Finite Group Databasebeta

← #402423

Subgroups — #402423

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1880C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic, complemented
3#15136C211yesmaximal, characteristic
4#13817C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11255C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11076C211yesmaximal, characteristic
9#11023C212yesmaximal
10#10931C212yesmaximal
11#10806C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#397—22no—
20C2×#391C2212yes—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25#1880—24no—
26#1803—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1439—21no—
36#1439—21no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41C4×#95—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—24no—
48C22×SD16—22no—
49C22×SD16—42no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83C2×#102—42no—
84#397—48no—
85#391—24no—
86#391—24no—
87#390—24no—
88#390—22no—
89#390—22no—
90#390—42no—
91#384—22no—
92#384—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—42no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95D412yes—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—24no—
107#340—22no—
108#340—24no—
109#340—42no—
110#336—22no—
111#336—42no—
112#336—42no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8C2×D412yes—
123C22×Q8#4114yes—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—22no—
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—22no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
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—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8#4112yes—
166C4×Q8—28no—
167C4×Q8—44no—
168C4×Q8—48no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4#4114yes—
194C2×C4⋊C4#4114yes—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—44no—
218C2×C42—82no—
219C2×C42C2×D412yes—
220C2×C42#4112yes—
221C4⋊C8—44no—
222#103—48no—
223#103—44no—
224#102—48no—
225#102—44no—
226#95—28no—
227#95—28no—
228#95—22no—
229#95—24no—
230#95—22no—
231#95—44no—
232#95—44no—
233#95—42no—
234#95—44no—
235#95—81no—
236C24—22no—
237C24—41no—
238C24—42no—
239C24—42no—
240C24—41no—
241C24—41no—
242C24—42no—
243C24—41no—
244C24—42no—
245C24—41no—
246C24—41no—
247C24C22×D411yescharacteristic
248C2×Q8—24no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8#12414yes—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4#12112yes—
333C22×C4#12112yes—
334C22×C4#13512yes—
335C22×C4AGL(1,ℤ/8)14yes—
336C22×C4#13712yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4112yes—
342C22×C4C2×#4112yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347SD16—416no—
348C2×C8—22no—
349C2×C8—24no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—22no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—84no—
380C4⋊C4—84no—
381C4⋊C4#12314yes—
382C22⋊C4—88no—
383C22⋊C4—84no—
384C22⋊C4—88no—
385C22⋊C4—84no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42—48no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—88no—
395C23—42no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—82no—
401C23—81no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—81no—
407C23—82no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—82no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23C2×#12111yescharacteristic
419C23#52512yes—
420C23#54612yes—
421C23#54611yescharacteristic
422C23#55011yescharacteristic
423C23C2×#13511yescharacteristic
424C23C2×AGL(1,ℤ/8)12yes—
425C23C2×#13711yescharacteristic
426C23#57711yescharacteristic
427C23#57712yes—
428C23#57811yescharacteristic
429Q8—24no—
430Q8—24no—
431Q8—44no—
432Q8—42no—
433Q8—42no—
434Q8—44no—
435D4—42no—
436D4—44no—
437D4—42no—
438D4—81no—
439D4—81no—
440D4—82no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4#46414yes—
494C2×C4C2×#12311yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4C2×#12411yescharacteristic
497C2×C4#52912yes—
498C2×C4#52912yes—
499C2×C4#52912yes—
500C2×C4#53211yescharacteristic
501C2×C4#53411yescharacteristic
502C2×C4#53511yescharacteristic
503C2×C4#53812yes—
504C2×C4#54012yes—
505C2×C4#55112yes—
506C2×C4#55112yes—
507C2×C4#55112yes—
508C2×C4#55311yescharacteristic
509C2×C4#55312yes—
510C2×C4#55312yes—
511C2×C4#55511yescharacteristic
512C8—42no—
513C8—82no—
514C8—81no—
515C22—84no—
516C22—82no—
517C22—84no—
518C22—82no—
519C22—82no—
520C22—161no—
521C22—161no—
522C22—162no—
523C22—162no—
524C22—161no—
525C22—161no—
526C22#241211yescharacteristic
527C22#241411yescharacteristic
528C22#245712yes—
529C22#246712yes—
530C22#285011yescharacteristic
531C22C2×#46411yescharacteristic
532C22#303811yescharacteristic
533C22#304311yescharacteristic
534C22#304611yescharacteristic
535C22#310612yes—
536C22#311212yes—
537C22#312112yes—
538C22#312512yes—
539C22#330112yes—
540C22#330211yescharacteristic
541C22#330612yes—
542C22#331012yes—
543C22#332012yes—
544C22#333012yes—
545C22#333111yescharacteristic
546C22#339411yescharacteristic
547C22#339712yes—
548C22#339811yescharacteristic
549C4—24no—
550C4—44no—
551C4—42no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—84no—
556C4—161no—
557C4—162no—
558C4#239511yescharacteristic
559C4#250411yescharacteristic
560C4#253112yes—
561C4#256111yescharacteristic
562C4#257812yes—
563C4#266111yescharacteristic
564C2—162no—
565C2#1484211yescharacteristic
566C2#2453311yescharacteristic
567C2#2454411yescharacteristic
568C2#2462911yescharacteristic
569C2#2464411yescharacteristic
570C2#2666512yes—
571C2#2667712yes—
572C2#2707112yes—
573C2#2708312yes—
574C2#3303111yescharacteristic
575C2#3303711yescharacteristic
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
2. C2×#1803C2semidirect
4. #13817C2semidirect
5. #11371C2semidirect
6. #11255C2semidirect
7. #11190C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18802
C2×#18032
#151362
#138172
#113712
#112552
#111902
#110762
#110232
#109312
#108062