Finite Group Databasebeta

← #366399

Subgroups — #366399

Order 512.

1274 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20844C211yesmaximal, characteristic
2#20700C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4#20385C211yesmaximal, characteristic
5C22×#340C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10727C211yesmaximal, characteristic
11#10723C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2280C2211yescharacteristic
22#1853—21no—
23#1853—21no—
24#1852—21no—
25#1852—21no—
26#1824—21no—
27#1824—21no—
28#1823—21no—
29#1823—21no—
30#1821C418yescomplemented
31#1752—21no—
32#1752—21no—
33C8×C22⋊C4—21no—
34C8×C22⋊C4—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340—21no—
45C2×#340—21no—
46C2×#340—21no—
47C2×#340—21no—
48C2×#340C2212yes—
49C2×#340C2212yes—
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2312yes—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—24no—
62C23×C8—42no—
63C23×C8D411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D411yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2×C414yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89#428—24no—
90#428—42no—
91C2×#107—22no—
92C2×#106—22no—
93#407—22no—
94#407—22no—
95#407—22no—
96#407—22no—
97#407C2×C414yes—
98#407C2×C414yes—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105#405—24no—
106#405—42no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#381—21no—
112#380D411yescharacteristic
113#380D411yescharacteristic
114#380D411yescharacteristic
115#380D411yescharacteristic
116#379—21no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#343—24no—
120#343—42no—
121#343C2×C418yescomplemented
122#343C2×C418yescomplemented
123#341—24no—
124#341—42no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139C25#4111yescharacteristic
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—41no—
152C23×C4—42no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4—82no—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C23×C4#4112yes—
164C23×C4#4112yes—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16C22×C412yes—
185C2×M16C22×C414yes—
186C2×M16C22×C412yes—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—22no—
197C22×C8—21no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—22no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—41no—
207C22×C8—41no—
208C22×C8—41no—
209C22×C8—41no—
210C22×C8—48no—
211C22×C8—48no—
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8#4114yes—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266#107—44no—
267#106—44no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C22⋊C8—42no—
273C22⋊C8—42no—
274#97—42no—
275#97—42no—
276C4×C8—42no—
277C4×C8—42no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24#11712yes—
289C24#12111yescharacteristic
290C24#12111yescharacteristic
291C24#12612yes—
292C24C2×#4111yescharacteristic
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—816no—
330C22×C4—88no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—88no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4#11712yes—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4#12412yes—
354C22×C4#12412yes—
355C22×C4#12612yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C42−1+412yes—
366M16—24no—
367M16—24no—
368M16—24no—
369M16—24no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—44no—
394C2×C8—44no—
395C2×C8—44no—
396C2×C8—44no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8#12312yes—
409C2×C8#12312yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C23—28no—
437C23—28no—
438C23—28no—
439C23—28no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23#38812yes—
459C23#38812yes—
460C23#47112yes—
461C23#47112yes—
462C23#47212yes—
463C23#47312yes—
464C23C2×#11711yescharacteristic
465C23C2×#12111yescharacteristic
466C23C2×#12611yescharacteristic
467C23#55112yes—
468C23#55212yes—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—84no—
502C2×C4—88no—
503C2×C4—84no—
504C2×C4—88no—
505C2×C4—88no—
506C2×C4—88no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#38812yes—
514C2×C4#38812yes—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C4×#4112yes—
518C2×C4#52012yes—
519C2×C4C2×#12311yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4C2×#12411yescharacteristic
523C2×C4#52912yes—
524C2×C4#53411yescharacteristic
525C2×C4#53611yescharacteristic
526C2×C4#53912yes—
527C2×C4#54612yes—
528C2×C4#54712yes—
529C2×C4#55112yes—
530C2×C4#55112yes—
531C2×C4#55212yes—
532C2×C4#55212yes—
533C8—22no—
534C8—22no—
535C8—42no—
536C8—44no—
537C8—41no—
538C8—41no—
539C8—42no—
540C8—82no—
541C22—216no—
542C22—216no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22#193014yes—
564C22#193114yes—
565C22C2×#38811yescharacteristic
566C22#234312yes—
567C22#248611yescharacteristic
568C22#249312yes—
569C22#253411yescharacteristic
570C22C2×#47111yescharacteristic
571C22C2×#47111yescharacteristic
572C22C2×#47211yescharacteristic
573C22C2×#47311yescharacteristic
574C4—22no—
575C4—22no—
576C4—81no—
577C4—81no—
578C4—84no—
579C4—81no—
580C4—84no—
581C4—81no—
582C4C4×#12311yescharacteristic
583C4#232211yescharacteristic
584C4#235912yes—
585C4#248311yescharacteristic
586C4#248411yescharacteristic
587C4#248812yes—
588C2—22no—
589C2—24no—
590C2—24no—
591C2—22no—
592C2#1399211yescharacteristic
593C2C2×#193011yescharacteristic
594C2C2×#193111yescharacteristic
595C2#2767211yescharacteristic
596C2#2767511yescharacteristic
597C2#2813411yescharacteristic
598C2#2813511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11303C2semidirect
7. #11290C2semidirect
8. #10896C2semidirect
9. #10827C2semidirect
30. #1821C4semidirect
121. #343C2×C4semidirect
122. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#208442
#207002
C2×#18212
#203852
C22×#3402
#113032
#112902
#108962
#108272
#107272
#107232