2214yescomplemented
89#1875C2214yes—
90#1874C2214yes—
91#1865C2214yescomplemented
92#1863C2214yescomplemented
93#1805C2214yescomplemented
94#1803C2214yescomplemented
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8C2311yescharacteristic
103C23×D4C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#536—24no—
109#536—42no—
110#535—24no—
111#535—42no—
112C2×C4⋊Q8—21no—
113C2×#127—21no—
114C2×#125—24no—
115C2×#125—22no—
116C2×#125—42no—
117C2×#125C2314yes—
118C2×#124—21no—
119C2×#124—21no—
120C2×#124—24no—
121C2×#124—22no—
122C2×#124—42no—
123C2×#124C2314yes—
124C2×#123—22no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×#121C2314yes—
128C2×#120—22no—
129#520—24no—
130#520—42no—
131#519—24no—
132#519—42no—
133C2×C4×Q8—21no—
134C2×C4×Q8—22no—
135C2×C4×Q8C2314yes—
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×#117—22no—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2312yes—
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—21no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2311yescharacteristic
150C22×C22⋊C4—21no—
151C22×C22⋊C4—21no—
152C22×C42C2311yescharacteristic
153#488—216no—
154#488—216no—
155#485—28no—
156#485—28no—
157#484—28no—
158#483—28no—
159#428—24no—
160#428—42no—
161#427—24no—
162#427—24no—
163C2×#107C2312yes—
164C2×#107C2311yescharacteristic
165C2×#107C2311yescharacteristic
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#106C2312yes—
169C2×#106C2311yescharacteristic
170C2×#106C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173#424—42no—
174C2×C4⋊C8C2314yes—
175#419—24no—
176#419—42no—
177#418—42no—
178#418—42no—
179C2×#103—24no—
180C2×#103D418yescomplemented
181C2×#103C2312yes—
182C2×#103C2314yes—
183C2×#103C2312yes—
184C2×#103C2314yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#102—24no—
188C2×#102—24no—
189C2×#102C2314yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2314yes—
193C2×#102C2312yes—
194C2×#102C2312yes—
195#408—24no—
196#408—42no—
197C2×C22⋊C8C2312yes—
198C2×C22⋊C8C2312yes—
199C2×C22⋊C8C2314yes—
200C2×C4×C8—24no—
201C2×C4×C8C2312yes—
202C2×C4×C8C2312yes—
203#400—24no—
204#399—24no—
205#398—24no—
206#396—24no—
207#396—24no—
208#396—24no—
209#394—24no—
210#394—24no—
211#391—21no—
212#391D412yes—
213#390—21no—
214#390—22no—
215#390—24no—
216#390—24no—
217#390D412yes—
218#390C2312yes—
219#390C2312yes—
220#389—22no—
221#389—22no—
222#389—24no—
223#389C2312yes—
224#389C2312yes—
225#389C2314yes—
226#389C2312yes—
227#389C2312yes—
228#388—24no—
229#388—24no—
230#387—24no—
231#386—22no—
232#386D412yes—
233#386D412yes—
234#385—21no—
235#385—21no—
236#385—21no—
237#385—21no—
238#385—22no—
239#385—24no—
240#384C2312yes—
241#382C2312yes—
242#380—22no—
243#380—24no—
244#380C2314yes—
245#379—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4—22no—
248C4×C22⋊C4C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C25—21no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C22×Q8—21no—
266C22×Q8—22no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C2×D414yes—
272C22×Q8C2×D411yescharacteristic
273C22×Q8C2412yes—
274C22×D4—24no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—22no—
278C22×D4—22no—
279C22×D4—22no—
280C22×D4—22no—
281C22×D4—22no—
282C22×D4C2×D412yes—
283C22×D4C2×D411yescharacteristic
284C22×D4C2412yes—
285C23×C4—21no—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2412yes—
292C23×C4C2411yescharacteristic
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8C2×D414yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2411yescharacteristic
309C22×C8C2411yescharacteristic
310C4⋊Q8—44no—
311#127—44no—
312#125—28no—
313#125—44no—
314#125—48no—
315#124—28no—
316#124—44no—
317#124—48no—
318#124—44no—
319#124—44no—
320#123—48no—
321C22⋊Q8—24no—
322C22⋊Q8—24no—
323C22⋊Q8—42no—
324C22⋊Q8—42no—
325#121—28no—
326#121—44no—
327#120—48no—
328C4×Q8—28no—
329C4×Q8—44no—
330C4×Q8—48no—
331C4×Q8—44no—
332C4×D4—44no—
333C4×D4—48no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—24no—
344#117—48no—
345#117—42no—
346#117#4114yes—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—42no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2×D414yes—
370C2×C4⋊C4C2411yescharacteristic
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2411yescharacteristic
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—44no—
392C2×C22⋊C4—44no—
393C2×C22⋊C4C2×D412yes—
394C2×C22⋊C4C2×D414yes—
395C2×C22⋊C4C2×D414yes—
396C2×C42—21no—
397C2×C42—22no—
398C2×C42—22no—
399C2×C42—24no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42C2×D412yes—
405C2×C42C2412yes—
406C2×C42C2412yes—
407#107—22no—
408#107—22no—
409#107—24no—
410#107—24no—
411#107—42no—
412#106—22no—
413#106—22no—
414#106—24no—
415#106—24no—
416#106—42no—
417C4⋊C8—44no—
418#103—28no—
419#103—24no—
420#103—24no—
421#103—44no—
422#103C2×D418yes—
423#103C2×D418yes—
424#102—24no—
425#102—24no—
426#102—28no—
427#102—44no—
428#102—42no—
429#102—42no—
430C22⋊C8—24no—
431C22⋊C8—24no—
432C22⋊C8—44no—
433C4×C8—24no—
434C4×C8—24no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—24no—
440#95—24no—
441#95—24no—
442#95C2×D414yes—
443#95C2×D414yes—
444C24—22no—
445C24—42no—
446C24—42no—
447C24—42no—
448C24—82no—
449C24—82no—
450C24C22×D412yes—
451C24C22×D411yescharacteristic
452#41—44no—
453#41—44no—
454#41—44no—
455#41—44no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—24no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—28no—
462C2×Q8—28no—
463C2×Q8—22no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—42no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—44no—
471C2×Q8C22×D414yes—
472C2×D4—24no—
473C2×D4—24no—
474C2×D4—24no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—22no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—48no—
485C2×D4—42no—
486C2×D4—44no—
487C2×D4—44no—
488C2×D4—42no—
489C2×D4—42no—
490C2×D4—44no—
491C2×D4—44no—
492C2×D4—84no—
493C2×D4—84no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—44no—
516C22×C4—42no—
517C22×C4—42no—
518C22×C4—42no—
519C22×C4—82no—
520C22×C4—82no—
521C22×C4—84no—
522C22×C4C22×D412yes—
523C22×C4C22×D414yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D411yescharacteristic
527C22×C4C22×D412yes—
528C22×C4C2×#4112yes—
529C22×C4H(5,𝔽2)12yes—
530C22×C4H(5,𝔽2)12yes—
531C22×C42−1+412yes—
532C22×C42−1+412yes—
533C22×C4C2511yescharacteristic
534M16—44no—
535M16—44no—
536C2×C8—28no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8—22no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—44no—
551C2×C8C22×D412yes—
552C2×C8C22×D412yes—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—22no—
568C4⋊C4—21no—
569C4⋊C4—44no—
570C4⋊C4—48no—
571C4⋊C4—48no—
572C4⋊C4—48no—
573C4⋊C4—44no—
574C4⋊C4—44no—
575C4⋊C4—42no—
576C4⋊C4—84no—
577C4⋊C4—84no—
578C4⋊C4—82no—
579C4⋊C4—82no—
580C4⋊C4—82no—
581C4⋊C4—82no—
582C4⋊C4C22×D414yes—
583C4⋊C4C2×#4114yes—
584C22⋊C4—24no—
585C22⋊C4—24no—
586C22⋊C4—44no—
587C22⋊C4—44no—
588C22⋊C4—48no—
589C22⋊C4—48no—
590C22⋊C4—48no—
591C22⋊C4—44no—
592C22⋊C4—44no—
593C22⋊C4—44no—
594C22⋊C4—42no—
595C22⋊C4—42no—
596C22⋊C4—42no—
597C22⋊C4—84no—
598C22⋊C4—84no—
599C22⋊C4—84no—
600C42—24no—
601C42—24no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—22no—
606C42—44no—
607C42—48no—
608C42—42no—
609C42—42no—
610C42—42no—
611C23—22no—
612C23—22no—
613C23—24no—
614C23—24no—
615C23—22no—
616C23—22no—
617C23—22no—
618C23—44no—
619C23—44no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—84no—
624C23—84no—
625C23—84no—
626C23—164no—
627C23—164no—
628C23#53612yes—
629C23D4212yes—
630C23#54612yes—
631C23C23×D411yescharacteristic
632Q8—24no—
633Q8—24no—
634Q8—48no—
635Q8—44no—
636Q8—42no—
637Q8—42no—
638D4—48no—
639D4—44no—
640D4—42no—
641D4—42no—
642D4—42no—
643D4—42no—
644D4—84no—
645D4—168no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—22no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—48no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—84no—
687C2×C4—84no—
688C2×C4—164no—
689C2×C4#53612yes—
690C2×C4#53714yes—
691C2×C4#54214yes—
692C2×C4D4212yes—
693C2×C4#54612yes—
694C2×C4#54714yes—
695C2×C4#56214yes—
696C2×C4C23×D411yescharacteristic
697C2×C4C22×#4111yescharacteristic
698C2×C4C2×H(5,𝔽2)11yescharacteristic
699C2×C4C2×H(5,𝔽2)11yescharacteristic
700C2×C4C2×2−1+411yescharacteristic
701C2×C4C2×2−1+411yescharacteristic
702C8—22no—
703C8—22no—
704C8—42no—
705C8—42no—
706C8—42no—
707C22—24no—
708C22—22no—
709C22—22no—
710C22—24no—
711C22—44no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—44no—
716C22—84no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—84no—
721C22—168no—
722C22#267214yes—
723C22C2×#53611yescharacteristic
724C22C2×D4211yescharacteristic
725C22C2×#54611yescharacteristic
726C22#358412yes—
727C22#358412yes—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—44no—
732C4—42no—
733C4—44no—
734C4—42no—
735C4—82no—
736C4—81no—
737C4—81no—
738C4—81no—
739C4—81no—
740C4—82no—
741C4#267214yes—
742C4C2×#53711yescharacteristic
743C4C2×#54211yescharacteristic
744C4C2×#54711yescharacteristic
745C4C2×#56211yescharacteristic
746C2—24no—
747C2—82no—
748C2—162no—
749C2C2×#267211yescharacteristic
750C2#6087912yes—
751C2#6105211yescharacteristic
752C2#6105411yescharacteristic
753C2#6121612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#488C2semidirect
2. C2×#3097C2semidirect
4. #35639C2semidirect
5. #30644C2semidirect
6. #30643C2semidirect
7. #30642C2semidirect
8. #30641C2semidirect
9. #23735C2semidirect
10. #23614C2semidirect
11. #23553C2semidirect
12. #23506C2semidirect
14. #23484C2semidirect
15. #21151C2semidirect
16. #21046C2semidirect
18. #20880C2semidirect
19. #20651C2semidirect
30. C2×#484C22semidirect
31. C2×#485C22semidirect
32. C2×#485C22semidirect
33. C2×#483C22semidirect
56. #2442C22semidirect
57. #2441C22semidirect
75. #2059C22semidirect
76. #2059C22semidirect
79. #2036C22semidirect
80. #2035C22semidirect
83. #1942C22semidirect
84. #1942C22semidirect
85. #1941C22semidirect
86. #1941C22semidirect
87. #1923C22semidirect
88. #1923C22semidirect
91. #1865C22semidirect
92. #1863C22semidirect
93. #1805C22semidirect
94. #1803C22semidirect
180. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4882
C2×#30972
#356772
#356392
#306442
#306432
#306422
#306412
#237352
#236142
#235532
#235062
#234862
#234842
#211512
#210462
#209142
#208802
#206512