Finite Group Databasebeta

← #414869

Subgroups — #414869

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#21322C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic
4C2×#1950C211yesmaximal, characteristic
5C2×#1803C211yesmaximal, characteristic
6C2×C4×#101C211yesmaximal, characteristic
7#11290C212yesmaximal
8#11249C212yesmaximal
9#11164C212yesmaximal
10#11128C212yesmaximal
11#10857C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14#3078—24no—
15#3078—24no—
16C2×#479—22no—
17C2×#477—21no—
18C2×#475—21no—
19C22×#108C2212yes—
20C2×#428C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C2×#419C2212yes—
24C22×#103C2212yes—
25C22×#101C2211yescharacteristic
26C22×#101C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×#384C2211yescharacteristic
30#1992—22no—
31#1990—21no—
32#1990—21no—
33#1985—22no—
34#1984—22no—
35#1950—24no—
36#1803—22no—
37#1803—22no—
38C4×#101—24no—
39C2×#343C2211yescharacteristic
40C2×#343C2212yes—
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#341C2212yes—
44#1736C2212yes—
45#1736C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48#1729C2212yes—
49C23×Q8C2311yescharacteristic
50C22×M16C2312yes—
51C22×M16C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2312yes—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8D411yescharacteristic
62C2×#125D412yes—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×C4×Q8—22no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—22no—
74C2×C4×Q8D414yes—
75C2×#117C2312yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83#479—216no—
84#477—28no—
85#475—28no—
86C2×#108—24no—
87C2×#108—24no—
88C2×#108—42no—
89#428—24no—
90#428—24no—
91C2×#107—22no—
92C2×#106—22no—
93#423—22no—
94#423D414yes—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8D412yes—
100C2×C4⋊C8D412yes—
101#419—24no—
102#419—24no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#101—22no—
109C2×#101—22no—
110C2×#101—22no—
111C2×#101—22no—
112C2×#101—22no—
113C2×#101Q814yescomplemented
114C4×M16—22no—
115C4×M16—22no—
116C4×M16—22no—
117C4×M16—22no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122#384—21no—
123#384—21no—
124#384—21no—
125#384—21no—
126#343—24no—
127#343—22no—
128#343—22no—
129#343—42no—
130#341—24no—
131#340—24no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336—24no—
137#336—24no—
138C22×Q8—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×Q8—21no—
151C22×Q8C2×D412yes—
152C22×Q8C2×Q812yes—
153C22×Q8#4112yes—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4#4111yescharacteristic
158C23×C4#4111yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—22no—
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—21no—
171C2×M16—22no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16C2×D414yes—
186C2×M16C2×Q812yes—
187C2×M16C2×Q812yes—
188C2×M16#4112yes—
189C2×M16#4112yes—
190C22×C8—22no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D414yes—
202C4⋊Q8—22no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206#125—24no—
207#125—24no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210C22⋊Q8—42no—
211C22⋊Q8—42no—
212C4×Q8—24no—
213C4×Q8—24no—
214C4×Q8—24no—
215C4×Q8—24no—
216C4×Q8—42no—
217C4×Q8—42no—
218C4×Q8—42no—
219C4×Q8—42no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×Q8—42no—
223C4×Q8—42no—
224#117—22no—
225#117—22no—
226#117—22no—
227#117—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—82no—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C4⋊C4C2×D412yes—
256C2×C22⋊C4—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—42no—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4112yes—
269C2×C42#4112yes—
270#108—48no—
271#107—28no—
272#106—28no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277#103—28no—
278#103—28no—
279#103—48no—
280#101—28no—
281#101—28no—
282#97—42no—
283#97—42no—
284C4×C8—42no—
285C4×C8—42no—
286C24C2×#4111yescharacteristic
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—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—81no—
319C2×Q8—81no—
320C2×Q8C22⋊Q814yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4#12111yescharacteristic
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4#12412yes—
351C22×C4#12412yes—
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+411yescharacteristic
364C22×C42−1+411yescharacteristic
365C22×C42−1+411yescharacteristic
366M16—22no—
367M16—22no—
368M16—22no—
369M16—22no—
370M16—22no—
371M16—24no—
372M16—24no—
373M16—22no—
374M16—22no—
375M16—22no—
376M16—42no—
377M16—41no—
378M16—41no—
379M16—42no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—42no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—82no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C42—42no—
433C42—42no—
434C42—44no—
435C42—44no—
436C42#12012yes—
437C42#12012yes—
438C42#12112yes—
439C42#12112yes—
440C42#12112yes—
441C42#12112yes—
442C42#12312yes—
443C42#12312yes—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23C2×#12111yescharacteristic
449C23C2×#12311yescharacteristic
450C23C2×#12411yescharacteristic
451C23#52911yescharacteristic
452C23#53311yescharacteristic
453C23#54111yescharacteristic
454C23#54211yescharacteristic
455Q8—24no—
456Q8—24no—
457Q8—42no—
458Q8—42no—
459Q8—42no—
460Q8—42no—
461Q8—42no—
462Q8—42no—
463Q8—84no—
464Q8—84no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
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—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—162no—
497C2×C4—162no—
498C2×C4#39412yes—
499C2×C4#39412yes—
500C2×C4#39412yes—
501C2×C4#39412yes—
502C2×C4#47112yes—
503C2×C4#47312yes—
504C2×C4C2×#12011yescharacteristic
505C2×C4C2×#12111yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×C22⋊Q811yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#52911yescharacteristic
511C2×C4#53311yescharacteristic
512C2×C4#53611yescharacteristic
513C2×C4#53711yescharacteristic
514C2×C4#54111yescharacteristic
515C2×C4#54211yescharacteristic
516C2×C4#54611yescharacteristic
517C2×C4#54611yescharacteristic
518C2×C4#54711yescharacteristic
519C2×C4#54711yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4#54811yescharacteristic
522C2×C4#54811yescharacteristic
523C2×C4#54811yescharacteristic
524C2×C4D4×Q811yescharacteristic
525C2×C4D4×Q811yescharacteristic
526C2×C4D4×Q811yescharacteristic
527C2×C4#55011yescharacteristic
528C2×C4#55211yescharacteristic
529C2×C4#55211yescharacteristic
530C2×C4#55411yescharacteristic
531C2×C4#55611yescharacteristic
532C8—44no—
533C8—42no—
534C8—42no—
535C8—42no—
536C8—42no—
537C8—44no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22C2×#39411yescharacteristic
551C22#242611yescharacteristic
552C22#242911yescharacteristic
553C22#248911yescharacteristic
554C22#249211yescharacteristic
555C22#262511yescharacteristic
556C22#285511yescharacteristic
557C22C2×#47111yescharacteristic
558C22C2×#47311yescharacteristic
559C22#306811yescharacteristic
560C22#306911yescharacteristic
561C4—22no—
562C4—22no—
563C4—22no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—162no—
569C4#205214yes—
570C4#240711yescharacteristic
571C4#240911yescharacteristic
572C4#241511yescharacteristic
573C4#241711yescharacteristic
574C4#242111yescharacteristic
575C4#242311yescharacteristic
576C4#242711yescharacteristic
577C4#242911yescharacteristic
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2#1517911yescharacteristic
583C2C2×#205211yescharacteristic
584C2#2353311yescharacteristic
585C2#2531411yescharacteristic
586C2#2531511yescharacteristic
587C2#2549111yescharacteristic
588C2#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
113. C2×#101Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#213222
#213022
C2×#19502
C2×#18032
C2×C4×#1012
#112902
#112492
#111642
#111282
#108572