Finite Group Databasebeta

← #363352

Subgroups — #363352

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11200C212yesmaximal
5#11199C212yesmaximal
6#10843C211yesmaximal, characteristic, complemented
7#10843C211yesmaximal, characteristic, complemented
8#10842C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10807C212yesmaximal, complemented
11#10806C212yesmaximal, complemented
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2314—24no—
18#2314C418yescomplemented
19C2×#388—22no—
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2212yes—
28#1750—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1441—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C24×C4C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117C2×C414yes—
52C2×#117D414yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78#388—216no—
79#387—216no—
80#384—22no—
81#384—22no—
82#384—22no—
83#384C2×C414yes—
84#382—22no—
85#382—22no—
86#382—22no—
87#382C2×C414yes—
88#380—24no—
89#380—42no—
90#380—42no—
91#380C2×C418yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4C2×C418yes—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111C25C2×D411yescharacteristic
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D412yes—
132C23×C4C2×D412yes—
133C23×C4C2411yescharacteristic
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8#4112yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162#117—28no—
163#117—28no—
164#117—28no—
165#117—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22⋊C418yes—
197C2×C4⋊C4C22×C414yes—
198C2×C4⋊C4C22×C414yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4C22⋊C418yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—28no—
213C2×C42—82no—
214C2×C42—82no—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C22⋊C8—28no—
224C22⋊C8—28no—
225#97—44no—
226C4×C8—44no—
227#95—28no—
228#95—28no—
229#95—28no—
230#95—44no—
231#95—44no—
232#95—82no—
233#95—82no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—21no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24C2×C22⋊C412yes—
245C24C22×D411yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
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—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C414yes—
320C22×C4C2×C22⋊C414yes—
321C22×C4#12314yes—
322C22×C4#12314yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)11yescharacteristic
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336C22×C42−1+411yescharacteristic
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C4⋊C4—216no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—216no—
385C22⋊C4—88no—
386C22⋊C4—88no—
387C42—22no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—88no—
392C42#11714yes—
393C42#11714yes—
394C42#12314yes—
395C42#12314yes—
396C23—22no—
397C23—24no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C22×C22⋊C411yescharacteristic
420C23C2×#12311yescharacteristic
421C23C2×#12311yescharacteristic
422C23#53211yescharacteristic
423C23#53211yescharacteristic
424C23#53511yescharacteristic
425C23#53611yescharacteristic
426C23#57612yes—
427C23#57712yes—
428C23#57712yes—
429C23#57812yes—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#38118yes—
487C2×C4C2×#11711yescharacteristic
488C2×C4C2×#11711yescharacteristic
489C2×C4C4×#4111yescharacteristic
490C2×C4C4×#4111yescharacteristic
491C2×C4#52011yescharacteristic
492C2×C4#52011yescharacteristic
493C2×C4C2×#12311yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4#52511yescharacteristic
496C2×C4#52511yescharacteristic
497C2×C4#53711yescharacteristic
498C2×C4#53711yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55211yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55311yescharacteristic
504C2×C4#55311yescharacteristic
505C2×C4#55311yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55511yescharacteristic
508C2×C4#55511yescharacteristic
509C2×C4#55511yescharacteristic
510C2×C4#55511yescharacteristic
511C2×C4#55611yescharacteristic
512C2×C4#55611yescharacteristic
513C2×C4#55611yescharacteristic
514C2×C4#55611yescharacteristic
515C8—42no—
516C8—42no—
517C8—42no—
518C8—42no—
519C22—44no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—44no—
524C22—82no—
525C22—81no—
526C22—81no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22C2×#38111yescharacteristic
531C22#229711yescharacteristic
532C22#231111yescharacteristic
533C22#249911yescharacteristic
534C22#249911yescharacteristic
535C22#250011yescharacteristic
536C22#250011yescharacteristic
537C22#289712yes—
538C22#289812yes—
539C22#309612yes—
540C22#309712yes—
541C22#309812yes—
542C22#309912yes—
543C22#317411yescharacteristic
544C22#317411yescharacteristic
545C22#317611yescharacteristic
546C22#317711yescharacteristic
547C22#317812yes—
548C22#317912yes—
549C22#322412yes—
550C22#322512yes—
551C22#324312yes—
552C22#324412yes—
553C4—28no—
554C4—42no—
555C4—44no—
556C4—84no—
557C4—84no—
558C4—162no—
559C4—162no—
560C4#234611yescharacteristic
561C4#234611yescharacteristic
562C4#234711yescharacteristic
563C4#234711yescharacteristic
564C4#248511yescharacteristic
565C4#248511yescharacteristic
566C4#248511yescharacteristic
567C4#248511yescharacteristic
568C2—82no—
569C2#1393911yescharacteristic
570C2#2070312yes—
571C2#2141311yescharacteristic
572C2#2141411yescharacteristic
573C2#2187112yes—
574C2#2788411yescharacteristic
575C2#2788511yescharacteristic
576C2#2788712yes—
577C2#2790011yescharacteristic
578C2#2790111yescharacteristic
579C2#2790412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #10843C2semidirect
7. #10843C2semidirect
8. #10842C2semidirect
9. #10841C2semidirect
10. #10807C2semidirect
11. #10806C2semidirect
18. #2314C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#17502
#143402
#112002
#111992
#108432
#108422
#108412
#108072
#108062