Finite Group Databasebeta

← #405930

Subgroups — #405930

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1894C211yesmaximal, characteristic
2C2×#1892C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×#1729C211yesmaximal, characteristic
5#14264C211yesmaximal, characteristic
6#14258C211yesmaximal, characteristic
7#11315C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#107C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#1894—24no—
26#1892—24no—
27#1808—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34#1729—24no—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1444—21no—
38#1442—21no—
39#1441—21no—
40#1441—21no—
41#1440—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—42no—
52C23×C8—22no—
53C23×C8—22no—
54C23×C8—24no—
55C23×C8—42no—
56C23×C8D414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C2×C22⋊Q8—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8Q814yes—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—42no—
69C2×#117—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42C2312yes—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95#391—42no—
96#391—42no—
97#389—42no—
98#389—42no—
99#380—22no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—42no—
121#336—24no—
122#336—24no—
123C25C2×D411yescharacteristic
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C2×Q812yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C2×M16—28no—
155C2×M16—48no—
156C2×M16—48no—
157C2×M16—44no—
158C2×M16—44no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—28no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—28no—
174C22×C8—44no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8#4114yes—
180C22×C8#4114yes—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185#117—44no—
186#117—44no—
187#117—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×Q812yes—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—81no—
234C2×C22⋊C4—81no—
235C2×C22⋊C4—81no—
236C2×C22⋊C4—81no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244#107—48no—
245#103—48no—
246#103—44no—
247#103—44no—
248#95—24no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—42no—
254#95—82no—
255#95—82no—
256#95—82no—
257#95—82no—
258C24—22no—
259C24—22no—
260C24—21no—
261C24—21no—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24#12112yes—
267C24#12112yes—
268C24C22⋊Q812yes—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4#12114yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4C22⋊Q814yes—
358C22×C4C22⋊Q814yes—
359C22×C4C22⋊Q812yes—
360C22×C4#12614yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371M16—416no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—416no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8#12314yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—48no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C23—24no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#39614yes—
444C23C2×#12111yescharacteristic
445C23C2×#12111yescharacteristic
446C23C2×C22⋊Q811yescharacteristic
447C23#53212yes—
448C23#53412yes—
449C23#57611yescharacteristic
450C23#57611yescharacteristic
451C23#57712yes—
452C23#57812yes—
453C23#57811yescharacteristic
454C23#57811yescharacteristic
455Q8—42no—
456Q8—44no—
457Q8—42no—
458Q8—81no—
459Q8—81no—
460Q8—82no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—28no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4#39614yes—
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×C22⋊Q811yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4#52511yescharacteristic
527C2×C4C2×#12611yescharacteristic
528C2×C4#52912yes—
529C2×C4#53411yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#53812yes—
532C2×C4#54612yes—
533C2×C4#54612yes—
534C2×C4#55112yes—
535C2×C4#55112yes—
536C2×C4#55312yes—
537C2×C4#55312yes—
538C2×C4#55412yes—
539C2×C4#55412yes—
540C8—24no—
541C8—44no—
542C8—42no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22C2×#39611yescharacteristic
555C22#254312yes—
556C22#256412yes—
557C22#261711yescharacteristic
558C22#285411yescharacteristic
559C22#304111yescharacteristic
560C22#304411yescharacteristic
561C22#304612yes—
562C22#305711yescharacteristic
563C22#305911yescharacteristic
564C22#306111yescharacteristic
565C22#306111yescharacteristic
566C22#308212yes—
567C22#308311yescharacteristic
568C22#308311yescharacteristic
569C22#317512yes—
570C22#317712yes—
571C22#317812yes—
572C22#317912yes—
573C22#319812yes—
574C22#320112yes—
575C22#320212yes—
576C22#320312yes—
577C4—24no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4—161no—
584C4—161no—
585C4—162no—
586C4#241111yescharacteristic
587C4#241211yescharacteristic
588C4#245012yes—
589C4#248711yescharacteristic
590C4#251012yes—
591C4#257911yescharacteristic
592C2—44no—
593C2#1492111yescharacteristic
594C2#2361111yescharacteristic
595C2#2361311yescharacteristic
596C2#2858612yes—
597C2#2858812yes—
598C2#2917112yes—
599C2#2917312yes—
600C2#2973911yescharacteristic
601C2#2974011yescharacteristic
602C2#3313111yescharacteristic
603C2#3313911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11315C2semidirect
8. #11219C2semidirect
9. #11124C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18942
C2×#18922
C2×#18082
C2×#17292
#142642
#142582
#113152
#112192
#111242
#110672
#107222