Finite Group Databasebeta

← #416056

Subgroups — #416056

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3C2×#1981C211yesmaximal, characteristic
4C2×#1960C211yesmaximal, characteristic
5C2×#1776C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic
7#11190C212yesmaximal
8#11190C212yesmaximal
9#11124C212yesmaximal
10#11122C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#97C2211yescharacteristic
22#2568—24no—
23#2568—24no—
24C2×#400—22no—
25C2×#391—21no—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#382—21no—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#1985—24no—
33#1981—24no—
34#1960—24no—
35#1776—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×Q8C2312yes—
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#107—22no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97D414yes—
101#400—216no—
102#391—28no—
103#391—24no—
104#391—24no—
105#391—24no—
106#391—24no—
107#382—28no—
108#382—22no—
109#382—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126#340—22no—
127#340—22no—
128#340—42no—
129#340—42no—
130#336—24no—
131#336—24no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8C2×D414yes—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8#4112yes—
182C22×C8#4112yes—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4112yes—
232C2×C4⋊C4#4112yes—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241C2×C42—42no—
242C2×C42—82no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42#4112yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249#107—28no—
250#106—28no—
251C4⋊C8—44no—
252#103—48no—
253#103—48no—
254#103—48no—
255#103—48no—
256#97—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—44no—
263#95—42no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—82no—
268#95—82no—
269C24C22×D411yescharacteristic
270C2×Q8—28no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—44no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—44no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#12012yes—
355C22×C4#12012yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12312yes—
359C22×C4#12312yes—
360C22×C4#13714yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C42−1+411yescharacteristic
374C22×C42−1+411yescharacteristic
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8#12414yes—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—48no—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C42—48no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—84no—
426C42#12114yes—
427C42#12114yes—
428C42#12314yes—
429C42#12414yes—
430C23C2×#12011yescharacteristic
431C23C2×#12111yescharacteristic
432C23C2×#12311yescharacteristic
433C23#54811yescharacteristic
434C23#54811yescharacteristic
435C23#55011yescharacteristic
436C23#55011yescharacteristic
437C23C2×#13712yes—
438C23#57512yes—
439C23#57712yes—
440C23#57812yes—
441Q8—416no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4#39414yes—
507C2×C4#39414yes—
508C2×C4C2×#12111yescharacteristic
509C2×C4C2×#12111yescharacteristic
510C2×C4C2×#12311yescharacteristic
511C2×C4#52511yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4C2×#12411yescharacteristic
514C2×C4C2×#12411yescharacteristic
515C2×C4#52911yescharacteristic
516C2×C4#52911yescharacteristic
517C2×C4#53211yescharacteristic
518C2×C4#53511yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53711yescharacteristic
521C2×C4#53711yescharacteristic
522C2×C4#54211yescharacteristic
523C2×C4#54211yescharacteristic
524C2×C4#54611yescharacteristic
525C2×C4#54611yescharacteristic
526C2×C4#54711yescharacteristic
527C2×C4#54711yescharacteristic
528C2×C4#55611yescharacteristic
529C2×C4#55611yescharacteristic
530C2×C4#55611yescharacteristic
531C2×C4#55611yescharacteristic
532C2×C4#55911yescharacteristic
533C2×C4#55911yescharacteristic
534C2×C4#55911yescharacteristic
535C2×C4#55911yescharacteristic
536C8—24no—
537C8—42no—
538C8—82no—
539C22C2×#39411yescharacteristic
540C22#240711yescharacteristic
541C22#240711yescharacteristic
542C22#244811yescharacteristic
543C22#244811yescharacteristic
544C22#254511yescharacteristic
545C22#254811yescharacteristic
546C22#300411yescharacteristic
547C22#300511yescharacteristic
548C22#301011yescharacteristic
549C22#301111yescharacteristic
550C22#305612yes—
551C22#306312yes—
552C22#309512yes—
553C22#309712yes—
554C22#337212yes—
555C22#337412yes—
556C22#337612yes—
557C22#337712yes—
558C22#339812yes—
559C22#340012yes—
560C22#340112yes—
561C22#340212yes—
562C4—24no—
563C4—24no—
564C4—44no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—162no—
572C4#244711yescharacteristic
573C4#244711yescharacteristic
574C4#254411yescharacteristic
575C4#254711yescharacteristic
576C4#259211yescharacteristic
577C4#259211yescharacteristic
578C4#262011yescharacteristic
579C4#285011yescharacteristic
580C2#1521711yescharacteristic
581C2#2348111yescharacteristic
582C2#2348411yescharacteristic
583C2#2419711yescharacteristic
584C2#2419911yescharacteristic
585C2#2420111yescharacteristic
586C2#2420311yescharacteristic
587C2#2633312yes—
588C2#2633512yes—
589C2#2865212yes—
590C2#2874712yes—
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×#25682
C2×#19852
C2×#19812
C2×#19602
C2×#17762
#139252
#111902
#111242
#111222
#108502