Finite Group Databasebeta

← #405885

Subgroups — #405885

Order 512.

1154 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20846C211yesmaximal, characteristic
2C2×#1853C211yesmaximal, characteristic
3#20628C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5#11366C212yesmaximal, complemented
6#11290C212yesmaximal, complemented
7#11238C212yesmaximal, complemented
8#11216C212yesmaximal, complemented
9#10739C211yesmaximal, characteristic
10#10734C211yesmaximal, characteristic
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2211yescharacteristic
16C2×#428C2211yescharacteristic
17C22×#106C2212yes—
18C2×#424C2212yes—
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#380C2211yescharacteristic
23#1854—21no—
24#1854—21no—
25#1853—22no—
26#1853—22no—
27#1853—21no—
28#1852—21no—
29#1798—22no—
30#1797—22no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37#1736—21no—
38#1736—21no—
39#1736C2212yes—
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1736C2212yes—
43C2×#340—21no—
44C2×#340—21no—
45C2×#340C2212yes—
46#1729C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C22×M16—21no—
50C22×M16—22no—
51C22×M16—42no—
52C22×M16—42no—
53C22×M16Q814yes—
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2312yes—
58C23×C8—21no—
59C23×C8C2312yes—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—24no—
63C2×#117—42no—
64C2×#117D411yescharacteristic
65C2×#117D411yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—42no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83#428—24no—
84#428—42no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—24no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—24no—
91C2×#106—42no—
92#424—24no—
93#424—42no—
94C2×C4⋊C8—24no—
95#407—22no—
96#407—22no—
97#407—22no—
98#407Q814yes—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8D412yes—
106C2×C22⋊C8D412yes—
107#380—21no—
108#380—21no—
109#380—21no—
110#380—21no—
111C2×#95—21no—
112#343—24no—
113#343—42no—
114#343—42no—
115#343—42no—
116#341—42no—
117#341—42no—
118#340—24no—
119#340—21no—
120#340—21no—
121#340—21no—
122#340—21no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#336—24no—
128C25#4111yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—44no—
139C23×C4—41no—
140C23×C4—41no—
141C23×C4—82no—
142C23×C4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×Q812yes—
148C23×C4C2×Q812yes—
149C23×C4#4111yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4C2411yescharacteristic
153C2×M16—24no—
154C2×M16—22no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—48no—
172C2×M16—48no—
173C2×M16—42no—
174C2×M16—42no—
175C2×M16C2×Q814yes—
176C2×M16C2×Q812yes—
177C2×M16C2×Q812yes—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8C2×D414yes—
192#117—22no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117—44no—
198#117—44no—
199#117—41no—
200#117—41no—
201#117#4114yes—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—82no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—44no—
248#107—42no—
249#107—42no—
250#106—48no—
251#106—42no—
252#106—42no—
253C4⋊C8—48no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#95—41no—
261#95—41no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—41no—
267C24—41no—
268C24#12111yescharacteristic
269C24#12111yescharacteristic
270C24#12512yes—
271C24#12512yes—
272C24C2×#4111yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—816no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C22⋊Q812yes—
331C22×C4#12312yes—
332C22×C4#12312yes—
333C22×C4#12512yes—
334C22×C4#12512yes—
335C22×C4C4⋊Q812yes—
336C22×C4C4⋊Q812yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4112yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C4H(5,𝔽2)12yes—
347M16—24no—
348M16—24no—
349M16—24no—
350M16—24no—
351M16—48no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—42no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—48no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—24no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4#12312yes—
401C4⋊C4#12312yes—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C42—24no—
409C42—42no—
410C42—42no—
411C42—42no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—21no—
424C23—21no—
425C23—24no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23#39812yes—
437C23#39812yes—
438C23#47112yes—
439C23#47112yes—
440C23C2×#12111yescharacteristic
441C23C2×#12511yescharacteristic
442C23C2×#12511yescharacteristic
443C23#55112yes—
444C23#55112yes—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—88no—
480C2×C4—88no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—84no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—162no—
491C2×C4—162no—
492C2×C4#39812yes—
493C2×C4#39812yes—
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×C22⋊Q811yescharacteristic
496C2×C4C2×#12311yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52912yes—
500C2×C4C2×C4⋊Q811yescharacteristic
501C2×C4#53312yes—
502C2×C4#53411yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53812yes—
505C2×C4#54312yes—
506C2×C4#54612yes—
507C2×C4#54612yes—
508C2×C4#55312yes—
509C2×C4#55312yes—
510C2×C4#55412yes—
511C2×C4#55412yes—
512C8—42no—
513C8—44no—
514C8—44no—
515C8—42no—
516C8—82no—
517C22—22no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—48no—
535C22—48no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22#209014yes—
541C22C2×#39811yescharacteristic
542C22#241111yescharacteristic
543C22#241211yescharacteristic
544C22#250512yes—
545C22#256912yes—
546C22C2×#47111yescharacteristic
547C22C2×#47111yescharacteristic
548C22#306811yescharacteristic
549C22#306911yescharacteristic
550C4—22no—
551C4—22no—
552C4—42no—
553C4—42no—
554C4—84no—
555C4—81no—
556C4—81no—
557C4—161no—
558C4—161no—
559C4#244412yes—
560C4#248711yescharacteristic
561C4#254912yes—
562C4#257911yescharacteristic
563C4#262611yescharacteristic
564C4#285511yescharacteristic
565C2—22no—
566C2—24no—
567C2—22no—
568C2—42no—
569C2#1492011yescharacteristic
570C2C2×#209011yescharacteristic
571C2#2367511yescharacteristic
572C2#2452111yescharacteristic
573C2#2452311yescharacteristic
574C2#2457111yescharacteristic
575C2#2457311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×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
5. #11366C2semidirect
6. #11290C2semidirect
7. #11238C2semidirect
8. #11216C2semidirect

Maximal subgroups

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

subgroupindex
#208462
C2×#18532
#206282
C2×#17362
#113662
#112902
#112382
#112162
#107392
#107342
#107272