Finite Group Databasebeta

← #410214

Subgroups — #410214

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic
3C2×#1825C211yesmaximal, characteristic
4C2×#1825C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic
6#14264C211yesmaximal, characteristic
7#11268C212yesmaximal, complemented
8#11267C212yesmaximal, complemented
9#11208C212yesmaximal, complemented
10#11207C212yesmaximal, complemented
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2317—24no—
18#2317—24no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#385—22no—
23C2×#384C2212yes—
24C2×#382C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27#1825—24no—
28#1825—24no—
29#1824—24no—
30#1823—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1442—21no—
45#1439—22no—
46C4⋊#95C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117D414yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D414yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2312yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#389—28no—
93#389—28no—
94#389—24no—
95#389—24no—
96#385—216no—
97#384—24no—
98#384—24no—
99#382—24no—
100#382—24no—
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—22no—
111C2×#95—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#336—24no—
123#336—24no—
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4—81no—
141C23×C4—81no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169#117—28no—
170#117—28no—
171#117—216no—
172#117—28no—
173#117—28no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—81no—
237C2×C22⋊C4—81no—
238C2×C22⋊C4—82no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42#4114yes—
247C2×C42#4114yes—
248C22⋊C8—44no—
249C22⋊C8—44no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—42no—
254#95—44no—
255#95—82no—
256#95—82no—
257C24—22no—
258C24—22no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24#12012yes—
266C24#12112yes—
267C24#12412yes—
268C24C22×D411yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—48no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—84no—
337C22×C4—84no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4#12012yes—
341C22×C4#12114yes—
342C22×C4#12114yes—
343C22×C4#12112yes—
344C22×C4#12314yes—
345C22×C4#12314yes—
346C22×C4#12314yes—
347C22×C4#12314yes—
348C22×C4#12412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C42−1+412yes—
358C22×C42−1+412yes—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—88no—
402C4⋊C4—88no—
403C4⋊C4—88no—
404C22⋊C4—216no—
405C22⋊C4—48no—
406C22⋊C4—84no—
407C22⋊C4—84no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—24no—
412C42—48no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23#39414yes—
436C23C2×#12011yescharacteristic
437C23C2×#12111yescharacteristic
438C23C2×#12312yes—
439C23C2×#12411yescharacteristic
440C23#53612yes—
441C23#57611yescharacteristic
442C23#57611yescharacteristic
443C23#57712yes—
444C23#57711yescharacteristic
445C23#57711yescharacteristic
446C23#57812yes—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4#39414yes—
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12111yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C2×C4C2×#12311yescharacteristic
513C2×C4#52511yescharacteristic
514C2×C4#52511yescharacteristic
515C2×C4#52512yes—
516C2×C4#53611yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53612yes—
519C2×C4#54712yes—
520C2×C4#54712yes—
521C2×C4#55212yes—
522C2×C4#55212yes—
523C2×C4#55212yes—
524C2×C4#55212yes—
525C2×C4#55212yes—
526C2×C4#55212yes—
527C8—82no—
528C8—81no—
529C8—81no—
530C22—24no—
531C22—24no—
532C22—24no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22C2×#39411yescharacteristic
542C22#248612yes—
543C22#250312yes—
544C22#261411yescharacteristic
545C22#284911yescharacteristic
546C22#301011yescharacteristic
547C22#301111yescharacteristic
548C22#301211yescharacteristic
549C22#301211yescharacteristic
550C22#304111yescharacteristic
551C22#304311yescharacteristic
552C22#304612yes—
553C22#309612yes—
554C22#309712yes—
555C22#309812yes—
556C22#309912yes—
557C22#313611yescharacteristic
558C22#313611yescharacteristic
559C22#313712yes—
560C22#324112yes—
561C22#324212yes—
562C22#324312yes—
563C22#324412yes—
564C4—24no—
565C4—44no—
566C4—44no—
567C4—84no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4—161no—
572C4—161no—
573C4#241312yes—
574C4#241311yescharacteristic
575C4#241311yescharacteristic
576C4#248412yes—
577C4#248411yescharacteristic
578C4#248411yescharacteristic
579C2—44no—
580C2#1507311yescharacteristic
581C2#2351011yescharacteristic
582C2#2351111yescharacteristic
583C2#2765512yes—
584C2#2765712yes—
585C2#2797012yes—
586C2#2797112yes—
587C2#2964711yescharacteristic
588C2#2964911yescharacteristic
589C2#3298711yescharacteristic
590C2#3298911yescharacteristic
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. #11268C2semidirect
8. #11267C2semidirect
9. #11208C2semidirect
10. #11207C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18242
C2×#18252
C2×#18232
#142642
#112682
#112672
#112082
#112072
#107132