Finite Group Databasebeta

← #410225

Subgroups — #410225

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1818C211yesmaximal, characteristic
6#14343C211yesmaximal, characteristic
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11220C212yesmaximal, complemented
10#11219C212yesmaximal, complemented
11#10731C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19#2317—24no—
20#2317—24no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#385—22no—
24C2×#384—21no—
25C2×#382—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28#1854—24no—
29#1824—24no—
30#1823—24no—
31#1818—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1445—21no—
43#1445—21no—
44#1440—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—24no—
60C2×#117—24no—
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⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2312yes—
85C2×#107—22no—
86C2×#107—24no—
87C2×#106—22no—
88C2×#106—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8D414yes—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—24no—
99#385—216no—
100#384—28no—
101#382—28no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—24no—
121#340—24no—
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—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
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—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169#117—216no—
170#117—28no—
171#117—28no—
172#117—44no—
173#117—44no—
174#117—44no—
175#117—44no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4—82no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241#107—44no—
242#106—44no—
243C22⋊C8—28no—
244C22⋊C8—44no—
245#95—28no—
246#95—28no—
247#95—42no—
248#95—42no—
249#95—82no—
250#95—82no—
251#95—82no—
252#95—82no—
253C24—22no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24#12012yes—
262C24#12112yes—
263C24#12312yes—
264C24C22×D411yescharacteristic
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
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—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—48no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4#12012yes—
337C22×C4#12114yes—
338C22×C4#12114yes—
339C22×C4#12112yes—
340C22×C4#12314yes—
341C22×C4#12312yes—
342C22×C4#12414yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4C2×#4111yescharacteristic
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C4⋊C4#12314yes—
394C4⋊C4#12314yes—
395C22⋊C4—216no—
396C22⋊C4—48no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C42—216no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C23—22no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23#39414yes—
427C23C2×#12011yescharacteristic
428C23C2×#12111yescharacteristic
429C23C2×#12311yescharacteristic
430C23#52512yes—
431C23#53612yes—
432C23#57611yescharacteristic
433C23#57611yescharacteristic
434C23#57711yescharacteristic
435C23#57711yescharacteristic
436C23#57712yes—
437C23#57812yes—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—816no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#39414yes—
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12311yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4#52512yes—
504C2×C4C2×#12411yescharacteristic
505C2×C4#53612yes—
506C2×C4#53611yescharacteristic
507C2×C4#53611yescharacteristic
508C2×C4#54712yes—
509C2×C4#54712yes—
510C2×C4#55212yes—
511C2×C4#55212yes—
512C2×C4#55212yes—
513C2×C4#55212yes—
514C2×C4#55212yes—
515C2×C4#55212yes—
516C8—42no—
517C8—42no—
518C8—82no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—44no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22—42no—
530C22C2×#39411yescharacteristic
531C22#241312yes—
532C22#248412yes—
533C22#261411yescharacteristic
534C22#284911yescharacteristic
535C22#301011yescharacteristic
536C22#301111yescharacteristic
537C22#301211yescharacteristic
538C22#301211yescharacteristic
539C22#305711yescharacteristic
540C22#305811yescharacteristic
541C22#306312yes—
542C22#309611yescharacteristic
543C22#309611yescharacteristic
544C22#309712yes—
545C22#311912yes—
546C22#312012yes—
547C22#312412yes—
548C22#312512yes—
549C22#324112yes—
550C22#324212yes—
551C22#324312yes—
552C22#324412yes—
553C4—24no—
554C4—44no—
555C4—44no—
556C4—82no—
557C4—82no—
558C4—84no—
559C4—162no—
560C4—161no—
561C4—161no—
562C4#241312yes—
563C4#248412yes—
564C4#248611yescharacteristic
565C4#248611yescharacteristic
566C4#250311yescharacteristic
567C4#250311yescharacteristic
568C2—44no—
569C2#1507311yescharacteristic
570C2#2348411yescharacteristic
571C2#2348611yescharacteristic
572C2#2461212yes—
573C2#2461612yes—
574C2#2761912yes—
575C2#2762212yes—
576C2#2964711yescharacteristic
577C2#2964911yescharacteristic
578C2#3301311yescharacteristic
579C2#3301711yescharacteristic
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. #11293C2semidirect
8. #11292C2semidirect
9. #11220C2semidirect
10. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18542
C2×#18242
C2×#18232
C2×#18182
#143432
#112932
#112922
#112202
#112192
#107312