Finite Group Databasebeta

← #441069

Subgroups — #441069

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic
3C2×#2043C212yesmaximal
4C2×#2026C211yesmaximal, characteristic
5#15218C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11010C212yesmaximal
8#11010C211yesmaximal, characteristic
9#10928C211yesmaximal, characteristic
10#10875C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#491—28no—
18C2×#491—24no—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×#103C2212yes—
25C22×#97C2211yescharacteristic
26#2546—24no—
27#2546—24no—
28C2×#398—22no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#389—22no—
32C2×#389C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34#2043—28no—
35#2026—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1444—22no—
42#1441—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C2×#125—24no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—42no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—24no—
69C2×C4×Q8—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4D412yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#491—232no—
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—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×#103—42no—
103C2×#103D418yes—
104C2×#97—24no—
105C2×#97—22no—
106#398—216no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391D414yes—
111#389—216no—
112#389—24no—
113#389—42no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—22no—
127#336—24no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
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—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8C2×D412yes—
149C22×Q8C2×D412yes—
150C22×Q8C2×D412yes—
151C22×Q8C2×D414yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D412yes—
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×C8C2×D414yes—
177C4⋊Q8—28no—
178C4⋊Q8—44no—
179C4⋊Q8—48no—
180C4⋊Q8—48no—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183C4⋊Q8—44no—
184C4⋊Q8—44no—
185#125—28no—
186#125—48no—
187#125—44no—
188C4×Q8—48no—
189C4×Q8—48no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
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—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4#4114yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42#4114yes—
245#103—216no—
246#103—216no—
247#103—48no—
248#103—48no—
249#103—48no—
250#103—48no—
251#97—28no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—44no—
256#95—44no—
257#95—82no—
258#95—82no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
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#12014yes—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
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—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
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—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4#12014yes—
362C22×C4#12412yes—
363C22×C4#12412yes—
364C22×C4#13714yes—
365C22×C4#13714yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C42−1+412yes—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—216no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—416no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—416no—
398C4⋊C4—48no—
399C4⋊C4—416no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—88no—
406C42—22no—
407C42—22no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—48no—
412C42—41no—
413C42—41no—
414C42—42no—
415C23C2×#12011yescharacteristic
416C23C2×#12411yescharacteristic
417C23#53511yescharacteristic
418C23D4212yes—
419C23#54812yes—
420C23C2×#13712yes—
421C23C2×#13712yes—
422C23#57712yes—
423C23#57812yes—
424Q8—28no—
425Q8—44no—
426Q8—44no—
427Q8—44no—
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—82no—
433Q8—82no—
434Q8—82no—
435Q8—82no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—44no—
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—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#49114yescomplemented
494C2×C4C2×#12011yescharacteristic
495C2×C4#53212yes—
496C2×C4#53211yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53611yescharacteristic
499C2×C4#53712yes—
500C2×C4#53812yes—
501C2×C4#54212yes—
502C2×C4D4212yes—
503C2×C4#54612yes—
504C2×C4#54612yes—
505C2×C4#54612yes—
506C2×C4#54712yes—
507C2×C4#54712yes—
508C2×C4#55512yes—
509C2×C4#56212yes—
510C8—44no—
511C8—82no—
512C22#240311yescharacteristic
513C22#243211yescharacteristic
514C22#243511yescharacteristic
515C22#249112yes—
516C22#267612yes—
517C22#300412yes—
518C22#301312yes—
519C22C2×#49111yescharacteristic
520C22#313712yes—
521C22#313911yescharacteristic
522C22#321812yes—
523C22#322111yescharacteristic
524C22#322311yescharacteristic
525C22#326612yes—
526C22#327012yes—
527C22#327612yes—
528C22#327812yes—
529C22#337212yes—
530C22#337212yes—
531C22#337412yes—
532C22#337412yes—
533C4—24no—
534C4—42no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—84no—
542C4—84no—
543C4#243012yes—
544C4#255512yes—
545C4#267212yes—
546C4#268612yes—
547C2#1567311yescharacteristic
548C2#2386812yes—
549C2#2565712yes—
550C2#2586111yescharacteristic
551C2#2586811yescharacteristic
552C2#2777712yes—
553C2#2778212yes—
554C2#3089412yes—
555C2#3089612yes—
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
493. C2×C4#491semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C2×#25462
C2×#20432
C2×#20262
#152182
#110672
#110102
#109282
#108752
#107892