Finite Group Databasebeta

← #363575

Subgroups — #363575

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2345C211yesmaximal, characteristic
2#21049C212yesmaximal
3#11204C212yesmaximal, complemented
4#11198C212yesmaximal
5#10913C212yesmaximal, complemented
6#10847C211yesmaximal, characteristic, complemented
7#10802C211yesmaximal, characteristic, complemented
8#10802C211yesmaximal, characteristic, complemented
9#10727C211yesmaximal, characteristic
10#10727C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic, complemented
12C23×C4⋊C4C2211yescharacteristic
13C2×#423C2212yes—
14C2×#407C2212yes—
15C2×#407C2212yes—
16C2×C4×M16C2212yes—
17#2345—22no—
18#2345—22no—
19#2345—22no—
20#2345—22no—
21C2×#389C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387C412yes—
24C2×#387C412yes—
25C2×#380C412yes—
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C412yes—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1924—22no—
32#1924—22no—
33#1923—22no—
34#1923—22no—
35C2×#343C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2212yes—
38C2×#343C2211yescharacteristic
39C2×#343C2211yescharacteristic
40C2×#343C2212yes—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45#1728C2212yes—
46#1728C2212yes—
47C24×C4C2×C412yes—
48C24×C4C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C22×M16C2312yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C42C2×C412yes—
69C22×C42C2×C412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#423—24no—
73#423—42no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76#407—24no—
77#407—24no—
78#407—42no—
79#407—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C4×M16—24no—
85C4×M16—42no—
86C2×#97—22no—
87C2×C4×C8—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#380—22no—
97#380—22no—
98#380—24no—
99#380—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4D412yes—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C2×#95C2×C412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C411yescharacteristic
118C2×#95C2×C411yescharacteristic
119#343—24no—
120#343—24no—
121#343—22no—
122#343—22no—
123#343—24no—
124#343—24no—
125#343—22no—
126#343—22no—
127#343—42no—
128#343—42no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C25C22×C411yescharacteristic
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16#4114yes—
184C2×M16#4114yes—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4#4112yes—
214C2×C4⋊C4#4112yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4C22⋊C411yescharacteristic
222C2×C22⋊C4C22⋊C411yescharacteristic
223C2×C22⋊C4C22⋊C411yescharacteristic
224C2×C22⋊C4C22⋊C412yes—
225C2×C22⋊C4C22⋊C412yes—
226C2×C22⋊C4C22⋊C411yescharacteristic
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—82no—
238C2×C42—82no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C4⋊C8—44no—
244C4⋊C8—44no—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—44no—
248C22⋊C8—44no—
249#97—44no—
250C4×C8—44no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—21no—
269C24—21no—
270C24—41no—
271C24—41no—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24C23×C411yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
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—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4#11714yes—
353C22×C4#11714yes—
354C22×C4C22×D412yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C42−1+411yescharacteristic
363C22×C42−1+411yescharacteristic
364M16—24no—
365M16—24no—
366M16—24no—
367M16—24no—
368M16—42no—
369M16—42no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379M16—42no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—216no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C22⋊C4—22no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4#10114yes—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—82no—
420C42—82no—
421C42—82no—
422C42—82no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23C2×#10112yes—
456C23#41312yes—
457C23C22×C22⋊C411yescharacteristic
458C23C2×#11711yescharacteristic
459C23C2×#11711yescharacteristic
460C23C4×#4111yescharacteristic
461C23C4×#4111yescharacteristic
462C23#52011yescharacteristic
463C23#52011yescharacteristic
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—88no—
520C2×C4—88no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4C2×#10112yes—
526C2×C4C2×#10112yes—
527C2×C4#52512yes—
528C2×C4#52512yes—
529C2×C4#52912yes—
530C2×C4#52912yes—
531C2×C4#54011yescharacteristic
532C2×C4#54111yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4#54211yescharacteristic
535C2×C4#54612yes—
536C2×C4#54612yes—
537C2×C4#54712yes—
538C2×C4#54712yes—
539C2×C4#55212yes—
540C2×C4#55312yes—
541C2×C4#55512yes—
542C2×C4#55612yes—
543C8—44no—
544C8—44no—
545C8—82no—
546C8—82no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22#228011yescharacteristic
578C22#229811yescharacteristic
579C22#231411yescharacteristic
580C22#233911yescharacteristic
581C22#233911yescharacteristic
582C22#234611yescharacteristic
583C22#234711yescharacteristic
584C22C22×#10111yescharacteristic
585C22C2×#41311yescharacteristic
586C22#288811yescharacteristic
587C22#288811yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—41no—
592C4—41no—
593C4—42no—
594C4—42no—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—82no—
599C4#247012yes—
600C4#247412yes—
601C4#247612yes—
602C4#247812yes—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—42no—
608C2—42no—
609C2#1394311yescharacteristic
610C2#2072711yescharacteristic
611C2#2072911yescharacteristic
612C2#2248111yescharacteristic
613C2#2248111yescharacteristic
614C2#2248211yescharacteristic
615C2#2248211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11204C2semidirect
5. #10913C2semidirect
6. #10847C2semidirect
7. #10802C2semidirect
8. #10802C2semidirect
11. C4×#343C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23452
#210492
#112042
#111982
#109132
#108472
#108022
#107272
C4×#3432