Finite Group Databasebeta

← #366808

Subgroups — #366808

Order 512.

1448 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C2×#1901C211yesmaximal, characteristic
3#20972C211yesmaximal, characteristic
4C2×#1855C211yesmaximal, characteristic
5#20700C211yesmaximal, characteristic
6#20389C211yesmaximal, characteristic
7C22×#343C211yesmaximal, characteristic
8#11365C212yesmaximal, complemented
9#11128C212yesmaximal, complemented
10#10942C212yesmaximal, complemented
11#10870C212yesmaximal, complemented
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#484—24no—
17C2×#484—21no—
18C2×#484—21no—
19C2×#484—22no—
20C2×#484C418yescomplemented
21C22×#108C2212yes—
22C22×#107C2212yes—
23C2×#419C2212yes—
24C22×#103C2212yes—
25C22×#101C2211yescharacteristic
26C22×#101C2211yescharacteristic
27C2×#407C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C2×#405C2212yes—
30#2280C2211yescharacteristic
31#1901—24no—
32#1894—22no—
33#1892—22no—
34#1855—24no—
35#1824—22no—
36#1823—22no—
37#1753—22no—
38#1752—21no—
39C8×C22⋊C4—21no—
40C2×#343—24no—
41C2×#343C2211yescharacteristic
42C2×#343C2212yes—
43C2×#343C2211yescharacteristic
44C2×#343C2212yes—
45C2×#341C2212yes—
46#1736C2212yes—
47#1736C2212yes—
48#1736C2212yes—
49#1729C2212yes—
50C23×Q8C2311yescharacteristic
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—21no—
54C22×M16—22no—
55C22×M16—21no—
56C22×M16—22no—
57C22×M16—24no—
58C22×M16—24no—
59C22×M16—22no—
60C22×M16—42no—
61C22×M16—42no—
62C22×M16D414yes—
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2312yes—
67C22×M16C2311yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C23×C8C2312yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8C2×C414yes—
79C2×#117C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C414yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93#484—28no—
94#484—28no—
95#484—216no—
96C2×#108—24no—
97C2×#108—24no—
98C2×#108—42no—
99C2×#107—22no—
100C2×#107—22no—
101C2×#107—24no—
102C2×#107C2×C418yes—
103#419—24no—
104#419—42no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103C2×C418yes—
110C2×#101—22no—
111C2×#101—22no—
112C2×#101—24no—
113C2×#101—42no—
114#407—22no—
115#407—22no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8C2×C414yes—
119C2×C22⋊C8D414yes—
120C2×C22⋊C8D411yescharacteristic
121C2×C22⋊C8D411yescharacteristic
122C2×C22⋊C8D412yes—
123#405—24no—
124#405—42no—
125C2×#97—22no—
126C2×C4×C8—22no—
127#381—21no—
128#380D412yes—
129#380D412yes—
130#379—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#343—22no—
134#343—22no—
135#343—24no—
136#343—24no—
137#341—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—22no—
142#336—22no—
143C25#4111yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8C22×C412yes—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—42no—
165C23×C4—84no—
166C23×C4C22×C412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4111yescharacteristic
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×M16—21no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—28no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—28no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—44no—
198C2×M16—44no—
199C2×M16—44no—
200C2×M16—48no—
201C2×M16—48no—
202C2×M16—44no—
203C2×M16—42no—
204C2×M16C2×D412yes—
205C2×M16C2×D412yes—
206C2×M16#4114yes—
207C2×M16#4114yes—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—24no—
214C22×C8—21no—
215C22×C8—22no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—44no—
219C22×C8—44no—
220C22×C8—42no—
221C22×C8C22×C414yes—
222C22×C8C2×D412yes—
223C22×C8C2×D412yes—
224C22×C8C2×D414yes—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—24no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C42—22no—
274#108—48no—
275#107—28no—
276#107—28no—
277#103—28no—
278#103—28no—
279#103—48no—
280#101—44no—
281#101—44no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8C22⋊C414yes—
285C22⋊C8C22⋊C414yes—
286C22⋊C8C22⋊C418yes—
287#97—44no—
288C4×C8—44no—
289#95—22no—
290#95—22no—
291C24—22no—
292C24—22no—
293C24—24no—
294C24—42no—
295C24C4×D412yes—
296C24#12111yescharacteristic
297C24#12111yescharacteristic
298C24#12412yes—
299C24C2×#4111yescharacteristic
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
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—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
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—44no—
351C22×C4—44no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—88no—
355C22×C4—82no—
356C22×C4—84no—
357C22×C4—88no—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C4×D412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12412yes—
368C22×C4#12712yes—
369C22×C4#12712yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380M16—22no—
381M16—24no—
382M16—22no—
383M16—24no—
384M16—22no—
385M16—24no—
386M16—48no—
387M16—48no—
388M16—42no—
389M16—42no—
390M16—42no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—42no—
396M16—42no—
397M16#12314yes—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—24no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—48no—
411C2×C8—48no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8C2×C22⋊C414yes—
420C2×C8C2×C22⋊C414yes—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—82no—
441C42—42no—
442C42—42no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—42no—
457C23—48no—
458C23—48no—
459C23#39012yes—
460C23#39012yes—
461C23#47314yes—
462C23C2×C4×D411yescharacteristic
463C23C2×#12111yescharacteristic
464C23C2×#12411yescharacteristic
465C23#52912yes—
466C23#53912yes—
467Q8—44no—
468Q8—44no—
469Q8—81no—
470Q8—81no—
471Q8—81no—
472Q8—81no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—88no—
500C2×C4—84no—
501C2×C4—88no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—84no—
508C2×C4#39012yes—
509C2×C4#39012yes—
510C2×C4C22×C22⋊C411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C4×#4112yes—
513C2×C4#52012yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12311yescharacteristic
516C2×C4#52511yescharacteristic
517C2×C4C2×#12711yescharacteristic
518C2×C4#53411yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#54612yes—
521C2×C4#54612yes—
522C2×C4#54612yes—
523C2×C4#54712yes—
524C2×C4#54712yes—
525C2×C4#54712yes—
526C2×C4#55112yes—
527C2×C4#55212yes—
528C8—24no—
529C8—24no—
530C8—24no—
531C8—42no—
532C8—44no—
533C8—81no—
534C8—81no—
535C22—21no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—28no—
542C22—21no—
543C22—21no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—44no—
548C22—48no—
549C22—44no—
550C22—44no—
551C22#197914yes—
552C22C2×#39011yescharacteristic
553C22C4×#12311yescharacteristic
554C22#232211yescharacteristic
555C22#242512yes—
556C22#248812yes—
557C22C2×#47312yes—
558C22#306812yes—
559C4—22no—
560C4—22no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4—84no—
565C4#233912yes—
566C4#235912yes—
567C4#241211yescharacteristic
568C4#241311yescharacteristic
569C4#250111yescharacteristic
570C4#253911yescharacteristic
571C2—21no—
572C2—22no—
573C2—24no—
574C2—21no—
575C2—42no—
576C2#1399611yescharacteristic
577C2C2×#197911yescharacteristic
578C2#2124911yescharacteristic
579C2#2526614yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11365C2semidirect
9. #11128C2semidirect
10. #10942C2semidirect
11. #10870C2semidirect
20. C2×#484C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C2×#19012
#209722
C2×#18552
#207002
#203892
C22×#3432
#113652
#111282
#109422
#108702