Finite Group Databasebeta

← #480691

Subgroups — #480691

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35618C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic, complemented
3#17368C211yesmaximal, characteristic, complemented
4#11556C212yesmaximal, complemented
5#11553C212yesmaximal, complemented
6#11551C212yesmaximal, complemented
7#11550C212yesmaximal, complemented
8#11511C212yesmaximal, complemented
9#11508C212yesmaximal, complemented
10C2×#537C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2975—21no—
13#2973—21no—
14#2504C411yescharacteristic
15#2504C2211yescharacteristic
16#2503C411yescharacteristic
17#2503C2211yescharacteristic
18#2500C411yescharacteristic
19#2500C2211yescharacteristic
20#2499C411yescharacteristic
21#2499C2211yescharacteristic
22#2444—21no—
23#2444—21no—
24#2444—21no—
25#2444—21no—
26#2421—21no—
27#2421—21no—
28#2420—21no—
29#2420—21no—
30#2317C411yescharacteristic
31#2311C2211yescharacteristic
32#2297C411yescharacteristic
33#2297C411yescharacteristic
34C2×#382C2211yescharacteristic
35#1467—21no—
36#1465—21no—
37#1465—21no—
38#1465—21no—
39#1464C2212yes—
40#1463C2212yes—
41#1463C2212yes—
42#1463C2212yes—
43#1461—21no—
44#1459—21no—
45#1458C2212yes—
46C2×#324C2212yes—
47#1324C2214yes—
48#1324C2214yes—
49#1319C2214yes—
50#1318C2214yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53#537—44no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×C411yescharacteristic
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121C2×C411yescharacteristic
61C2×#121C2×C411yescharacteristic
62#520—44no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42C2×C411yescharacteristic
80#436—44no—
81#435—44no—
82#432—22no—
83#431—22no—
84#407—22no—
85#407—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400—21no—
92#397—21no—
93#397—21no—
94#397—21no—
95#397—21no—
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#394—21no—
101#394—21no—
102#394—21no—
103#394—21no—
104#391—21no—
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#391C2×C411yescharacteristic
108#391C2311yescharacteristic
109#390—21no—
110#390C2×C411yescharacteristic
111#390C2×C411yescharacteristic
112#390C2×C411yescharacteristic
113#390C2311yescharacteristic
114#389—21no—
115#389—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386—21no—
125#386—21no—
126#386—21no—
127#386—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#385—21no—
135#385—21no—
136#384C2×C411yescharacteristic
137#384C2×C411yescharacteristic
138#382—21no—
139#382—21no—
140#382C2×C411yescharacteristic
141#382C2×C411yescharacteristic
142#382C2×C411yescharacteristic
143#382C2×C411yescharacteristic
144#382C2311yescharacteristic
145#382C2311yescharacteristic
146#381—21no—
147#381—21no—
148#381—21no—
149#381—21no—
150#380—21no—
151#380—21no—
152C4×C22⋊C4—21no—
153C4×C22⋊C4—21no—
154C4×C22⋊C4C2×C411yescharacteristic
155C4×C22⋊C4C2×C411yescharacteristic
156C4×C22⋊C4C2×C411yescharacteristic
157C4×C22⋊C4C2×C411yescharacteristic
158C2×#95—21no—
159C2×#95—21no—
160#324—22no—
161#324—22no—
162#324—22no—
163#324—22no—
164#324C2314yes—
165#324C2314yes—
166#323—22no—
167#323—22no—
168C2×#41—24no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—42no—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—42no—
199C22⋊Q8—44no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202#121—42no—
203#121—44no—
204#121—44no—
205#121—44no—
206#121—42no—
207C4×Q8—42no—
208C4×Q8—42no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2×D411yescharacteristic
266C2×C42—21no—
267C2×C42—42no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280C22⋊C8—24no—
281C22⋊C8—24no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284C22⋊C8—42no—
285C22⋊C8—42no—
286#97—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312C24—21no—
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316#41—416no—
317#41—88no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—81no—
341C2×D4—81no—
342C2×D4—81no—
343C2×D4—81no—
344C2×D4#9914yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394M16—162no—
395M16—162no—
396M16—162no—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—162no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—42no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23C2×#9912yes—
475C23C2×#9912yes—
476C23C22×C22⋊C411yescharacteristic
477Q8—44no—
478Q8—82no—
479D4—44no—
480D4—44no—
481D4—44no—
482D4—82no—
483D4—84no—
484D4—162no—
485D4—162no—
486C2×C4—28no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—84no—
541C2×C4—161no—
542C2×C4—161no—
543C2×C4—162no—
544C2×C4C2×#9912yes—
545C2×C4#41312yes—
546C2×C4#56812yes—
547C8—84no—
548C8—84no—
549C22—21no—
550C22—21no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22#287911yescharacteristic
574C22C22×#9911yescharacteristic
575C22C2×#41311yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—81no—
595C2—21no—
596C2—21no—
597C2—42no—
598C2—42no—
599C2—81no—
600C2—161no—
601C2#1739211yescharacteristic
602C2#3336111yescharacteristic
603C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17370C2semidirect
3. #17368C2semidirect
4. #11556C2semidirect
5. #11553C2semidirect
6. #11551C2semidirect
7. #11550C2semidirect
8. #11511C2semidirect
9. #11508C2semidirect

Maximal subgroups

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

subgroupindex
#356182
#173702
#173682
#115562
#115532
#115512
#115502
#115112
#115082