Finite Group Databasebeta

← #393321

Subgroups — #393321

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2C4×#518C211yesmaximal, characteristic
3C2×#1799C211yesmaximal, characteristic
4#13923C212yesmaximal, complemented
5#10918C212yesmaximal, complemented
6#10918C212yesmaximal, complemented
7#10846C212yesmaximal, complemented
8#10745C212yesmaximal
9#10745C212yesmaximal
10C2×#518C2212yes—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#99C2212yes—
14#2579—24no—
15#2579C418yescomplemented
16C2×#398—21no—
17C2×#396—21no—
18C2×#387—21no—
19C2×#386C2211yescharacteristic
20C2×#386C2211yescharacteristic
21C2×#385—21no—
22C2×#385C2211yescharacteristic
23C2×#385C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26#2275—24no—
27#2274—21no—
28#2273—21no—
29#2273C2212yes—
30C42×D4—24no—
31C4×#117—21no—
32C2×C4×C22⋊C4—21no—
33C2×C4×C22⋊C4C2212yes—
34#1799—24no—
35C2×#342C2214yes—
36C2×#342C2214yes—
37C2×#342C2212yes—
38#1441—24no—
39#1436—24no—
40C22⋊#95C2214yes—
41C22⋊#95C2214yes—
42C23×D4C2311yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47#518—24no—
48#518—28no—
49#518Q818yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—28no—
53C2×C4×D4—24no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×#117—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2×C414yescomplemented
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2311yescharacteristic
75C22×C22⋊C4C2311yescharacteristic
76C22×C22⋊C4C2311yescharacteristic
77C22×C22⋊C4C2314yes—
78C22×C42—22no—
79C22×C42D411yescharacteristic
80C22×C42D414yes—
81C2×#99—24no—
82C2×#99—24no—
83C2×#99—24no—
84#398—28no—
85#396—28no—
86#387—28no—
87#386—22no—
88#386—22no—
89#386C2×C414yes—
90#386C2×C414yes—
91#385—28no—
92#385—22no—
93#385—22no—
94#385C2×C414yescomplemented
95#385C2×C414yescomplemented
96#380—22no—
97#380—24no—
98#380—24no—
99#380C2×C414yescomplemented
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—28no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—28no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—28no—
113#376—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—28no—
121C2×#95—22no—
122C2×#95—28no—
123C2×#95D414yes—
124C43—22no—
125#342—24no—
126#342—28no—
127#342—28no—
128C25C2×D411yescharacteristic
129C25C2411yescharacteristic
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4C22×C412yes—
134C22×D4C2×D412yes—
135C22×D4C2×Q814yes—
136C22×D4#4114yes—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—44no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C22⋊C414yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D418yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4#4114yes—
170C4×D4—232no—
171C4×D4—28no—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—232no—
176#117—28no—
177#117—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—84no—
200C2×C4⋊C4C22⋊C414yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4C22⋊C414yes—
227C2×C22⋊C4C22×C412yes—
228C2×C22⋊C4C22×C412yes—
229C2×C22⋊C4C22×C412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×Q818yes—
232C2×C22⋊C4#4118yes—
233C2×C42—24no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—84no—
242C2×C42—84no—
243C2×C42C22⋊C414yes—
244#99—48no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—416no—
255#95—44no—
256#95—88no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24C2×C22⋊C412yes—
265C24C22⋊Q814yes—
266C24C23×C411yescharacteristic
267C24C22×D411yescharacteristic
268C24C22×D412yes—
269C24C22×D411yescharacteristic
270C24C22×Q812yes—
271C24C2×#4112yes—
272C24C2×#4112yes—
273C24C2×#4114yes—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—28no—
278C2×D4—28no—
279C2×D4—28no—
280C2×D4—24no—
281C2×D4—28no—
282C2×D4C4×Q818yes—
283C2×D4#12414yes—
284C2×D4C4⋊Q814yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—28no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—28no—
294C22×C4—21no—
295C22×C4—216no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—28no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—48no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—48no—
319C22×C4—48no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—48no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—82no—
349C22×C4—84no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C4×D414yes—
356C22×C4C4×D414yes—
357C22×C4#12011yescharacteristic
358C22×C4#12012yes—
359C22×C4#12011yescharacteristic
360C22×C4#12114yes—
361C22×C4#12114yes—
362C22×C4C22⋊Q814yes—
363C22×C4C22⋊Q8116yes—
364C22×C4#12312yes—
365C22×C4#12312yes—
366C4⋊C4—216no—
367C4⋊C4—28no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—48no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C4⋊C4—88no—
375C22⋊C4—216no—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—28no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—48no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—84no—
393C42—216no—
394C42—24no—
395C42—24no—
396C42—48no—
397C42—84no—
398C42—84no—
399C42—84no—
400C23—24no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—24no—
405C23—22no—
406C23—24no—
407C23—22no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—44no—
415C23—48no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—48no—
420C23—42no—
421C23—42no—
422C23—81no—
423C23—84no—
424C23—81no—
425C23#39114yes—
426C23C22×C22⋊C411yescharacteristic
427C23C2×C4×D412yes—
428C23C2×C4×Q812yes—
429C23C4×#4112yes—
430C23C2×#12011yescharacteristic
431C23C2×#12112yes—
432C23C2×C22⋊Q812yes—
433C23C2×C22⋊Q814yes—
434C23C2×#12311yescharacteristic
435C23#52514yes—
436C23#52512yes—
437C23C2×#12411yescharacteristic
438C23#52914yes—
439C23C2×C4⋊Q811yescharacteristic
440C23#53212yes—
441C23#53314yes—
442D4—28no—
443D4—28no—
444D4—216no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—216no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—48no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—48no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—416no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—84no—
490C2×C4—88no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—816no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—84no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4#38612yes—
505C2×C4#38612yes—
506C2×C4#38614yes—
507C2×C4#39114yes—
508C22—28no—
509C22—28no—
510C22—24no—
511C22—24no—
512C22—21no—
513C22—22no—
514C22—22no—
515C22—21no—
516C22—21no—
517C22—22no—
518C22—22no—
519C22—21no—
520C22—24no—
521C22—42no—
522C22—42no—
523C22—41no—
524C22—44no—
525C22—41no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—48no—
530C22—42no—
531C22—42no—
532C22—82no—
533C22—84no—
534C22C2×#38611yescharacteristic
535C22C2×#39111yescharacteristic
536C22#229711yescharacteristic
537C22C4×C22⋊Q814yes—
538C22#284812yes—
539C22#285012yes—
540C22#285512yes—
541C22#285712yes—
542C22#302512yes—
543C22#302612yes—
544C4—24no—
545C4—22no—
546C4—22no—
547C4—24no—
548C4—416no—
549C4—44no—
550C4—42no—
551C4—42no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—164no—
556C2—21no—
557C2—22no—
558C2—21no—
559C2—22no—
560C2—22no—
561C2—24no—
562C2—44no—
563C2#1470211yescharacteristic
564C2#2095911yescharacteristic
565C2#2096011yescharacteristic
566C2#3297714yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13923C2semidirect
5. #10918C2semidirect
6. #10918C2semidirect
7. #10846C2semidirect
15. #2579C4semidirect
67. C22×C4⋊C4C2×C4semidirect
94. #385C2×C4semidirect
95. #385C2×C4semidirect
99. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
C4×#5182
C2×#17992
#139232
#109182
#108462
#107452