Finite Group Databasebeta

← #404966

Subgroups — #404966

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2356C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4C2×#1863C212yesmaximal
5#15151C211yesmaximal, characteristic
6#11208C212yesmaximal
7#11202C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11077C211yesmaximal, characteristic
10#10907C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#398—21no—
18C2×#398—21no—
19#2356—24no—
20#2356—24no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#383—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376—21no—
33#2072—24no—
34#1947—24no—
35#1863—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1445—21no—
45#1445—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85#398—44no—
86#398—44no—
87#391—22no—
88#391—22no—
89#391—22no—
90#391D414yes—
91#389—24no—
92#389—24no—
93#387—28no—
94#387—28no—
95#384—22no—
96#384—22no—
97#383—28no—
98#382—22no—
99#382—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106#376—28no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95D411yescharacteristic
121C2×#95D411yescharacteristic
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#336—24no—
128#336—22no—
129#336—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8C2×D412yes—
138C22×Q8C2×D412yes—
139C22×Q8#4112yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
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#4114yes—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4114yes—
214C2×C4⋊C4#4112yes—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238#107—48no—
239C4⋊C8—28no—
240#103—216no—
241#103—48no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—42no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—44no—
257C24C22×D411yescharacteristic
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8#12114yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4#12012yes—
338C22×C4#12012yes—
339C22×C4#12112yes—
340C22×C4#12112yes—
341C22×C4#13512yes—
342C22×C4AGL(1,ℤ/8)12yes—
343C22×C4AGL(1,ℤ/8)12yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4112yes—
352C22×C42−1+412yes—
353C22×C42−1+412yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C4⋊C4—216no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—44no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4#12114yes—
392C42—216no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—48no—
401C42—41no—
402C42—41no—
403C42—42no—
404C42—48no—
405C23C2×#12011yescharacteristic
406C23C2×#12111yescharacteristic
407C23#52511yescharacteristic
408C23#54712yes—
409C23#54812yes—
410C23C2×#13511yescharacteristic
411C23C2×AGL(1,ℤ/8)11yescharacteristic
412C23C2×AGL(1,ℤ/8)11yescharacteristic
413C23#57511yescharacteristic
414C23#57812yes—
415C23#57812yes—
416Q8—28no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—44no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#44914yes—
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4#52511yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52512yes—
495C2×C4#52511yescharacteristic
496C2×C4#52912yes—
497C2×C4#53211yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53611yescharacteristic
500C2×C4#53612yes—
501C2×C4#54112yes—
502C2×C4#54712yes—
503C2×C4#54712yes—
504C2×C4#55212yes—
505C2×C4#55212yes—
506C2×C4#55212yes—
507C2×C4#55612yes—
508C8—44no—
509C8—82no—
510C22#240611yescharacteristic
511C22#241512yes—
512C22#243211yescharacteristic
513C22#247512yes—
514C22#284711yescharacteristic
515C22C2×#44911yescharacteristic
516C22#300611yescharacteristic
517C22#301111yescharacteristic
518C22#301111yescharacteristic
519C22#303511yescharacteristic
520C22#303811yescharacteristic
521C22#304412yes—
522C22#310611yescharacteristic
523C22#311511yescharacteristic
524C22#312712yes—
525C22#333612yes—
526C22#334312yes—
527C22#334312yes—
528C22#335212yes—
529C22#336412yes—
530C22#336712yes—
531C22#336712yes—
532C22#337712yes—
533C4—24no—
534C4—42no—
535C4—42no—
536C4—44no—
537C4—42no—
538C4—44no—
539C4—42no—
540C4—81no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4—162no—
545C4#239211yescharacteristic
546C4#241312yes—
547C4#247812yes—
548C4#259911yescharacteristic
549C4#261111yescharacteristic
550C4#262111yescharacteristic
551C2#1490711yescharacteristic
552C2#2409411yescharacteristic
553C2#2411011yescharacteristic
554C2#2468712yes—
555C2#2469412yes—
556C2#2565011yescharacteristic
557C2#2566111yescharacteristic
558C2#2737112yes—
559C2#2738212yes—
560C2#3290411yescharacteristic
561C2#3291511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#23562
C2×#20722
C2×#19472
C2×#18632
#151512
#112082
#112022
#111902
#110772
#109072
#108062