Finite Group Databasebeta

← #374332

Subgroups — #374332

Order 512.

1934 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1960C211yesmaximal, characteristic, complemented
4C2×#1956C211yesmaximal, characteristic, complemented
5C2×#1873C212yesmaximal
6C2×#1872C212yesmaximal
7C2×#1750C211yesmaximal, characteristic
8#11052C212yesmaximal, complemented
9#11050C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2621—24no—
25#2621C418yescomplemented
26C2×#398—22no—
27C2×#397—22no—
28#2317—24no—
29#2317—24no—
30C2×#391C2212yes—
31C2×#389—22no—
32C2×#385—22no—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38#1960—24no—
39#1956—24no—
40#1873—28no—
41#1872—28no—
42#1750—24no—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#336C2212yes—
48C23×Q8C2312yes—
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yescomplemented
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8C2×C418yes—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—24no—
72C2×#117—42no—
73C2×#117C2×C414yes—
74C2×#117D414yes—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—42no—
84C22×C4⋊C4D412yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D412yes—
93C22×C42—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—42no—
105C2×#103—42no—
106C2×#103—42no—
107C2×#103—42no—
108C2×C22⋊C8—24no—
109C2×C22⋊C8—24no—
110C2×C22⋊C8D418yes—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117#398—216no—
118#397—216no—
119#391—24no—
120#391C2×C418yescomplemented
121#389—216no—
122#385—216no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384D414yes—
127#382—22no—
128#382—22no—
129#382—22no—
130#382—22no—
131#382C2×C414yescomplemented
132#382D414yes—
133C2×#95—22no—
134C2×#95—24no—
135C2×#95—22no—
136C2×#95—22no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141#336—24no—
142C25C2×D411yescharacteristic
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8C22×C414yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—21no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—84no—
173C23×C4C22×C412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D414yes—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C2×D414yes—
194C22×C8C2×D414yes—
195C4⋊Q8—28no—
196C4⋊Q8—28no—
197C4⋊Q8—44no—
198C4⋊Q8—44no—
199C22⋊Q8—216no—
200C22⋊Q8—216no—
201C22⋊Q8—48no—
202C22⋊Q8—48no—
203#117—28no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—48no—
208#117—48no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—84no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4C22⋊C418yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4C22⋊C418yes—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—42no—
264C2×C42—42no—
265C2×C42C22×C412yes—
266C2×C42C22×C412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273#103—48no—
274#103—48no—
275#103—48no—
276#103—48no—
277C22⋊C8—216no—
278#97—44no—
279C4×C8—44no—
280#95—28no—
281#95—28no—
282#95—216no—
283#95—28no—
284C24—21no—
285C24—21no—
286C24—21no—
287C24—21no—
288C24—24no—
289C24—24no—
290C24C2×C22⋊C412yes—
291C24#12412yes—
292C24#12712yes—
293C24C22×D411yescharacteristic
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—24no—
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—42no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—48no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C414yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C4×D418yes—
382C22×C4#12014yes—
383C22×C4#12118yes—
384C22×C4#12314yes—
385C22×C4#12412yes—
386C22×C4#12712yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—216no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—416no—
426C4⋊C4—416no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—216no—
430C22⋊C4—48no—
431C22⋊C4—416no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—48no—
437C42C4×D414yes—
438C42C4×D414yes—
439C42#12114yes—
440C42#12114yes—
441C23—22no—
442C23—28no—
443C23—28no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23#39014yes—
458C23C22×C22⋊C411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×#12311yescharacteristic
461C23C2×#12411yescharacteristic
462C23C2×#12711yescharacteristic
463C23#53511yescharacteristic
464C23#53611yescharacteristic
465C23#57714yes—
466C23#57814yes—
467Q8—44no—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—82no—
472Q8—82no—
473Q8—82no—
474Q8—82no—
475C2×C4—22no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#38618yes—
524C2×C4#39014yes—
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×C4×D412yes—
528C2×C4#52012yes—
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12112yes—
532C2×C4#53712yes—
533C2×C4D4212yes—
534C2×C4D4212yes—
535C2×C4#54712yes—
536C2×C4#54712yes—
537C2×C4#54812yes—
538C2×C4#54812yes—
539C2×C4#55312yes—
540C2×C4#55312yes—
541C8—44no—
542C8—44no—
543C22—24no—
544C22—28no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22C2×#38611yescharacteristic
553C22C2×#39011yescharacteristic
554C22#231111yescharacteristic
555C22#243311yescharacteristic
556C22#243611yescharacteristic
557C22#250111yescharacteristic
558C22#250411yescharacteristic
559C22#289814yes—
560C22#300912yes—
561C22#301112yes—
562C22#309714yes—
563C22#313714yes—
564C22#315412yes—
565C22#315612yes—
566C22#322212yes—
567C22#322312yes—
568C22#324214yes—
569C4—24no—
570C4—28no—
571C4—44no—
572C4—84no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—84no—
578C4#236112yes—
579C4#236112yes—
580C4#241612yes—
581C4#241612yes—
582C2—44no—
583C2#1413711yescharacteristic
584C2#2091312yes—
585C2#2120512yes—
586C2#2187312yes—
587C2#2573211yescharacteristic
588C2#2573511yescharacteristic
589C2#2591412yes—
590C2#2792212yes—
591C2#2798512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1960C2semidirect
4. C2×#1956C2semidirect
8. #11052C2semidirect
9. #11050C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
25. #2621C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
120. #391C2×C4semidirect
131. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
C2×#23172
C2×#19602
C2×#19562
C2×#18732
C2×#18722
C2×#17502
#110522
#110502
#108432
#108422