Finite Group Databasebeta

← #6382201

Subgroups — #6382201

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3036C211yesmaximal, characteristic
2#34206C211yesmaximal, characteristic, complemented
3#34186C211yesmaximal, characteristic, complemented
4#23112C214yesmaximal, complemented
5#23109C214yesmaximal, complemented
6#23108C214yesmaximal, complemented
7#23105C214yesmaximal, complemented
8#21235C212yesmaximal, complemented
9#21199C211yesmaximal, characteristic, complemented
10#21170C211yesmaximal, characteristic, complemented
11#21062C212yesmaximal, complemented
12#20932C212yesmaximal, complemented
13#20438C211yesmaximal, characteristic, complemented
14#20415C212yesmaximal, complemented
15C2×#1762C211yesmaximal, characteristic, complemented
16C22×AGL(1,ℤ/8)C2212yes—
17C2×#532C2211yescharacteristic
18C2×#520C2212yes—
19C22×C4×D4C2211yescharacteristic
20#3036C4116yescomplemented
21C2×#463C2214yes—
22C2×#461C2212yes—
23C2×#459C2212yes—
24C2×#428C2212yes—
25C2×#423C2211yescharacteristic
26C2×#417C2212yes—
27C2×#416C2212yes—
28C2×#416C2212yes—
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C2×#405C2212yes—
32C22×#97C2211yescharacteristic
33#2367C2214yescomplemented
34#2366C2212yescomplemented
35#2365C2212yescomplemented
36#2357—22no—
37#2357C2214yes—
38#2354—21no—
39#2354—21no—
40#2354—21no—
41#2354C2212yes—
42#2354C2212yes—
43#2352—22no—
44D4×C4⋊C4—21no—
45#2325—22no—
46#2312—22no—
47#2311C2212yes—
48#2309C2212yes—
49C4×#121—22no—
50#2285C2211yescharacteristic
51#2285C2211yescharacteristic
52#2280—22no—
53#2274C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55#1973C2214yes—
56#1972C2214yes—
57#1970C2214yes—
58#1968C2214yes—
59#1961C2212yescomplemented
60#1960C2212yescomplemented
61#1959C2212yescomplemented
62#1959C2212yescomplemented
63#1949C2214yescomplemented
64#1948C2212yescomplemented
65#1947C2212yescomplemented
66#1928C2214yescomplemented
67#1927C2214yescomplemented
68#1926C2214yescomplemented
69#1925C2214yescomplemented
70#1880C2214yes—
71#1880C2214yes—
72#1879C2214yes—
73#1879C2214yes—
74#1771C2214yescomplemented
75#1771C2214yescomplemented
76#1763C2214yescomplemented
77#1762C2214yes—
78#1762C2214yes—
79#1762C2214yescomplemented
80C4×#103C2214yescomplemented
81C4×#102C2214yescomplemented
82#1736C2212yes—
83C2×#340C2211yescharacteristic
84C2×#340C2211yescharacteristic
85#1729C2212yes—
86C22×#41C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4—21no—
89C2×AGL(1,ℤ/8)—44no—
90C2×AGL(1,ℤ/8)—44no—
91C2×AGL(1,ℤ/8)C2×C4116yescomplemented
92C22×SD16C2314yes—
93C22×D8C2314yes—
94C22×M16C2312yes—
95C23×C8C2311yescharacteristic
96C23×C8C2311yescharacteristic
97#532—24no—
98#532C2×C418yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#127C2311yescharacteristic
101C2×#124C2312yes—
102C2×#121—21no—
103C2×#121—21no—
104#520—28no—
105#520—44no—
106C2×C4×Q8C2314yes—
107C2×C4×D4—22no—
108C2×C4×D4—24no—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4C2×C418yescomplemented
115C2×C4×D4C2314yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117—22no—
120C2×#117—22no—
121C2×#117C2312yes—
122C2×#117C2312yes—
123C22×C4⋊C4—22no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C22⋊C4—21no—
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130#463C2×C4132yescomplemented
131#461C2×C4116yescomplemented
132#459C2×C4116yescomplemented
133#428—44no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136#423C2×C418yescomplemented
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139#417C2×C4116yescomplemented
140#416—28no—
141#416—44no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2312yes—
145C2×#103C2312yes—
146C2×#103C2314yes—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×#102C2312yes—
153C2×#102C2312yes—
154C2×#102C2312yes—
155C2×#102C2314yes—
156#405—44no—
157C2×#97—24no—
158C2×#97C2312yes—
159C2×#97C2312yes—
160C2×#97C2314yes—
161C2×C4×C8C2314yes—
162#391C2314yes—
163#390—22no—
164#390—22no—
165#390—22no—
166#390C2314yes—
167#390C2314yes—
168#390C2314yes—
169#389—22no—
170#388—24no—
171#388—24no—
172#388—24no—
173#388—24no—
174#388—24no—
175#388—24no—
176#387—24no—
177#386—24no—
178#386—24no—
179#385—24no—
180#384D411yescharacteristic
181#384D411yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382D412yes—
185#382C2312yes—
186#381—24no—
187#380—21no—
188#380—21no—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380D412yes—
194#380D412yes—
195#379—24no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4—24no—
210C4×C22⋊C4—24no—
211C4×C22⋊C4—24no—
212C2×#95—21no—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#336C2314yes—
218#336C2314yes—
219C25—21no—
220C2×#41—24no—
221C2×#41—24no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41C22×C418yes—
226C22×Q8C2412yes—
227C22×D4—22no—
228C22×D4—21no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C22×C414yes—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—22no—
243C23×C4—28no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—24no—
247C23×C4—82no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2411yescharacteristic
252AGL(1,ℤ/8)—432no—
253C2×SD16—44no—
254C2×SD16—44no—
255C2×SD16C22×C4116yes—
256C2×D8—44no—
257C2×D8—44no—
258C2×D8C22×C4116yes—
259C2×M16—24no—
260C2×M16—24no—
261C2×M16C22×C418yes—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8#4114yes—
268C22×C8C2412yes—
269C22×C8C2414yes—
270C22×C8C2412yes—
271C4⋊Q8—22no—
272C4⋊Q8C22×C414yes—
273#127—22no—
274#127C22×C414yes—
275#124—24no—
276#124C22×C418yes—
277#121—28no—
278#121—28no—
279C4×Q8—28no—
280C4×Q8—44no—
281C4×D4—28no—
282C4×D4—28no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—44no—
289C4×D4—416no—
290C4×D4—84no—
291C4×D4C22×C418yes—
292C4×D4C22×C418yes—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—48no—
298#117—84no—
299#117—84no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—48no—
315C2×C4⋊C4C22×C414yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4#4114yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4C2×D414yes—
342C2×C42—24no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—22no—
348C2×C42—82no—
349C2×C42C22×C414yes—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42#4114yes—
353C2×C42C2414yes—
354C2×C42C2411yescharacteristic
355C2×C42C2411yescharacteristic
356#107—44no—
357#106—44no—
358C4⋊C8C22×C418yes—
359C4⋊C8C22×C418yes—
360#103—24no—
361#103—24no—
362#103—42no—
363#103—42no—
364#103C22×C4116yes—
365#102—28no—
366#102—24no—
367#102—24no—
368#102—28no—
369#102—42no—
370#102—42no—
371#102C22×C4116yes—
372#97—24no—
373#97—24no—
374#97—44no—
375C4×C8—44no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—24no—
384#95C2×D414yes—
385#95C2×D414yes—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—42no—
390C24—42no—
391C24—81no—
392C24—81no—
393C24—81no—
394C24—81no—
395C24C22×D411yescharacteristic
396#41—216no—
397#41—216no—
398#41—48no—
399#41—48no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C23×C414yes—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—81no—
431C2×D4—81no—
432C2×D4C23×C414yes—
433C2×D4C23×C414yes—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—82no—
456C22×C4—82no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—816no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—81no—
466C22×C4C4×D414yes—
467C22×C4C4×D414yes—
468C22×C4C23×C412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4C2511yescharacteristic
482SD16—416no—
483D8—416no—
484M16—216no—
485C2×C8—28no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8C23×C418yes—
495C2×C8C2×#4114yes—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—48no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4C23×C414yes—
521C4⋊C4C2×#4114yes—
522C22⋊C4—28no—
523C22⋊C4—28no—
524C22⋊C4—28no—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—84no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—24no—
536C42—24no—
537C42—48no—
538C42—44no—
539C42—44no—
540C42—82no—
541C42—84no—
542C42—84no—
543C42C4×D414yes—
544C42C4×D414yes—
545C42C23×C412yes—
546C42C23×C412yes—
547C42C2×#4112yes—
548C42C2×#4112yes—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—24no—
554C23—24no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23—84no—
572C23—84no—
573C23—84no—
574C23C2×C4×D412yes—
575C23#54612yes—
576C23#55012yes—
577C23C23×D411yescharacteristic
578Q8—24no—
579Q8—24no—
580Q8—42no—
581Q8—42no—
582D4—24no—
583D4—24no—
584D4—24no—
585D4—24no—
586D4—48no—
587D4—416no—
588D4—48no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—84no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—416no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—416no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—168no—
635C2×C4C2×C4×D412yes—
636C2×C4C2×C4×D412yes—
637C2×C4C2×C4×D412yes—
638C2×C4C2×C4×D412yes—
639C2×C4C2×C4×D412yes—
640C2×C4C4×#4114yes—
641C2×C4#51814yes—
642C2×C4#54612yes—
643C2×C4#55012yes—
644C2×C4#55314yes—
645C2×C4#55514yes—
646C2×C4C24×C411yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×H(5,𝔽2)11yescharacteristic
652C8—28no—
653C8—24no—
654C8—42no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#236314yes—
677C22C22×C4×D411yescharacteristic
678C22C2×#54611yescharacteristic
679C22C2×#55011yescharacteristic
680C22#358511yescharacteristic
681C22#358511yescharacteristic
682C22#358511yescharacteristic
683C22#358611yescharacteristic
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—44no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—44no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#236314yes—
697C4C2×C4×#4111yescharacteristic
698C4C2×#51811yescharacteristic
699C4C2×#55311yescharacteristic
700C4C2×#55511yescharacteristic
701C2—24no—
702C2—84no—
703C2—84no—
704C2C2×#236311yescharacteristic
705C2#6016811yescharacteristic
706C2#6017011yescharacteristic
707C2#6121811yescharacteristic
708C2#6122011yescharacteristic
709C2#6141411yescharacteristic
710C2#6141511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3036C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34206C2semidirect
3. #34186C2semidirect
4. #23112C2semidirect
5. #23109C2semidirect
6. #23108C2semidirect
7. #23105C2semidirect
8. #21235C2semidirect
9. #21199C2semidirect
10. #21170C2semidirect
11. #21062C2semidirect
12. #20932C2semidirect
13. #20438C2semidirect
14. #20415C2semidirect
15. C2×#1762C2semidirect
20. #3036C4semidirect
33. #2367C22semidirect
34. #2366C22semidirect
35. #2365C22semidirect
59. #1961C22semidirect
60. #1960C22semidirect
61. #1959C22semidirect
62. #1959C22semidirect
63. #1949C22semidirect
64. #1948C22semidirect
65. #1947C22semidirect
66. #1928C22semidirect
67. #1927C22semidirect
68. #1926C22semidirect
69. #1925C22semidirect
74. #1771C22semidirect
75. #1771C22semidirect
76. #1763C22semidirect
79. #1762C22semidirect
80. C4×#103C22semidirect
81. C4×#102C22semidirect
91. C2×AGL(1,ℤ/8)C2×C4semidirect
114. C2×C4×D4C2×C4semidirect
130. #463C2×C4semidirect
131. #461C2×C4semidirect
132. #459C2×C4semidirect
136. #423C2×C4semidirect
139. #417C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30362
#342062
#341862
#231122
#231092
#231082
#231052
#212352
#211992
#211702
#210622
#209322
#204382
#204152
C2×#17622