Finite Group Databasebeta

← #396574

Subgroups — #396574

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1877C211yesmaximal, characteristic
2C2×#1877C212yesmaximal
3C2×#1802C211yesmaximal, characteristic
4#15185C211yesmaximal, characteristic, complemented
5D4×#95C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11255C211yesmaximal, characteristic, complemented
9#11190C212yesmaximal, complemented
10#11077C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×D4C2212yes—
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#384C2211yescharacteristic
23C22×#95—21no—
24#1877—28no—
25#1877—24no—
26#1802—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—21no—
36#1445—21no—
37#1442—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C22⋊#95—22no—
42C22⋊#95—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—22no—
47C24×C4—21no—
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—24no—
52C22×SD16D414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—24no—
58C2×C4×D4—24no—
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—22no—
62C2×C4×D4—42no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C2×C4×D4D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—22no—
86C2×#102—24no—
87C2×#102—24no—
88#391—24no—
89#391—22no—
90#391—22no—
91#391—42no—
92#384—22no—
93#384—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—44no—
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—22no—
111#340—24no—
112#340—42no—
113#336—24no—
114#336—22no—
115#336—42no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8C2×D412yes—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4C2×D412yes—
133C22×D4C2×D412yes—
134C22×D4#4112yes—
135C22×D4#4114yes—
136C22×D4#4114yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—44no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×SD16—28no—
160C2×SD16—216no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—48no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8C2×D412yes—
181C4×D4—28no—
182C4×D4—44no—
183C4×D4—48no—
184C4×D4—48no—
185C4×D4—48no—
186C4×D4—48no—
187C4×D4—48no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—81no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4#4114yes—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42C2×D412yes—
228C2×C42#4112yes—
229C4⋊C8—44no—
230#103—48no—
231#103—44no—
232#102—48no—
233#102—44no—
234#95—22no—
235#95—22no—
236#95—24no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—81no—
244#95—82no—
245#95—82no—
246#95—82no—
247C24—22no—
248C24—24no—
249C24—22no—
250C24—24no—
251C24—22no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24—42no—
257C24—41no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4#12114yes—
291C2×D4#12314yes—
292C2×D4#12318yes—
293C2×D4#12414yes—
294C2×D4#12618yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—48no—
318C22×C4—416no—
319C22×C4—416no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—84no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#13512yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4#13714yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)12yes—
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C42−1+411yescharacteristic
370SD16—416no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—42no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C22⋊C4—48no—
405C22⋊C4—416no—
406C42—22no—
407C42—22no—
408C42—24no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—28no—
417C23—28no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23C2×#12111yescharacteristic
437C23#52512yes—
438C23D4211yescharacteristic
439C23#54612yes—
440C23#54811yescharacteristic
441C23C2×#13511yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13712yes—
444C23#57611yescharacteristic
445C23#57711yescharacteristic
446C23#57712yes—
447Q8—42no—
448Q8—44no—
449Q8—42no—
450Q8—81no—
451Q8—81no—
452Q8—82no—
453D4—24no—
454D4—28no—
455D4—24no—
456D4—28no—
457D4—24no—
458D4#39718yes—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—816no—
491C2×C4—832no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—81no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4#46314yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12312yes—
516C2×C4C2×#12311yescharacteristic
517C2×C4C2×#12411yescharacteristic
518C2×C4C2×#12612yes—
519C2×C4#52912yes—
520C2×C4#53211yescharacteristic
521C2×C4#53411yescharacteristic
522C2×C4#53711yescharacteristic
523C2×C4#53812yes—
524C2×C4#53912yes—
525C2×C4#54612yes—
526C2×C4#54711yescharacteristic
527C2×C4#55112yes—
528C2×C4#55112yes—
529C2×C4#55212yes—
530C2×C4#55311yescharacteristic
531C2×C4#55312yes—
532C8—42no—
533C8—81no—
534C8—82no—
535C22—28no—
536C22—44no—
537C22—44no—
538C22—48no—
539C22—48no—
540C22—44no—
541C22—44no—
542C22—48no—
543C22—48no—
544C22—44no—
545C22—44no—
546C22#241211yescharacteristic
547C22#241611yescharacteristic
548C22#241812yes—
549C22#248112yes—
550C22#285011yescharacteristic
551C22C2×#46311yescharacteristic
552C22#303911yescharacteristic
553C22#304211yescharacteristic
554C22#304511yescharacteristic
555C22#310712yes—
556C22#311012yes—
557C22#311812yes—
558C22#312512yes—
559C22#326612yes—
560C22#327411yescharacteristic
561C22#328411yescharacteristic
562C22#329612yes—
563C22#331012yes—
564C22#331412yes—
565C22#333112yes—
566C22#336111yescharacteristic
567C22#337212yes—
568C22#337311yescharacteristic
569C4—24no—
570C4—44no—
571C4—42no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4—82no—
576C4—161no—
577C4—162no—
578C4C2×#39711yescharacteristic
579C4#241912yes—
580C4#249811yescharacteristic
581C4#251012yes—
582C4#254311yescharacteristic
583C4#263911yescharacteristic
584C2—48no—
585C2#1473511yescharacteristic
586C2#2452711yescharacteristic
587C2#2454811yescharacteristic
588C2#2473111yescharacteristic
589C2#2474811yescharacteristic
590C2#2490712yes—
591C2#2492112yes—
592C2#2756512yes—
593C2#2757312yes—
594C2#3303211yescharacteristic
595C2#3303411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #15185C2semidirect
6. #11373C2semidirect
7. #11316C2semidirect
8. #11255C2semidirect
9. #11190C2semidirect
10. #11077C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18772
C2×#18022
#151852
D4×#952
#113732
#113162
#112552
#111902
#110772
#107882