Finite Group Databasebeta

← #416954

Subgroups — #416954

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic
7#11293C212yesmaximal
8#11292C212yesmaximal
9#11190C212yesmaximal
10#11190C212yesmaximal
11#10849C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#97C2211yescharacteristic
19#2582—24no—
20#2582—24no—
21C2×#398—22no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1985—24no—
30#1836—24no—
31#1805—24no—
32#1763—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×Q8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#97—22no—
90C2×#97—22no—
91C2×#97D414yes—
92#398—216no—
93#391—28no—
94#391—24no—
95#391—24no—
96#389—28no—
97#389—24no—
98#389—24no—
99#384—22no—
100#384D414yes—
101#382—22no—
102#382—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340—24no—
122#336—24no—
123#336—24no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8C2×D412yes—
138C22×Q8#4114yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D414yes—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4—81no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4114yes—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C2×C42#4112yes—
236C4⋊C8—44no—
237#103—216no—
238#103—44no—
239#103—44no—
240#97—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—44no—
245#95—42no—
246#95—42no—
247#95—42no—
248#95—44no—
249#95—82no—
250#95—82no—
251C24C22×D411yescharacteristic
252C2×Q8—28no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—44no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×Q8#12414yes—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4#12012yes—
331C22×C4#12012yes—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#12312yes—
335C22×C4#12312yes—
336C22×C4#13714yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4112yes—
344C22×C4C2×#4112yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—48no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—48no—
393C42—84no—
394C42#12114yes—
395C42#12114yes—
396C42#12314yes—
397C42#12414yes—
398C23C2×#12011yescharacteristic
399C23C2×#12111yescharacteristic
400C23C2×#12311yescharacteristic
401C23#54812yes—
402C23#54812yes—
403C23C2×#13712yes—
404C23#57512yes—
405C23#57611yescharacteristic
406C23#57711yescharacteristic
407C23#57711yescharacteristic
408C23#57811yescharacteristic
409Q8—24no—
410Q8—24no—
411Q8—416no—
412Q8—42no—
413Q8—44no—
414Q8—44no—
415Q8—42no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4#39414yes—
477C2×C4#39414yes—
478C2×C4C2×#12111yescharacteristic
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4#52512yes—
482C2×C4C2×#12411yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4#52912yes—
485C2×C4#53211yescharacteristic
486C2×C4#53612yes—
487C2×C4#53611yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#54112yes—
490C2×C4#54712yes—
491C2×C4#54812yes—
492C2×C4#55212yes—
493C2×C4#55212yes—
494C2×C4#55212yes—
495C2×C4#55612yes—
496C8—44no—
497C8—81no—
498C8—81no—
499C22C2×#39411yescharacteristic
500C22#242112yes—
501C22#243211yescharacteristic
502C22#243611yescharacteristic
503C22#253612yes—
504C22#300411yescharacteristic
505C22#300511yescharacteristic
506C22#301211yescharacteristic
507C22#301311yescharacteristic
508C22#305612yes—
509C22#306211yescharacteristic
510C22#306311yescharacteristic
511C22#309512yes—
512C22#309811yescharacteristic
513C22#309911yescharacteristic
514C22#336912yes—
515C22#337012yes—
516C22#337112yes—
517C22#337112yes—
518C22#337212yes—
519C22#337412yes—
520C22#337512yes—
521C22#337712yes—
522C4—24no—
523C4—24no—
524C4—44no—
525C4—44no—
526C4—84no—
527C4—84no—
528C4—82no—
529C4—82no—
530C4—161no—
531C4—161no—
532C4#241512yes—
533C4#248912yes—
534C4#249911yescharacteristic
535C4#250311yescharacteristic
536C4#262111yescharacteristic
537C4#285011yescharacteristic
538C2#1523611yescharacteristic
539C2#2348111yescharacteristic
540C2#2348511yescharacteristic
541C2#2500812yes—
542C2#2500912yes—
543C2#2565711yescharacteristic
544C2#2565811yescharacteristic
545C2#2594111yescharacteristic
546C2#2594211yescharacteristic
547C2#2819812yes—
548C2#2820012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#19852
C2×#18362
C2×#18052
C2×#17632
#139252
#112932
#112922
#111902
#108492