Finite Group Databasebeta

← #374586

Subgroups — #374586

Order 512.

1828 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#478C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×C4×#102C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic, complemented
6#11220C212yesmaximal, complemented
7#11219C212yesmaximal, complemented
8#10850C211yesmaximal, characteristic, complemented
9#10834C211yesmaximal, characteristic, complemented
10#10788C212yesmaximal
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C2×#478—24no—
16C2×#478—24no—
17C2×#478C418yescomplemented
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2212yes—
24C22×C4×C8C2211yescharacteristic
25#2349—24no—
26#2349—24no—
27C2×#389C2212yes—
28C2×#385—22no—
29C2×#382C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C22×#95—21no—
34#1991—24no—
35C4×#102—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44#1436—21no—
45#1436—21no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C24×C4—22no—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125C2×C414yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—24no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—42no—
68C2×C4×D4—42no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D414yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42C2312yes—
85C22×C42C2311yescharacteristic
86#478—232no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107C2×C414yescomplemented
90C2×#106—22no—
91C2×#106—22no—
92C2×#106C2×C414yescomplemented
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yescomplemented
96C2×#102—22no—
97C2×#102—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102C2×C414yes—
103C2×#102C2×C414yes—
104C2×#102Q818yes—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108#389—24no—
109#389—24no—
110#385—216no—
111#382—22no—
112#382D414yes—
113#380—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—28no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95D411yescharacteristic
128C2×#95D411yescharacteristic
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336—42no—
135#336—42no—
136C25—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4C22×C412yes—
143C22×D4C2×D412yes—
144C22×D4C2×D412yes—
145C22×D4C2×Q814yes—
146C22×D4#4114yes—
147C23×C4—21no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—28no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4—82no—
163C23×C4—82no—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C2×Q814yes—
186#125—28no—
187#125—28no—
188#125—44no—
189#125—44no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—48no—
194C4×D4—48no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4C22⋊C4116yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C22×C412yes—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—42no—
253C2×C42—82no—
254C2×C42C22×C412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260#107—28no—
261#106—28no—
262C4⋊C8—28no—
263#102—28no—
264#102—28no—
265#102—216no—
266C4×C8—44no—
267#95—28no—
268#95—22no—
269#95—24no—
270#95—24no—
271#95—22no—
272#95—44no—
273#95—42no—
274#95—44no—
275#95—82no—
276#95—82no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—24no—
281C24—24no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C22×D411yescharacteristic
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4C2×C22⋊C414yes—
301C2×D4C4×Q818yes—
302C2×D4C22⋊Q818yes—
303C2×D4#12414yes—
304C2×D4C4⋊Q814yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
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—216no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—416no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—416no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
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—88no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#13514yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q812yes—
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4111yescharacteristic
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—88no—
420C4⋊C4C2×C22⋊C414yes—
421C22⋊C4—216no—
422C22⋊C4—416no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—84no—
428C42C2×C22⋊C414yes—
429C42C4×D414yes—
430C42#12114yes—
431C42#12314yes—
432C23—28no—
433C23—28no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23C2×C4×D411yescharacteristic
454C23C2×#12011yescharacteristic
455C23C2×#12111yescharacteristic
456C23#54612yes—
457C23D4×Q812yes—
458C23C2×#13512yes—
459C23#57512yes—
460C23#57612yes—
461C23#57712yes—
462D4—24no—
463D4—28no—
464D4—28no—
465D4—24no—
466D4—24no—
467D4#39118yes—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—816no—
507C2×C4—832no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#38614yes—
522C2×C4#38614yes—
523C2×C4C22×C22⋊C411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×Q812yes—
526C2×C4#52012yes—
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×C22⋊Q812yes—
529C2×C4C2×#12311yescharacteristic
530C2×C4C2×#12411yescharacteristic
531C2×C4C2×C4⋊Q811yescharacteristic
532C2×C4#53511yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53712yes—
535C2×C4#54612yes—
536C2×C4D4×Q812yes—
537C2×C4#55212yes—
538C2×C4#55212yes—
539C2×C4#55412yes—
540C2×C4#55412yes—
541C8—42no—
542C8—42no—
543C8—44no—
544C22—28no—
545C22—48no—
546C22—48no—
547C22—44no—
548C22—48no—
549C22—44no—
550C22—48no—
551C22—48no—
552C22—48no—
553C22C2×#38611yescharacteristic
554C22#236012yes—
555C22#241712yes—
556C22#243411yescharacteristic
557C22#243511yescharacteristic
558C22C4×#13511yescharacteristic
559C22#294311yescharacteristic
560C22#294612yes—
561C22#300212yes—
562C22#301212yes—
563C22#303512yes—
564C22#304111yescharacteristic
565C22#304211yescharacteristic
566C22#329612yes—
567C22#329712yes—
568C22#332212yes—
569C22#332312yes—
570C22#338012yes—
571C22#338112yes—
572C22#338812yes—
573C22#338912yes—
574C4—24no—
575C4—24no—
576C4—44no—
577C4—44no—
578C4—82no—
579C4—84no—
580C4—82no—
581C4—82no—
582C4—84no—
583C4C2×#39111yescharacteristic
584C4#231111yescharacteristic
585C4#236012yes—
586C4#241712yes—
587C4#250211yescharacteristic
588C4#250311yescharacteristic
589C2—48no—
590C2#1413911yescharacteristic
591C2#2092411yescharacteristic
592C2#2092911yescharacteristic
593C2#2296612yes—
594C2#2297212yes—
595C2#2482012yes—
596C2#2482212yes—
597C2#2584312yes—
598C2#2589012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
3. C2×#1991C2semidirect
5. #14160C2semidirect
6. #11220C2semidirect
7. #11219C2semidirect
8. #10850C2semidirect
9. #10834C2semidirect
17. C2×#478C4semidirect
59. C2×#125C2×C4semidirect
89. C2×#107C2×C4semidirect
92. C2×#106C2×C4semidirect
95. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4782
C2×#23492
C2×#19912
C2×C4×#1022
#141602
#112202
#112192
#108502
#108342
#107882