Finite Group Databasebeta

← #412530

Subgroups — #412530

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2064C212yesmaximal
2C2×#1950C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4#20793C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×#1801C211yesmaximal, characteristic
7C4×#419C211yesmaximal, characteristic
8#11242C212yesmaximal
9#11216C212yesmaximal
10#11062C212yesmaximal
11#10857C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#424C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C2×#419C2211yescharacteristic
21C2×#419C2211yescharacteristic
22C22×#103C2212yes—
23C22×#101C2212yes—
24C2×C4×M16C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#2064—28no—
28#1950—24no—
29#1948—21no—
30#1948—21no—
31#1947—21no—
32#1947—21no—
33#1843—22no—
34#1842—22no—
35#1803—22no—
36#1803—22no—
37#1801—24no—
38#1763—21no—
39#1763—21no—
40C4×#103—21no—
41C4×#103—21no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2211yescharacteristic
48#1736C2211yescharacteristic
49C2×#340C2212yes—
50C23×Q8C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2311yescharacteristic
53C22×M16C2312yes—
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8Q814yes—
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73C2×#108—24no—
74C2×#108—24no—
75C2×#108—42no—
76#428—24no—
77#428—24no—
78C2×#107—22no—
79C2×#106—22no—
80#424—24no—
81#424—24no—
82#424—42no—
83#424—42no—
84#423—22no—
85#423—22no—
86#423—22no—
87#423D414yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8D412yes—
97C2×C4⋊C8D412yes—
98#419—22no—
99#419—22no—
100#419D414yes—
101#419Q814yes—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—42no—
109C2×#101—24no—
110C2×#101—24no—
111C2×#101—42no—
112C4×M16—22no—
113C4×M16—22no—
114C2×#97—21no—
115C2×C4×C8—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—21no—
127C43—21no—
128#343—24no—
129#343—24no—
130#343—42no—
131#341—24no—
132#340—24no—
133#340—22no—
134#340—22no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8C2×D412yes—
151C22×Q8C2×Q812yes—
152C22×Q8C2×Q812yes—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4#4111yescharacteristic
157C23×C4#4111yescharacteristic
158C23×C4C2411yescharacteristic
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—22no—
173C2×M16—21no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16C2×D414yes—
181C2×M16C2×D412yes—
182C2×M16C2×Q812yes—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C2×D414yes—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202#117—21no—
203#117—21no—
204#117—21no—
205#117—21no—
206#117C2×D412yes—
207#117C2×Q812yes—
208#117#4112yes—
209#117#4112yes—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×Q812yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—24no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—41no—
246C2×C42—41no—
247C2×C42—42no—
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×Q812yes—
254C2×C42#4112yes—
255C2×C42#4112yes—
256#108—48no—
257#107—44no—
258#106—44no—
259C4⋊C8—24no—
260C4⋊C8—24no—
261C4⋊C8—24no—
262C4⋊C8—24no—
263C4⋊C8—48no—
264C4⋊C8—48no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267#103—24no—
268#103—24no—
269#103—24no—
270#103—24no—
271#103—48no—
272#101—48no—
273#97—42no—
274C4×C8—42no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C24C2×#4111yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8C4⋊Q814yes—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4#12111yescharacteristic
334C22×C4#12111yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4C22⋊Q812yes—
338C22×C4C22⋊Q812yes—
339C22×C4#12612yes—
340C22×C4#12612yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C22×Q811yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+411yescharacteristic
354C22×C42−1+411yescharacteristic
355M16—24no—
356M16—22no—
357M16—24no—
358M16—22no—
359M16—22no—
360M16—22no—
361M16—41no—
362M16—42no—
363M16—41no—
364M16—41no—
365M16—42no—
366M16—41no—
367M16—42no—
368M16—42no—
369M16—42no—
370M16—42no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4#12112yes—
409C4⋊C4#12112yes—
410C4⋊C4C22⋊Q812yes—
411C4⋊C4C22⋊Q812yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—44no—
421C42—44no—
422C42—42no—
423C42—42no—
424C42—81no—
425C42—81no—
426C42#12112yes—
427C42#12112yes—
428C42#12112yes—
429C42#12112yes—
430C42C22⋊Q812yes—
431C42C22⋊Q812yes—
432C42C22⋊Q812yes—
433C42C22⋊Q812yes—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23C2×#12111yescharacteristic
439C23C2×C22⋊Q811yescharacteristic
440C23C2×#12611yescharacteristic
441C23#54711yescharacteristic
442C23#55211yescharacteristic
443C23#55311yescharacteristic
444C23#55411yescharacteristic
445Q8—24no—
446Q8—24no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—162no—
491C2×C4—162no—
492C2×C4#39612yes—
493C2×C4#39612yes—
494C2×C4#39612yes—
495C2×C4#39612yes—
496C2×C4#47112yes—
497C2×C4#47312yes—
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12111yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×C22⋊Q811yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4C2×C4⋊Q811yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4#53311yescharacteristic
507C2×C4#53311yescharacteristic
508C2×C4#53711yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#54011yescharacteristic
511C2×C4#54111yescharacteristic
512C2×C4#54411yescharacteristic
513C2×C4#54411yescharacteristic
514C2×C4D4211yescharacteristic
515C2×C4D4211yescharacteristic
516C2×C4#54611yescharacteristic
517C2×C4#54711yescharacteristic
518C2×C4#54711yescharacteristic
519C2×C4#54811yescharacteristic
520C2×C4D4×Q811yescharacteristic
521C2×C4D4×Q811yescharacteristic
522C2×C4#55011yescharacteristic
523C2×C4#55411yescharacteristic
524C2×C4#55511yescharacteristic
525C2×C4#55611yescharacteristic
526C8—44no—
527C8—42no—
528C8—44no—
529C8—42no—
530C8—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22C2×#39611yescharacteristic
544C22#241611yescharacteristic
545C22#241711yescharacteristic
546C22#244011yescharacteristic
547C22#244111yescharacteristic
548C22#249611yescharacteristic
549C22#249711yescharacteristic
550C22C2×#47111yescharacteristic
551C22C2×#47311yescharacteristic
552C22#306811yescharacteristic
553C22#306911yescharacteristic
554C4—22no—
555C4—22no—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—44no—
560C4—42no—
561C4—42no—
562C4—82no—
563C4—162no—
564C4#206414yescomplemented
565C4#242811yescharacteristic
566C4#242911yescharacteristic
567C4#243911yescharacteristic
568C4#244011yescharacteristic
569C4#253711yescharacteristic
570C4#253811yescharacteristic
571C4#262911yescharacteristic
572C4#285711yescharacteristic
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2#1513111yescharacteristic
578C2C2×#206411yescharacteristic
579C2#2360411yescharacteristic
580C2#2477211yescharacteristic
581C2#2477311yescharacteristic
582C2#2485711yescharacteristic
583C2#2485811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
564. C4#2064semidirect

Maximal subgroups

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

subgroupindex
C2×#20642
C2×#19502
#211712
#207932
C2×#18032
C2×#18012
C4×#4192
#112422
#112162
#110622
#108572