Finite Group Databasebeta

← #385423

Subgroups — #385423

Order 512.

942 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21178C211yesmaximal, characteristic, complemented
2C2×#1937C211yesmaximal, characteristic
3#21072C211yesmaximal, characteristic
4#20886C211yesmaximal, characteristic
5C2×#1861C211yesmaximal, characteristic
6#20852C211yesmaximal, characteristic
7C2×#1848C211yesmaximal, characteristic, complemented
8#11341C211yesmaximal, characteristic, complemented
9#11332C211yesmaximal, characteristic, complemented
10#11272C211yesmaximal, characteristic, complemented
11#11240C211yesmaximal, characteristic, complemented
12#11108C211yesmaximal, characteristic
13#11080C211yesmaximal, characteristic
14#10894C211yesmaximal, characteristic, complemented
15#10818C211yesmaximal, characteristic, complemented
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C22×#108C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#421C2211yescharacteristic
28C2×#421C2211yescharacteristic
29C22×#104C2211yescharacteristic
30C2×#416C2211yescharacteristic
31#2879C2211yescharacteristic
32C2×#408C2211yescharacteristic
33#2873C2211yescharacteristic
34C22×C4×C8C2211yescharacteristic
35#1950—24no—
36#1937C414yescomplemented
37#1937C414yescomplemented
38#1929—24no—
39#1866—24no—
40#1861—24no—
41#1855—24no—
42#1848—24no—
43C2×#344C2211yescharacteristic
44C2×#344C2211yescharacteristic
45C2×#344C2211yescharacteristic
46C2×#344C2211yescharacteristic
47#1738C2211yescharacteristic
48#1738C2211yescharacteristic
49C2×#341C2211yescharacteristic
50C2×#341C2211yescharacteristic
51#1735C2211yescharacteristic
52#1733C2211yescharacteristic
53#1733C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C22×#41C2311yescharacteristic
60#568—22no—
61#568—41no—
62#568—41no—
63#568—41no—
64C2×#131—22no—
65C2×#131—41no—
66C22×M16—22no—
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C22×M16C2311yescharacteristic
74C22×M16C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#430—22no—
82#430—41no—
83#430—41no—
84#430—41no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—22no—
89C2×#108—22no—
90C2×#108—22no—
91C2×#108—22no—
92C2×#108—22no—
93C2×#108—22no—
94C2×#108—22no—
95C2×#108—41no—
96C2×#108—41no—
97C2×#108—41no—
98C2×#108—41no—
99#424—22no—
100#424—22no—
101#424—41no—
102#424C2×C414yes—
103#423—22no—
104#423—22no—
105#423—41no—
106#423—41no—
107#423D414yes—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—41no—
110C2×C4⋊C8—41no—
111#421—22no—
112#421—41no—
113#421—41no—
114#421—41no—
115C2×#104—22no—
116C2×#104—22no—
117C2×#104—22no—
118C2×#104—22no—
119C2×#104C2×C412yes—
120C2×#104C2×C412yes—
121#416C2×C414yes—
122#416D414yes—
123C2×#103Q812yes—
124C2×#102Q812yes—
125#408—22no—
126#408—22no—
127#408—41no—
128#408C2×C414yes—
129#407—22no—
130C2×C22⋊C8—22no—
131#405—22no—
132#405—41no—
133C4×M16—22no—
134C2×C4×C8—22no—
135C2×C4×C8—22no—
136C2×C4×C8C2×C412yes—
137C2×C4×C8C2×C412yes—
138#344—22no—
139#344—41no—
140#344—41no—
141#344—41no—
142#341—22no—
143#341—22no—
144#341C2×C414yes—
145#341D414yes—
146#340—22no—
147#339—22no—
148#339—22no—
149#339—22no—
150#339—22no—
151#339C2×C412yes—
152#339C2×C412yes—
153#336—22no—
154#336—22no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41C22×C412yes—
166C2×#41C2×D412yes—
167C22×Q8C2×Q811yescharacteristic
168C22×D4—21no—
169C22×D4C2×Q811yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4C2411yescharacteristic
173#131—44no—
174#131—44no—
175#131—44no—
176#131—44no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—42no—
220C2×M16—44no—
221C2×M16—44no—
222C2×M16C22×C412yes—
223C2×M16C22×C412yes—
224C2×M16C2×D412yes—
225C2×M16C2×D412yes—
226C2×M16C2×D412yes—
227C2×M16C2×D412yes—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—42no—
245C22×C8C22×C412yes—
246C22×C8C22×C412yes—
247C22×C8C2×D412yes—
248C22×C8C2×D412yes—
249C22×C8C2×Q812yes—
250C22×C8#4112yes—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117C22×C412yes—
262#117C2×D412yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C2×Q812yes—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279#108—44no—
280#108—44no—
281#108—44no—
282#108—44no—
283#108—44no—
284#108—44no—
285C4⋊C8—28no—
286C4⋊C8—44no—
287C4⋊C8—44no—
288C4⋊C8—44no—
289#104—24no—
290#104—24no—
291#104—44no—
292#104—44no—
293#103C4⋊C418yes—
294#102C4⋊C418yes—
295C22⋊C8—28no—
296C22⋊C8—44no—
297#97—42no—
298C4×C8—24no—
299C4×C8—24no—
300C4×C8—42no—
301C24—21no—
302C242−1+411yescharacteristic
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—22no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C2×C4⋊C412yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4C2×C4⋊C412yes—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—22no—
378M16—48no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—48no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8C2×C4⋊C414yes—
437C2×C8C4×D414yes—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4C2×C4⋊C414yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42C4×D412yes—
461C42C4×D412yes—
462C23—22no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—82no—
467C23#52011yescharacteristic
468C23#53611yescharacteristic
469C23#53911yescharacteristic
470C23#54711yescharacteristic
471C23D4×Q811yescharacteristic
472C23#56311yescharacteristic
473C23#56311yescharacteristic
474Q8—22no—
475Q8—22no—
476Q8—41no—
477Q8—41no—
478D4—22no—
479D4—22no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—41no—
485D4—41no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4C22×C4⋊C411yescharacteristic
506C2×C4C2×C4×D411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4#52011yescharacteristic
509C2×C4#52011yescharacteristic
510C2×C4#52011yescharacteristic
511C2×C4#53411yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#53611yescharacteristic
516C2×C4#53911yescharacteristic
517C2×C4#54211yescharacteristic
518C2×C4#54211yescharacteristic
519C2×C4#54411yescharacteristic
520C2×C4#54411yescharacteristic
521C2×C4D4211yescharacteristic
522C2×C4D4211yescharacteristic
523C2×C4D4211yescharacteristic
524C2×C4#54711yescharacteristic
525C2×C4#54711yescharacteristic
526C2×C4#54811yescharacteristic
527C2×C4D4×Q811yescharacteristic
528C2×C4#55311yescharacteristic
529C2×C4#55411yescharacteristic
530C2×C4#55411yescharacteristic
531C2×C4#56011yescharacteristic
532C2×C4#56311yescharacteristic
533C8—24no—
534C8—24no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—42no—
539C8—42no—
540C8—42no—
541C8—81no—
542C8—81no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—44no—
552C22—161no—
553C22—161no—
554C22#232211yescharacteristic
555C22D4×C4⋊C411yescharacteristic
556C22#238311yescharacteristic
557C22#267411yescharacteristic
558C22#267711yescharacteristic
559C22#271811yescharacteristic
560C22#272011yescharacteristic
561C4—24no—
562C4—42no—
563C4—42no—
564C4—44no—
565C4—161no—
566C4—161no—
567C4#231711yescharacteristic
568C4#232611yescharacteristic
569C4D4×C4⋊C411yescharacteristic
570C4#236111yescharacteristic
571C4#266911yescharacteristic
572C4#268011yescharacteristic
573C4#272911yescharacteristic
574C4#279011yescharacteristic
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—21no—
581C2—82no—
582C2#1440511yescharacteristic
583C2#2269412yes—
subgroupG/H
centerC2×C4D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22D4×C4⋊C4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21178C2semidirect
7. C2×#1848C2semidirect
8. #11341C2semidirect
9. #11332C2semidirect
10. #11272C2semidirect
11. #11240C2semidirect
14. #10894C2semidirect
15. #10818C2semidirect
36. #1937C4semidirect
37. #1937C4semidirect

Maximal subgroups

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

subgroupindex
#211782
C2×#19372
#210722
#208862
C2×#18612
#208522
C2×#18482
#113412
#113322
#112722
#112402
#111082
#110802
#108942
#108182