Finite Group Databasebeta

← #375910

Subgroups — #375910

Order 512.

1500 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
1#34270C211yesmaximal, characteristic
2C2×#2030C211yesmaximal, characteristic
3#14263C212yesmaximal
4#13776C211yesmaximal, characteristic
5#11390C211yesmaximal, characteristic
6#11344C211yesmaximal, characteristic
7#11070C211yesmaximal, characteristic
8#10936C212yesmaximal
9#10918C212yesmaximal
10#10912C212yesmaximal
11#10801C211yesmaximal, characteristic
12C2×#543C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14#2368—24no—
15#2360—24no—
16#2329—22no—
17#2315C2211yescharacteristic
18C4×C22⋊Q8—22no—
19C2×#385C2212yes—
20C2×#385C2212yes—
21C2×#381C2212yes—
22C2×#381C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25#2273C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2211yescharacteristic
28#2030C418yescomplemented
29C2×#342C2211yescharacteristic
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2212yes—
34C2×#342C2211yescharacteristic
35C2×#342C2212yes—
36#1444—24no—
37#1442—24no—
38#1436—22no—
39#1436—22no—
40C22⋊#95C2211yescharacteristic
41C22⋊#95C2212yes—
42C22⋊#95C2211yescharacteristic
43C22⋊#95C2212yes—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47#543—22no—
48#543—24no—
49#543C2×C414yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8—24no—
52C2×C4×Q8—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4D414yes—
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2312yes—
63C22×C22⋊C4C2311yescharacteristic
64C22×C22⋊C4C2312yes—
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42D412yes—
71#391—24no—
72#391—22no—
73#389—24no—
74#389—22no—
75#387—24no—
76#387—24no—
77#387—24no—
78#385—22no—
79#385—22no—
80#385—22no—
81#385—24no—
82#385—22no—
83#385—24no—
84#385—42no—
85#381—22no—
86#381—24no—
87#381—22no—
88#381—24no—
89#381—42no—
90#381C2×C414yes—
91#381C2×C418yes—
92#381Q814yescomplemented
93#380—22no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4—24no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—42no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—24no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95D412yes—
114C2×#95D414yes—
115#342—24no—
116#342—24no—
117#342—22no—
118#342—42no—
119#342—42no—
120#342C2×C414yes—
121#342C2×C418yes—
122C25C2411yescharacteristic
123C22×Q8—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D414yes—
149C23×C4C2×D412yes—
150C23×C4C2×D414yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C22⋊Q8—28no—
158C22⋊Q8—216no—
159C22⋊Q8—28no—
160C22⋊Q8—216no—
161C4×Q8—84no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—84no—
178C2×C4⋊C4—84no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C22×C414yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C22×C414yes—
214C2×C22⋊C4C2×Q812yes—
215C2×C22⋊C4C2×Q812yes—
216C2×C22⋊C4#4112yes—
217C2×C22⋊C4#4114yes—
218C2×C22⋊C4#4112yes—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—82no—
225C2×C42—82no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—28no—
230#95—216no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—28no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—84no—
240#95—82no—
241#95—82no—
242#95—84no—
243C24—42no—
244C24—42no—
245C24C23×C411yescharacteristic
246C24C22×D411yescharacteristic
247C24C22×D412yes—
248C24C22×Q811yescharacteristic
249C24C2×#4111yescharacteristic
250C24C2×#4112yes—
251C24C2×#4111yescharacteristic
252C24C2×#4111yescharacteristic
253C24C2×#4111yescharacteristic
254C24H(5,𝔽2)12yes—
255C242−1+412yes—
256C2×Q8—28no—
257C2×Q8—82no—
258C2×Q8—82no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—82no—
341C22×C4—84no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—84no—
345C22×C4—82no—
346C22×C4C4×D414yes—
347C22×C4C4×D418yes—
348C22×C4C4×Q814yescomplemented
349C22×C4#12012yes—
350C22×C4#12012yes—
351C22×C4C22⋊Q814yes—
352C4⋊C4—216no—
353C4⋊C4—28no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C4⋊C4—48no—
357C4⋊C4—84no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—88no—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C42—44no—
386C42—82no—
387C42—84no—
388C42—84no—
389C42—84no—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—24no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—44no—
404C23—42no—
405C23—44no—
406C23C2×C4×D411yescharacteristic
407C23C2×C4×D412yes—
408C23C2×C4×Q811yescharacteristic
409C23C4×#4111yescharacteristic
410C23#52012yes—
411C23C2×#12011yescharacteristic
412C23C2×C22⋊Q811yescharacteristic
413C23#52511yescharacteristic
414C23#52511yescharacteristic
415C23#52512yes—
416C23#52911yescharacteristic
417C23#53311yescharacteristic
418C23#53511yescharacteristic
419C23#53611yescharacteristic
420C23#54212yes—
421C23#54612yes—
422C23#54612yes—
423C23#54812yes—
424C23D4×Q812yes—
425C23#55212yes—
426C23#55412yes—
427C23#55612yes—
428C23#55912yes—
429Q8—88no—
430C2×C4—28no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—28no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—84no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—84no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—84no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—82no—
503C22—22no—
504C22—22no—
505C22—24no—
506C22—24no—
507C22—21no—
508C22—21no—
509C22—22no—
510C22—22no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—22no—
516C22—24no—
517C22—22no—
518C22—24no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22C4×#12011yescharacteristic
530C22C4×C22⋊Q811yescharacteristic
531C22#232611yescharacteristic
532C22#236012yes—
533C22#236712yes—
534C22#240411yescharacteristic
535C22#240911yescharacteristic
536C22#245812yes—
537C22#246512yes—
538C22#258011yescharacteristic
539C22#259911yescharacteristic
540C22#302612yes—
541C22#302812yes—
542C4—44no—
543C4—44no—
544C4—42no—
545C4—42no—
546C4—44no—
547C4—48no—
548C4—82no—
549C4—82no—
550C4—84no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—162no—
555C2—21no—
556C2—21no—
557C2—22no—
558C2—22no—
559C2—21no—
560C2—22no—
561C2—22no—
562C2—21no—
563C2#1417211yescharacteristic
564C2#2148112yes—
565C2#2398711yescharacteristic
566C2#2398911yescharacteristic
567C2#2435312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
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
28. #2030C4semidirect
92. #381Q8semidirect
348. C22×C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
#342702
C2×#20302
#142632
#137762
#113902
#113442
#110702
#109362
#109182
#109122
#108012