Finite Group Databasebeta

← #413621

Subgroups — #413621

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C212yesmaximal
2C2×#2071C212yesmaximal
3C2×#1804C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#14354C211yesmaximal, characteristic, complemented
6#13785C211yesmaximal, characteristic
7#11358C212yesmaximal, complemented
8#11357C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C2×#390C2212yes—
19C2×#384C2212yes—
20C2×#384C2211yescharacteristic
21C2×#382C2212yes—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#2073—28no—
26#2071—28no—
27#1804—24no—
28#1802—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38#1444—21no—
39#1444—21no—
40#1444—22no—
41#1437—22no—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C22⋊#95—21no—
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4D414yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×#102—24no—
78C2×#102—22no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84#390—24no—
85#390D418yes—
86#384—22no—
87#384—24no—
88#384—24no—
89#384—22no—
90#382—22no—
91#382—24no—
92#382—24no—
93#382—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96#376—22no—
97#376—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107C2×#95D412yes—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336—24no—
116#336—24no—
117C25—21no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D412yes—
127C22×D4C2×D414yes—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—24no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C4×D4—28no—
161C4×D4—28no—
162C4×D4—44no—
163C4×D4—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—81no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—42no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42C2×D414yes—
218C2×C42C2×D412yes—
219C2×C42#4114yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C4⋊C8—44no—
226C4⋊C8—44no—
227#102—48no—
228#102—44no—
229#102—44no—
230#95—24no—
231#95—22no—
232#95—22no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—82no—
239#95—82no—
240C24—24no—
241C24—22no—
242C24—41no—
243C24—41no—
244C24—42no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24—41no—
251C24C22×D411yescharacteristic
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4#12714yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4#12114yes—
323C22×C4#12114yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4#12312yes—
327C22×C4#12312yes—
328C22×C4#12412yes—
329C22×C4#12412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4112yes—
338C22×C42−1+412yes—
339C22×C42−1+412yes—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C22⋊C4—48no—
380C22⋊C4—88no—
381C22⋊C4—88no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—84no—
387C42#12314yes—
388C42#12314yes—
389C42#12614yes—
390C42#12614yes—
391C23—24no—
392C23—42no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—44no—
402C23—42no—
403C23—44no—
404C23—82no—
405C23—81no—
406C23—81no—
407C23—82no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23C2×#12111yescharacteristic
415C23C2×#12112yes—
416C23C2×#12311yescharacteristic
417C23C2×#12411yescharacteristic
418C23#53612yes—
419C23#57612yes—
420C23#57611yescharacteristic
421C23#57712yes—
422C23#57711yescharacteristic
423C23#57711yescharacteristic
424C23#57811yescharacteristic
425D4—24no—
426D4—24no—
427D4—44no—
428D4—44no—
429D4—42no—
430D4—42no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—816no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4#39714yes—
486C2×C4#39714yes—
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12311yescharacteristic
489C2×C4C2×#12611yescharacteristic
490C2×C4C2×#12611yescharacteristic
491C2×C4C2×#12711yescharacteristic
492C2×C4#53212yes—
493C2×C4#53211yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#54112yes—
497C2×C4#54712yes—
498C2×C4#54712yes—
499C2×C4#54712yes—
500C2×C4#54712yes—
501C2×C4#55212yes—
502C2×C4#55212yes—
503C2×C4#55612yes—
504C2×C4#55612yes—
505C8—82no—
506C8—81no—
507C8—81no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22C2×#39711yescharacteristic
520C22#248612yes—
521C22#254712yes—
522C22#262111yescharacteristic
523C22#285611yescharacteristic
524C22#304112yes—
525C22#304111yescharacteristic
526C22#304212yes—
527C22#304211yescharacteristic
528C22#304512yes—
529C22#304511yescharacteristic
530C22#304612yes—
531C22#304611yescharacteristic
532C22#309612yes—
533C22#309811yescharacteristic
534C22#309911yescharacteristic
535C22#313611yescharacteristic
536C22#313612yes—
537C22#313711yescharacteristic
538C22#324112yes—
539C22#324112yes—
540C22#324312yes—
541C22#324412yes—
542C4—24no—
543C4—24no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—82no—
548C4—161no—
549C4—162no—
550C4—161no—
551C4#249612yes—
552C4#249612yes—
553C4#249911yescharacteristic
554C4#249911yescharacteristic
555C4#250111yescharacteristic
556C4#250111yescharacteristic
557C2—84no—
558C2#1515011yescharacteristic
559C2#2363211yescharacteristic
560C2#2363411yescharacteristic
561C2#2765412yes—
562C2#2765512yes—
563C2#2872812yes—
564C2#2873012yes—
565C2#2987211yescharacteristic
566C2#2987411yescharacteristic
567C2#3316611yescharacteristic
568C2#3316811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14354C2semidirect
7. #11358C2semidirect
8. #11357C2semidirect
9. #10850C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×#20712
C2×#18042
C2×#18022
#143542
#137852
#113582
#113572
#108502
#108492
#108332