Finite Group Databasebeta

← #461515

Subgroups — #461515

Order 512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 578 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#13931C211yesmaximal, characteristic, complemented
5#11373C211yesmaximal, characteristic, complemented
6#11322C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11077C212yesmaximal, complemented
9#11076C212yesmaximal
10#11076C211yesmaximal, characteristic
11#11023C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#397—21no—
21C2×#397—21no—
22C2×#391C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#384C2211yescharacteristic
26#1880—24no—
27#1803—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1445—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40#1434—21no—
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C22×SD16—22no—
46C22×SD16—22no—
47C22×SD16—24no—
48C22×SD16—22no—
49C22×SD16—42no—
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
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⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
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×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8—42no—
74C2×C4⋊C8—42no—
75C2×C4⋊C8—42no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#102—22no—
82C2×#102—42no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86#397—44no—
87#397—44no—
88#391—24no—
89#391—42no—
90#390—24no—
91#390—22no—
92#390—22no—
93#390—42no—
94#384—22no—
95#384—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—22no—
110#340—42no—
111#340—42no—
112#336—22no—
113#336—42no—
114C25—21no—
115C22×Q8—21no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8C2×D412yes—
124C22×Q8#4114yes—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4#4112yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C2×SD16—28no—
146C2×SD16—48no—
147C2×SD16—48no—
148C2×SD16—44no—
149C2×SD16—44no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8#4112yes—
167C4×Q8—28no—
168C4×Q8—44no—
169C4×Q8—48no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
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—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42C2×D412yes—
220C2×C42#4112yes—
221C4⋊C8—44no—
222C4⋊C8—48no—
223#103—48no—
224#103—44no—
225#102—44no—
226#102—48no—
227#95—22no—
228#95—42no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—82no—
236#95—81no—
237#95SD1614yescomplemented
238C24—22no—
239C24—41no—
240C24—42no—
241C24—42no—
242C24—41no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—41no—
247C24—41no—
248C24—42no—
249C24C22×D411yescharacteristic
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#12314yes—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
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—
284C2×D4—42no—
285C2×D4—42no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
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—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4C2×SD1614yes—
337C22×C4C2×SD1612yes—
338C22×C4AGL(1,ℤ/8)12yes—
339C22×C4C22×D411yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)11yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349SD16—416no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—84no—
383C4⋊C4—84no—
384C22⋊C4—88no—
385C22⋊C4—84no—
386C22⋊C4—88no—
387C22⋊C4—84no—
388C42—22no—
389C42—22no—
390C42—24no—
391C42—48no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C23—42no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—81no—
412C23—81no—
413C23—82no—
414C23—81no—
415C23—81no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C2×#12111yescharacteristic
420C23#53712yes—
421C23#54612yes—
422C23#54611yescharacteristic
423C23#55011yescharacteristic
424C23C22×SD1611yescharacteristic
425C23C2×AGL(1,ℤ/8)11yescharacteristic
426C23#57512yes—
427C23#57611yescharacteristic
428C23#57711yescharacteristic
429C23#57812yes—
430Q8—24no—
431Q8—24no—
432Q8—42no—
433Q8—44no—
434Q8—44no—
435Q8—42no—
436D4—42no—
437D4—42no—
438D4—44no—
439D4—81no—
440D4—82no—
441D4—81no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
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—82no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#46014yes—
495C2×C4C2×#12311yescharacteristic
496C2×C4#52511yescharacteristic
497C2×C4#53411yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53511yescharacteristic
500C2×C4#53611yescharacteristic
501C2×C4#53912yes—
502C2×C4#53912yes—
503C2×C4#53912yes—
504C2×C4#54212yes—
505C2×C4#54212yes—
506C2×C4#55112yes—
507C2×C4#55112yes—
508C2×C4#55112yes—
509C2×C4#55311yescharacteristic
510C2×C4#55511yescharacteristic
511C2×C4#56012yes—
512C2×C4#56012yes—
513C8—42no—
514C8—81no—
515C8—82no—
516C22—82no—
517C22—82no—
518C22—84no—
519C22—82no—
520C22—84no—
521C22—161no—
522C22—161no—
523C22—162no—
524C22—162no—
525C22—161no—
526C22—161no—
527C22#241211yescharacteristic
528C22#241411yescharacteristic
529C22#262011yescharacteristic
530C22#269312yes—
531C22#270512yes—
532C22C2×#46011yescharacteristic
533C22#304011yescharacteristic
534C22#304411yescharacteristic
535C22#304511yescharacteristic
536C22#324612yes—
537C22#325112yes—
538C22#325812yes—
539C22#326212yes—
540C22#329511yescharacteristic
541C22#329812yes—
542C22#330812yes—
543C22#330912yes—
544C22#331512yes—
545C22#332411yescharacteristic
546C22#332612yes—
547C22#339211yescharacteristic
548C22#339911yescharacteristic
549C22#340212yes—
550C4—24no—
551C4—44no—
552C4—44no—
553C4—42no—
554C4—82no—
555C4—82no—
556C4—162no—
557C4—161no—
558C4—162no—
559C4#248211yescharacteristic
560C4#262311yescharacteristic
561C4#263811yescharacteristic
562C4#273712yes—
563C4#274612yes—
564C4#281511yescharacteristic
565C2—162no—
566C2#1639511yescharacteristic
567C2#2452811yescharacteristic
568C2#2454711yescharacteristic
569C2#2462811yescharacteristic
570C2#2464511yescharacteristic
571C2#2982611yescharacteristic
572C2#2984011yescharacteristic
573C2#3155212yes—
574C2#3156312yes—
575C2#3190012yes—
576C2#3192812yes—
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. #13931C2semidirect
5. #11373C2semidirect
6. #11322C2semidirect
7. #11316C2semidirect
8. #11077C2semidirect
237. #95SD16semidirect

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
#139312
#113732
#113222
#113162
#110772
#110762
#110232