Finite Group Databasebeta

← #470305

Subgroups — #470305

Order 512.

1306 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3C2×#2043C211yesmaximal, characteristic
4#14699C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11122C212yesmaximal
8#11122C212yesmaximal
9#11067C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#106C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2862—24no—
20#2862—24no—
21C2×#391—21no—
22C2×#391—21no—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#382—21no—
29C2×#382—21no—
30C2×C43C2211yescharacteristic
31#2058—24no—
32#2043—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46#1434—21no—
47C23×Q8—21no—
48C23×Q8C2312yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—42no—
78C2×#106—42no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91#391—28no—
92#391—28no—
93#391—24no—
94#391—24no—
95#391—24no—
96#391—24no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#387—44no—
102#382—28no—
103#382—28no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C43—22no—
119C43—22no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—22no—
125#336—22no—
126#336—42no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8C2×D414yes—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8#4114yes—
175C4⋊Q8—48no—
176C4⋊Q8—48no—
177C4⋊Q8—48no—
178C4⋊Q8—48no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4#4112yes—
217C2×C4⋊C4#4112yes—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42SD1614yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#106—48no—
245#103—48no—
246#103—48no—
247#103—48no—
248#103—48no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263C24C22×D411yescharacteristic
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—48no—
271C2×Q8—48no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—48no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
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—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—84no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4#12012yes—
374C22×C4#12012yes—
375C22×C4C2×SD1612yes—
376C22×C4C2×SD1612yes—
377C22×C4C2×SD1612yes—
378C22×C4#13512yes—
379C22×C4#13712yes—
380C22×C4#13712yes—
381C22×C4#13712yes—
382C22×C4#13712yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C42−1+412yes—
392C22×C42−1+411yescharacteristic
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—84no—
426C42—28no—
427C42—24no—
428C42—24no—
429C42—28no—
430C42—44no—
431C42—41no—
432C42—41no—
433C42—42no—
434C42—41no—
435C42—42no—
436C42—41no—
437C42—44no—
438C23C2×#12011yescharacteristic
439C23#52511yescharacteristic
440C23#53211yescharacteristic
441C23#53511yescharacteristic
442C23#53511yescharacteristic
443C23#53611yescharacteristic
444C23#53711yescharacteristic
445C23C22×SD1611yescharacteristic
446C23C2×#13511yescharacteristic
447C23C2×#13711yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450C23C2×#13711yescharacteristic
451C23#57511yescharacteristic
452C23#57511yescharacteristic
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—816no—
458Q8—82no—
459Q8—82no—
460Q8—82no—
461Q8—82no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4#44814yes—
545C2×C4#45214yes—
546C2×C4#53412yes—
547C2×C4#53412yes—
548C2×C4#53812yes—
549C2×C4#54212yes—
550C2×C4#54212yes—
551C2×C4#54212yes—
552C2×C4#54611yescharacteristic
553C2×C4#54712yes—
554C2×C4#54711yescharacteristic
555C2×C4#54812yes—
556C2×C4#54811yescharacteristic
557C2×C4#55011yescharacteristic
558C2×C4#55112yes—
559C2×C4#55512yes—
560C2×C4#56212yes—
561C2×C4#56312yes—
562C8—44no—
563C8—82no—
564C22#240711yescharacteristic
565C22#243211yescharacteristic
566C22#261011yescharacteristic
567C22#261311yescharacteristic
568C22#263211yescharacteristic
569C22#266211yescharacteristic
570C22#282011yescharacteristic
571C22C2×#44811yescharacteristic
572C22C2×#45211yescharacteristic
573C22#300411yescharacteristic
574C22#300511yescharacteristic
575C22#310111yescharacteristic
576C22#310711yescharacteristic
577C22#311311yescharacteristic
578C22#311611yescharacteristic
579C22#316711yescharacteristic
580C22#317211yescharacteristic
581C22#318111yescharacteristic
582C22#318511yescharacteristic
583C22#320611yescharacteristic
584C22#320611yescharacteristic
585C22#321111yescharacteristic
586C22#321111yescharacteristic
587C22#321511yescharacteristic
588C22#321511yescharacteristic
589C22#321811yescharacteristic
590C22#321811yescharacteristic
591C22#322911yescharacteristic
592C22#323111yescharacteristic
593C22#323611yescharacteristic
594C22#323711yescharacteristic
595C22#324611yescharacteristic
596C22#324811yescharacteristic
597C22#325211yescharacteristic
598C22#325511yescharacteristic
599C4—24no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4—161no—
610C4—161no—
611C4#268612yes—
612C4#271712yes—
613C4#273012yes—
614C4#273112yes—
615C2#1668111yescharacteristic
616C2#2415111yescharacteristic
617C2#2415711yescharacteristic
618C2#2567211yescharacteristic
619C2#2567711yescharacteristic
620C2#2955411yescharacteristic
621C2#2955911yescharacteristic
622C2#2962411yescharacteristic
623C2#2962611yescharacteristic
624C2#3006011yescharacteristic
625C2#3006711yescharacteristic
626C2#3033911yescharacteristic
627C2#3034411yescharacteristic
628C2#3282011yescharacteristic
629C2#3282411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#28622
C2×#20582
C2×#20432
#146992
#113842
#111902
#111222
#110672
#110102
#107952