Finite Group Databasebeta

← #412536

Subgroups — #412536

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2064C212yesmaximal
2#21170C211yesmaximal, characteristic
3#21143C211yesmaximal, characteristic
4C2×#1848C211yesmaximal, characteristic
5#20647C211yesmaximal, characteristic
6C2×#1801C211yesmaximal, characteristic
7C2×C4×#103C211yesmaximal, characteristic
8#11249C212yesmaximal
9#11242C212yesmaximal
10#11128C212yesmaximal
11#10847C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#108C2212yes—
14C22×#108C2212yes—
15C2×#428C2212yes—
16C2×#424C2212yes—
17C2×#423C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×#419C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#101C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2064—28no—
28#1949—22no—
29#1948—21no—
30#1947—21no—
31#1939—22no—
32#1939—22no—
33#1848—24no—
34#1805—22no—
35#1803—21no—
36#1803—21no—
37#1801—24no—
38C4×#103—22no—
39C4×#103—22no—
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47#1729C2212yes—
48C23×Q8C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—22no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8Q814yes—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C2×#108—24no—
71C2×#108—24no—
72C2×#108—24no—
73C2×#108—24no—
74C2×#108—42no—
75C2×#108—42no—
76#428—24no—
77#428—42no—
78C2×#107—22no—
79C2×#106—22no—
80#424—24no—
81#424—24no—
82#423—22no—
83#423—22no—
84#423—22no—
85#423—22no—
86#423—22no—
87#423D414yes—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95#419—24no—
96#419—42no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103D414yes—
104C2×#103Q814yes—
105C2×#101—24no—
106C2×#101—24no—
107C2×#101—42no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8D412yes—
111C2×C4×C8D412yes—
112#384D411yescharacteristic
113#384D411yescharacteristic
114#382D412yes—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C2×#95—21no—
124#343—24no—
125#343—24no—
126#343—42no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—22no—
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—21no—
151C23×C4—21no—
152C23×C4#4111yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16C2×D414yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8C2×Q812yes—
188C22×C8#4112yes—
189C22×C8#4112yes—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
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—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×Q812yes—
229C2×C4⋊C4C2×Q812yes—
230C2×C22⋊C4—22no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—41no—
234C2×C42—41no—
235C2×C42—42no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—82no—
239C2×C42C2×D412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#108—48no—
248#108—48no—
249#107—44no—
250#106—44no—
251C4⋊C8—24no—
252C4⋊C8—24no—
253C4⋊C8—48no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258#103—28no—
259#103—28no—
260#103—44no—
261#103—44no—
262#101—48no—
263C4×C8—24no—
264C4×C8—24no—
265#95—22no—
266#95—22no—
267#95—41no—
268#95—41no—
269C24C2×#4111yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
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—41no—
286C2×Q8—41no—
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×Q8C4⋊Q814yes—
298C22×C4—22no—
299C22×C4—22no—
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—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—81no—
326C22×C4#12111yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4C22⋊Q812yes—
331C22×C4C22⋊Q812yes—
332C22×C4#12612yes—
333C22×C4#12612yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C22×Q811yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+411yescharacteristic
348M16—24no—
349M16—24no—
350M16—42no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16—42no—
356M16—42no—
357M16—42no—
358M16—42no—
359M16—42no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8#12114yes—
382C2×C8C22⋊Q814yes—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C42—42no—
408C42—42no—
409C42—44no—
410C42—44no—
411C42—82no—
412C42—82no—
413C42#12112yes—
414C42#12112yes—
415C42#12112yes—
416C42#12112yes—
417C42C22⋊Q812yes—
418C42C22⋊Q812yes—
419C42C22⋊Q812yes—
420C42C22⋊Q812yes—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23C2×#12111yescharacteristic
426C23C2×C22⋊Q811yescharacteristic
427C23C2×#12611yescharacteristic
428C23#54711yescharacteristic
429C23#55411yescharacteristic
430C23#55511yescharacteristic
431C23#55611yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8—42no—
437Q8—42no—
438Q8—42no—
439Q8—42no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—42no—
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—82no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#39612yes—
479C2×C4#39612yes—
480C2×C4#39612yes—
481C2×C4#39612yes—
482C2×C4#47112yes—
483C2×C4#47312yes—
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×C22⋊Q811yescharacteristic
488C2×C4C2×C22⋊Q811yescharacteristic
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4C2×C4⋊Q811yescharacteristic
491C2×C4#53211yescharacteristic
492C2×C4#53311yescharacteristic
493C2×C4#53311yescharacteristic
494C2×C4#53711yescharacteristic
495C2×C4#53711yescharacteristic
496C2×C4#54011yescharacteristic
497C2×C4#54111yescharacteristic
498C2×C4#54411yescharacteristic
499C2×C4#54411yescharacteristic
500C2×C4D4211yescharacteristic
501C2×C4D4211yescharacteristic
502C2×C4#54611yescharacteristic
503C2×C4#54711yescharacteristic
504C2×C4#54711yescharacteristic
505C2×C4#54811yescharacteristic
506C2×C4D4×Q811yescharacteristic
507C2×C4D4×Q811yescharacteristic
508C2×C4#55011yescharacteristic
509C2×C4#55211yescharacteristic
510C2×C4#55311yescharacteristic
511C2×C4#55411yescharacteristic
512C8—24no—
513C8—24no—
514C8—44no—
515C8—82no—
516C8—82no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22C2×#39611yescharacteristic
530C22#243911yescharacteristic
531C22#244011yescharacteristic
532C22#249611yescharacteristic
533C22#249711yescharacteristic
534C22#253711yescharacteristic
535C22#253811yescharacteristic
536C22C2×#47111yescharacteristic
537C22C2×#47311yescharacteristic
538C22#306711yescharacteristic
539C22#306711yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—82no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4#206414yescomplemented
550C4#241611yescharacteristic
551C4#241711yescharacteristic
552C4#242811yescharacteristic
553C4#242911yescharacteristic
554C4#244011yescharacteristic
555C4#244111yescharacteristic
556C4#262911yescharacteristic
557C4#285711yescharacteristic
558C2—21no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2#1513111yescharacteristic
563C2C2×#206411yescharacteristic
564C2#2360111yescharacteristic
565C2#2823911yescharacteristic
566C2#2824111yescharacteristic
567C2#2829611yescharacteristic
568C2#2829711yescharacteristic
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
549. C4#2064semidirect

Maximal subgroups

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

subgroupindex
C2×#20642
#211702
#211432
C2×#18482
#206472
C2×#18012
C2×C4×#1032
#112492
#112422
#111282
#108472