Finite Group Databasebeta

← #7505169

Subgroups — #7505169

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35555C211yesmaximal, characteristic, complemented
2#35307C211yesmaximal, characteristic
3#31596C214yesmaximal, complemented
4#31586C214yesmaximal, complemented
5#31582C214yesmaximal, complemented
6#31576C214yesmaximal, complemented
7#23737C211yesmaximal, characteristic
8C2×#2095C211yesmaximal, characteristic, complemented
9#23660C211yesmaximal, characteristic, complemented
10#23633C211yesmaximal, characteristic, complemented
11#23619C211yesmaximal, characteristic, complemented
12#23512C211yesmaximal, characteristic
13#21205C211yesmaximal, characteristic
14#21158C211yesmaximal, characteristic, complemented
15#21067C211yesmaximal, characteristic
16#21066C211yesmaximal, characteristic, complemented
17#21027C211yesmaximal, characteristic, complemented
18#20872C211yesmaximal, characteristic
19#20648C211yesmaximal, characteristic
20C2×#536C2211yescharacteristic
21C2×C4×#41C2211yescharacteristic
22C2×#428C2211yescharacteristic
23C2×#427C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#417C2211yescharacteristic
29C2×#419C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#405C2211yescharacteristic
36#2855—22no—
37#2625—22no—
38#2570C2214yes—
39#2552C2214yes—
40#2507—22no—
41#2506—22no—
42#2473—24no—
43#2447C2214yes—
44#2446C2214yes—
45#2419—24no—
46#2317C2211yescharacteristic
47#2317C2211yescharacteristic
48#2316C2211yescharacteristic
49#2316C2211yescharacteristic
50#2311C2211yescharacteristic
51#2310C2211yescharacteristic
52#2296C2211yescharacteristic
53#2296C2211yescharacteristic
54C2×#391C2211yescharacteristic
55C2×#384C2211yescharacteristic
56#2274C2211yescharacteristic
57#2097C2212yes—
58#2095C2212yes—
59#2095C2214yes—
60#2095C2214yes—
61#2093C2212yes—
62#2092C2212yes—
63#2085C2214yes—
64#2083C2214yes—
65#2074C2214yes—
66#2072C2214yes—
67#2068C2214yes—
68#2066C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1960C2212yes—
72#1959C2212yes—
73#1958C2212yes—
74#1956C2212yes—
75#1943C2214yes—
76#1942C2214yes—
77#1928C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1925C2214yes—
81#1917C2214yes—
82#1916C2214yes—
83#1865C2212yes—
84#1864C2212yes—
85#1863C2212yes—
86#1862C2212yes—
87#1803C2214yes—
88#1802C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×Q8C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C22×M16C2311yescharacteristic
101C23×C8C2311yescharacteristic
102C23×C8C2311yescharacteristic
103#536—24no—
104#536—42no—
105C2×C4⋊Q8C2312yes—
106C2×#125—24no—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C4×#41—24no—
111C4×#41—42no—
112C2×C4×Q8C2312yes—
113C2×C4×D4—22no—
114C2×C4×D4C2312yes—
115C2×#117—21no—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—22no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127#428—42no—
128#427—42no—
129C2×#107—24no—
130C2×#107—42no—
131C2×#107C2312yes—
132C2×#107C2312yes—
133C2×#107C2312yes—
134C2×#107C2314yes—
135C2×#107C2312yes—
136C2×#106C2312yes—
137C2×#106C2312yes—
138#423—42no—
139C2×C4⋊C8C2314yes—
140#419—42no—
141#419—42no—
142#417—42no—
143#417—42no—
144#416—24no—
145#416—24no—
146C2×#103—24no—
147C2×#103—42no—
148C2×#103C2311yescharacteristic
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2311yescharacteristic
152C2×#103C2312yes—
153C2×#103C2312yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2314yes—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#102C2311yescharacteristic
160C2×#102C2311yescharacteristic
161C2×#102C2312yes—
162C2×#102C2312yes—
163C2×#102C2312yes—
164#405—42no—
165C2×#97C2312yes—
166C2×C4×C8C2312yes—
167#400—24no—
168#400—24no—
169#398—24no—
170#398—24no—
171#398—24no—
172#398—24no—
173#397—24no—
174#397—24no—
175#391—22no—
176#391—21no—
177#391—21no—
178#391C2314yes—
179#391C2312yes—
180#391C2314yes—
181#390C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—24no—
187#389D412yes—
188#389D412yes—
189#389C2312yes—
190#389C2314yes—
191#389C2314yes—
192#389C2312yes—
193#387—24no—
194#387—24no—
195#387—24no—
196#387—24no—
197#386—22no—
198#385—22no—
199#385—24no—
200#385—22no—
201#385—22no—
202#385D414yes—
203#384—21no—
204#384—21no—
205#384C2314yes—
206#384C2312yes—
207#384C2312yes—
208#382C2312yes—
209#382C2312yes—
210#381—22no—
211#381—24no—
212#380—22no—
213#380—22no—
214#380—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2312yes—
231#336C2312yes—
232#336C2312yes—
233C2×#41—22no—
234C2×#41—22no—
235C2×#41—22no—
236C2×#41—22no—
237C2×#41—22no—
238C2×#41#4114yes—
239C22×Q8—21no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8C2412yes—
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4C2×D412yes—
257C23×C4C2411yescharacteristic
258C2×M16—22no—
259C2×M16—22no—
260C2×M16—22no—
261C2×M16—22no—
262C2×M16—22no—
263C2×M16—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8—22no—
269C22×C8#4114yes—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2411yescharacteristic
274C22×C8C2411yescharacteristic
275C4⋊Q8—24no—
276C4⋊Q8—42no—
277#125—88no—
278#124—24no—
279#124—42no—
280#123—48no—
281C22⋊Q8—48no—
282C4×Q8—24no—
283C4×Q8—42no—
284C4×D4—24no—
285C4×D4—42no—
286C4×D4—48no—
287#117—22no—
288#117—22no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—44no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—84no—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—21no—
340C2×C42—21no—
341C2×C42—21no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350#107—24no—
351#107—24no—
352#107—44no—
353#107—42no—
354#107—42no—
355#106—42no—
356#106—42no—
357C4⋊C8—44no—
358#103—24no—
359#103—24no—
360#103—24no—
361#103—22no—
362#103—22no—
363#103—44no—
364#103—42no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103—42no—
369#102—24no—
370#102—22no—
371#102—22no—
372#102—42no—
373#102—42no—
374#97—42no—
375C4×C8—42no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—24no—
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—21no—
391C24C22×D411yescharacteristic
392#41—28no—
393#41—28no—
394#41—44no—
395#41—44no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—42no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—44no—
414C2×Q8C2×#4112yes—
415C2×D4—24no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4C2×#4112yes—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—42no—
437C22×C4—44no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D411yescharacteristic
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C42−1+412yes—
461C22×C42−1+412yes—
462C22×C4C2511yescharacteristic
463M16—44no—
464M16—44no—
465C2×C8—28no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—44no—
480C2×C8C2×#4112yes—
481C2×C8C2×#4112yes—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—28no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—44no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C4⋊C4—84no—
512C4⋊C4—84no—
513C22⋊C4—24no—
514C22⋊C4—24no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—84no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—42no—
531C42—42no—
532C42—42no—
533C42—84no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—42no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23#53612yes—
542C23#54612yes—
543C23#55012yes—
544C23C23×D411yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—42no—
550Q8—42no—
551Q8—42no—
552Q8—42no—
553Q8—84no—
554D4—22no—
555D4—22no—
556D4—48no—
557D4—41no—
558D4—41no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—48no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—88no—
595C2×C4—84no—
596C2×C4—162no—
597C2×C4—162no—
598C2×C4—162no—
599C2×C4#53612yes—
600C2×C4#53914yes—
601C2×C4#53914yes—
602C2×C4#54612yes—
603C2×C4#55012yes—
604C2×C4#55214yes—
605C2×C4#56214yes—
606C2×C4#57814yes—
607C2×C4C22×#4111yescharacteristic
608C2×C4C22×#4111yescharacteristic
609C2×C4C2×H(5,𝔽2)11yescharacteristic
610C2×C4C2×H(5,𝔽2)11yescharacteristic
611C2×C4C2×2−1+411yescharacteristic
612C2×C4C2×2−1+411yescharacteristic
613C8—22no—
614C8—22no—
615C8—42no—
616C8—42no—
617C8—42no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—22no—
622C22—44no—
623C22—42no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22#269414yes—
628C22C2×#53611yescharacteristic
629C22C2×#54611yescharacteristic
630C22C2×#55011yescharacteristic
631C22C2×#57812yes—
632C22#358512yes—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—81no—
642C4—81no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4#269414yes—
647C4C2×#53911yescharacteristic
648C4C2×#53911yescharacteristic
649C4C2×#55211yescharacteristic
650C4C2×#56211yescharacteristic
651C2—24no—
652C2—82no—
653C2C2×#269411yescharacteristic
654C2#6089312yes—
655C2#6119412yes—
656C2#6141111yescharacteristic
657C2#6141711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35555C2semidirect
3. #31596C2semidirect
4. #31586C2semidirect
5. #31582C2semidirect
6. #31576C2semidirect
8. C2×#2095C2semidirect
9. #23660C2semidirect
10. #23633C2semidirect
11. #23619C2semidirect
14. #21158C2semidirect
16. #21066C2semidirect
17. #21027C2semidirect

Maximal subgroups

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

subgroupindex
#355552
#353072
#315962
#315862
#315822
#315762
#237372
C2×#20952
#236602
#236332
#236192
#235122
#212052
#211582
#210672
#210662
#210272
#208722
#206482