Finite Group Databasebeta

← #411270

Subgroups — #411270

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1968C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
3#15192C211yesmaximal, characteristic, complemented
4#15147C211yesmaximal, characteristic
5#11257C212yesmaximal, complemented
6#11077C211yesmaximal, characteristic, complemented
7#11076C211yesmaximal, characteristic
8#11023C212yesmaximal
9#10926C212yesmaximal, complemented
10#10925C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2212yes—
21C2×#397—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1968—24no—
28C8×C4⋊C4—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1445—21no—
35#1443—22no—
36#1442—21no—
37#1440—22no—
38#1439—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—24no—
49C22×SD16—42no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42C2312yes—
64C22×C42C2311yescharacteristic
65C2×#106—22no—
66C2×#106—42no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×C4×C8—24no—
80C2×C4×C8—24no—
81C2×C4×C8—42no—
82C2×C4×C8—42no—
83C2×C4×C8D414yes—
84#397—44no—
85#391—24no—
86#391—22no—
87#391—22no—
88#391—42no—
89#390—24no—
90#390—22no—
91#390—22no—
92#390—42no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—24no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8#4112yes—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4#4112yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C2×SD16—28no—
145C2×SD16—48no—
146C2×SD16—48no—
147C2×SD16—44no—
148C2×SD16—44no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8#4114yes—
164C22×C8#4114yes—
165C22×C8#4114yes—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
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⋊C4SD1614yescomplemented
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4—82no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#106—44no—
213C4⋊C8—28no—
214#103—48no—
215#103—44no—
216#102—48no—
217#102—44no—
218C4×C8—28no—
219C4×C8—48no—
220#95—24no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—42no—
229#95—81no—
230#95—81no—
231#95—82no—
232#95—82no—
233C24—22no—
234C24—41no—
235C24—42no—
236C24—41no—
237C24—42no—
238C24—41no—
239C24—42no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4C2×SD1614yes—
331C22×C4C2×SD1612yes—
332C22×C4#13514yes—
333C22×C4#13512yes—
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344SD16—416no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8#12314yes—
361C2×C8#12414yes—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—81no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—82no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—82no—
412C23C2×#12111yescharacteristic
413C23#53212yes—
414C23#54611yescharacteristic
415C23#54611yescharacteristic
416C23#55012yes—
417C23C22×SD1611yescharacteristic
418C23C2×#13512yes—
419C23C2×#13511yescharacteristic
420C23#57611yescharacteristic
421C23#57712yes—
422C23#57811yescharacteristic
423Q8—44no—
424Q8—42no—
425Q8—42no—
426Q8—82no—
427Q8—81no—
428Q8—81no—
429D4—44no—
430D4—42no—
431D4—42no—
432D4—82no—
433D4—81no—
434D4—81no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#46514yes—
488C2×C4C2×#12311yescharacteristic
489C2×C4#52511yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4C2×#12411yescharacteristic
492C2×C4#52912yes—
493C2×C4#52912yes—
494C2×C4#52912yes—
495C2×C4#53511yescharacteristic
496C2×C4#53511yescharacteristic
497C2×C4#53812yes—
498C2×C4#53812yes—
499C2×C4#55112yes—
500C2×C4#55112yes—
501C2×C4#55311yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#55312yes—
504C2×C4#55312yes—
505C2×C4#55912yes—
506C8—24no—
507C8—24no—
508C8—44no—
509C22—82no—
510C22—84no—
511C22—84no—
512C22—82no—
513C22—82no—
514C22—161no—
515C22—161no—
516C22—161no—
517C22—161no—
518C22—162no—
519C22—162no—
520C22#253511yescharacteristic
521C22#254811yescharacteristic
522C22#255412yes—
523C22#255412yes—
524C22#285411yescharacteristic
525C22C2×#46511yescharacteristic
526C22#305011yescharacteristic
527C22#305711yescharacteristic
528C22#306111yescharacteristic
529C22#316712yes—
530C22#316812yes—
531C22#317812yes—
532C22#317912yes—
533C22#329511yescharacteristic
534C22#329612yes—
535C22#329811yescharacteristic
536C22#330112yes—
537C22#330211yescharacteristic
538C22#330511yescharacteristic
539C22#339212yes—
540C22#339312yes—
541C22#339412yes—
542C22#339612yes—
543C4—24no—
544C4—42no—
545C4—44no—
546C4—44no—
547C4—82no—
548C4—161no—
549C4—161no—
550C4—162no—
551C4—162no—
552C4#239811yescharacteristic
553C4#248211yescharacteristic
554C4#250411yescharacteristic
555C4#259712yes—
556C4#259712yes—
557C4#263111yescharacteristic
558C2—162no—
559C2#1511211yescharacteristic
560C2#2814111yescharacteristic
561C2#2814411yescharacteristic
562C2#2873811yescharacteristic
563C2#2874011yescharacteristic
564C2#2887412yes—
565C2#2887612yes—
566C2#2887712yes—
567C2#2887912yes—
568C2#3312811yescharacteristic
569C2#3313111yescharacteristic
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
2. C2×C8×C4⋊C4C2semidirect
3. #15192C2semidirect
5. #11257C2semidirect
6. #11077C2semidirect
9. #10926C2semidirect
11. #10820C2semidirect
190. C2×C4⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#19682
C2×C8×C4⋊C42
#151922
#151472
#112572
#110772
#110762
#110232
#109262
#109252
#108202