Finite Group Databasebeta

← #7613899

Subgroups — #7613899

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3400C212yesmaximal
2#59828C211yesmaximal, characteristic, complemented
3#59827C211yesmaximal, characteristic
4#31215C214yesmaximal, complemented
5#28471C212yesmaximal, complemented
6#28274C212yesmaximal, complemented
7#28255C211yesmaximal, characteristic, complemented
8#28252C211yesmaximal, characteristic
9#26266C214yesmaximal, complemented
10#24730C212yesmaximal, complemented
11#24729C212yesmaximal, complemented
12#24191C212yesmaximal, complemented
13#24190C212yesmaximal
14#22671C212yesmaximal, complemented
15#21712C211yesmaximal, characteristic, complemented
16C4×#494C211yesmaximal, characteristic, complemented
17C4×#492C211yesmaximal, characteristic
18C2×#559C2212yes—
19C2×#557C2212yes—
20#3400—216no—
21C2×#497C2214yes—
22C2×#494C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#464C2214yes—
25C2×#462C2212yes—
26C2×#462C2212yes—
27C2×#460C2212yes—
28C2×#461C2214yes—
29C2×#461C2212yes—
30C2×#446C2212yes—
31C2×#440C2214yes—
32C2×C4×Q16C2212yes—
33C2×C4×Q16C2212yes—
34C2×C4×SD16C2212yes—
35C2×C4×SD16C2212yes—
36#2726—22no—
37#2724—22no—
38#2582C2212yes—
39#2582C2212yes—
40#2552—21no—
41#2547—21no—
42#2547C2212yes—
43#2546C2212yes—
44#2506—22no—
45#2486—22no—
46#2453—22no—
47#2447—22no—
48#2447C2214yes—
49#2446C2214yes—
50#2365—21no—
51Q8×C4⋊C4C2212yes—
52D4×C4⋊C4—21no—
53D4×C4⋊C4C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×#127C2211yescharacteristic
59#2066C2214yes—
60#2066C2214yes—
61#2056C2214yes—
62#2055C2214yes—
63#2043C2214yes—
64#2042C2214yes—
65#2016C2214yes—
66#2015C2214yes—
67#1984C2212yes—
68#1971C2214yes—
69#1969C2212yes—
70#1969C2212yes—
71#1968C2212yes—
72#1968C2212yes—
73#1957C2211yescharacteristic
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1944C2214yes—
78#1939C2212yes—
79#1939C2212yes—
80#1878C2214yes—
81#1877C2214yes—
82#1863C2214yes—
83#1862C2214yes—
84#1845C2211yescharacteristic
85#1844C2211yescharacteristic
86#1842C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1802C2212yes—
91#1770C2212yes—
92#1770C2212yes—
93C42×C8C2211yescharacteristic
94C22×Q16C2312yes—
95C22×Q16C2314yes—
96C22×Q16C2312yes—
97C22×SD16C2312yes—
98C22×SD16C2314yes—
99C22×SD16C2312yes—
100#559—28no—
101#559—44no—
102#557—28no—
103#557—44no—
104C2×C4⋊Q8D411yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2314yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—22no—
111C2×#125—24no—
112C2×#124—22no—
113C2×#124C2314yes—
114C2×#123—24no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2314yes—
120C2×C4×Q8C2312yes—
121C2×C4×D4—22no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4C2312yes—
125C2×C4×D4C2314yes—
126C22×C4⋊C4—22no—
127#497—216no—
128#494—24no—
129#492D418yes—
130#464—216no—
131#462—28no—
132#462—44no—
133#461—216no—
134#461—44no—
135#460—28no—
136#446—28no—
137#440—216no—
138C4×Q16—28no—
139C4×Q16—44no—
140C4×SD16—28no—
141C4×SD16—44no—
142C2×#107C2312yes—
143C2×#107C2312yes—
144C2×#106C2312yes—
145C2×#106C2312yes—
146C2×C4⋊C8C2312yes—
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2314yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2312yes—
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2312yes—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102C2314yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#97C2314yes—
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#400—24no—
170#398—24no—
171#398—24no—
172#398—24no—
173#398—24no—
174#397—24no—
175#397—24no—
176#397—24no—
177#391—22no—
178#391—22no—
179#391D412yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2314yes—
184#391C2314yes—
185#390—21no—
186#390—21no—
187#390C2312yes—
188#389—22no—
189#389—22no—
190#389—22no—
191#389—22no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—24no—
199#389D412yes—
200#389D412yes—
201#389C2314yes—
202#389C2314yes—
203#387—24no—
204#387—24no—
205#385—24no—
206#385—24no—
207#384—22no—
208#384D411yescharacteristic
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#381—24no—
214#380—22no—
215#380—22no—
216#380—22no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C43C2311yescharacteristic
227#340C2314yes—
228#340C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8C2×D412yes—
232C22×Q8C2412yes—
233C22×Q8C2412yes—
234C22×Q8C2412yes—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—22no—
239C23×C4—24no—
240C2×Q16—28no—
241C2×Q16—24no—
242C2×Q16—24no—
243C2×Q16—24no—
244C2×Q16—24no—
245C2×Q16—42no—
246C2×Q16—42no—
247C2×Q16—44no—
248C2×Q16—44no—
249C2×SD16—28no—
250C2×SD16—24no—
251C2×SD16—24no—
252C2×SD16—42no—
253C2×SD16—42no—
254C2×SD16—44no—
255C2×SD16—44no—
256C2×SD16C2×D418yes—
257C2×SD16C2×D418yes—
258C22×C8C2414yes—
259C22×C8C2411yescharacteristic
260C22×C8C2411yescharacteristic
261C22×C8C2411yescharacteristic
262C22×C8C2411yescharacteristic
263C4⋊Q8—28no—
264C4⋊Q8—24no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268C4⋊Q8—22no—
269C4⋊Q8—22no—
270C4⋊Q8—44no—
271C4⋊Q8C2×D414yes—
272#127—22no—
273#127C2×D414yes—
274#125—416no—
275#125—48no—
276#124—28no—
277#124—44no—
278#124—48no—
279#123—416no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—28no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×Q8—48no—
291C4×D4—28no—
292C4×D4—24no—
293C4×D4—24no—
294C4×D4—44no—
295C4×D4—48no—
296C4×D4—44no—
297C4×D4—82no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—84no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2412yes—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2414yes—
346C2×C42C2411yescharacteristic
347C2×C42C2411yescharacteristic
348C2×C42C2411yescharacteristic
349#107—24no—
350#107—42no—
351#106—24no—
352#106—42no—
353C4⋊C8—28no—
354C4⋊C8—24no—
355C4⋊C8—24no—
356C4⋊C8—42no—
357#103—28no—
358#103—24no—
359#103—24no—
360#103—24no—
361#103—42no—
362#103—42no—
363#103—42no—
364#103—44no—
365#103—44no—
366#102—28no—
367#102—24no—
368#102—42no—
369#97—28no—
370C4×C8—22no—
371C4×C8—22no—
372C4×C8—22no—
373C4×C8—41no—
374C4×C8C2×D414yes—
375C4×C8#4114yes—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95C2×D414yes—
389#95C2×D414yes—
390C24—22no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8C22×D414yes—
412C2×D4—24no—
413C2×D4—24no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4C22×D414yes—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—48no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D411yescharacteristic
441C22×C4C22×D411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C2511yescharacteristic
446Q16—48no—
447Q16—48no—
448Q16—88no—
449SD16—216no—
450SD16—48no—
451SD16—88no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—24no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—24no—
460C2×C8—21no—
461C2×C8—24no—
462C2×C8C22×D412yes—
463C2×C8C22×D412yes—
464C2×C8C2×#4112yes—
465C2×C8C2×#4112yes—
466C2×C82−1+412yes—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—42no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—84no—
501C4⋊C4—84no—
502C4⋊C4—81no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—82no—
509C42—22no—
510C42—22no—
511C42—24no—
512C42—24no—
513C42—21no—
514C42—21no—
515C42—41no—
516C42—48no—
517C42—44no—
518C42—44no—
519C42—44no—
520C42#13714yes—
521C42C22×D412yes—
522C42C2×#4112yes—
523C42H(5,𝔽2)12yes—
524C42H(5,𝔽2)12yes—
525C23—24no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23C23×D411yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—42no—
536Q8—48no—
537Q8—42no—
538Q8—42no—
539Q8—42no—
540Q8—42no—
541Q8—42no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546D4—24no—
547D4—24no—
548D4—42no—
549D4—48no—
550D4—42no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—21no—
567C2×C4—24no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—88no—
589C2×C4—168no—
590C2×C4D4212yes—
591C2×C4D4212yes—
592C2×C4#55012yes—
593C2×C4#55012yes—
594C2×C4C2×#13712yes—
595C2×C4C2×#13712yes—
596C2×C4C2×#13712yes—
597C2×C4#57712yes—
598C2×C4#57812yes—
599C2×C4C23×D411yescharacteristic
600C2×C4C22×#4111yescharacteristic
601C2×C4C2×H(5,𝔽2)11yescharacteristic
602C2×C4C2×H(5,𝔽2)11yescharacteristic
603C2×C4C2×2−1+411yescharacteristic
604C2×C4#58511yescharacteristic
605C8—22no—
606C8—22no—
607C8—44no—
608C8—41no—
609C8#54714yes—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22C2×D4211yescharacteristic
618C22C2×#55011yescharacteristic
619C22#349711yescharacteristic
620C22C22×#13711yescharacteristic
621C22C2×#57711yescharacteristic
622C22C2×#57811yescharacteristic
623C22#358411yescharacteristic
624C4—22no—
625C4—22no—
626C4—22no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—82no—
635C4—82no—
636C4—82no—
637C4#327614yes—
638C4#340014yescomplemented
639C4C2×#54711yescharacteristic
640C4#349611yescharacteristic
641C4#350611yescharacteristic
642C4#354811yescharacteristic
643C2—84no—
644C2#5981511yescharacteristic
645C2C2×#327611yescharacteristic
646C2#6108511yescharacteristic
647C2C2×#340011yescharacteristic
648C2#6140811yescharacteristic
649C2#6236611yescharacteristic
650C2#6238211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127C23
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
2. #59828C2semidirect
4. #31215C2semidirect
5. #28471C2semidirect
6. #28274C2semidirect
7. #28255C2semidirect
9. #26266C2semidirect
10. #24730C2semidirect
11. #24729C2semidirect
12. #24191C2semidirect
14. #22671C2semidirect
15. #21712C2semidirect
16. C4×#494C2semidirect
638. C4#3400semidirect

Maximal subgroups

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

subgroupindex
C2×#34002
#598282
#598272
#312152
#284712
#282742
#282552
#282522
#262662
#247302
#247292
#241912
#241902
#226712
#217122
C4×#4942
C4×#4922