Finite Group Databasebeta

← #371120

Subgroups — #371120

Order 512.

896 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1868C211yesmaximal, characteristic
2#20427C211yesmaximal, characteristic, complemented
3#20421C211yesmaximal, characteristic
4#11369C211yesmaximal, characteristic, complemented
5#11264C211yesmaximal, characteristic, complemented
6#11111C211yesmaximal, characteristic
7#11092C211yesmaximal, characteristic
8#10945C211yesmaximal, characteristic
9#10930C211yesmaximal, characteristic
10#10920C211yesmaximal, characteristic, complemented
11#10900C211yesmaximal, characteristic, complemented
12#10895C211yesmaximal, characteristic, complemented
13#10883C211yesmaximal, characteristic, complemented
14#10805C211yesmaximal, characteristic, complemented
15#10782C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17#2911C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C22×#104C2211yescharacteristic
22#2896C2211yescharacteristic
23#2896C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#410C2211yescharacteristic
26C2×#408C2211yescharacteristic
27#2873C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C22×#97C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1868C418yescomplemented
32#1768—22no—
33#1767—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×#338C2211yescharacteristic
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#428—22no—
79#428—22no—
80#427—41no—
81#424—22no—
82#424—22no—
83#424—22no—
84#424—41no—
85#424C2×C414yes—
86#424C2×C414yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—41no—
90C2×#104—21no—
91C2×#104—21no—
92C2×#104—22no—
93C2×#104Q814yes—
94#419—21no—
95#419—21no—
96#418—21no—
97#418—21no—
98#417—22no—
99#417—22no—
100#416—22no—
101#416—22no—
102#416—41no—
103#416—41no—
104C2×#103—21no—
105C2×#102—21no—
106#410—22no—
107#410C2×C414yes—
108C2×#99—22no—
109#408—41no—
110#408C2×C414yes—
111C2×C22⋊C8—41no—
112#405—22no—
113#405—41no—
114C4×M16—41no—
115C4×M16—41no—
116C4×M16—41no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—41no—
120C2×#97—41no—
121C2×#97D412yes—
122C2×C4×C8D412yes—
123C4×C4⋊C4—22no—
124C4×C22⋊C4—22no—
125#376—21no—
126C43—21no—
127#344—22no—
128#344—22no—
129#344C2×C414yes—
130#344C2×C414yes—
131#342—22no—
132#341—22no—
133#341—22no—
134#341—41no—
135#340—22no—
136#339—21no—
137#339—21no—
138#339—22no—
139#339—22no—
140#339—41no—
141#339—41no—
142#338—41no—
143H(3,ℤ/4)—41no—
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×Q812yes—
160C22×Q8#4111yescharacteristic
161C22×D4—21no—
162C22×D4#4111yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2411yescharacteristic
166#131—28no—
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—42no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16C22×C412yes—
207C2×M16C2×Q812yes—
208C2×M16#4112yes—
209C22×C8—21no—
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×C8C22×C412yes—
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8#4112yes—
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—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117C22×C412yes—
248#117C22×C412yes—
249#117#4112yes—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4#4112yes—
254C2×C4⋊C4#4112yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4#4112yes—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—24no—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×Q812yes—
272C2×C42#4112yes—
273#107—42no—
274#106—42no—
275C4⋊C8—28no—
276C4⋊C8—28no—
277C4⋊C8—44no—
278#104—24no—
279#104—24no—
280#103—24no—
281#103—42no—
282#103—42no—
283#102—24no—
284#102—42no—
285#102—42no—
286#99—28no—
287C22⋊C8—44no—
288#97—42no—
289#97—42no—
290#97—42no—
291#97—42no—
292C4×C8—42no—
293C4×C8—42no—
294#95—24no—
295C24—21no—
296C24C2×#4111yescharacteristic
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8C4×Q812yes—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4C4×Q812yes—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—82no—
348C22×C4C23×C411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362M16—22no—
363M16—22no—
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—48no—
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—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
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—48no—
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—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8C4×D414yes—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4#11714yes—
440C42—22no—
441C42—22no—
442C42—21no—
443C42—21no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—84no—
455C42C22⋊Q812yes—
456C42C22⋊Q812yes—
457C23—22no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—82no—
462C23C2×#11711yescharacteristic
463C23#52511yescharacteristic
464C23#52911yescharacteristic
465C23#55311yescharacteristic
466C23#55311yescharacteristic
467C23#55311yescharacteristic
468C23#55411yescharacteristic
469Q8—22no—
470Q8—22no—
471Q8—41no—
472Q8—41no—
473D4—22no—
474D4—22no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—41no—
480D4—41no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×Q811yescharacteristic
506C2×C4C4×#4111yescharacteristic
507C2×C4C4×#4111yescharacteristic
508C2×C4#51811yescharacteristic
509C2×C4#51811yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4#52911yescharacteristic
514C2×C4#52911yescharacteristic
515C2×C4#53311yescharacteristic
516C2×C4#53411yescharacteristic
517C2×C4#53511yescharacteristic
518C2×C4#53811yescharacteristic
519C2×C4#53811yescharacteristic
520C2×C4#55111yescharacteristic
521C2×C4#55111yescharacteristic
522C2×C4#55111yescharacteristic
523C2×C4#55311yescharacteristic
524C2×C4#55311yescharacteristic
525C2×C4#55311yescharacteristic
526C2×C4#55411yescharacteristic
527C2×C4#55411yescharacteristic
528C2×C4#55411yescharacteristic
529C2×C4#55911yescharacteristic
530C2×C4#55911yescharacteristic
531C2×C4#55911yescharacteristic
532C8—24no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—42no—
538C8—42no—
539C8—81no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—161no—
551C22#229011yescharacteristic
552C22#234511yescharacteristic
553C22#235711yescharacteristic
554C22#257911yescharacteristic
555C22#258711yescharacteristic
556C22#259511yescharacteristic
557C22#259811yescharacteristic
558C4—44no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—161no—
566C4C4×C22⋊Q811yescharacteristic
567C4#232111yescharacteristic
568C4#236811yescharacteristic
569C4#237411yescharacteristic
570C4#258011yescharacteristic
571C4#258911yescharacteristic
572C4#259711yescharacteristic
573C4#260011yescharacteristic
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2—21no—
579C2—21no—
580C2—82no—
581C2#1406711yescharacteristic
582C2#2938712yes—
subgroupG/H
centerC2×C4#535
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2598
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. #20427C2semidirect
4. #11369C2semidirect
5. #11264C2semidirect
10. #10920C2semidirect
11. #10900C2semidirect
12. #10895C2semidirect
13. #10883C2semidirect
14. #10805C2semidirect
31. #1868C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18682
#204272
#204212
#113692
#112642
#111112
#110922
#109452
#109302
#109202
#109002
#108952
#108832
#108052
#107822