Finite Group Databasebeta

← #402290

Subgroups — #402290

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2073C212yesmaximal
2C2×#2055C212yesmaximal
3C2×#1877C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#15240C211yesmaximal, characteristic, complemented
6#13820C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11077C211yesmaximal, characteristic, complemented
10#10932C212yesmaximal, complemented
11#10850C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#390C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4—21no—
27#2073—28no—
28#2055—28no—
29#1877—24no—
30#1802—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1445—21no—
39#1445—21no—
40#1445—22no—
41#1437—22no—
42C4⋊#95C2211yescharacteristic
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—22no—
52C22×SD16—42no—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×D4—24no—
57C2×C4×D4—22no—
58C2×C4×D4—42no—
59C2×C4×D4D414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8—42no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—22no—
87#391—24no—
88#391—22no—
89#391—22no—
90#391—24no—
91#390—24no—
92#390D418yes—
93#384—22no—
94#384—22no—
95#382—24no—
96#382—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C4×C22⋊C4—44no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D411yescharacteristic
110C2×#95D411yescharacteristic
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—22no—
117#336—24no—
118#336—22no—
119C25—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8#4112yes—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C2×D412yes—
139C22×D4C2×D414yes—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×SD16—28no—
156C2×SD16—48no—
157C2×SD16—48no—
158C2×SD16—44no—
159C2×SD16—44no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8#4112yes—
177C4×D4—28no—
178C4×D4—44no—
179C4×D4—48no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—82no—
212C2×C22⋊C4—82no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—82no—
222C2×C42C2×D412yes—
223C2×C42C2×D414yes—
224C2×C42#4112yes—
225C2×C42#4114yes—
226C4⋊C8—48no—
227C4⋊C8—44no—
228#103—48no—
229#103—44no—
230#102—48no—
231#102—44no—
232#95—22no—
233#95—22no—
234#95—42no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—81no—
244C24—24no—
245C24—22no—
246C24—42no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24—41no—
251C24—42no—
252C24—41no—
253C24—42no—
254C24—41no—
255C24C22×D411yescharacteristic
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—24no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4#12714yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
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—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—48no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—84no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4#12114yes—
345C22×C4#12114yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#13512yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+411yescharacteristic
359C22×C42−1+411yescharacteristic
360SD16—416no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4#12314yes—
394C22⋊C4—48no—
395C22⋊C4—416no—
396C22⋊C4—88no—
397C42—22no—
398C42—24no—
399C42—22no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—88no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23—44no—
409C23—44no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—81no—
419C23—82no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23—82no—
427C23—82no—
428C23C2×#12111yescharacteristic
429C23C2×#12112yes—
430C23#54612yes—
431C23#54711yescharacteristic
432C23#54811yescharacteristic
433C23C2×#13511yescharacteristic
434C23C2×AGL(1,ℤ/8)11yescharacteristic
435C23#57512yes—
436C23#57711yescharacteristic
437C23#57712yes—
438C23#57811yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—44no—
442Q8—81no—
443Q8—81no—
444Q8—82no—
445D4—24no—
446D4—24no—
447D4—42no—
448D4—42no—
449D4—44no—
450D4—44no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—816no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
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—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4#46314yes—
505C2×C4C2×#12311yescharacteristic
506C2×C4#52511yescharacteristic
507C2×C4#52912yes—
508C2×C4#52912yes—
509C2×C4C2×#12711yescharacteristic
510C2×C4#53211yescharacteristic
511C2×C4#53212yes—
512C2×C4#53411yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#53912yes—
515C2×C4#54212yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#54712yes—
519C2×C4#54712yes—
520C2×C4#55112yes—
521C2×C4#55211yescharacteristic
522C2×C4#55611yescharacteristic
523C8—42no—
524C8—81no—
525C8—82no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—84no—
533C22—82no—
534C22—82no—
535C22—84no—
536C22—84no—
537C22#241211yescharacteristic
538C22#241511yescharacteristic
539C22#242512yes—
540C22#242712yes—
541C22#285611yescharacteristic
542C22C2×#46311yescharacteristic
543C22#303512yes—
544C22#303612yes—
545C22#304011yescharacteristic
546C22#304212yes—
547C22#304211yescharacteristic
548C22#304611yescharacteristic
549C22#304612yes—
550C22#329612yes—
551C22#331012yes—
552C22#331712yes—
553C22#332712yes—
554C22#334011yescharacteristic
555C22#335012yes—
556C22#335311yescharacteristic
557C22#336411yescharacteristic
558C22#336811yescharacteristic
559C22#337112yes—
560C4—24no—
561C4—44no—
562C4—44no—
563C4—44no—
564C4—42no—
565C4—84no—
566C4—82no—
567C4—162no—
568C4—161no—
569C4#239511yescharacteristic
570C4#250111yescharacteristic
571C4#252112yes—
572C4#253911yescharacteristic
573C4#255312yes—
574C4#266311yescharacteristic
575C2—84no—
576C2#1483911yescharacteristic
577C2#2452711yescharacteristic
578C2#2454311yescharacteristic
579C2#2468111yescharacteristic
580C2#2469711yescharacteristic
581C2#2520512yes—
582C2#2521612yes—
583C2#2532912yes—
584C2#2533212yes—
585C2#3316411yescharacteristic
586C2#3316811yescharacteristic
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
5. #15240C2semidirect
7. #11357C2semidirect
8. #11322C2semidirect
9. #11077C2semidirect
10. #10932C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20732
C2×#20552
C2×#18772
C2×#18022
#152402
#138202
#113572
#113222
#110772
#109322
#108502