Finite Group Databasebeta

← #402292

Subgroups — #402292

Order 512.

1464 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2071C212yesmaximal
2C2×#2055C212yesmaximal
3C2×#1877C211yesmaximal, characteristic
4C2×#1802C211yesmaximal, characteristic
5#15199C211yesmaximal, characteristic, complemented
6#13820C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11322C211yesmaximal, characteristic, complemented
9#11077C211yesmaximal, characteristic, complemented
10#10932C212yesmaximal, complemented
11#10849C212yesmaximal, 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×#384C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4—21no—
27#2071—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#1444—21no—
39#1444—22no—
40#1444—21no—
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—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2312yes—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—22no—
89#391—24no—
90#391—22no—
91#391—22no—
92#391—24no—
93#390—24no—
94#390D418yes—
95#384—22no—
96#384—24no—
97#384—22no—
98#384—24no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—42no—
101C4×C22⋊C4—44no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95D411yescharacteristic
110C2×#95D412yes—
111C2×#95D411yescharacteristic
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—21no—
145C23×C4—22no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
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—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—81no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42—82no—
228C2×C42C2×D412yes—
229C2×C42C2×D414yes—
230C2×C42#4112yes—
231C2×C42#4114yes—
232C4⋊C8—48no—
233C4⋊C8—44no—
234#103—48no—
235#103—44no—
236#102—48no—
237#102—44no—
238#95—24no—
239#95—22no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—44no—
244#95—42no—
245#95—42no—
246#95—81no—
247#95SD1614yescomplemented
248C24—24no—
249C24—22no—
250C24—42no—
251C24—41no—
252C24—41no—
253C24—41no—
254C24—42no—
255C24—41no—
256C24—41no—
257C24—42no—
258C24—41no—
259C24C22×D411yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4#12714yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—48no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—48no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—84no—
344C22×C4—88no—
345C22×C4—88no—
346C22×C4#12112yes—
347C22×C4#12114yes—
348C22×C4#12114yes—
349C22×C4#12112yes—
350C22×C4C2×SD1612yes—
351C22×C4C2×SD1614yes—
352C22×C4#13714yes—
353C22×C4#13712yes—
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C2×#4112yes—
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364SD16—416no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—84no—
399C4⋊C4#12314yes—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—88no—
403C42—22no—
404C42—24no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—88no—
411C23—24no—
412C23—42no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—82no—
425C23—81no—
426C23—82no—
427C23—81no—
428C23—81no—
429C23—81no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23C2×#12111yescharacteristic
435C23C2×#12112yes—
436C23#54612yes—
437C23#54711yescharacteristic
438C23#54811yescharacteristic
439C23C22×SD1611yescharacteristic
440C23C2×#13712yes—
441C23C2×#13711yescharacteristic
442C23#57611yescharacteristic
443C23#57612yes—
444C23#57711yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—44no—
448Q8—81no—
449Q8—81no—
450Q8—82no—
451D4—24no—
452D4—24no—
453D4—42no—
454D4—42no—
455D4—44no—
456D4—44no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—816no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
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—81no—
508C2×C4—82no—
509C2×C4#46118yes—
510C2×C4#46114yes—
511C2×C4C2×#12311yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4#52912yes—
514C2×C4#52912yes—
515C2×C4C2×#12711yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53212yes—
518C2×C4#53411yescharacteristic
519C2×C4#53611yescharacteristic
520C2×C4#53912yes—
521C2×C4#54212yes—
522C2×C4#54612yes—
523C2×C4#54612yes—
524C2×C4#54712yes—
525C2×C4#54712yes—
526C2×C4#55112yes—
527C2×C4#55211yescharacteristic
528C2×C4#55611yescharacteristic
529C8—42no—
530C8—81no—
531C8—82no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—82no—
536C22—84no—
537C22—82no—
538C22—84no—
539C22—82no—
540C22—82no—
541C22—84no—
542C22—84no—
543C22#241211yescharacteristic
544C22#241511yescharacteristic
545C22#242512yes—
546C22#242712yes—
547C22#285611yescharacteristic
548C22C2×#46111yescharacteristic
549C22C2×#46112yes—
550C22#303911yescharacteristic
551C22#303912yes—
552C22#304111yescharacteristic
553C22#304112yes—
554C22#304511yescharacteristic
555C22#304512yes—
556C22#329512yes—
557C22#331412yes—
558C22#331612yes—
559C22#333312yes—
560C22#333511yescharacteristic
561C22#334612yes—
562C22#335711yescharacteristic
563C22#336211yescharacteristic
564C22#336911yescharacteristic
565C22#337512yes—
566C4—28no—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—42no—
571C4—84no—
572C4—82no—
573C4—162no—
574C4—161no—
575C4#239511yescharacteristic
576C4#250111yescharacteristic
577C4#252112yes—
578C4#253911yescharacteristic
579C4#255312yes—
580C4#266311yescharacteristic
581C2—84no—
582C2#1483911yescharacteristic
583C2#2453011yescharacteristic
584C2#2454611yescharacteristic
585C2#2468011yescharacteristic
586C2#2469611yescharacteristic
587C2#2520212yes—
588C2#2522312yes—
589C2#2532212yes—
590C2#2533612yes—
591C2#3316211yescharacteristic
592C2#3316611yescharacteristic
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. #15199C2semidirect
7. #11357C2semidirect
8. #11322C2semidirect
9. #11077C2semidirect
10. #10932C2semidirect
11. #10849C2semidirect
247. #95SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#20712
C2×#20552
C2×#18772
C2×#18022
#151992
#138202
#113572
#113222
#110772
#109322
#108492