Finite Group Databasebeta

← #364256

Subgroups — #364256

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C22×#340C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#11227C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117—21no—
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18#2344—24no—
19#2344C418yescomplemented
20C2×#389C2212yes—
21C2×#387—22no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#384C2212yes—
25C2×#381—21no—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1850—24no—
31#1823—24no—
32C2×#340—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44#1444—22no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C2×#117—28no—
58C2×#117—44no—
59C2×#117—44no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4C2×C414yes—
73C22×C4⋊C4D414yes—
74C22×C4⋊C4Q814yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C42—22no—
86C22×C42C2312yes—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—24no—
89C2×C4⋊C8—42no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93#389—24no—
94#389C2×C418yescomplemented
95#387—216no—
96#385—44no—
97#385—44no—
98#384—24no—
99#384—42no—
100#381—28no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380C2×C414yes—
105C4×C4⋊C4—24no—
106C4×C4⋊C4C2×C418yescomplemented
107C4×C22⋊C4C2×C414yes—
108C4×C22⋊C4D414yes—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#340—42no—
126#336—24no—
127C25C2×D411yescharacteristic
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×Q812yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—48no—
178C22×C8—48no—
179C22×C8—44no—
180C22×C8—44no—
181C22×C8#4114yes—
182#117—432no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—28no—
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—22no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
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—48no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C4⋊C4116yescomplemented
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C4⋊C4#4114yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4C22⋊C414yes—
245C2×C22⋊C4C22⋊C414yes—
246C2×C22⋊C4D814yes—
247C2×C22⋊C4SD1614yes—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—28no—
253C2×C42—44no—
254C2×C42—44no—
255C2×C42C22×C414yes—
256C2×C42C2×D414yes—
257C4⋊C8—48no—
258C22⋊C8—44no—
259#95—28no—
260#95—24no—
261#95—28no—
262#95—44no—
263#95—42no—
264#95—44no—
265#95C22⋊C418yes—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24C2×C22⋊C412yes—
271C24#12112yes—
272C24C22⋊Q812yes—
273C24C2×D812yes—
274C24C2×SD1612yes—
275C24AGL(1,ℤ/8)12yes—
276C24#13712yes—
277C24C22×D411yescharacteristic
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—216no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—48no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4C2×C22⋊C414yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C414yes—
357C22×C4C2×C4⋊C414yes—
358C22×C4#11714yes—
359C22×C4#12114yes—
360C22×C4#12112yes—
361C22×C4C22⋊Q814yes—
362C22×C4C22⋊Q812yes—
363C22×C4C2×D814yes—
364C22×C4C2×SD1614yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—416no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—216no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—416no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4C2×C4⋊C414yes—
419C4⋊C4C2×C4⋊C414yes—
420C22⋊C4—28no—
421C22⋊C4—48no—
422C22⋊C4—84no—
423C22⋊C4#102116yes—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—416no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23#38014yes—
446C23C2×#10214yes—
447C23#41714yes—
448C23#45914yes—
449C23#46114yes—
450C23#47414yes—
451C23#47614yes—
452C23C22×C22⋊C411yescharacteristic
453C23C2×#12111yescharacteristic
454C23C2×C22⋊Q811yescharacteristic
455C23#54612yes—
456C23#54712yes—
457C23C22×D811yescharacteristic
458C23C22×SD1611yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×#13711yescharacteristic
461C23#57712yes—
462C23#57812yes—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—432no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—816no—
508C2×C4—82no—
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—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4#38014yes—
527C2×C4C2×#10218yes—
528C2×C4C22×C4⋊C411yescharacteristic
529C2×C4C2×#11711yescharacteristic
530C2×C4C4×#4112yes—
531C2×C4#52012yes—
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×C22⋊Q811yescharacteristic
534C2×C4#52511yescharacteristic
535C2×C4#52511yescharacteristic
536C2×C4#53312yes—
537C2×C4#53511yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#54412yes—
540C2×C4#54612yes—
541C2×C4#54712yes—
542C2×C4#55112yes—
543C2×C4#55112yes—
544C2×C4#55212yes—
545C2×C4#55212yes—
546C8—44no—
547C8—81no—
548C8—81no—
549C22—24no—
550C22—24no—
551C22—28no—
552C22—28no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—28no—
558C22—28no—
559C22#180218yes—
560C22C2×#38011yescharacteristic
561C22#233912yes—
562C22#241111yescharacteristic
563C22#241311yescharacteristic
564C22#243812yes—
565C22C22×#10211yescharacteristic
566C22C2×#41711yescharacteristic
567C22#289812yes—
568C22C2×#45911yescharacteristic
569C22C2×#46111yescharacteristic
570C22#304311yescharacteristic
571C22#304411yescharacteristic
572C22C2×#47411yescharacteristic
573C22C2×#47611yescharacteristic
574C22#308212yes—
575C22#329412yes—
576C22#329812yes—
577C22#331012yes—
578C22#331212yes—
579C22#333412yes—
580C22#333512yes—
581C22#334312yes—
582C22#334412yes—
583C4—24no—
584C4—28no—
585C4—44no—
586C4—44no—
587C4—44no—
588C4—84no—
589C4—84no—
590C4—161no—
591C4—161no—
592C4#229611yescharacteristic
593C4#231711yescharacteristic
594C4#234312yes—
595C4#241111yescharacteristic
596C4#241311yescharacteristic
597C4#243812yes—
598C2—28no—
599C2#1396211yescharacteristic
600C2C2×#180211yescharacteristic
601C2#2065111yescharacteristic
602C2#2248512yes—
603C2#2249112yes—
604C2#2447611yescharacteristic
605C2#2448011yescharacteristic
606C2#2457611yescharacteristic
607C2#2458011yescharacteristic
608C2#2598812yes—
609C2#2599012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11227C2semidirect
7. #11207C2semidirect
8. #10849C2semidirect
9. #10806C2semidirect
19. #2344C4semidirect
94. #389C2×C4semidirect
106. C4×C4⋊C4C2×C4semidirect
227. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23442
C2×#18502
C2×#18232
C22×#3402
#138802
#112272
#112072
#108492
#108062
#107302
#107212