Finite Group Databasebeta

← #359917

Subgroups — #359917

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3C2×#1821C212yesmaximal
4#20682C211yesmaximal, characteristic
5#11204C214yesmaximal, complemented
6#10913C214yesmaximal, complemented
7#10727C212yesmaximal
8C23×C4⋊C4C412yes—
9C23×C4⋊C4C2211yescharacteristic
10C23×C4⋊C4C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432—21no—
14C2×#432—21no—
15C2×#431—21no—
16C2×#431—21no—
17C2×#423C2212yes—
18C2×#407C2212yes—
19C2×#407C2212yes—
20C2×C4×M16C2212yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#389—21no—
25C2×#389—21no—
26C2×#389—21no—
27C2×#389—21no—
28C2×#389—21no—
29C2×#389C412yes—
30C2×#389C412yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C22×#95C412yes—
34#1821—28no—
35#1817—21no—
36#1817—21no—
37#1817—21no—
38#1817—21no—
39C2×#343C2218yes—
40C2×#343C2218yes—
41#1728C2212yes—
42#1728C2212yes—
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C412yes—
46C24×C4C2×C412yes—
47C24×C4C2311yescharacteristic
48C22×M16C2314yes—
49C22×M16C2314yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—28no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—48no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C412yes—
68C22×C4⋊C4C2×C414yes—
69C22×C4⋊C4C2×C414yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#432—28no—
87#432—28no—
88#431—28no—
89#431—28no—
90#423—28no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93#407—24no—
94#407—24no—
95#407—24no—
96#407D418yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C4×M16—28no—
102C2×#97—22no—
103C2×C4×C8—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—22no—
117#389—22no—
118#389—22no—
119#389—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95C2×C414yes—
125C2×#95C2×C414yes—
126#343—216no—
127#343—216no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132C25C22×C411yescharacteristic
133C23×C4—24no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—28no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—28no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
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×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C414yes—
165C23×C4C22×C414yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D414yes—
171C23×C4#4114yes—
172C23×C4C2411yescharacteristic
173C2×M16—24no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—24no—
177C2×M16—28no—
178C2×M16—24no—
179C2×M16—28no—
180C2×M16C2×D418yes—
181C22×C8—24no—
182C22×C8—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—48no—
215C2×C4⋊C4—48no—
216C2×C4⋊C4#4118yes—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C4⋊C8—28no—
252C4⋊C8—28no—
253C22⋊C8—28no—
254C22⋊C8—28no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257#97—28no—
258C4×C8—28no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24C4×D414yes—
272C24C23×C411yescharacteristic
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—216no—
288C22×C4—216no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
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—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—24no—
316C22×C4—28no—
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—416no—
326C22×C4—416no—
327C22×C4—44no—
328C22×C4—432no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—816no—
343C22×C4#10014yes—
344C22×C4#10114yes—
345C22×C4C2×C22⋊C411yescharacteristic
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×C4#11718yes—
356C22×C4C4×D414yes—
357C22×C4#12012yes—
358C22×C4#12012yes—
359C22×C4#12114yes—
360C22×C4#12114yes—
361C22×C4#12312yes—
362C22×C4#12312yes—
363C22×C4C22×D412yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4112yes—
367C22×C4C2×#4114yes—
368C22×C42−1+414yes—
369M16—216no—
370M16—48no—
371M16—48no—
372M16—48no—
373C2×C8—28no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C4⋊C4—28no—
380C4⋊C4—28no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C23—28no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—21no—
417C23—21no—
418C23—28no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—28no—
428C23—28no—
429C23—21no—
430C23—21no—
431C23#38612yes—
432C23#38612yes—
433C23C2×#10012yes—
434C23C2×#10112yes—
435C23#41312yes—
436C23#41312yes—
437C23C22×C22⋊C411yescharacteristic
438C23C2×#11712yes—
439C23C2×C4×D412yes—
440C23#51912yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—28no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—28no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—416no—
485C2×C4—416no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—88no—
495C2×C4—88no—
496C2×C4—88no—
497C2×C4#38612yes—
498C2×C4#38612yes—
499C2×C4C2×#10012yes—
500C2×C4C2×#10012yes—
501C2×C4C2×#10112yes—
502C2×C4C2×#10112yes—
503C2×C4C2×#12011yescharacteristic
504C2×C4C2×#12112yes—
505C2×C4C2×#12311yescharacteristic
506C2×C4#52512yes—
507C2×C4#52512yes—
508C2×C4#53612yes—
509C2×C4#53612yes—
510C2×C4#54718yes—
511C2×C4#55218yes—
512C8—48no—
513C8—84no—
514C22—232no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
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#190014yes—
534C22#190114yes—
535C22#228011yescharacteristic
536C22C2×#38611yescharacteristic
537C22#231011yescharacteristic
538C22#234414yes—
539C22C22×#10011yescharacteristic
540C22C22×#10111yescharacteristic
541C22C2×#41311yescharacteristic
542C22C2×#41311yescharacteristic
543C4—22no—
544C4—22no—
545C4—22no—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—48no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—84no—
555C4#240612yes—
556C4#241314yes—
557C4#248412yes—
558C2—24no—
559C2—24no—
560C2—24no—
561C2#1387011yescharacteristic
562C2#2072611yescharacteristic
563C2#2072711yescharacteristic
564C2C2×#190011yescharacteristic
565C2C2×#190111yescharacteristic
566C2#2182911yescharacteristic
567C2#2183011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11204C2semidirect
6. #10913C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C22×#3892
C2×#18212
#206822
#112042
#109132
#107272