Finite Group Databasebeta

← #440754

Subgroups — #440754

Order 512.

1324 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×#2446C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#14161C211yesmaximal, characteristic
5#11346C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11190C212yesmaximal
8#11010C212yesmaximal
9#10932C211yesmaximal, characteristic
10#10932C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19#2446—24no—
20#2446—24no—
21C2×#400—21no—
22C2×#398—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#387—21no—
28C2×#387—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2024—24no—
32#2024—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446—21no—
42#1446—21no—
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92#400—28no—
93#398—28no—
94#391—22no—
95#391—22no—
96#391—22no—
97#391—22no—
98#391D414yes—
99#391D414yes—
100#389—22no—
101#389—22no—
102#387—28no—
103#387—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—42no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—22no—
126#336—22no—
127#336—42no—
128#336—42no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—21no—
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—22no—
142C22×Q8—22no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8C2×D412yes—
148C22×Q8C2×D412yes—
149C22×Q8#4114yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
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—22no—
175C22×C8—22no—
176C22×C8—22no—
177C4⋊Q8—48no—
178C4⋊Q8—48no—
179C4⋊Q8—48no—
180C4⋊Q8—44no—
181C4⋊Q8—44no—
182C4×Q8—48no—
183C4×Q8—44no—
184C4×Q8—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4Q1614yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4114yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240#103—48no—
241#103—48no—
242#103—48no—
243#103—48no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—22no—
252#95—22no—
253#95—42no—
254#95—82no—
255#95—82no—
256#95—82no—
257#95—82no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—48no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
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—84no—
292C2×Q8—84no—
293C2×Q8—88no—
294C2×Q8—88no—
295C2×Q8#12414yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4#12012yes—
356C22×C4#12012yes—
357C22×C4C2×Q1612yes—
358C22×C4C2×Q1612yes—
359C22×C4C2×Q1612yes—
360C22×C4#13712yes—
361C22×C4#13712yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+412yes—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
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—42no—
382C2×C8—42no—
383C2×C8—42no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C42—24no—
406C42—24no—
407C42—48no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—84no—
415C23C2×#12011yescharacteristic
416C23#52511yescharacteristic
417C23#53511yescharacteristic
418C23#54612yes—
419C23#54812yes—
420C23C22×Q1611yescharacteristic
421C23C2×#13711yescharacteristic
422C23C2×#13711yescharacteristic
423C23#57511yescharacteristic
424C23#57712yes—
425C23#57712yes—
426Q8—28no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—816no—
433Q8—82no—
434Q8—82no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
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—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
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—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4#45114yes—
501C2×C4C2×#12411yescharacteristic
502C2×C4#52912yes—
503C2×C4#53211yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53812yes—
507C2×C4#53912yes—
508C2×C4#54212yes—
509C2×C4#54612yes—
510C2×C4#54612yes—
511C2×C4#54612yes—
512C2×C4#54712yes—
513C2×C4#55112yes—
514C2×C4#55212yes—
515C2×C4#55912yes—
516C2×C4#56212yes—
517C8—82no—
518C8—82no—
519C22#240411yescharacteristic
520C22#243211yescharacteristic
521C22#243511yescharacteristic
522C22#245812yes—
523C22#269612yes—
524C22C2×#45111yescharacteristic
525C22#300711yescharacteristic
526C22#301011yescharacteristic
527C22#301011yescharacteristic
528C22#310411yescharacteristic
529C22#311311yescharacteristic
530C22#312312yes—
531C22#320911yescharacteristic
532C22#321511yescharacteristic
533C22#322212yes—
534C22#329912yes—
535C22#331612yes—
536C22#332112yes—
537C22#332912yes—
538C22#336312yes—
539C22#336912yes—
540C22#337212yes—
541C22#337612yes—
542C4—24no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—42no—
548C4—81no—
549C4—82no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—84no—
554C4#248812yes—
555C4#255712yes—
556C4#268712yes—
557C4#271012yes—
558C2#1566811yescharacteristic
559C2#2393411yescharacteristic
560C2#2394211yescharacteristic
561C2#2565111yescharacteristic
562C2#2566211yescharacteristic
563C2#2588811yescharacteristic
564C2#2589511yescharacteristic
565C2#2669412yes—
566C2#2671112yes—
567C2#3164012yes—
568C2#3166312yes—
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×#24462
C2×#20242
#141612
#113462
#112932
#111902
#110102
#109322
#107892