Finite Group Databasebeta

← #461736

Subgroups — #461736

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2055C212yesmaximal
2C2×#1880C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic, complemented
4#15136C211yesmaximal, characteristic
5#13935C211yesmaximal, characteristic, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11322C211yesmaximal, characteristic, complemented
8#11227C212yesmaximal, complemented
9#11190C212yesmaximal, complemented
10#11076C211yesmaximal, characteristic
11#11066C212yesmaximal
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#397—22no—
21C2×#391C2212yes—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26#2055—28no—
27#1880—24no—
28#1803—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1441—22no—
37#1439—21no—
38#1439—21no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—42no—
49C22×SD16—42no—
50C22×SD16D414yes—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8—42no—
77C2×C4⋊C8—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—42no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—42no—
87C2×#102—42no—
88#397—48no—
89#391—24no—
90#391—42no—
91#390—24no—
92#390—22no—
93#390—22no—
94#390—42no—
95#389—24no—
96#389—42no—
97#384—22no—
98#384—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95D412yes—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109#340—24no—
110#340—24no—
111#340—22no—
112#340—42no—
113#340—42no—
114#336—22no—
115#336—42no—
116C25—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C2×D412yes—
125C22×Q8C2×D412yes—
126C22×Q8#4112yes—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4C2×D412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C2×SD16—28no—
148C2×SD16—48no—
149C2×SD16—44no—
150C2×SD16—44no—
151C2×SD16—48no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8C2×D412yes—
169C4×Q8—28no—
170C4×Q8—44no—
171C4×Q8—48no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
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—22no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4#4114yes—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4—82no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—44no—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224C4⋊C8—44no—
225C4⋊C8—48no—
226#103—44no—
227#103—48no—
228#102—48no—
229#102—44no—
230#95—22no—
231#95—24no—
232#95—22no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—44no—
239#95—81no—
240C24—22no—
241C24—41no—
242C24—42no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24—41no—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8#12114yes—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4#13512yes—
339C22×C4AGL(1,ℤ/8)14yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C42−1+412yes—
350C22×C42−1+411yescharacteristic
351SD16—416no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—48no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4#12314yes—
387C22⋊C4—88no—
388C22⋊C4—84no—
389C22⋊C4—88no—
390C22⋊C4—84no—
391C42—22no—
392C42—22no—
393C42—24no—
394C42—48no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C23—44no—
400C23—42no—
401C23—44no—
402C23—42no—
403C23—42no—
404C23—82no—
405C23—81no—
406C23—82no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—82no—
419C23—81no—
420C23—82no—
421C23—82no—
422C23C2×#12111yescharacteristic
423C23#53712yes—
424C23D4211yescharacteristic
425C23#54612yes—
426C23#54811yescharacteristic
427C23C2×#13511yescharacteristic
428C23C2×AGL(1,ℤ/8)12yes—
429C23C2×#13711yescharacteristic
430C23#57712yes—
431C23#57711yescharacteristic
432C23#57811yescharacteristic
433Q8—24no—
434Q8—24no—
435Q8—42no—
436Q8—44no—
437Q8—44no—
438Q8—42no—
439D4—42no—
440D4—44no—
441D4—42no—
442D4—81no—
443D4—82no—
444D4—81no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—42no—
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—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#46414yes—
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4#53412yes—
501C2×C4#53411yescharacteristic
502C2×C4#53511yescharacteristic
503C2×C4#53611yescharacteristic
504C2×C4#53711yescharacteristic
505C2×C4#53812yes—
506C2×C4#54012yes—
507C2×C4#54212yes—
508C2×C4#54212yes—
509C2×C4#54711yescharacteristic
510C2×C4#54712yes—
511C2×C4#55112yes—
512C2×C4#55212yes—
513C2×C4#55311yescharacteristic
514C2×C4#56212yes—
515C2×C4#56312yes—
516C8—42no—
517C8—82no—
518C8—81no—
519C22—82no—
520C22—84no—
521C22—82no—
522C22—84no—
523C22—82no—
524C22—162no—
525C22—162no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22#241211yescharacteristic
531C22#241611yescharacteristic
532C22#262011yescharacteristic
533C22#267112yes—
534C22#270212yes—
535C22C2×#46411yescharacteristic
536C22#303811yescharacteristic
537C22#304311yescharacteristic
538C22#304611yescharacteristic
539C22#324712yes—
540C22#325412yes—
541C22#325812yes—
542C22#326312yes—
543C22#326512yes—
544C22#327611yescharacteristic
545C22#328411yescharacteristic
546C22#330112yes—
547C22#330612yes—
548C22#332012yes—
549C22#333012yes—
550C22#336411yescharacteristic
551C22#336812yes—
552C22#336911yescharacteristic
553C4—24no—
554C4—44no—
555C4—44no—
556C4—42no—
557C4—44no—
558C4—82no—
559C4—161no—
560C4—162no—
561C4—162no—
562C4#248611yescharacteristic
563C4#261711yescharacteristic
564C4#262411yescharacteristic
565C4#271712yes—
566C4#274112yes—
567C4#281411yescharacteristic
568C2—162no—
569C2#1640111yescharacteristic
570C2#2453311yescharacteristic
571C2#2454411yescharacteristic
572C2#2472911yescharacteristic
573C2#2474211yescharacteristic
574C2#2982711yescharacteristic
575C2#2984111yescharacteristic
576C2#3060312yes—
577C2#3061712yes—
578C2#3181612yes—
579C2#3182012yes—
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
3. C2×#1803C2semidirect
5. #13935C2semidirect
6. #11371C2semidirect
7. #11322C2semidirect
8. #11227C2semidirect
9. #11190C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20552
C2×#18802
C2×#18032
#151362
#139352
#113712
#113222
#112272
#111902
#110762
#110662