Finite Group Databasebeta

← #415932

Subgroups — #415932

Order 512.

1706 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic
2C22×#479C211yesmaximal, characteristic
3C2×#2546C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#14160C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11122C212yesmaximal
9#11010C212yesmaximal
10#11010C212yesmaximal
11#10835C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2212yes—
14C22×#125C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C2×#491—24no—
17C2×#491—24no—
18C2×#491—24no—
19C2×#479—24no—
20C2×#479—24no—
21C2×#479—24no—
22C22×#107C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C22×#103C2211yescharacteristic
26C22×#103C2211yescharacteristic
27C22×#103C2212yes—
28C22×#103C2212yes—
29C22×#103C2212yes—
30C22×#103C2212yes—
31C22×#103C2212yes—
32C22×#97C2211yescharacteristic
33#2546—24no—
34#2546—24no—
35C2×#398—22no—
36C2×#391C2212yes—
37C2×#389—22no—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#1981—24no—
41#1763—24no—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1441—22no—
46#1436—22no—
47C4⋊#95C2212yes—
48C4⋊#95C2212yes—
49C23×Q8C2312yes—
50C23×Q8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8—42no—
65C2×C4⋊Q8D414yes—
66C2×#125—22no—
67C2×#125—22no—
68C2×#125—22no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125D414yes—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8—22no—
76C2×C4×Q8—22no—
77C2×C4×Q8D414yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C42—21no—
90C22×C42—21no—
91C22×C42C2312yes—
92C22×C42C2311yescharacteristic
93#491—232no—
94#479—232no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107—22no—
98C2×#106—22no—
99C2×#106—22no—
100C2×#106—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8D414yes—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—22no—
110C2×#103—22no—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—24no—
114C2×#103—22no—
115C2×#103—22no—
116C2×#103—24no—
117C2×#103—24no—
118C2×#103—24no—
119C2×#103—24no—
120C2×#103—24no—
121C2×#103—24no—
122C2×#103—24no—
123C2×#103—42no—
124C2×#103—42no—
125C2×#97—22no—
126C2×#97—22no—
127C2×#97D414yes—
128#398—216no—
129#391—24no—
130#391—24no—
131#389—216no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—21no—
139C2×#95—24no—
140C2×#95D412yes—
141#340—24no—
142#336—42no—
143#336—42no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—21no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8C2×D414yes—
166C22×Q8C2×D412yes—
167C22×Q8#4114yes—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4C2×D414yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8C2×D414yes—
193C22×C8C2×D412yes—
194C22×C8C2×D412yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—48no—
198C4⋊Q8—48no—
199C4⋊Q8—48no—
200C4⋊Q8—48no—
201C4⋊Q8—44no—
202C4⋊Q8—44no—
203#125—28no—
204#125—28no—
205#125—44no—
206#125—44no—
207C4×Q8—28no—
208C4×Q8—28no—
209C4×Q8—44no—
210C4×Q8—44no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—48no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4#4114yes—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—42no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42—82no—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272#107—28no—
273#106—28no—
274C4⋊C8—28no—
275#103—216no—
276#103—28no—
277#103—28no—
278#103—216no—
279#103—216no—
280#103—48no—
281#103—48no—
282#97—28no—
283#95—216no—
284#95—24no—
285#95—42no—
286#95—48no—
287#95—84no—
288C24C22×D411yescharacteristic
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—24no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—41no—
306C2×Q8—42no—
307C2×Q8—41no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8#12414yes—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4#12012yes—
376C22×C4#12012yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12712yes—
380C22×C4#12712yes—
381C22×C4#13718yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C2×C8—24no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—216no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—432no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—48no—
437C42—84no—
438C42#12014yes—
439C42#12014yes—
440C42#12114yes—
441C42#12114yes—
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23C2×#12711yescharacteristic
445C23#54612yes—
446C23#54812yes—
447C23C2×#13714yes—
448C23#57712yes—
449C23#57812yes—
450Q8—24no—
451Q8—24no—
452Q8—42no—
453Q8—44no—
454Q8—44no—
455Q8—42no—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—44no—
460Q8—82no—
461Q8—82no—
462Q8—82no—
463Q8—82no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4#39214yes—
510C2×C4#39214yes—
511C2×C4C2×#12011yescharacteristic
512C2×C4C2×#12011yescharacteristic
513C2×C4C2×#12111yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4#52512yes—
516C2×C4C2×#12411yescharacteristic
517C2×C4#52912yes—
518C2×C4#53211yescharacteristic
519C2×C4#53511yescharacteristic
520C2×C4#53611yescharacteristic
521C2×C4#53712yes—
522C2×C4#54212yes—
523C2×C4#54612yes—
524C2×C4#54612yes—
525C2×C4#54712yes—
526C2×C4#54812yes—
527C2×C4#54812yes—
528C2×C4#55012yes—
529C8—44no—
530C8—42no—
531C8—42no—
532C22C2×#39211yescharacteristic
533C22#242612yes—
534C22#243211yescharacteristic
535C22#243511yescharacteristic
536C22#254012yes—
537C22#300412yes—
538C22#301312yes—
539C22#303912yes—
540C22#304612yes—
541C22#315212yes—
542C22#315411yescharacteristic
543C22#315611yescharacteristic
544C22#331214yes—
545C22#332014yes—
546C22#337214yes—
547C22#337414yes—
548C4—24no—
549C4—24no—
550C4—48no—
551C4—44no—
552C4—84no—
553C4—84no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4#240712yes—
558C4#242712yes—
559C4#243111yescharacteristic
560C4#243611yescharacteristic
561C4#262011yescharacteristic
562C4#285011yescharacteristic
563C2#1521511yescharacteristic
564C2#2342711yescharacteristic
565C2#2343211yescharacteristic
566C2#2529014yes—
567C2#2565712yes—
568C2#2589412yes—
569C2#2842312yes—
570C2#2843312yes—
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
C22×#4912
C22×#4792
C2×#25462
C2×#19812
C2×#17632
#141602
#111242
#111222
#110102
#108352