Finite Group Databasebeta

← #7545724

Subgroups — #7545724

Order 512.

2123 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3368C212yesmaximal, complemented
2#59828C211yesmaximal, characteristic, complemented
3#36128C211yesmaximal, characteristic
4#30892C214yesmaximal, complemented
5#28749C211yesmaximal, characteristic, complemented
6#28441C211yesmaximal, characteristic, complemented
7#27775C212yesmaximal, complemented
8#26913C214yesmaximal, complemented
9#26467C212yesmaximal, complemented
10#25935C212yesmaximal, complemented
11#25366C212yesmaximal, complemented
12#25303C212yesmaximal, complemented
13#24188C212yesmaximal, complemented
14#23190C212yesmaximal, complemented
15#23153C211yesmaximal, characteristic, complemented
16C4×#474C211yesmaximal, characteristic, complemented
17C4×#461C211yesmaximal, characteristic, complemented
18C2×#557C2212yes—
19C2×#550C2212yes—
20C2×#532—21no—
21#3368—216no—
22C2×#492C2212yes—
23C2×#490C2214yes—
24C2×#486C2212yes—
25C2×#478C2214yes—
26C2×#474C2211yescharacteristic
27C2×#460C2214yes—
28C2×#460C2212yes—
29C2×#460C2212yes—
30C2×#461C2211yescharacteristic
31C2×#446C2212yes—
32C2×#440C2214yes—
33C2×C4×SD16C2212yes—
34#2867C2211yescharacteristic
35#2856—21no—
36#2726—22no—
37#2582C2212yes—
38#2552C2212yes—
39#2548C2212yes—
40#2546—21no—
41#2541C2212yes—
42#2541C2212yes—
43#2506—22no—
44#2453C2214yes—
45#2446—22no—
46#2431—22no—
47#2428—22no—
48#2428—22no—
49#2366C2212yes—
50#2366C2212yes—
51#2365C2212yes—
52Q8×C4⋊C4—21no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C42×D4C2211yescharacteristic
57#2067C2214yes—
58#2055C2214yes—
59#2041C2214yes—
60#2041C2214yes—
61#2025C2214yes—
62#2025C2214yes—
63#2014C2214yes—
64#2014C2214yes—
65#1988C2211yescharacteristic
66#1984C2212yes—
67#1982C2212yes—
68#1982C2212yes—
69#1981C2211yescharacteristic
70#1973C2214yes—
71#1957C2212yes—
72#1957C2212yes—
73#1948C2211yescharacteristic
74#1947C2211yescharacteristic
75#1939C2212yes—
76#1936C2214yes—
77#1927C2212yes—
78#1918C2212yes—
79#1917C2214yes—
80#1880C2214yes—
81#1880C2214yes—
82#1877C2214yes—
83#1877C2212yes—
84#1864C2214yes—
85#1862C2212yes—
86#1802C2211yescharacteristic
87#1802C2211yescharacteristic
88C4×#107C2212yes—
89#1770C2212yes—
90C4×C4⋊C8C2211yescharacteristic
91C4×C4⋊C8C2211yescharacteristic
92C4×#103C2212yes—
93C4×#102C2212yes—
94C22×#41—22no—
95C22×SD16C2314yes—
96C22×SD16C2312yes—
97C22×SD16C2314yes—
98#557—28no—
99#557—44no—
100#550—28no—
101#550—44no—
102#532—48no—
103C2×C4⋊Q8—22no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127—21no—
108C2×#127—21no—
109C2×#127—21no—
110C2×#127—21no—
111C2×#127C2312yes—
112C2×#127C2312yes—
113C2×#127C2314yes—
114C2×#127C2312yes—
115C2×#125—24no—
116C2×#125C2314yes—
117C2×#124—22no—
118C2×#121—22no—
119C2×#121—22no—
120C2×#121—22no—
121C2×#121—24no—
122C2×#121—22no—
123C2×#121—22no—
124C2×C4×Q8—22no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2314yes—
128C2×C4×Q8C2312yes—
129C2×C4×D4—22no—
130C2×C4×D4—22no—
131C2×C4×D4C2311yescharacteristic
132C2×C4×D4C2314yes—
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C22×C42—21no—
136#492—28no—
137#490—216no—
138#486—28no—
139#478—216no—
140#474D418yescomplemented
141#461—24no—
142#460—28no—
143#460—216no—
144#460—44no—
145#446—28no—
146#440—216no—
147C4×SD16—28no—
148C2×#107C2312yes—
149C2×#107C2312yes—
150C2×#106C2312yes—
151C2×C4⋊C8C2314yes—
152C2×C4⋊C8C2311yescharacteristic
153C2×C4⋊C8C2311yescharacteristic
154C2×C4⋊C8C2311yescharacteristic
155C2×C4⋊C8C2311yescharacteristic
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2312yes—
165C2×#102C2314yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#97C2314yes—
169C2×C4×C8C2312yes—
170C2×C4×C8C2312yes—
171#400—24no—
172#398—24no—
173#398—24no—
174#398—24no—
175#394—24no—
176#392—24no—
177#392—24no—
178#391—22no—
179#391—21no—
180#391D412yes—
181#391C2312yes—
182#391C2314yes—
183#390—22no—
184#390—22no—
185#390—22no—
186#390C2314yes—
187#390C2314yes—
188#389—22no—
189#389—24no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389D412yes—
196#389D412yes—
197#389C2312yes—
198#389C2314yes—
199#387—24no—
200#387—24no—
201#386—24no—
202#386—24no—
203#384—22no—
204#384D411yescharacteristic
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207C4×C4⋊C4—22no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C4⋊C4C2312yes—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C43C2311yescharacteristic
221#340C2312yes—
222#340C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2312yes—
231C2×#41—48no—
232C2×#41—48no—
233C2×#41—84no—
234C22×Q8—21no—
235C22×Q8C2×D412yes—
236C22×Q8C2412yes—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—24no—
241C22×D4—24no—
242C22×D4—22no—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4C2414yes—
246C22×D4C2411yescharacteristic
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C2×SD16—24no—
252C2×SD16—28no—
253C2×SD16—24no—
254C2×SD16—28no—
255C2×SD16—28no—
256C2×SD16—44no—
257C2×SD16—42no—
258C2×SD16—44no—
259C2×SD16—44no—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—48no—
268C4⋊Q8C2×D414yes—
269C4⋊Q8#4114yescomplemented
270#127—28no—
271#127—24no—
272#127—24no—
273#127—24no—
274#127—24no—
275#127—42no—
276#127—42no—
277#127—44no—
278#127—44no—
279#127—82no—
280#127—82no—
281#127—82no—
282#125—28no—
283#125—44no—
284#125—416no—
285#124—48no—
286#121—416no—
287#121—48no—
288#121—48no—
289#121—48no—
290#121—84no—
291#121—84no—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—28no—
295C4×Q8—24no—
296C4×Q8—24no—
297C4×Q8—44no—
298C4×Q8—48no—
299C4×Q8—44no—
300C4×D4—28no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—41no—
305C4×D4—41no—
306C4×D4—44no—
307C4×D4—48no—
308C4×D4—48no—
309C4×D4C2×D414yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—24no—
337C2×C22⋊C4—24no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C42—21no—
344C2×C42—21no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—21no—
349C2×C42—44no—
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2414yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358#107—42no—
359#107C2×D418yes—
360#106—24no—
361C4⋊C8—28no—
362C4⋊C8—22no—
363C4⋊C8—22no—
364C4⋊C8—22no—
365C4⋊C8C2×D414yes—
366#103—24no—
367#103—24no—
368#103—42no—
369#102—24no—
370#102—28no—
371#102—28no—
372#102—28no—
373#102—44no—
374#102—42no—
375#102—44no—
376#102—44no—
377#102C2×D418yes—
378#97—28no—
379C4×C8—24no—
380C4×C8—24no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—22no—
388#95—22no—
389#95—24no—
390#95—24no—
391#95C2×D414yes—
392#95C2×D414yes—
393C24—21no—
394C24—24no—
395#41—832no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8C2×#4114yes—
408C2×D4—24no—
409C2×D4—21no—
410C2×D4—24no—
411C2×D4—21no—
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—24no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—44no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—84no—
435C2×D4—84no—
436C2×D4—84no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4—82no—
442C2×D4—82no—
443C2×D4C22×D412yes—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—84no—
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C2511yescharacteristic
472SD16—416no—
473SD16—48no—
474SD16—88no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8C22×D414yes—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—21no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4—84no—
516C4⋊C4C22×D414yes—
517C4⋊C4C22×D412yes—
518C22⋊C4—24no—
519C22⋊C4—48no—
520C22⋊C4—42no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—42no—
524C22⋊C4—84no—
525C22⋊C4—84no—
526C42—22no—
527C42—22no—
528C42—24no—
529C42—24no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—48no—
534C42—44no—
535C42—42no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42AGL(1,ℤ/8)14yes—
540C42C22×D412yes—
541C42C2×#4112yes—
542C42H(5,𝔽2)12yes—
543C422−1+412yes—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23C23×D411yescharacteristic
559Q8—24no—
560Q8—24no—
561Q8—42no—
562Q8—48no—
563Q8—48no—
564Q8—42no—
565D4—22no—
566D4—22no—
567D4—44no—
568D4—44no—
569D4—41no—
570D4—44no—
571D4—41no—
572D4—44no—
573D4—88no—
574D4—816no—
575D4—88no—
576D4—88no—
577D4—164no—
578D4—164no—
579D4—168no—
580D4—164no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—48no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—84no—
619C2×C4—816no—
620C2×C4—84no—
621C2×C4D4212yes—
622C2×C4D4212yes—
623C2×C4#54812yes—
624C2×C4#54812yes—
625C2×C4C2×AGL(1,ℤ/8)12yes—
626C2×C4C2×AGL(1,ℤ/8)12yes—
627C2×C4C2×AGL(1,ℤ/8)12yes—
628C2×C4#57712yes—
629C2×C4#57712yes—
630C2×C4C23×D411yescharacteristic
631C2×C4C22×#4111yescharacteristic
632C2×C4C2×H(5,𝔽2)11yescharacteristic
633C2×C4C2×2−1+411yescharacteristic
634C2×C4#58511yescharacteristic
635C2×C4#58511yescharacteristic
636C8—24no—
637C8—42no—
638C8—42no—
639C8—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22C2×D4211yescharacteristic
655C22C2×#54811yescharacteristic
656C22#348711yescharacteristic
657C22C22×AGL(1,ℤ/8)11yescharacteristic
658C22C2×#57711yescharacteristic
659C22C2×#57711yescharacteristic
660C22#358511yescharacteristic
661C4—22no—
662C4—22no—
663C4—22no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—41no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4#327214yes—
675C4#336814yescomplemented
676C4D4×#4111yescharacteristic
677C4#349711yescharacteristic
678C4#351411yescharacteristic
679C4#355411yescharacteristic
680C2—82no—
681C2—164no—
682C2#5908011yescharacteristic
683C2C2×#327211yescharacteristic
684C2#6109511yescharacteristic
685C2C2×#336811yescharacteristic
686C2#6134111yescharacteristic
687C2#6189011yescharacteristic
688C2#6193311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36128C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3368C2semidirect
2. #59828C2semidirect
4. #30892C2semidirect
5. #28749C2semidirect
6. #28441C2semidirect
7. #27775C2semidirect
8. #26913C2semidirect
9. #26467C2semidirect
10. #25935C2semidirect
11. #25366C2semidirect
12. #25303C2semidirect
13. #24188C2semidirect
14. #23190C2semidirect
15. #23153C2semidirect
16. C4×#474C2semidirect
17. C4×#461C2semidirect
140. #474D4semidirect
269. C4⋊Q8#41semidirect
675. C4#3368semidirect

Maximal subgroups

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

subgroupindex
C2×#33682
#598282
#361282
#308922
#287492
#284412
#277752
#269132
#264672
#259352
#253662
#253032
#241882
#231902
#231532
C4×#4742
C4×#4612