Finite Group Databasebeta

← #403809

Subgroups — #403809

Order 512.

1434 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic, complemented
2C2×#2042C212yesmaximal
3C2×#2024C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#14273C211yesmaximal, characteristic
6#11207C212yesmaximal, complemented
7#11052C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal
11#10796C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18#2331—28no—
19C2×#391—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#387—22no—
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×#376C2211yescharacteristic
30#2042—28no—
31#2024—24no—
32#2023—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1439—22no—
42#1439—22no—
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8D414yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—24no—
55C2×#124—42no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C42—22no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83#391—216no—
84#391—22no—
85#391—22no—
86#391—24no—
87#390—22no—
88#390—22no—
89#390—24no—
90#389—24no—
91#389D418yes—
92#387—216no—
93#384—42no—
94#382—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97#376—24no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95D412yes—
106C2×#95D412yes—
107#340—24no—
108#340—24no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—22no—
113#336—24no—
114#336—24no—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—82no—
129C22×Q8—82no—
130C22×Q8#4112yes—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4#4112yes—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C4⋊Q8—216no—
166C4⋊Q8—216no—
167#124—48no—
168#124—48no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—28no—
217C2×C42—82no—
218C2×C42—82no—
219C2×C42C2×D414yes—
220C2×C42C2×D412yes—
221C2×C42#4112yes—
222C2×C42#4114yes—
223#103—48no—
224#103—48no—
225#102—48no—
226#102—48no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—28no—
231#95—24no—
232#95—24no—
233#95—42no—
234#95—42no—
235#95—44no—
236C24—22no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24C22×D411yescharacteristic
248C2×Q8—216no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—84no—
266C2×Q8—84no—
267C2×Q8—88no—
268C2×Q8—88no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—42no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
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—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12014yes—
337C22×C4#12112yes—
338C22×C4#12112yes—
339C22×C4#12414yes—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D412yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4112yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+411yescharacteristic
349C22×C42−1+411yescharacteristic
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—216no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4—88no—
378C4⋊C4#12118yes—
379C22⋊C4—416no—
380C22⋊C4—88no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—88no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—81no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—81no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23C2×#12011yescharacteristic
414C23C2×#12111yescharacteristic
415C23#52511yescharacteristic
416C23#52511yescharacteristic
417C23C2×#12411yescharacteristic
418C23#53511yescharacteristic
419C23#53611yescharacteristic
420C23#57612yes—
421C23#57712yes—
422C23#57712yes—
423C23#57812yes—
424Q8—44no—
425Q8—44no—
426Q8—816no—
427Q8—82no—
428Q8—82no—
429D4—44no—
430D4—44no—
431D4—82no—
432D4—82no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—28no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
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—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4C2×#12112yes—
489C2×C4#52512yes—
490C2×C4#52912yes—
491C2×C4#52912yes—
492C2×C4#54212yes—
493C2×C4#54212yes—
494C2×C4#54612yes—
495C2×C4#54612yes—
496C2×C4#54811yescharacteristic
497C2×C4#54811yescharacteristic
498C2×C4#54812yes—
499C2×C4#54812yes—
500C2×C4#55012yes—
501C2×C4#55211yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55912yes—
504C8—82no—
505C8—82no—
506C22—84no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—84no—
511C22—162no—
512C22—161no—
513C22—161no—
514C22—162no—
515C22—161no—
516C22—161no—
517C22#239811yescharacteristic
518C22#240411yescharacteristic
519C22#243611yescharacteristic
520C22#253611yescharacteristic
521C22#254811yescharacteristic
522C22#263211yescharacteristic
523C22#284711yescharacteristic
524C22#301212yes—
525C22#301312yes—
526C22#304111yescharacteristic
527C22#304211yescharacteristic
528C22#304311yescharacteristic
529C22#304411yescharacteristic
530C22#311812yes—
531C22#312112yes—
532C22#312412yes—
533C22#312512yes—
534C22#313612yes—
535C22#313712yes—
536C22#322412yes—
537C22#322512yes—
538C22#324312yes—
539C22#324412yes—
540C4—44no—
541C4—44no—
542C4—42no—
543C4—48no—
544C4—42no—
545C4—82no—
546C4—84no—
547C4—161no—
548C4—161no—
549C4#242612yes—
550C4#242612yes—
551C4#245812yes—
552C4#245812yes—
553C2—162no—
554C2#1488311yescharacteristic
555C2#2382712yes—
556C2#2394712yes—
557C2#2591612yes—
558C2#2821111yescharacteristic
559C2#2821211yescharacteristic
560C2#2875811yescharacteristic
561C2#2875911yescharacteristic
562C2#3006812yes—
563C2#3291711yescharacteristic
564C2#3291811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2331C2semidirect
3. C2×#2024C2semidirect
6. #11207C2semidirect
7. #11052C2semidirect
9. #10932C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20422
C2×#20242
C2×#20232
#142732
#112072
#110522
#110512
#109322
#109312
#107962