Finite Group Databasebeta

← #376652

Subgroups — #376652

Order 512.

887 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2080C211yesmaximal, characteristic
2#21075C211yesmaximal, characteristic
3C4×#416C211yesmaximal, characteristic
4#11305C211yesmaximal, characteristic, complemented
5#11236C211yesmaximal, characteristic, complemented
6#11176C211yesmaximal, characteristic
7#10944C211yesmaximal, characteristic
8#10895C211yesmaximal, characteristic, complemented
9#10882C211yesmaximal, characteristic, complemented
10#10816C211yesmaximal, characteristic, complemented
11#10814C211yesmaximal, characteristic, complemented
12#10804C211yesmaximal, characteristic, complemented
13#10772C211yesmaximal, characteristic
14#10758C211yesmaximal, characteristic
15#10706C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17#2911C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#421C2211yescharacteristic
20#2896C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#2080C418yescomplemented
30#1931—22no—
31#1930—22no—
32C4×#103—22no—
33C4×#102—22no—
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36#1742C2211yescharacteristic
37#1740C2211yescharacteristic
38#1740C2211yescharacteristic
39#1740C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43#1735C2211yescharacteristic
44#1735C2211yescharacteristic
45#1735C2211yescharacteristic
46#1733C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55#1729C2211yescharacteristic
56C22×#41C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—41no—
67C2×C4×Q8—21no—
68C2×C4×D4—21no—
69C2×C4×D4—41no—
70C2×#117—21no—
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#428—22no—
81#428—22no—
82#427—41no—
83#424—41no—
84#424C2×C414yescomplemented
85C2×C4⋊C8—41no—
86#421—41no—
87#421C2×C414yes—
88C2×#104—41no—
89#419—21no—
90#418—21no—
91#417—22no—
92#416—22no—
93#416—22no—
94#416—21no—
95#416—21no—
96#416—41no—
97#416C2×C414yes—
98C2×#103—21no—
99C2×#103D412yes—
100C2×#102—21no—
101C2×#102D412yes—
102#413—22no—
103#413C2×C414yes—
104C2×#101—21no—
105C2×#100—21no—
106#410—22no—
107#410—22no—
108C2×#99—41no—
109#405—22no—
110#405—22no—
111#405—22no—
112#405—41no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—22no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—22no—
119C2×C4×C8—41no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—41no—
122#376—21no—
123C43—21no—
124#344—22no—
125#344C2×C414yescomplemented
126#343—22no—
127#342—41no—
128#342—41no—
129#342—41no—
130#341—22no—
131#341C2×C414yescomplemented
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#339—21no—
137#339—22no—
138#339—22no—
139#339—22no—
140#339—21no—
141#339—41no—
142H(3,ℤ/4)—22no—
143H(3,ℤ/4)—22no—
144H(3,ℤ/4)—41no—
145H(3,ℤ/4)C2×C414yescomplemented
146#336—41no—
147C2×#41—21no—
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×#41C22×C412yes—
158C2×#41#4112yes—
159C22×Q8C2×D411yescharacteristic
160C22×D4—21no—
161C22×D4C2×D411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—41no—
165C23×C4C2411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
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×M16C22×C412yes—
188C2×M16C22×C412yes—
189C2×M16C22×C412yes—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8C22×C412yes—
205C22×C8C2×D412yes—
206C22×C8#4112yes—
207C22×C8#4111yescharacteristic
208C22×C8#4111yescharacteristic
209C4×Q8—24no—
210C4×Q8—42no—
211C4×D4—24no—
212C4×D4—42no—
213C4×D4—44no—
214C4×D4—44no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—42no—
240#117—42no—
241#117C22×C412yes—
242#117C22×C412yes—
243#117#4112yes—
244#117#4112yes—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4#4112yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—41no—
261C2×C42—21no—
262C2×C42—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×C42—42no—
272C2×C42—41no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42#4111yescharacteristic
276C2×C42#4112yes—
277C2×C42#4112yes—
278C2×C42#4111yescharacteristic
279#107—42no—
280#106—42no—
281C4⋊C8—44no—
282#104—44no—
283#103—24no—
284#103—22no—
285#103—22no—
286#103—42no—
287#102—24no—
288#102—22no—
289#102—22no—
290#102—42no—
291#101—24no—
292#100—24no—
293#99—44no—
294#97—42no—
295#97—42no—
296C4×C8—44no—
297C4×C8—42no—
298C4×C8—42no—
299#95—42no—
300C24—21no—
301C24H(5,𝔽2)11yescharacteristic
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8C4×D412yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4C4×D412yes—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—84no—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371M16—22no—
372M16—22no—
373M16—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8#12312yes—
415C2×C8#12312yes—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—44no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—44no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C42—22no—
447C42—22no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—84no—
466C42#12612yes—
467C42#12612yes—
468C23—22no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—82no—
473C23#52011yescharacteristic
474C23#53811yescharacteristic
475C23#54011yescharacteristic
476C23#54611yescharacteristic
477C23#55111yescharacteristic
478C23#55111yescharacteristic
479C23#55311yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—41no—
483Q8—41no—
484D4—22no—
485D4—22no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—41no—
491D4—41no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—88no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C4×#4111yescharacteristic
522C2×C4C4×#4111yescharacteristic
523C2×C4C4×#4111yescharacteristic
524C2×C4C4×#4111yescharacteristic
525C2×C4#51811yescharacteristic
526C2×C4C2×#12311yescharacteristic
527C2×C4#52511yescharacteristic
528C2×C4C2×#12611yescharacteristic
529C2×C4#52911yescharacteristic
530C2×C4#52911yescharacteristic
531C2×C4#52911yescharacteristic
532C2×C4#52911yescharacteristic
533C2×C4#52911yescharacteristic
534C2×C4#53911yescharacteristic
535C2×C4#54211yescharacteristic
536C2×C4#54611yescharacteristic
537C2×C4#54611yescharacteristic
538C2×C4#54711yescharacteristic
539C2×C4#55111yescharacteristic
540C2×C4#55111yescharacteristic
541C2×C4#55211yescharacteristic
542C2×C4#55211yescharacteristic
543C2×C4#55311yescharacteristic
544C2×C4#55311yescharacteristic
545C2×C4#55511yescharacteristic
546C2×C4#55511yescharacteristic
547C2×C4#55611yescharacteristic
548C8—22no—
549C8—22no—
550C8—42no—
551C8—42no—
552C8—42no—
553C8—42no—
554C8—81no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—42no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22#233511yescharacteristic
567C22#235911yescharacteristic
568C22#237311yescharacteristic
569C22#242011yescharacteristic
570C22#248011yescharacteristic
571C22#253111yescharacteristic
572C22#257611yescharacteristic
573C4—24no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—161no—
583C4C4×#12311yescharacteristic
584C4C4×#12611yescharacteristic
585C4#236411yescharacteristic
586C4#237211yescharacteristic
587C4#241911yescharacteristic
588C4#247411yescharacteristic
589C4#249511yescharacteristic
590C4#257711yescharacteristic
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2#1418411yescharacteristic
599C2#2498512yes—
subgroupG/H
centerC2×C4#555
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2420
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
4. #11305C2semidirect
5. #11236C2semidirect
8. #10895C2semidirect
9. #10882C2semidirect
10. #10816C2semidirect
11. #10814C2semidirect
12. #10804C2semidirect
15. #10706C2semidirect
29. #2080C4semidirect
84. #424C2×C4semidirect
125. #344C2×C4semidirect
131. #341C2×C4semidirect
145. H(3,ℤ/4)C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20802
#210752
C4×#4162
#113052
#112362
#111762
#109442
#108952
#108822
#108162
#108142
#108042
#107722
#107582
#107062