Finite Group Databasebeta

← #413473

Subgroups — #413473

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1968C212yesmaximal
2C2×#1968C211yesmaximal, characteristic
3C2×#1941C212yesmaximal, complemented
4C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
5#15193C211yesmaximal, characteristic, complemented
6#15148C211yesmaximal, characteristic
7#11077C211yesmaximal, characteristic, complemented
8#11076C211yesmaximal, characteristic
9#11067C212yesmaximal, complemented
10#11066C212yesmaximal
11#10861C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C4×C8C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#399—21no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1968—28no—
31#1968—24no—
32#1941—28no—
33C8×C4⋊C4—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1445—22no—
39#1445—21no—
40#1444—21no—
41#1439—22no—
42#1439—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×D4C2311yescharacteristic
47C22×SD16—22no—
48C22×SD16—24no—
49C22×SD16—24no—
50C22×SD16—22no—
51C22×SD16Q814yes—
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#106—24no—
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—22no—
73C2×#106D418yes—
74C2×C4⋊C8—24no—
75C2×C4⋊C8—22no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86C2×C4×C8—24no—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8D414yes—
91#399—44no—
92#391—24no—
93#391—22no—
94#391—22no—
95#391—42no—
96#390—24no—
97#390—22no—
98#390—22no—
99#390—42no—
100#389—24no—
101#389—24no—
102#389—42no—
103#389—42no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95D412yes—
117#340—24no—
118#340—24no—
119#340—42no—
120#336—24no—
121C25—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8C2×Q812yes—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C2×Q812yes—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C2×SD16—28no—
153C2×SD16—216no—
154C2×SD16—44no—
155C2×SD16—48no—
156C2×SD16—44no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D414yes—
170C22×C8C2×Q812yes—
171C22×C8#4114yes—
172C22×C8#4114yes—
173C22×C8#4112yes—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4SD1614yescomplemented
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C2×D412yes—
225C2×C42#4112yes—
226#106—216no—
227#106—44no—
228C4⋊C8—28no—
229#103—48no—
230#103—44no—
231#102—48no—
232#102—44no—
233C4×C8—28no—
234C4×C8—48no—
235#95—24no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—42no—
241#95—42no—
242#95—42no—
243#95—42no—
244#95—81no—
245#95—81no—
246C24—22no—
247C24—41no—
248C24—42no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24—41no—
255C24—42no—
256C24—41no—
257C24C22×D411yescharacteristic
258C2×Q8—24no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—42no—
294C22×C4—22no—
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—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
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—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4C2×SD1614yes—
344C22×C4C2×SD1612yes—
345C22×C4#13514yes—
346C22×C4#13512yes—
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×Q811yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4111yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+411yescharacteristic
357SD16—416no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8C22⋊Q814yes—
371C2×C8C22⋊Q818yes—
372C2×C8#12314yes—
373C2×C8#12318yes—
374C2×C8#12714yes—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C42—22no—
398C42—22no—
399C42—24no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C23—42no—
405C23—42no—
406C23—44no—
407C23—44no—
408C23—42no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23—82no—
421C23—82no—
422C23—81no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23C2×#12111yescharacteristic
428C23#53212yes—
429C23D4212yes—
430C23D4×Q811yescharacteristic
431C23#55011yescharacteristic
432C23C22×SD1611yescharacteristic
433C23C2×#13512yes—
434C23C2×#13511yescharacteristic
435C23#57711yescharacteristic
436C23#57711yescharacteristic
437C23#57712yes—
438Q8—42no—
439Q8—42no—
440Q8—44no—
441Q8—81no—
442Q8—82no—
443Q8—81no—
444D4—42no—
445D4—42no—
446D4—44no—
447D4—81no—
448D4—82no—
449D4—81no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4#46514yes—
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4C2×C22⋊Q812yes—
505C2×C4C2×#12311yescharacteristic
506C2×C4C2×#12312yes—
507C2×C4#52511yescharacteristic
508C2×C4#52912yes—
509C2×C4C2×#12711yescharacteristic
510C2×C4#53511yescharacteristic
511C2×C4#53711yescharacteristic
512C2×C4#54012yes—
513C2×C4#54212yes—
514C2×C4#54612yes—
515C2×C4#54812yes—
516C2×C4#55012yes—
517C2×C4#55412yes—
518C2×C4#55411yescharacteristic
519C2×C4#55611yescharacteristic
520C2×C4#55912yes—
521C8—42no—
522C8—44no—
523C8#39918yescomplemented
524C22—82no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22—161no—
533C22—162no—
534C22—162no—
535C22#244512yes—
536C22#245212yes—
537C22#253811yescharacteristic
538C22#254111yescharacteristic
539C22#285411yescharacteristic
540C22C2×#46511yescharacteristic
541C22#305011yescharacteristic
542C22#305711yescharacteristic
543C22#306011yescharacteristic
544C22#316712yes—
545C22#316812yes—
546C22#317312yes—
547C22#317412yes—
548C22D4×SD1612yes—
549C22#328212yes—
550C22#328212yes—
551C22#328412yes—
552C22Q8×SD1611yescharacteristic
553C22#338111yescharacteristic
554C22#338112yes—
555C22#339211yescharacteristic
556C22#339411yescharacteristic
557C22#339412yes—
558C4—24no—
559C4—44no—
560C4—44no—
561C4—42no—
562C4—82no—
563C4—84no—
564C4—84no—
565C4—161no—
566C4—161no—
567C4C2×#39911yescharacteristic
568C4#244812yes—
569C4#245412yes—
570C4#254011yescharacteristic
571C4#258011yescharacteristic
572C4#262411yescharacteristic
573C2—162no—
574C2#1514811yescharacteristic
575C2#2625012yes—
576C2#2625112yes—
577C2#2648012yes—
578C2#2648112yes—
579C2#2826411yescharacteristic
580C2#2826611yescharacteristic
581C2#2848211yescharacteristic
582C2#2848311yescharacteristic
583C2#3312811yescharacteristic
584C2#3313711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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×#1941C2semidirect
4. C2×C8×C4⋊C4C2semidirect
5. #15193C2semidirect
7. #11077C2semidirect
9. #11067C2semidirect
11. #10861C2semidirect
202. C2×C4⋊C4SD16semidirect
523. C8#399semidirect

Maximal subgroups

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

subgroupindex
C2×#19682
C2×#19412
C2×C8×C4⋊C42
#151932
#151482
#110772
#110762
#110672
#110662
#108612