Finite Group Databasebeta

← #386906

Subgroups — #386906

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 576 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
5#20852C211yesmaximal, characteristic
6#11332C211yesmaximal, characteristic, complemented
7#11275C211yesmaximal, characteristic, complemented
8#11272C211yesmaximal, characteristic, complemented
9#11241C211yesmaximal, characteristic, complemented
10#11114C211yesmaximal, characteristic
11#11104C211yesmaximal, characteristic
12#10901C211yesmaximal, characteristic, complemented
13#10864C211yesmaximal, characteristic, complemented
14#10859C211yesmaximal, characteristic, complemented
15#10780C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C22×#108C2211yescharacteristic
20C22×#108C2211yescharacteristic
21C22×#108C2211yescharacteristic
22C2×#427C2211yescharacteristic
23#2905C2211yescharacteristic
24C2×#424C2211yescharacteristic
25C2×#421C2211yescharacteristic
26C2×#421C2211yescharacteristic
27C22×#104C2211yescharacteristic
28C2×#416C2211yescharacteristic
29#2879C2211yescharacteristic
30C2×#408C2211yescharacteristic
31#2873C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#1950—24no—
34#1937C418yescomplemented
35#1929—24no—
36#1866—24no—
37#1855—24no—
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#344C2211yescharacteristic
42#1738C2211yescharacteristic
43#1738C2211yescharacteristic
44C2×#341C2211yescharacteristic
45C2×#341C2211yescharacteristic
46C2×#341C2211yescharacteristic
47#1733C2211yescharacteristic
48#1733C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×#339C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1728C2211yescharacteristic
55#1728C2211yescharacteristic
56C22×#41C2311yescharacteristic
57#568—22no—
58#568—41no—
59#568—41no—
60#568—41no—
61C2×#131—22no—
62C2×#131—41no—
63C22×M16—22no—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#430—22no—
79#430—41no—
80#430—41no—
81#430—41no—
82C2×#108—22no—
83C2×#108—22no—
84C2×#108—22no—
85C2×#108—22no—
86C2×#108—22no—
87C2×#108—22no—
88C2×#108—41no—
89C2×#108—41no—
90C2×#108—41no—
91C2×#108—41no—
92C2×#108D414yes—
93#427—41no—
94#427—41no—
95C2×#107—21no—
96C2×#106—21no—
97#424—21no—
98#424—21no—
99#424—22no—
100#424—41no—
101#424C2×C414yes—
102#423—22no—
103C2×C4⋊C8—22no—
104#421—22no—
105#421—41no—
106#421—41no—
107#421—41no—
108C2×#104—22no—
109C2×#104—22no—
110C2×#104—41no—
111C2×#104—41no—
112C2×#104C2×C414yes—
113#416C2×C414yes—
114#416D414yes—
115C2×#103Q812yes—
116C2×#102Q812yes—
117#408—21no—
118#408—21no—
119#408—22no—
120#408—41no—
121#408C2×C414yes—
122#407—22no—
123C2×C22⋊C8—22no—
124#405—22no—
125#405—41no—
126C4×M16—22no—
127C2×C4×C8—41no—
128C2×C4×C8—41no—
129C2×C4×C8C2×C414yes—
130#344—22no—
131#344—41no—
132#344—41no—
133#344—41no—
134#341—22no—
135#341—22no—
136#341—22no—
137#341—22no—
138#341C2×C414yes—
139#339—22no—
140#339—22no—
141#339—41no—
142#339—41no—
143#339C2×C414yes—
144#336—22no—
145#336—22no—
146#336—41no—
147#336—41no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41C22×C412yes—
159C2×#41C2×D412yes—
160C22×Q8C2×Q811yescharacteristic
161C22×D4—21no—
162C22×D4C2×Q811yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2411yescharacteristic
166#131—44no—
167#131—44no—
168#131—44no—
169#131—44no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
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—42no—
213C2×M16—44no—
214C2×M16—44no—
215C2×M16C22×C412yes—
216C2×M16C22×C412yes—
217C2×M16C2×D412yes—
218C2×M16C2×D412yes—
219C2×M16#4112yes—
220C2×M16#4112yes—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
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—42no—
238C22×C8C22×C412yes—
239C22×C8C22×C412yes—
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2×Q812yes—
243C22×C8#4111yescharacteristic
244C22×C8#4111yescharacteristic
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117C22×C412yes—
256#117C2×D412yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4C2×Q812yes—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42C22×C412yes—
272#108—44no—
273#108—44no—
274#108—44no—
275#108—44no—
276#108—44no—
277#107—42no—
278#106—42no—
279C4⋊C8—24no—
280C4⋊C8—24no—
281C4⋊C8—44no—
282#104—44no—
283#104—44no—
284#104—44no—
285#103C4⋊C418yes—
286#102C4⋊C418yes—
287C22⋊C8—24no—
288C22⋊C8—24no—
289C22⋊C8—44no—
290#97—42no—
291C4×C8—44no—
292C4×C8—42no—
293C24—21no—
294C242−1+411yescharacteristic
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8C2×C4⋊C412yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4C2×C4⋊C412yes—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—44no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)11yescharacteristic
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357M16—22no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—48no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
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—48no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
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×C8C2×C4⋊C414yes—
430C2×C8C4×D412yes—
431C2×C8C4×D412yes—
432C4⋊C4—22no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4C2×C4⋊C414yes—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C23—22no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—82no—
459C23#52011yescharacteristic
460C23#54111yescharacteristic
461C23#54211yescharacteristic
462C23#54711yescharacteristic
463C23D4×Q811yescharacteristic
464C23#56211yescharacteristic
465C23#56311yescharacteristic
466Q8—22no—
467Q8—22no—
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—22no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—41no—
477D4—41no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4C22×C4⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C4×#4111yescharacteristic
499C2×C4#51811yescharacteristic
500C2×C4#52011yescharacteristic
501C2×C4#52011yescharacteristic
502C2×C4#53411yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53611yescharacteristic
505C2×C4#53711yescharacteristic
506C2×C4#53811yescharacteristic
507C2×C4#53811yescharacteristic
508C2×C4#53911yescharacteristic
509C2×C4#53911yescharacteristic
510C2×C4#54411yescharacteristic
511C2×C4#54411yescharacteristic
512C2×C4D4211yescharacteristic
513C2×C4#54611yescharacteristic
514C2×C4#54611yescharacteristic
515C2×C4#54611yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4#55411yescharacteristic
518C2×C4#55411yescharacteristic
519C2×C4#55411yescharacteristic
520C2×C4#55611yescharacteristic
521C2×C4#55911yescharacteristic
522C2×C4#56011yescharacteristic
523C2×C4#56211yescharacteristic
524C8—22no—
525C8—22no—
526C8—24no—
527C8—42no—
528C8—42no—
529C8—42no—
530C8—42no—
531C8—42no—
532C8—42no—
533C8—81no—
534C8—81no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—44no—
544C22—161no—
545C22—161no—
546C22#233411yescharacteristic
547C22D4×C4⋊C411yescharacteristic
548C22#237611yescharacteristic
549C22#268911yescharacteristic
550C22#270411yescharacteristic
551C22#271011yescharacteristic
552C22#272911yescharacteristic
553C4—44no—
554C4—42no—
555C4—44no—
556C4—161no—
557C4—161no—
558C4#231711yescharacteristic
559C4#232511yescharacteristic
560C4#234911yescharacteristic
561C4#236711yescharacteristic
562C4#267811yescharacteristic
563C4#268111yescharacteristic
564C4#278811yescharacteristic
565C4#280711yescharacteristic
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—82no—
573C2#1444211yescharacteristic
574C2#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
6. #11332C2semidirect
7. #11275C2semidirect
8. #11272C2semidirect
9. #11241C2semidirect
12. #10901C2semidirect
13. #10864C2semidirect
14. #10859C2semidirect
34. #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
#208522
#113322
#112752
#112722
#112412
#111142
#111042
#109012
#108642
#108592
#107802