Finite Group Databasebeta

← #464152

Subgroups — #464152

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2043C212yesmaximal
3C2×#2043C211yesmaximal, characteristic
4C2×#1863C211yesmaximal, characteristic
5#15194C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11067C212yesmaximal
9#11010C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10789C211yesmaximal, characteristic
12C22×C4⋊Q8C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23#2546—24no—
24#2546—24no—
25C2×#398—22no—
26C2×#391C2212yes—
27C2×#389—21no—
28C2×#389—21no—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#2043—28no—
33#2043—24no—
34#1863—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1444—21no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2312yes—
47C23×Q8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8D414yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—42no—
63C2×C4×Q8—42no—
64C2×C4×Q8D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—42no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103—42no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103D414yes—
103#398—216no—
104#391—24no—
105#391—24no—
106#389—28no—
107#389—28no—
108#389—24no—
109#389—22no—
110#389—22no—
111#389—42no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—22no—
124#340—42no—
125#336—24no—
126#336—22no—
127#336—22no—
128#336—42no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
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—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8C2×D414yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×D412yes—
152C22×Q8#4112yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4⋊Q8—48no—
185C4⋊Q8—48no—
186C4⋊Q8—48no—
187C4⋊Q8—48no—
188C4×Q8—44no—
189C4×Q8—44no—
190C4×Q8—48no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4Q1614yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4#4114yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245#107—44no—
246#103—28no—
247#103—48no—
248#103—48no—
249#103—48no—
250#103—48no—
251#103—44no—
252#103—48no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—81no—
260#95—82no—
261#95—82no—
262C24C22×D411yescharacteristic
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
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—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C2×Q8#12114yes—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4C2×Q1614yes—
364C22×C4C2×Q1612yes—
365C22×C4#13714yes—
366C22×C4#13712yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+412yes—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—28no—
395C4⋊C4—28no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—416no—
406C4⋊C4—416no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—88no—
416C4⋊C4—84no—
417C42—22no—
418C42—22no—
419C42—24no—
420C42—48no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C23C2×#12011yescharacteristic
426C23#53512yes—
427C23D4211yescharacteristic
428C23#54611yescharacteristic
429C23#54812yes—
430C23C22×Q1611yescharacteristic
431C23C2×#13711yescharacteristic
432C23C2×#13712yes—
433C23#57712yes—
434C23#57711yescharacteristic
435C23#57811yescharacteristic
436Q8—24no—
437Q8—24no—
438Q8—42no—
439Q8—42no—
440Q8—44no—
441Q8—44no—
442Q8—44no—
443Q8—44no—
444Q8—44no—
445Q8—44no—
446Q8—82no—
447Q8—82no—
448Q8—82no—
449Q8—82no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4#45114yes—
508C2×C4C2×#12111yescharacteristic
509C2×C4#53211yescharacteristic
510C2×C4#53411yescharacteristic
511C2×C4#53411yescharacteristic
512C2×C4#53412yes—
513C2×C4#53511yescharacteristic
514C2×C4#53611yescharacteristic
515C2×C4#53712yes—
516C2×C4#53812yes—
517C2×C4#53912yes—
518C2×C4#54012yes—
519C2×C4D4211yescharacteristic
520C2×C4#54612yes—
521C2×C4#54611yescharacteristic
522C2×C4#54712yes—
523C2×C4#55312yes—
524C2×C4#56212yes—
525C2×C4#56212yes—
526C8—42no—
527C8—81no—
528C8—82no—
529C22#240811yescharacteristic
530C22#243211yescharacteristic
531C22#261311yescharacteristic
532C22#267612yes—
533C22#271112yes—
534C22C2×#45111yescharacteristic
535C22#300411yescharacteristic
536C22#301011yescharacteristic
537C22#301311yescharacteristic
538C22#320912yes—
539C22#321812yes—
540C22#322112yes—
541C22#322512yes—
542C22#326612yes—
543C22#327811yescharacteristic
544C22D4×Q1611yescharacteristic
545C22#329911yescharacteristic
546C22#332012yes—
547C22#332811yescharacteristic
548C22#336312yes—
549C22#337212yes—
550C22#337212yes—
551C22#337412yes—
552C4—24no—
553C4—44no—
554C4—44no—
555C4—44no—
556C4—42no—
557C4—84no—
558C4—84no—
559C4—82no—
560C4—82no—
561C4—161no—
562C4#254311yescharacteristic
563C4#261811yescharacteristic
564C4#266111yescharacteristic
565C4#267112yes—
566C4#270812yes—
567C4#281611yescharacteristic
568C2#1650111yescharacteristic
569C2#2424311yescharacteristic
570C2#2425111yescharacteristic
571C2#2565111yescharacteristic
572C2#2565711yescharacteristic
573C2#2962711yescharacteristic
574C2#2963111yescharacteristic
575C2#3088012yes—
576C2#3089412yes—
577C2#3208412yes—
578C2#3209812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#25462
C2×#20432
C2×#18632
#151942
#113462
#111242
#110672
#110102
#107892