Finite Group Databasebeta

← #161939

Subgroups — #161939

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic
3C2×#1797C211yesmaximal, characteristic
4#7711C214yesmaximal, complemented
5#7668C212yesmaximal, complemented
6#7665C212yesmaximal, complemented
7#7598C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#423—21no—
11C22×C4⋊C8—21no—
12C2×#407—21no—
13C22×C22⋊C8—21no—
14#2857C411yescharacteristic
15#2857C411yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18#2855—21no—
19#2855—21no—
20#2851C412yes—
21#2851C2212yes—
22#2848—21no—
23#2848—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28#2296C2212yes—
29#2285C411yescharacteristic
30#2280C412yes—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1271C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—22no—
47C23×C8—22no—
48#533—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125C2×C412yes—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#117C2×C412yes—
61C2×#117C2×C412yes—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#423—44no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—44no—
81#407—44no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—44no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2312yes—
98#389C2312yes—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382C2×C412yes—
112#381—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123#379—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#336—22no—
131#336—22no—
132#336C2314yes—
133#336C2314yes—
134C25C22⋊C411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—42no—
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×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—24no—
170C22×C8—24no—
171C22×C8—44no—
172C22×C8—44no—
173C22×C8—44no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176#125—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—42no—
185C4×Q8—44no—
186C4×Q8—44no—
187#117—28no—
188#117—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2411yescharacteristic
241C4⋊C8—44no—
242C4⋊C8—44no—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249#95—22no—
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—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24C2×C22⋊C411yescharacteristic
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
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—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
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—22no—
315C22×C4—22no—
316C22×C4—22no—
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—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—416no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4#10114yes—
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4#13114yes—
349C22×C4AGL(1,ℤ/8)12yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4#13712yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—88no—
357C2×C8—44no—
358C2×C8—44no—
359C2×C8—44no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—88no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C22⋊C4—44no—
383C22⋊C4—44no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—44no—
404C42—44no—
405C42#9914yes—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—28no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23#40812yes—
430C23C2×#10112yes—
431C23#41712yes—
432C23#41712yes—
433C23#42112yes—
434C23C22×C22⋊C411yescharacteristic
435Q8—84no—
436Q8—84no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4#40812yes—
498C2×C4C2×#9912yes—
499C2×C4C2×#9912yes—
500C2×C4C2×#9912yes—
501C2×C4C2×#10112yes—
502C2×C4C2×#10112yes—
503C2×C4#41712yes—
504C2×C4#41712yes—
505C2×C4#42112yes—
506C2×C4C2×#13112yes—
507C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
508C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
509C2×C4C2×#13711yescharacteristic
510C2×C4C2×#13711yescharacteristic
511C8—84no—
512C8—84no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—416no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22#156114yes—
535C22C2×#40811yescharacteristic
536C22C22×#9911yescharacteristic
537C22C22×#10111yescharacteristic
538C22C2×#41711yescharacteristic
539C22C2×#41711yescharacteristic
540C22C2×#42111yescharacteristic
541C22C2×#42111yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4#147214yes—
559C4#151314yes—
560C2—24no—
561C2—42no—
562C2—42no—
563C2C2×#147211yescharacteristic
564C2#1746611yescharacteristic
565C2#1748111yescharacteristic
566C2C2×#151311yescharacteristic
567C2#1759211yescharacteristic
568C2#1767211yescharacteristic
569C2C2×#156111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7711C2semidirect
5. #7668C2semidirect
6. #7665C2semidirect
7. #7598C2semidirect
41. #1271C22semidirect
42. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#206262
C2×#17972
#77112
#76682
#76652
#75982