Finite Group Databasebeta

← #160452

Subgroups — #160452

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4#7579C214yesmaximal, complemented
5#7576C212yesmaximal, complemented
6#7573C212yesmaximal, complemented
7#7564C214yesmaximal, complemented
8C22×#117C411yescharacteristic
9C22×#117C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C22×C22⋊C8—21no—
13C22×C4×C8—21no—
14#2296—21no—
15#2296—21no—
16#2296—21no—
17#2296—21no—
18#2296C412yes—
19#2296C412yes—
20#2296C2212yes—
21#2296C2212yes—
22C2×#389C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#385—21no—
27C2×#385—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4—21no—
30#1817C2212yes—
31#1797—21no—
32#1797C2212yes—
33#1779—21no—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1271C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C23×C8—22no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—24no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—44no—
53C2×#117C818yescomplemented
54C2×#117C2×C412yes—
55C2×#117C2×C412yes—
56C2×#117C2×C412yes—
57C2×#117C2×C412yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#431—44no—
84C2×C4⋊C8—22no—
85C2×C22⋊C8—28no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—44no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389C2×C412yes—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2312yes—
110#389C2312yes—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25C22⋊C411yescharacteristic
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C2×C814yes—
158C23×C4M1614yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C22×C8—28no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—44no—
172C22×C8—44no—
173#117—28no—
174#117—28no—
175#117—24no—
176#117—24no—
177#117—44no—
178#117—44no—
179#117—44no—
180#117—44no—
181#117C2×C818yes—
182#117C2×C818yescomplemented
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×C814yescomplemented
204C2×C4⋊C4C2×C814yescomplemented
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C4⋊C4C22×C412yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—44no—
215C2×C42—28no—
216C2×C42—24no—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2411yescharacteristic
249C4⋊C8—44no—
250C22⋊C8—28no—
251C22⋊C8—28no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C4×C8—44no—
256C4×C8—44no—
257C4×C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24C22⋊C814yes—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—216no—
282C22×C4—216no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
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—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—416no—
329C22×C4—416no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C22⋊C814yes—
341C22×C4C22⋊C814yes—
342C22×C4C22⋊C814yes—
343C22×C4#9914yes—
344C22×C4#10114yes—
345C22×C4#10414yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C22×C812yes—
356C22×C4C22×C812yes—
357C22×C4C22×C812yes—
358C22×C4C2×M1612yes—
359C22×C4C2×M1612yes—
360C22×C4C2×M1612yes—
361C22×C4#13514yes—
362C22×C4C23×C411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C2×C8—28no—
366C2×C8—44no—
367C2×C8—416no—
368C2×C8—44no—
369C2×C8—44no—
370C2×C8—44no—
371C2×C8—44no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4C22×C814yes—
389C4⋊C4C22×C814yes—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C42—28no—
395C42—28no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—44no—
413C42—44no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—28no—
429C23—28no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23C2×C22⋊C812yes—
436C23C2×C22⋊C812yes—
437C23C2×C22⋊C812yes—
438C23C2×#9912yes—
439C23C2×#10112yes—
440C23#41612yes—
441C23C2×#10412yes—
442C23C22×C22⋊C411yescharacteristic
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—84no—
500C2×C4—84no—
501C2×C4#324116yes—
502C2×C4C2×C22⋊C812yes—
503C2×C4C2×C22⋊C812yes—
504C2×C4C2×C22⋊C812yes—
505C2×C4C2×#9912yes—
506C2×C4C2×#9912yes—
507C2×C4C2×#10112yes—
508C2×C4C2×#10112yes—
509C2×C4#41612yes—
510C2×C4C2×#10412yes—
511C2×C4C2×#10412yes—
512C2×C4#42112yes—
513C2×C4C23×C811yescharacteristic
514C2×C4C22×M1611yescharacteristic
515C2×C4C2×#13512yes—
516C2×C4#57512yes—
517C8—48no—
518C8—84no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—416no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22C2×#32414yes—
540C22#147614yes—
541C22C22×C22⋊C811yescharacteristic
542C22C22×#9911yescharacteristic
543C22C22×#10111yescharacteristic
544C22C2×#41611yescharacteristic
545C22#289611yescharacteristic
546C22C22×#10411yescharacteristic
547C22C2×#42111yescharacteristic
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—44no—
558C4—44no—
559C4—41no—
560C4—41no—
561C4—82no—
562C4—82no—
563C4C2×#32414yes—
564C4C2×#32414yes—
565C2—24no—
566C2—24no—
567C2—42no—
568C2C22×#32411yescharacteristic
569C2C2×#147611yescharacteristic
570C2#1743711yescharacteristic
571C2#1754211yescharacteristic
572C2#1757211yescharacteristic
573C2#1764811yescharacteristic
574C2#1771511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7579C2semidirect
5. #7576C2semidirect
6. #7573C2semidirect
7. #7564C2semidirect
41. #1271C22semidirect
42. #1271C22semidirect
53. C2×#117C8semidirect
182. #117C2×C8semidirect
203. C2×C4⋊C4C2×C8semidirect
204. C2×C4⋊C4C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#207142
C2×C4×C22⋊C82
#75792
#75762
#75732
#75642