Finite Group Databasebeta

← #392295

Subgroups — #392295

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2275C211yesmaximal, characteristic
2C2×#1832C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic, complemented
6#10850C211yesmaximal, characteristic, complemented
7#10849C211yesmaximal, characteristic, complemented
8#10807C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×#382C2211yescharacteristic
21#2275—24no—
22#2275C418yescomplemented
23C2×C4×C4⋊C4C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4—22no—
27C2×#376C2211yescharacteristic
28C22×#95—21no—
29#1832—24no—
30#1804—24no—
31#1802—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—21no—
42#1442—22no—
43#1442—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×D4C2311yescharacteristic
47C24×C4—22no—
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C418yes—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4Q814yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C42—22no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90#384—22no—
91#384—22no—
92#382—22no—
93#382—22no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4C2×C418yes—
97C4×C4⋊C4C2×C414yes—
98C4×C22⋊C4—216no—
99C4×C22⋊C4—28no—
100#376C2×C414yes—
101#376D414yes—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—28no—
105C2×#95—21no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D411yescharacteristic
112C2×#95D411yescharacteristic
113C2×#95D412yes—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—24no—
122#336—24no—
123C25—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C22×C412yes—
130C22×D4C2×D412yes—
131C22×D4C2×Q812yes—
132C22×D4#4114yes—
133C22×D4#4114yes—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—28no—
137C23×C4—28no—
138C23×C4—28no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C4×D4—216no—
174C4×D4—216no—
175C4×D4—216no—
176C4×D4—216no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—28no—
180C4×D4C4⋊C4116yes—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4#4114yes—
211C2×C4⋊C4#4114yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—28no—
223C2×C42—24no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42—82no—
227C2×C42—82no—
228C2×C42C22×C414yes—
229C2×C42C22×C412yes—
230C2×C42C2×D414yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×Q812yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C4⋊C8—44no—
237C4⋊C8—44no—
238#102—48no—
239#102—44no—
240#102—44no—
241#95—28no—
242#95—24no—
243#95—42no—
244#95—42no—
245#95—82no—
246#95—82no—
247#95—82no—
248#95—82no—
249#95C22⋊C414yes—
250#95C22⋊C418yes—
251#95C22⋊C414yes—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—24no—
257C24—42no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—41no—
263C24C22×D411yescharacteristic
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4C2×C4⋊C414yes—
278C2×D4#11718yes—
279C2×D4#12414yes—
280C2×D4#12414yes—
281C2×D4#12518yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—232no—
307C22×C4—216no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—88no—
332C22×C4—88no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—88no—
341C22×C4C2×C22⋊C414yes—
342C22×C4C2×C22⋊C414yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4C23×C411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
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—41no—
375C2×C8—41no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4C2×C4⋊C414yes—
395C4⋊C4#11718yes—
396C22⋊C4—232no—
397C22⋊C4—216no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—84no—
403C42—88no—
404C42C2×C4⋊C414yes—
405C42#11714yes—
406C42#12114yes—
407C42C22⋊Q814yes—
408C23—24no—
409C23—24no—
410C23—28no—
411C23—24no—
412C23—28no—
413C23—44no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C22×C22⋊C411yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×C22⋊Q811yescharacteristic
434C23#52512yes—
435C23#52512yes—
436C23#57612yes—
437C23#57611yescharacteristic
438C23#57712yes—
439C23#57711yescharacteristic
440C23#57711yescharacteristic
441C23#57811yescharacteristic
442D4—24no—
443D4—24no—
444D4—28no—
445D4—28no—
446D4—24no—
447D4#38218yes—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—816no—
485C2×C4—832no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4#38014yes—
504C2×C4#38014yes—
505C2×C4C22×C4⋊C411yescharacteristic
506C2×C4C2×#11712yes—
507C2×C4C2×#11711yescharacteristic
508C2×C4C2×#11712yes—
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4#52512yes—
512C2×C4#52512yes—
513C2×C4C2×#12411yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4C2×#12512yes—
516C2×C4#52912yes—
517C2×C4#52912yes—
518C2×C4#52912yes—
519C2×C4#52912yes—
520C2×C4#53211yescharacteristic
521C2×C4#53211yescharacteristic
522C2×C4#53312yes—
523C8—81no—
524C8—81no—
525C8—82no—
526C22—28no—
527C22—48no—
528C22—48no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—48no—
534C22—48no—
535C22—44no—
536C22—44no—
537C22C2×#38011yescharacteristic
538C22#228012yes—
539C22#285011yescharacteristic
540C22#285011yescharacteristic
541C22#285112yes—
542C22#289711yescharacteristic
543C22#289712yes—
544C22#289811yescharacteristic
545C22#304111yescharacteristic
546C22#304211yescharacteristic
547C22#304511yescharacteristic
548C22#304611yescharacteristic
549C22#308112yes—
550C22#308311yescharacteristic
551C22#308411yescharacteristic
552C22#311812yes—
553C22#311812yes—
554C22#311912yes—
555C22#312012yes—
556C22#312212yes—
557C22#312312yes—
558C22#312412yes—
559C22#312512yes—
560C4—24no—
561C4—24no—
562C4—44no—
563C4—48no—
564C4—44no—
565C4—44no—
566C4—161no—
567C4—161no—
568C4—162no—
569C4C2×#38211yescharacteristic
570C4#228312yes—
571C4#228511yescharacteristic
572C4#285011yescharacteristic
573C4#285011yescharacteristic
574C4#285112yes—
575C2—48no—
576C2#1468211yescharacteristic
577C2#2065011yescharacteristic
578C2#2065211yescharacteristic
579C2#2074412yes—
580C2#2074612yes—
581C2#3303411yescharacteristic
582C2#3303611yescharacteristic
583C2#3303811yescharacteristic
584C2#3304011yescharacteristic
585C2#3307412yes—
586C2#3307612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
2. C2×#1832C2semidirect
5. #14348C2semidirect
6. #10850C2semidirect
7. #10849C2semidirect
8. #10807C2semidirect
9. #10806C2semidirect
22. #2275C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22752
C2×#18322
C2×#18042
C2×#18022
#143482
#108502
#108492
#108072
#108062
#107882