Finite Group Databasebeta

← #412563

Subgroups — #412563

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2064C212yesmaximal
2C2×#1950C211yesmaximal, characteristic
3#21145C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6C2×#1803C211yesmaximal, characteristic
7C2×C4×#101C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11216C212yesmaximal
10#11145C212yesmaximal
11#10857C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#428C2212yes—
15C2×#428C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×#419C2212yes—
20C22×#103C2212yes—
21C22×#101C2211yescharacteristic
22C22×#101C2211yescharacteristic
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27#2064—28no—
28#1950—24no—
29#1940—22no—
30#1939—22no—
31#1847—22no—
32#1845—21no—
33#1844—21no—
34#1805—22no—
35#1803—22no—
36#1803—22no—
37#1803—21no—
38#1803—21no—
39C4×#101—24no—
40C2×#343C2212yes—
41C2×#343C2211yescharacteristic
42C2×#343C2211yescharacteristic
43C2×#343C2212yes—
44C2×#341C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8Q814yes—
63C2×#117C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C2×#108—24no—
72C2×#108—24no—
73C2×#108—24no—
74#428—24no—
75#428—24no—
76#428—42no—
77#428—42no—
78C2×#107—22no—
79C2×#107—22no—
80C2×#106—22no—
81C2×#106—22no—
82#424—24no—
83#424—42no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8D412yes—
92C2×C4⋊C8D412yes—
93#419—24no—
94#419—42no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—42no—
100C2×#101—22no—
101C2×#101—22no—
102C2×#101—22no—
103C2×#101—22no—
104C2×#101D414yes—
105C2×#101Q814yes—
106C4×M16—22no—
107C4×M16—22no—
108C4×M16—22no—
109C4×M16D414yes—
110C2×#97—21no—
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#382D412yes—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125#343—24no—
126#343—22no—
127#343—22no—
128#343—42no—
129#341—24no—
130#340—24no—
131#340—22no—
132#340—22no—
133#336—24no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8C2×D412yes—
146C22×Q8C2×Q812yes—
147C22×Q8C2×Q812yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C23×C4C2411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16C2×D412yes—
177C2×M16C2×D412yes—
178C2×M16C2×D412yes—
179C2×M16C2×D412yes—
180C2×M16C2×Q812yes—
181C2×M16C2×Q812yes—
182C2×M16#4112yes—
183C2×M16#4112yes—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8C2×D414yes—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—24no—
199C4×Q8—24no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
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—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C4⋊C4C2×Q812yes—
236C2×C22⋊C4—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#108—48no—
251#107—44no—
252#107—44no—
253#106—44no—
254#106—44no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—48no—
258C4⋊C8—42no—
259C4⋊C8—42no—
260#103—44no—
261#103—44no—
262#103—48no—
263#101—28no—
264#101—28no—
265#97—24no—
266#97—42no—
267C4×C8—24no—
268C4×C8—42no—
269#95—22no—
270#95—22no—
271C24C2×#4111yescharacteristic
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8C4⋊Q814yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12111yescharacteristic
326C22×C4#12111yescharacteristic
327C22×C4C22⋊Q812yes—
328C22×C4C22⋊Q812yes—
329C22×C4#12612yes—
330C22×C4#12612yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×Q811yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345M16—22no—
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—22no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16—42no—
363M16#12114yes—
364M16C22⋊Q814yes—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—22no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42—44no—
415C42#12112yes—
416C42#12112yes—
417C42#12112yes—
418C42#12112yes—
419C42C22⋊Q812yes—
420C42C22⋊Q812yes—
421C42C22⋊Q812yes—
422C42C22⋊Q812yes—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23C2×#12111yescharacteristic
428C23C2×C22⋊Q811yescharacteristic
429C23C2×#12611yescharacteristic
430C23#53211yescharacteristic
431C23#53311yescharacteristic
432C23#54011yescharacteristic
433C23#54111yescharacteristic
434Q8—24no—
435Q8—24no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—42no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—162no—
474C2×C4—162no—
475C2×C4#39612yes—
476C2×C4#39612yes—
477C2×C4#39612yes—
478C2×C4#39612yes—
479C2×C4#47112yes—
480C2×C4#47312yes—
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12111yescharacteristic
483C2×C4C2×#12111yescharacteristic
484C2×C4C2×C22⋊Q811yescharacteristic
485C2×C4C2×C22⋊Q811yescharacteristic
486C2×C4C2×C22⋊Q811yescharacteristic
487C2×C4C2×C4⋊Q811yescharacteristic
488C2×C4#53311yescharacteristic
489C2×C4#53711yescharacteristic
490C2×C4#53711yescharacteristic
491C2×C4#54411yescharacteristic
492C2×C4#54411yescharacteristic
493C2×C4D4211yescharacteristic
494C2×C4D4211yescharacteristic
495C2×C4#54611yescharacteristic
496C2×C4#54711yescharacteristic
497C2×C4#54711yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4D4×Q811yescharacteristic
501C2×C4D4×Q811yescharacteristic
502C2×C4#55011yescharacteristic
503C2×C4#55211yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4#55411yescharacteristic
506C2×C4#55411yescharacteristic
507C2×C4#55511yescharacteristic
508C2×C4#55611yescharacteristic
509C8—24no—
510C8—24no—
511C8—44no—
512C8—42no—
513C8—42no—
514C8—82no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22C2×#39611yescharacteristic
528C22#242811yescharacteristic
529C22#242911yescharacteristic
530C22#249611yescharacteristic
531C22#249711yescharacteristic
532C22#262911yescharacteristic
533C22#285711yescharacteristic
534C22C2×#47111yescharacteristic
535C22C2×#47311yescharacteristic
536C22#306811yescharacteristic
537C22#306911yescharacteristic
538C4—22no—
539C4—22no—
540C4—22no—
541C4—44no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—162no—
546C4#206414yescomplemented
547C4#241611yescharacteristic
548C4#241711yescharacteristic
549C4#243911yescharacteristic
550C4#244011yescharacteristic
551C4#244011yescharacteristic
552C4#244111yescharacteristic
553C4#253711yescharacteristic
554C4#253811yescharacteristic
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2#1513111yescharacteristic
560C2C2×#206411yescharacteristic
561C2#2360411yescharacteristic
562C2#2540611yescharacteristic
563C2#2540711yescharacteristic
564C2#2549111yescharacteristic
565C2#2549211yescharacteristic
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
546. 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
#211452
#208002
#206472
C2×#18032
C2×C4×#1012
#112492
#112162
#111452
#108572