Finite Group Databasebeta

← #359999

Subgroups — #359999

Order 512.

1792 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3C2×#1799C212yesmaximal
4#14339C211yesmaximal, characteristic, complemented
5#11197C212yesmaximal, complemented
6#10966C212yesmaximal
7#10960C212yesmaximal
8#10912C212yesmaximal, complemented
9#10846C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2214yes—
14#2322—24no—
15#2322—24no—
16#2313—28no—
17C2×#389C412yescomplemented
18C2×#388C412yescomplemented
19C2×#387—24no—
20C2×#387C412yescomplemented
21C2×#386C2211yescharacteristic
22C2×#385C412yescomplemented
23C2×#385C2211yescharacteristic
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1799—28no—
32C2×#342C2214yes—
33C2×#342C2214yes—
34C2×#342C2214yes—
35#1441—24no—
36C22⋊#95C2212yes—
37C23×D4C2311yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×#123—24no—
43C2×#123—22no—
44C2×#123—22no—
45C2×#123—24no—
46C2×C4×D4—24no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4D414yes—
61C22×C22⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C412yes—
72C2×#99—28no—
73C2×#99—28no—
74C2×#99—44no—
75#389—28no—
76#388—28no—
77#387—232no—
78#387—28no—
79#386—22no—
80#386—22no—
81#385—22no—
82#385—22no—
83#385—28no—
84#381—22no—
85#381—22no—
86#381—28no—
87#381—44no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380D418yes—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—28no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—24no—
98C2×#95—22no—
99C2×#95—24no—
100C2×#95—24no—
101C2×#95C2×C412yes—
102C2×#95C2×C412yes—
103C2×#95C2×C412yes—
104C2×#95C2×C412yes—
105C2×#95C2×C412yes—
106#342—28no—
107#342—28no—
108#342—44no—
109C25C22×C411yescharacteristic
110C25C2411yescharacteristic
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4#4114yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—82no—
132C23×C4—82no—
133C23×C4—82no—
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C412yes—
140C23×C4C22×C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D414yes—
150C23×C4#4114yes—
151#123—216no—
152#123—216no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—44no—
156C4×D4—44no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22⋊C418yes—
175C2×C4⋊C4C22⋊C418yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4C2×D418yes—
206C2×C22⋊C4#4114yes—
207C2×C42—22no—
208C2×C42—24no—
209C2×C42—28no—
210C2×C42—24no—
211C2×C42—84no—
212C2×C42—82no—
213#99—416no—
214#95—28no—
215#95—216no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—28no—
222#95—48no—
223#95—44no—
224#95—84no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24#11712yes—
234C24C23×C411yescharacteristic
235C24C22×D411yescharacteristic
236C24C22×D411yescharacteristic
237C24C22×D412yes—
238C24C2×#4112yes—
239C24C2×#4112yes—
240C24C2×#4112yes—
241C24H(5,𝔽2)12yes—
242C242−1+412yes—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—84no—
321C22×C4—84no—
322C22×C4—84no—
323C22×C4—84no—
324C22×C4—84no—
325C22×C4—88no—
326C22×C4C2×C22⋊C414yes—
327C22×C4C2×C22⋊C414yes—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4#11712yes—
331C22×C4#11714yes—
332C22×C4C4×D414yes—
333C22×C4C4×D414yes—
334C22×C4#12014yes—
335C22×C4#12114yes—
336C22×C4#12114yes—
337C22×C4#12314yes—
338C4⋊C4—216no—
339C4⋊C4—216no—
340C4⋊C4—216no—
341C4⋊C4—44no—
342C4⋊C4—84no—
343C4⋊C4—88no—
344C4⋊C4—88no—
345C22⋊C4—216no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—41no—
351C22⋊C4—42no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—44no—
363C22⋊C4—84no—
364C42—44no—
365C42—84no—
366C42—88no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—41no—
390C23—82no—
391C23—84no—
392C23—82no—
393C23C22×C22⋊C411yescharacteristic
394C23C2×#11711yescharacteristic
395C23C2×#11711yescharacteristic
396C23C2×C4×D412yes—
397C23#52012yes—
398C23C2×#12011yescharacteristic
399C23C2×#12112yes—
400C23C2×#12311yescharacteristic
401C23#52512yes—
402C23#52512yes—
403C23#53511yescharacteristic
404C23#53611yescharacteristic
405C23#53712yes—
406C23#54612yes—
407C23#54712yes—
408C23#54812yes—
409C23#55012yes—
410C23#55212yes—
411C23#55312yes—
412C23#55512yes—
413C23#55612yes—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—44no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—41no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—48no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—88no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4—88no—
470C2×C4—88no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4#38614yes—
474C2×C4#38614yes—
475C22—22no—
476C22—24no—
477C22—24no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—24no—
482C22—24no—
483C22—21no—
484C22—21no—
485C22—22no—
486C22—22no—
487C22—21no—
488C22—22no—
489C22—22no—
490C22—21no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—44no—
501C22—84no—
502C22—84no—
503C22—82no—
504C22—81no—
505C22—81no—
506C22#228011yescharacteristic
507C22C2×#38611yescharacteristic
508C22#231111yescharacteristic
509C22#235212yes—
510C22#235412yes—
511C22#240712yes—
512C22#241412yes—
513C22#241512yes—
514C22#248512yes—
515C22#288312yes—
516C22#288412yes—
517C4—24no—
518C4—24no—
519C4—44no—
520C4—42no—
521C4—44no—
522C4—44no—
523C4—88no—
524C4—82no—
525C4—81no—
526C4—81no—
527C4—164no—
528C2—22no—
529C2—22no—
530C2—22no—
531C2—22no—
532C2—22no—
533C2—22no—
534C2—82no—
535C2#1387111yescharacteristic
536C2#2072512yes—
537C2#2098411yescharacteristic
538C2#2098611yescharacteristic
539C2#2187712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2313C2semidirect
4. #14339C2semidirect
5. #11197C2semidirect
8. #10912C2semidirect
9. #10846C2semidirect
17. C2×#389C4semidirect
18. C2×#388C4semidirect
20. C2×#387C4semidirect
22. C2×#385C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23222
C2×#23132
C2×#17992
#143392
#111972
#109662
#109602
#109122
#108462