Finite Group Databasebeta

← #371122

Subgroups — #371122

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1861C211yesmaximal, characteristic
2#20438C211yesmaximal, characteristic, complemented
3#20422C211yesmaximal, characteristic
4#11374C211yesmaximal, characteristic, complemented
5#11262C211yesmaximal, characteristic, complemented
6#11084C211yesmaximal, characteristic
7#11081C211yesmaximal, characteristic
8#10944C211yesmaximal, characteristic
9#10930C211yesmaximal, characteristic
10#10920C211yesmaximal, characteristic, complemented
11#10898C211yesmaximal, characteristic, complemented
12#10894C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10804C211yesmaximal, characteristic, complemented
15#10748C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#421C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#410C2211yescharacteristic
25C2×#408C2211yescharacteristic
26#2873C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C22×#97C2211yescharacteristic
30#2285C2211yescharacteristic
31#1861C418yescomplemented
32#1771—22no—
33#1771—22no—
34#1765—22no—
35#1764—22no—
36C2×#344C2211yescharacteristic
37C2×#344C2211yescharacteristic
38C2×#344C2211yescharacteristic
39C2×#344C2211yescharacteristic
40#1740C2211yescharacteristic
41#1738C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45#1733C2211yescharacteristic
46#1733C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1729C2211yescharacteristic
56C22×#41C2311yescharacteristic
57#568—22no—
58#568—41no—
59#568—41no—
60#568C2×C414yes—
61C2×#131—22no—
62C2×#131—41no—
63C22×M16—41no—
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#430—41no—
79#430C2×C414yes—
80C2×#108—41no—
81#424—22no—
82#424—22no—
83#423—22no—
84#423C2×C414yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—41no—
87#421—22no—
88#421—22no—
89#421—41no—
90#421C2×C414yes—
91C2×#104—22no—
92C2×#104—21no—
93C2×#104—21no—
94C2×#104—22no—
95C2×#104—41no—
96C2×#104C2×C414yes—
97#416—21no—
98#416—21no—
99#416—22no—
100C2×#103Q812yes—
101C2×#102Q812yes—
102#410—22no—
103#410—41no—
104C2×#99—22no—
105#408—22no—
106#408—22no—
107C2×C22⋊C8—41no—
108#405—22no—
109#405—41no—
110#405—41no—
111#405—41no—
112#405—41no—
113#405—41no—
114C4×M16—41no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—22no—
118C2×#97—22no—
119C2×#97—21no—
120C2×#97D412yes—
121C2×C4×C8—21no—
122C2×C4×C8D412yes—
123#384—21no—
124#382—21no—
125#380—22no—
126C4×C4⋊C4—22no—
127#344—22no—
128#344—22no—
129#344—41no—
130#344C2×C414yes—
131#342—22no—
132#341—21no—
133#341—21no—
134#341—22no—
135#341—41no—
136#340—22no—
137#339—22no—
138#339—21no—
139#339—21no—
140#339—22no—
141#339—41no—
142#339C2×C414yes—
143H(3,ℤ/4)—41no—
144H(3,ℤ/4)—41no—
145#336—22no—
146#336—22no—
147#336—41no—
148#336—41no—
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×#41—21no—
159C2×#41C22×C412yes—
160C2×#41#4112yes—
161C22×Q8C2×Q811yescharacteristic
162C22×D4—21no—
163C22×D4C2×Q811yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C2411yescharacteristic
167#131—28no—
168#131—44no—
169#131—44no—
170#131—44no—
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—42no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C22×C412yes—
207C2×M16C22×C412yes—
208C2×M16#4112yes—
209C2×M16#4112yes—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—42no—
226C22×C8C2×D412yes—
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×Q812yes—
230C22×C8#4112yes—
231C22×C8#4112yes—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—21no—
250#117—21no—
251#117C22×C412yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4C2×Q812yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4#4112yes—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42C22×C412yes—
271C2×C42C2×D412yes—
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4112yes—
275#108—44no—
276C4⋊C8—28no—
277C4⋊C8—44no—
278#104—28no—
279#104—24no—
280#104—24no—
281#104—44no—
282#103—22no—
283#103—22no—
284#102—22no—
285#102—22no—
286#99—44no—
287C22⋊C8—44no—
288#97—24no—
289#97—24no—
290#97—42no—
291#97—42no—
292#97—42no—
293C4×C8—42no—
294C4×C8—42no—
295C4×C8—42no—
296#95—24no—
297C24—21no—
298C24C2×#4111yescharacteristic
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C4×Q812yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4C4×Q812yes—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—82no—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16—22no—
374M16—22no—
375M16—22no—
376M16—48no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—48no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
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×C8C4×D414yes—
423C2×C8C22⋊Q812yes—
424C2×C8C22⋊Q812yes—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—84no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
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—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42#11712yes—
461C42#11712yes—
462C23—22no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—82no—
467C23C4×#4111yescharacteristic
468C23#52511yescharacteristic
469C23#52911yescharacteristic
470C23#55111yescharacteristic
471C23#55311yescharacteristic
472C23#55411yescharacteristic
473C23#55911yescharacteristic
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—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×Q811yescharacteristic
512C2×C4C4×#4111yescharacteristic
513C2×C4#51811yescharacteristic
514C2×C4#51811yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4#52511yescharacteristic
517C2×C4#52511yescharacteristic
518C2×C4#52911yescharacteristic
519C2×C4#52911yescharacteristic
520C2×C4#53311yescharacteristic
521C2×C4#53411yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53811yescharacteristic
524C2×C4#53811yescharacteristic
525C2×C4#55111yescharacteristic
526C2×C4#55111yescharacteristic
527C2×C4#55311yescharacteristic
528C2×C4#55311yescharacteristic
529C2×C4#55311yescharacteristic
530C2×C4#55311yescharacteristic
531C2×C4#55311yescharacteristic
532C2×C4#55411yescharacteristic
533C2×C4#55411yescharacteristic
534C2×C4#55411yescharacteristic
535C2×C4#55911yescharacteristic
536C2×C4#55911yescharacteristic
537C8—24no—
538C8—22no—
539C8—22no—
540C8—42no—
541C8—42no—
542C8—42no—
543C8—42no—
544C8—42no—
545C8—81no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—82no—
555C22—82no—
556C22—161no—
557C22#229011yescharacteristic
558C22#236811yescharacteristic
559C22#237411yescharacteristic
560C22#258011yescharacteristic
561C22#258711yescharacteristic
562C22#259511yescharacteristic
563C22#260011yescharacteristic
564C4—42no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—161no—
572C4C4×C22⋊Q811yescharacteristic
573C4#232111yescharacteristic
574C4#234511yescharacteristic
575C4#235711yescharacteristic
576C4#257911yescharacteristic
577C4#258911yescharacteristic
578C4#259711yescharacteristic
579C4#259811yescharacteristic
580C2—21no—
581C2—21no—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—82no—
587C2#1406711yescharacteristic
588C2#2927112yes—
subgroupG/H
centerC2×C4#535
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2580
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
2. #20438C2semidirect
4. #11374C2semidirect
5. #11262C2semidirect
10. #10920C2semidirect
11. #10898C2semidirect
12. #10894C2semidirect
13. #10883C2semidirect
14. #10804C2semidirect
31. #1861C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18612
#204382
#204222
#113742
#112622
#110842
#110812
#109442
#109302
#109202
#108982
#108942
#108832
#108042
#107482