Finite Group Databasebeta

← #368271

Subgroups — #368271

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3C2×#1986C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11340C212yesmaximal
7#11276C212yesmaximal
8#10899C212yesmaximal
9#10891C212yesmaximal
10#10857C211yesmaximal, characteristic
11#10852C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2360—22no—
18#2360—22no—
19#2355—21no—
20#2355—21no—
21Q8×C4⋊C4—21no—
22Q8×C4⋊C4—21no—
23#2330—22no—
24#2317C2212yes—
25C4×C4⋊Q8—21no—
26C4×C4⋊Q8—21no—
27C2×#389—22no—
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30#2274C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1989—22no—
35#1988—21no—
36#1987—21no—
37#1986C418yescomplemented
38#1776—21no—
39#1776—21no—
40C4×#107—21no—
41C4×#106—21no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44#1736C2211yescharacteristic
45#1736C2212yes—
46#1736C2212yes—
47#1736C2211yescharacteristic
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51C2×#339C2212yes—
52#1444—22no—
53C4⋊#95C2212yes—
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57#533—22no—
58#533—22no—
59#533C2×C414yes—
60#533Q814yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8D411yescharacteristic
63C2×#125D412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×#117C2311yescharacteristic
71C2×#117C2312yes—
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4D412yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83#428—22no—
84#428—22no—
85#428—22no—
86#428Q814yescomplemented
87C2×#107—21no—
88C2×#107—21no—
89C2×#106—21no—
90C2×#106—21no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C4×M16—22no—
96C4×M16C2×C414yes—
97C2×#97—21no—
98C2×C4×C8—21no—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—24no—
104#389—24no—
105#389—22no—
106#389—22no—
107#389—48no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384—22no—
112#384—24no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—42no—
118#382—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2×C414yes—
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—24no—
138#376—21no—
139C2×#95—21no—
140C2×#95—24no—
141C2×#95—21no—
142C43—21no—
143#343—24no—
144#343—42no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#339—24no—
154#339—42no—
155#339C2×C418yes—
156#339C2×C418yes—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D414yes—
164C23×C4C2×D411yescharacteristic
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—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16C22×C412yes—
181C2×M16C22×C412yes—
182C2×M16C2×Q812yes—
183C2×M16#4112yes—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8C4⋊C414yescomplemented
190C4⋊Q8C4⋊C414yescomplemented
191#125—24no—
192#125C4⋊C418yescomplemented
193C22⋊Q8—24no—
194C22⋊Q8—24no—
195C22⋊Q8—24no—
196C22⋊Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—84no—
202C4×Q8—84no—
203#117—22no—
204#117—22no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—21no—
210#117—21no—
211#117C22×C414yes—
212#117C2×Q814yes—
213#117C2×Q812yes—
214#117#4112yes—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4—84no—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×Q812yes—
256C2×C4⋊C4#4112yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—44no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42C22×C414yes—
278C2×C42C22×C412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×Q812yes—
284C2×C42#4112yes—
285C2×C42#4112yes—
286#107—24no—
287#107—24no—
288#106—24no—
289#106—24no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292#97—24no—
293C4×C8—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—48no—
304#95—41no—
305#95—41no—
306C24C22×D411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—24no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4C4×D412yes—
346C22×C4C4×D412yes—
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12011yescharacteristic
351C22×C4C22⋊Q812yes—
352C22×C4C22⋊Q812yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366C22×C42−1+411yescharacteristic
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—22no—
372M16—41no—
373M16—41no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16C4×Q814yescomplemented
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—216no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—28no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4—84no—
419C4⋊C4—816no—
420C4⋊C4C2×C4⋊C414yes—
421C4⋊C4C2×C4⋊C414yes—
422C4⋊C4C22⋊Q812yes—
423C4⋊C4C22⋊Q812yes—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—24no—
435C42—24no—
436C42—42no—
437C42—42no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—82no—
445C42—84no—
446C42—84no—
447C42—82no—
448C42C2×C4⋊C412yes—
449C42C2×C4⋊C412yes—
450C42C4×D412yes—
451C42C4×D412yes—
452C42C22⋊Q812yes—
453C42C22⋊Q812yes—
454C42#12312yes—
455C42#12312yes—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23C2×C4×D411yescharacteristic
461C23C2×#12011yescharacteristic
462C23C2×C22⋊Q811yescharacteristic
463C23#54611yescharacteristic
464C23#54711yescharacteristic
465C23#54811yescharacteristic
466C23D4×Q811yescharacteristic
467Q8—48no—
468Q8—48no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4#38512yes—
520C2×C4#38512yes—
521C2×C4#38512yes—
522C2×C4#38512yes—
523C2×C4#45312yes—
524C2×C4#45612yes—
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×Q811yescharacteristic
528C2×C4C4×#4111yescharacteristic
529C2×C4#51911yescharacteristic
530C2×C4#52011yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×C22⋊Q811yescharacteristic
533C2×C4C2×#12311yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52911yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53711yescharacteristic
539C2×C4#54111yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4#54811yescharacteristic
542C2×C4#54811yescharacteristic
543C2×C4D4×Q811yescharacteristic
544C2×C4#55011yescharacteristic
545C2×C4#55611yescharacteristic
546C2×C4#55611yescharacteristic
547C2×C4#55711yescharacteristic
548C2×C4#55711yescharacteristic
549C2×C4#55711yescharacteristic
550C2×C4#55711yescharacteristic
551C2×C4Q8211yescharacteristic
552C2×C4#55911yescharacteristic
553C8—24no—
554C8—42no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22C2×#38511yescharacteristic
569C22#236511yescharacteristic
570C22#236711yescharacteristic
571C22#240711yescharacteristic
572C22#240911yescharacteristic
573C22#245111yescharacteristic
574C22#245311yescharacteristic
575C22C2×#45311yescharacteristic
576C22C2×#45611yescharacteristic
577C22#301911yescharacteristic
578C22#302011yescharacteristic
579C4—22no—
580C4—22no—
581C4—22no—
582C4—48no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—81no—
589C4—84no—
590C4—82no—
591C4—81no—
592C4#185914yes—
593C4C4×C22⋊Q811yescharacteristic
594C4#232311yescharacteristic
595C4Q8×C4⋊C411yescharacteristic
596C4#235111yescharacteristic
597C4#244611yescharacteristic
598C4#244811yescharacteristic
599C4#258111yescharacteristic
600C4#258211yescharacteristic
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2#1401711yescharacteristic
606C2C2×#185911yescharacteristic
607C2#2086411yescharacteristic
608C2#2414411yescharacteristic
609C2#2414711yescharacteristic
610C2#2432011yescharacteristic
611C2#2432311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1986C4semidirect
86. #428Q8semidirect
189. C4⋊Q8C4⋊C4semidirect
190. C4⋊Q8C4⋊C4semidirect
192. #125C4⋊C4semidirect
378. M16C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342142
#213122
C2×#19862
C4×#4282
#139102
#113402
#112762
#108992
#108912
#108572
#108522