Finite Group Databasebeta

← #404967

Subgroups — #404967

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#2072C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4C2×#1865C212yesmaximal
5#15151C211yesmaximal, characteristic
6#11207C212yesmaximal
7#11202C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11077C211yesmaximal, characteristic
10#10908C212yesmaximal
11#10807C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#398—21no—
18C2×#398—21no—
19#2355—24no—
20#2355—24no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—21no—
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#382—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2072—24no—
34#1948—24no—
35#1865—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—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4D411yescharacteristic
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C42—21no—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87#398—44no—
88#398—44no—
89#391—22no—
90#391—22no—
91#391—22no—
92#391D414yes—
93#389—24no—
94#389—24no—
95#387—28no—
96#387—28no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384—22no—
101#382—28no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4D414yes—
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—21no—
143C23×C4—21no—
144C23×C4—22no—
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—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
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—24no—
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—24no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4SD1614yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4#4114yes—
219C2×C4⋊C4#4112yes—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42#4112yes—
241#106—48no—
242C4⋊C8—28no—
243#103—216no—
244#103—48no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—42no—
257#95—44no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8#12114yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
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#12012yes—
335C22×C4#12012yes—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C2×SD1612yes—
339C22×C4C2×SD1612yes—
340C22×C4C2×SD1612yes—
341C22×C4AGL(1,ℤ/8)12yes—
342C22×C4#13712yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
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—
367C4⋊C4—216no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—84no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—88no—
392C4⋊C4#12114yes—
393C42—216no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—24no—
401C42—41no—
402C42—41no—
403C42—42no—
404C42—84no—
405C42—84no—
406C23C2×#12011yescharacteristic
407C23C2×#12111yescharacteristic
408C23#52511yescharacteristic
409C23#54712yes—
410C23#54812yes—
411C23C22×SD1611yescharacteristic
412C23C2×AGL(1,ℤ/8)11yescharacteristic
413C23C2×#13711yescharacteristic
414C23#57511yescharacteristic
415C23#57712yes—
416C23#57812yes—
417Q8—28no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422Q8—44no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—82no—
473C2×C4—82no—
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#44814yes—
488C2×C4#46014yes—
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4#52511yescharacteristic
492C2×C4#52511yescharacteristic
493C2×C4#52512yes—
494C2×C4#52511yescharacteristic
495C2×C4#52912yes—
496C2×C4#53211yescharacteristic
497C2×C4#53611yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53612yes—
500C2×C4#54112yes—
501C2×C4#54712yes—
502C2×C4#54712yes—
503C2×C4#55212yes—
504C2×C4#55212yes—
505C2×C4#55212yes—
506C2×C4#55612yes—
507C8—44no—
508C8—82no—
509C22#240611yescharacteristic
510C22#241512yes—
511C22#243211yescharacteristic
512C22#247512yes—
513C22#284711yescharacteristic
514C22C2×#44811yescharacteristic
515C22#300611yescharacteristic
516C22#300911yescharacteristic
517C22#301111yescharacteristic
518C22C2×#46011yescharacteristic
519C22#303711yescharacteristic
520C22#304612yes—
521C22#310311yescharacteristic
522C22#311211yescharacteristic
523C22#312012yes—
524C22#333512yes—
525C22#334412yes—
526C22#334912yes—
527C22#335312yes—
528C22#335912yes—
529C22#337112yes—
530C22#337312yes—
531C22#337412yes—
532C4—24no—
533C4—24no—
534C4—42no—
535C4—42no—
536C4—44no—
537C4—42no—
538C4—44no—
539C4—81no—
540C4—82no—
541C4—82no—
542C4—84no—
543C4—162no—
544C4#239211yescharacteristic
545C4#241312yes—
546C4#247812yes—
547C4#259911yescharacteristic
548C4#261111yescharacteristic
549C4#262111yescharacteristic
550C2#1490711yescharacteristic
551C2#2409711yescharacteristic
552C2#2410711yescharacteristic
553C2#2467812yes—
554C2#2469112yes—
555C2#2564711yescharacteristic
556C2#2565911yescharacteristic
557C2#2736212yes—
558C2#2737512yes—
559C2#3290311yescharacteristic
560C2#3291611yescharacteristic
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×#23552
C2×#20722
C2×#19482
C2×#18652
#151512
#112072
#112022
#111902
#110772
#109082
#108072