Finite Group Databasebeta

← #354176

Subgroups — #354176

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2290C211yesmaximal, characteristic
2C2×C4×#99C212yesmaximal
3#13753C211yesmaximal, characteristic, complemented
4C42×C22⋊C4C211yesmaximal, characteristic, complemented
5#10960C214yesmaximal
6#10912C214yesmaximal, complemented
7C4×#342C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117C411yescharacteristic, complemented
10C23×C42C2211yescharacteristic
11C22×#99C2214yes—
12#2290—21no—
13#2290—21no—
14#2290—22no—
15#2290—21no—
16#2290—21no—
17#2290—22no—
18C2×#388C412yescomplemented
19C2×#387C412yescomplemented
20C2×#386C2212yes—
21C2×#385C2212yes—
22C2×C4×C4⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4C411yescharacteristic, complemented
24C2×C4×C22⋊C4C2214yes—
25C2×C4×C22⋊C4C2212yes—
26C2×C4×C22⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×#376—21no—
29C2×#376C411yescharacteristic, complemented
30C2×C43—21no—
31C4×#99C4116yescomplemented
32C2×#342C2218yes—
33C2×#342C2214yes—
34#1434—22no—
35C22⋊#95C2214yes—
36C4×#95—21no—
37C4×#95—22no—
38C4×#95—21no—
39C23×D4C2311yescharacteristic
40C24×C4C2×C411yescharacteristic
41C24×C4C2312yes—
42C24×C4C2311yescharacteristic
43C2×C4×D4—21no—
44C2×C4×D4—21no—
45C2×C4×D4—21no—
46C2×C4×D4—21no—
47C2×C4×D4—24no—
48C2×C4×D4C2×C418yes—
49C2×#117—22no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—44no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C412yes—
55C22×C4⋊C4C2×C412yes—
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C412yes—
58C22×C22⋊C4C2×C412yes—
59C22×C22⋊C4C2312yes—
60C22×C22⋊C4C2314yes—
61C22×C22⋊C4C2311yescharacteristic
62C22×C22⋊C4C2314yes—
63C22×C42—24no—
64C22×C42—22no—
65C22×C42—22no—
66C22×C42—44no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C412yes—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D412yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D412yes—
79C2×#99—216no—
80C2×#99C2×C4116yes—
81#388—22no—
82#388—22no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#386—22no—
90#386—22no—
91#386—22no—
92#386—22no—
93#385—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C4×C22⋊C4—21no—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—24no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—216no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4C2×C418yes—
111C4×C22⋊C4C2×C418yes—
112#376—22no—
113#376—21no—
114#376—21no—
115#376—44no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—24no—
120C2×#95—24no—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125C2×#95C2×C412yes—
126C43—44no—
127#342—216no—
128#342C2×C4116yes—
129C25C22×C411yescharacteristic
130C25C2411yescharacteristic
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—28no—
135C22×D4C22×C414yes—
136C23×C4—24no—
137C23×C4—28no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—42no—
143C23×C4—44no—
144C23×C4—42no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C4214yescomplemented
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C412yes—
151C23×C4C22×C412yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C414yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4#4114yes—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—28no—
172#117—48no—
173#117—44no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C4214yescomplemented
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4C4214yescomplemented
206C2×C22⋊C4C22×C418yes—
207C2×C22⋊C4C22×C414yes—
208C2×C22⋊C4C22×C418yes—
209C2×C22⋊C4#41116yes—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—216no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—28no—
222C2×C42—28no—
223C2×C42—42no—
224C2×C42—44no—
225C2×C42—42no—
226C2×C42—44no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42—48no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—48no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C4214yescomplemented
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C412yes—
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C412yes—
247C2×C42C22⋊C411yescharacteristic
248#99—232no—
249#95—216no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—28no—
261#95—28no—
262#95—48no—
263#95—48no—
264C24—21no—
265C24—21no—
266C24—24no—
267C24—41no—
268C24—41no—
269C24—42no—
270C24—41no—
271C24—42no—
272C24—41no—
273C24C2×C4212yes—
274C24C23×C411yescharacteristic
275C24C23×C412yes—
276C24C22×D411yescharacteristic
277C24C22×D411yescharacteristic
278C24C2×#4112yes—
279C24C2×#4118yes—
280C2×D4—28no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—28no—
284C2×D4—44no—
285C2×D4—48no—
286C2×D4—44no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—216no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—48no—
322C22×C4—48no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—432no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—48no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—44no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4C2×C4212yes—
357C22×C4C2×C4212yes—
358C22×C4C2×C4212yes—
359C22×C4C2×C4212yes—
360C22×C4C2×C4212yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4#11712yes—
370C22×C4#11712yes—
371C22×C4C4×D414yes—
372C22×C4C4×D414yes—
373C22×C4C4×D414yes—
374C22×C4C4×D414yes—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—28no—
384C22⋊C4—216no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—48no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4#117116yescomplemented
397C42—24no—
398C42—24no—
399C42—24no—
400C42—22no—
401C42—24no—
402C42—22no—
403C42—24no—
404C42—22no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—432no—
409C42—42no—
410C42—416no—
411C42—416no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42#9914yescomplemented
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—21no—
424C23—22no—
425C23—22no—
426C23—24no—
427C23—42no—
428C23—42no—
429C23—48no—
430C23—42no—
431C23—42no—
432C23—48no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—84no—
443C23—84no—
444C23C22×C4211yescharacteristic
445C23C22×C22⋊C411yescharacteristic
446C23C2×#11711yescharacteristic
447C23C2×#11714yes—
448C23C2×C4×D412yes—
449C23C2×C4×D412yes—
450C23C4×#4114yes—
451C23C4×#4114yes—
452C23#52514yes—
453C23#52918yes—
454C23#53214yes—
455D4—48no—
456D4—48no—
457C2×C4—216no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—216no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—416no—
494C2×C4—48no—
495C2×C4—416no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—48no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—88no—
517C2×C4—88no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4C4×C22⋊C412yes—
521C2×C4C4×C22⋊C412yes—
522C2×C4C4×C22⋊C412yes—
523C2×C4C4×C22⋊C412yes—
524C2×C4C2×#9914yes—
525C2×C4C2×#9912yes—
526C2×C4#41014yes—
527C2×C4#41012yes—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—44no—
543C22—44no—
544C22—48no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—84no—
549C22—84no—
550C22—82no—
551C22—82no—
552C22C2×C4×C22⋊C411yescharacteristic
553C22C4×#11712yes—
554C22#229014yes—
555C22#229314yes—
556C22#229712yes—
557C22#229812yes—
558C22C22×#9911yescharacteristic
559C22C2×#41011yescharacteristic
560C22C2×#41012yes—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—416no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—88no—
573C4—82no—
574C4—82no—
575C4C4×#9914yes—
576C2—24no—
577C2—24no—
578C2—22no—
579C2—22no—
580C2—82no—
581C2#1376111yescharacteristic
582C2C2×C4×#9911yescharacteristic
583C2C4×#41011yescharacteristic
584C2#2142012yes—
585C2#2142112yes—
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
socleC23C22×C22⋊C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13753C2semidirect
4. C42×C22⋊C4C2semidirect
6. #10912C2semidirect
7. C4×#342C2semidirect
9. C22×#117C4semidirect
18. C2×#388C4semidirect
19. C2×#387C4semidirect
22. C2×C4×C4⋊C4C4semidirect
23. C2×C4×C22⋊C4C4semidirect
29. C2×#376C4semidirect
31. C4×#99C4semidirect
148. C23×C4C42semidirect
184. C2×C4⋊C4C42semidirect
205. C2×C22⋊C4C42semidirect
237. C2×C42C42semidirect
396. C22⋊C4#117semidirect
415. C42#99semidirect

Maximal subgroups

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

subgroupindex
C2×#22902
C2×C4×#992
#137532
C42×C22⋊C42
#109602
#109122
C4×#3422