Finite Group Databasebeta

← #7541809

Subgroups — #7541809

Order 512.

2157 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3335C212yesmaximal, complemented
2#59815C211yesmaximal, characteristic
3#36559C211yesmaximal, characteristic, complemented
4#30561C214yesmaximal, complemented
5#28687C211yesmaximal, characteristic, complemented
6#28509C211yesmaximal, characteristic, complemented
7#26232C212yesmaximal, complemented
8#26130C214yesmaximal, complemented
9#25699C212yesmaximal, complemented
10#25387C212yesmaximal, complemented
11#25339C212yesmaximal, complemented
12#25014C212yesmaximal, complemented
13#24118C212yesmaximal, complemented
14#23173C211yesmaximal, characteristic, complemented
15D4×#106C212yesmaximal, complemented
16C4×#475C211yesmaximal, characteristic, complemented
17C4×#460C211yesmaximal, characteristic, complemented
18C2×#554C2212yes—
19C2×#550—21no—
20C2×#547C2212yes—
21#3335—216no—
22C2×#498C2212yes—
23C2×#481C2214yes—
24C2×#483C2214yes—
25C2×#475C2211yescharacteristic
26C2×#476C2212yes—
27C2×#476C2212yes—
28C2×#465C2214yes—
29C2×#460C2211yescharacteristic
30C2×C4×SD16C2212yes—
31C2×C8×D4C2212yes—
32C2×C8×D4C2212yes—
33C22×#106C2214yes—
34#2868C2211yescharacteristic
35#2859—21no—
36#2682—22no—
37#2566—22no—
38#2563C2212yes—
39#2541C2212yes—
40#2538—21no—
41#2538C2212yes—
42#2538C2212yes—
43#2537C2212yes—
44#2452—22no—
45#2452—22no—
46#2445—22no—
47#2416C2214yes—
48#2407—22no—
49D4×C4⋊C4C2212yes—
50D4×C4⋊C4C2212yes—
51D4×C4⋊C4C2212yes—
52C4×C4⋊Q8C2211yescharacteristic
53C4×#127—21no—
54C4×#127C2211yescharacteristic
55C4×#127C2211yescharacteristic
56C42×Q8C2211yescharacteristic
57#2094C2214yes—
58#2084C2214yes—
59#2077C2214yes—
60#2035C2214yes—
61#2034C2214yes—
62#2034C2214yes—
63#2034C2214yes—
64#2000C2214yes—
65#1990C2212yes—
66#1988C2211yescharacteristic
67#1982C2211yescharacteristic
68#1957C2212yes—
69#1954C2212yes—
70#1954C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2211yescharacteristic
74#1948C2211yescharacteristic
75#1926C2212yes—
76#1919C2212yes—
77#1893C2214yes—
78#1880C2212yes—
79#1874C2214yes—
80#1869C2214yes—
81#1865C2212yes—
82#1853C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1844C2212yes—
86#1807C2214yes—
87#1803C2211yescharacteristic
88#1803C2211yescharacteristic
89C4×#106C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×#41—22no—
95C22×SD16C2312yes—
96C23×C8C2314yes—
97#554—28no—
98#554—44no—
99#550—48no—
100#547—28no—
101#547—44no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127—22no—
107C2×#127D411yescharacteristic
108C2×#127C2311yescharacteristic
109C2×#125—22no—
110C2×#123—24no—
111C2×C22⋊Q8—22no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8C2314yes—
114C2×#121—22no—
115C2×#121—24no—
116C2×#121C2314yes—
117C2×#121C2314yes—
118C2×C4×Q8—21no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—22no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—22no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2312yes—
132C22×C4⋊C4C2314yes—
133C22×C4⋊C4C2314yes—
134#498—28no—
135#483—216no—
136#481—216no—
137#476—28no—
138#476—44no—
139#475D418yescomplemented
140#465—216no—
141#460—24no—
142C4×SD16—28no—
143C8×D4—28no—
144C8×D4—28no—
145C2×#106—216no—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149C2×#106C2312yes—
150C2×#106C2312yes—
151C2×#106C2314yes—
152C2×#106C2312yes—
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#103C2312yes—
160C2×#102C2312yes—
161C2×#102C2314yes—
162C2×#102C2312yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×C22⋊C8C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C22⋊C8C2314yes—
168C2×C4×C8C2312yes—
169C2×C4×C8C2312yes—
170#396—24no—
171#396—24no—
172#396—24no—
173#394—24no—
174#394—24no—
175#393—24no—
176#392—24no—
177#391—21no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—21no—
183#390—21no—
184#390D412yes—
185#390D412yes—
186#390D412yes—
187#390C2312yes—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—22no—
193#389—21no—
194#389C2312yes—
195#386—24no—
196#385—24no—
197#385—24no—
198#385—24no—
199#384—21no—
200#384—21no—
201#384D411yescharacteristic
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2312yes—
206#382—22no—
207#380—22no—
208#380C2314yes—
209#380C2314yes—
210#380C2314yes—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2311yescharacteristic
219C43—21no—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#336C2312yes—
230#336C2312yes—
231C2×#41—48no—
232C2×#41—48no—
233C2×#41—84no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2411yescharacteristic
237C22×D4—24no—
238C22×D4—24no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C2×D412yes—
242C22×D4C2×D412yes—
243C22×D4C2×D412yes—
244C22×D4C2412yes—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4C2414yes—
249C2×SD16—24no—
250C2×SD16—24no—
251C2×SD16—42no—
252C22×C8—28no—
253C22×C8—28no—
254C22×C8—44no—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22×C8C2412yes—
259C4⋊Q8—24no—
260C4⋊Q8—22no—
261C4⋊Q8—24no—
262C4⋊Q8—82no—
263C4⋊Q8—82no—
264C4⋊Q8C2×D414yes—
265#127—22no—
266#127—22no—
267#127—48no—
268#127SD1618yescomplemented
269#125—48no—
270#123—416no—
271C22⋊Q8—28no—
272C22⋊Q8—416no—
273C22⋊Q8—44no—
274C22⋊Q8—48no—
275#121—28no—
276#121—44no—
277#121—48no—
278#121—416no—
279#121—44no—
280#121—44no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—41no—
285C4×Q8—41no—
286C4×Q8—44no—
287C4×Q8C2×D414yes—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—42no—
295C4×D4—42no—
296C4×D4—48no—
297C4×D4—48no—
298C4×D4—44no—
299C4×D4—82no—
300C4×D4—84no—
301C4×D4—84no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C42—21no—
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—22no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359#106—28no—
360#106—24no—
361#106—28no—
362#106—28no—
363#106—42no—
364#106—44no—
365#106C2×D418yes—
366C4⋊C8—22no—
367C4⋊C8—22no—
368C4⋊C8C2×D414yes—
369C4⋊C8#4114yescomplemented
370#103—24no—
371#103—42no—
372#103C2×D418yes—
373#102—28no—
374#102—24no—
375#102—24no—
376#102—42no—
377#102—44no—
378C22⋊C8—28no—
379C22⋊C8—28no—
380C22⋊C8—44no—
381C4×C8—24no—
382C4×C8—24no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—22no—
387#95—22no—
388#95—24no—
389#95—24no—
390#95—24no—
391C24—22no—
392C24C22×D414yes—
393#41—832no—
394C2×Q8—24no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8C22×D412yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4—81no—
426C2×D4—81no—
427C2×D4—84no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4C2×SD1618yes—
431C2×D4C2×SD1618yes—
432C2×D4C2×SD1618yes—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D414yes—
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C2511yescharacteristic
464SD16—48no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—416no—
476C2×C8C22×D414yes—
477C2×C8C2×#4114yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—82no—
509C4⋊C4—82no—
510C4⋊C4C22×D412yes—
511C4⋊C4C22×D414yes—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—44no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—82no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C42—24no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C42—41no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—44no—
534C42—42no—
535C42—42no—
536C42C2×SD1614yes—
537C42C22×D412yes—
538C42C2×#4112yes—
539C42H(5,𝔽2)12yes—
540C422−1+412yes—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C22×SD1618yes—
555C23C23×D411yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—41no—
559Q8—48no—
560Q8—41no—
561Q8—164no—
562D4—216no—
563D4—216no—
564D4—48no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—42no—
569D4—84no—
570D4—816no—
571D4—84no—
572D4—168no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4—88no—
615C2×C4—168no—
616C2×C4D4212yes—
617C2×C4D4212yes—
618C2×C4#54712yes—
619C2×C4#54712yes—
620C2×C4C22×SD1612yes—
621C2×C4C22×SD1612yes—
622C2×C4C22×SD1612yes—
623C2×C4#57712yes—
624C2×C4#57712yes—
625C2×C4C23×D411yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58511yescharacteristic
630C2×C4#58511yescharacteristic
631C8—24no—
632C8—24no—
633C8—42no—
634C8—42no—
635C22—28no—
636C22—28no—
637C22—28no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C2×D4211yescharacteristic
650C22C2×#54711yescharacteristic
651C22#348511yescharacteristic
652C22C23×SD1611yescharacteristic
653C22C2×#57711yescharacteristic
654C22C2×#57711yescharacteristic
655C22#358511yescharacteristic
656C4—22no—
657C4—22no—
658C4—22no—
659C4—41no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—82no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—84no—
669C4D4×SD1614yes—
670C4#333514yescomplemented
671C4D4×#4111yescharacteristic
672C4#349711yescharacteristic
673C4#351711yescharacteristic
674C4#355411yescharacteristic
675C2—48no—
676C2—162no—
677C2#5906611yescharacteristic
678C2C2×D4×SD1611yescharacteristic
679C2#6109511yescharacteristic
680C2C2×#333511yescharacteristic
681C2#6127911yescharacteristic
682C2#6169711yescharacteristic
683C2#6179211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#59815C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3335C2semidirect
3. #36559C2semidirect
4. #30561C2semidirect
5. #28687C2semidirect
6. #28509C2semidirect
7. #26232C2semidirect
8. #26130C2semidirect
9. #25699C2semidirect
10. #25387C2semidirect
11. #25339C2semidirect
12. #25014C2semidirect
13. #24118C2semidirect
14. #23173C2semidirect
15. D4×#106C2semidirect
16. C4×#475C2semidirect
17. C4×#460C2semidirect
139. #475D4semidirect
268. #127SD16semidirect
369. C4⋊C8#41semidirect
670. C4#3335semidirect

Maximal subgroups

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

subgroupindex
C2×#33352
#598152
#365592
#305612
#286872
#285092
#262322
#261302
#256992
#253872
#253392
#250142
#241182
#231732
D4×#1062
C4×#4752
C4×#4602