Finite Group Databasebeta

← #162763

Subgroups — #162763

Order 512.

1164 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#7695C214yesmaximal, complemented
5#7688C212yesmaximal, complemented
6#7687C212yesmaximal, complemented
7#7623C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#407—21no—
11C2×#407—21no—
12C2×C4×M16—21no—
13#2857C411yescharacteristic
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2855—21no—
18#2855C2212yes—
19#2851—21no—
20#2851—21no—
21#2848—21no—
22#2848C412yes—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C412yes—
26C4×C22⋊Q8C2212yes—
27#2296—21no—
28#2285C411yescharacteristic
29#2280—21no—
30#1817—21no—
31#1817—21no—
32#1817C2212yes—
33#1817C2212yes—
34#1749—21no—
35#1749C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8C2212yes—
38#1728—21no—
39#1728C2212yes—
40#1728C2212yes—
41#1272C2218yescomplemented
42#1271C2218yescomplemented
43C24×C4C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C22×M16—22no—
47C22×M16—22no—
48#533—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×#117—21no—
61C2×#117—21no—
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#407—28no—
76#407—44no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C4×M16—44no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#391—21no—
85#391—21no—
86#391C2×C412yes—
87#391C2312yes—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2312yes—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#382—21no—
109#381—22no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2×C412yes—
120#379—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336C2314yes—
132#336C2314yes—
133C25C22⋊C411yescharacteristic
134C22×Q8—21no—
135C22×Q8C22×C412yes—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
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×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—28no—
164C2×M16—44no—
165C2×M16—44no—
166C2×M16—44no—
167C2×M16—44no—
168C2×M16—44no—
169C22×C8—24no—
170C22×C8—24no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173#125—44no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C4×Q8—44no—
181C4×Q8—44no—
182#117—44no—
183#117—44no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2411yescharacteristic
236C22⋊C8—28no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240#97—44no—
241C4×C8—44no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—41no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C411yescharacteristic
261C24C2×C22⋊C411yescharacteristic
262C2×Q8—24no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
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—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
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—416no—
324C22×C4—416no—
325C22×C4—416no—
326C22×C4#9914yes—
327C22×C4#10414yes—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4#13714yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341M16—416no—
342M16—88no—
343C2×C8—28no—
344C2×C8—44no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—44no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—44no—
370C22⋊C4—44no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—44no—
388C42—44no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23C2×#9912yes—
415C23#41912yes—
416C23C2×#10412yes—
417C23C22×C22⋊C411yescharacteristic
418Q8—84no—
419Q8—84no—
420C2×C4—28no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—21no—
438C2×C4—21no—
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—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4C2×#9912yes—
477C2×C4C2×#9912yes—
478C2×C4#41312yes—
479C2×C4#41912yes—
480C2×C4C2×#10412yes—
481C2×C4C2×#10412yes—
482C2×C4#42112yes—
483C2×C4#56812yes—
484C2×C4C2×#13712yes—
485C2×C4#57512yes—
486C8—48no—
487C8—84no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—416no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22#287911yescharacteristic
511C22C22×#9911yescharacteristic
512C22C2×#41311yescharacteristic
513C22C2×#41911yescharacteristic
514C22#289611yescharacteristic
515C22C22×#10411yescharacteristic
516C22C2×#42111yescharacteristic
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—22no—
523C4—22no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—41no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4#151114yes—
533C2—42no—
534C2—42no—
535C2—42no—
536C2#1740011yescharacteristic
537C2#1749311yescharacteristic
538C2#1750111yescharacteristic
539C2#1754211yescharacteristic
540C2C2×#151111yescharacteristic
541C2#1764911yescharacteristic
542C2#1774711yescharacteristic
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. #7695C2semidirect
5. #7688C2semidirect
6. #7687C2semidirect
7. #7623C2semidirect
41. #1272C22semidirect
42. #1271C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#206822
C4×#4072
#76952
#76882
#76872
#76232