Finite Group Databasebeta

← #413804

Subgroups — #413804

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2330C211yesmaximal, characteristic, complemented
2C2×#2026C211yesmaximal, characteristic, complemented
3C2×#2025C211yesmaximal, characteristic
4C2×#1983C212yesmaximal
5#14273C211yesmaximal, characteristic
6#11067C212yesmaximal, complemented
7#11066C212yesmaximal
8#11050C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#10849C212yesmaximal, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19#2330—28no—
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389—22no—
25C2×#389C2212yes—
26C2×#387—22no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#2026—24no—
32#2025—24no—
33#1983—28no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1439—22no—
42#1439—22no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—24no—
48C2×#125—24no—
49C2×#125—22no—
50C2×#125D414yes—
51C2×#124—24no—
52C2×#124—22no—
53C2×#124—24no—
54C2×#124Q814yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
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×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—24no—
84C2×#102—42no—
85C2×#102—42no—
86#391—22no—
87#391—22no—
88#391—42no—
89#390—22no—
90#390—22no—
91#390—42no—
92#389—216no—
93#389—24no—
94#389—24no—
95#387—216no—
96#384—24no—
97#384—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4D418yes—
100#376—24no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95D412yes—
107C2×#95D412yes—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—22no—
114#336—42no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C2×Q812yes—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4C2×Q812yes—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
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—
160#125—216no—
161#125—216no—
162#124—216no—
163#124—48no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—28no—
213C2×C42—82no—
214C2×C42—82no—
215C2×C42C2×D412yes—
216C2×C42C2×D414yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4114yes—
220C2×C42#4112yes—
221C4⋊C8—48no—
222#103—48no—
223#103—48no—
224#102—48no—
225#102—48no—
226#95—28no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—42no—
231#95—42no—
232#95—44no—
233C24—22no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24C22×D411yescharacteristic
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—42no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4#12014yes—
329C22×C4#12114yes—
330C22×C4C22⋊Q814yes—
331C22×C4#12414yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C4C22×Q811yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C42−1+411yescharacteristic
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—232no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C4⋊C4—88no—
371C4⋊C4#12118yes—
372C22⋊C4—416no—
373C22⋊C4—88no—
374C22⋊C4—84no—
375C22⋊C4—84no—
376C42—24no—
377C42—24no—
378C42—42no—
379C42—42no—
380C42—88no—
381C42C22⋊Q818yes—
382C42#12418yes—
383C23—44no—
384C23—44no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—81no—
389C23—81no—
390C23—82no—
391C23—82no—
392C23—81no—
393C23—82no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—81no—
406C23C2×#12011yescharacteristic
407C23C2×#12111yescharacteristic
408C23C2×C22⋊Q811yescharacteristic
409C23#52511yescharacteristic
410C23C2×#12411yescharacteristic
411C23#53511yescharacteristic
412C23#53711yescharacteristic
413C23#57612yes—
414C23#57712yes—
415C23#57712yes—
416C23#57812yes—
417Q8—44no—
418Q8—44no—
419Q8—82no—
420Q8—82no—
421D4—44no—
422D4—44no—
423D4—82no—
424D4—82no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#39318yes—
480C2×C4C2×#12112yes—
481C2×C4C2×C22⋊Q812yes—
482C2×C4#52512yes—
483C2×C4C2×#12412yes—
484C2×C4#53512yes—
485C2×C4#53712yes—
486C2×C4D4×Q811yescharacteristic
487C2×C4#55012yes—
488C2×C4#55011yescharacteristic
489C2×C4#55411yescharacteristic
490C2×C4#55412yes—
491C2×C4#55611yescharacteristic
492C2×C4#55612yes—
493C2×C4#55912yes—
494C2×C4#55912yes—
495C2×C4#55912yes—
496C8—82no—
497C8—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—84no—
502C22—84no—
503C22—161no—
504C22—161no—
505C22—162no—
506C22—162no—
507C22—161no—
508C22—161no—
509C22C2×#39311yescharacteristic
510C22#253811yescharacteristic
511C22#254811yescharacteristic
512C22#258011yescharacteristic
513C22#259211yescharacteristic
514C22#261511yescharacteristic
515C22#284711yescharacteristic
516C22#301212yes—
517C22#301312yes—
518C22#304512yes—
519C22#304612yes—
520C22#308312yes—
521C22#308412yes—
522C22#312412yes—
523C22#312512yes—
524C22#313612yes—
525C22#313712yes—
526C22#322011yescharacteristic
527C22#322111yescharacteristic
528C22#322211yescharacteristic
529C22#322311yescharacteristic
530C22#325712yes—
531C22#325812yes—
532C4—28no—
533C4—44no—
534C4—48no—
535C4—82no—
536C4—81no—
537C4—84no—
538C4—81no—
539C4—161no—
540C4—161no—
541C4#253812yes—
542C4#254812yes—
543C4#258012yes—
544C4#259212yes—
545C2—162no—
546C2#1515711yescharacteristic
547C2#2345812yes—
548C2#2829312yes—
549C2#2876011yescharacteristic
550C2#2876111yescharacteristic
551C2#2926811yescharacteristic
552C2#2926911yescharacteristic
553C2#2936312yes—
554C2#2967511yescharacteristic
555C2#2967611yescharacteristic
556C2#3291912yes—
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×#2330C2semidirect
2. C2×#2026C2semidirect
6. #11067C2semidirect
8. #11050C2semidirect
10. #10849C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23302
C2×#20262
C2×#20252
C2×#19832
#142732
#110672
#110662
#110502
#110492
#108492
#108332