Finite Group Databasebeta

← #363574

Subgroups — #363574

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, 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
12C22×#117C412yes—
13C23×C4⋊C4C2211yescharacteristic
14C2×#423C2212yes—
15C2×#407C2212yes—
16C2×#407C2212yes—
17C2×C4×M16C2212yes—
18#2344—22no—
19#2344—22no—
20#2344—22no—
21#2344—22no—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387C412yes—
25C2×#387C412yes—
26C2×#381C412yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
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—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—48no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C412yes—
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C42C2×C412yes—
74C22×C42C2×C412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#423—24no—
78#423—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81#407—24no—
82#407—24no—
83#407—42no—
84#407—42no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C4×M16—24no—
90C4×M16—42no—
91C2×#97—22no—
92C2×C4×C8—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#387—24no—
98#387—24no—
99#387—24no—
100#387—24no—
101#381—24no—
102#381—24no—
103#380—22no—
104#380—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C411yescharacteristic
119C2×#95C2×C411yescharacteristic
120#343—24no—
121#343—24no—
122#343—22no—
123#343—22no—
124#343—24no—
125#343—24no—
126#343—22no—
127#343—22no—
128#343—42no—
129#343—42no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134C25C22×C411yescharacteristic
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2411yescharacteristic
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—22no—
184C2×M16#4114yes—
185C2×M16#4114yes—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190#117—48no—
191#117—48no—
192#117—48no—
193#117—48no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4#4112yes—
221C2×C4⋊C4#4112yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C22⋊C412yes—
232C2×C22⋊C4C22⋊C412yes—
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C4⋊C8—44no—
251C4⋊C8—44no—
252C22⋊C8—44no—
253C22⋊C8—44no—
254C22⋊C8—44no—
255C22⋊C8—44no—
256#97—44no—
257C4×C8—44no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C22⋊C412yes—
266#95C22⋊C412yes—
267C24—21no—
268C24—21no—
269C24—22no—
270C24—22no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—21no—
275C24—41no—
276C24—41no—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279C24C23×C411yescharacteristic
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
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—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—44no—
327C22×C4—416no—
328C22×C4—416no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C412yes—
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4#11714yes—
358C22×C4#11714yes—
359C22×C4C22×D412yes—
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+411yescharacteristic
368C22×C42−1+411yescharacteristic
369M16—24no—
370M16—24no—
371M16—24no—
372M16—24no—
373M16—42no—
374M16—42no—
375M16—42no—
376M16—42no—
377M16—42no—
378M16—42no—
379M16—42no—
380M16—42no—
381M16—42no—
382M16—42no—
383M16—42no—
384M16—42no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C4⋊C4—216no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C22⋊C4—22no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4#10014yes—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42—44no—
428C42—44no—
429C42—44no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—24no—
435C23—24no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23C2×#10012yes—
463C23#41312yes—
464C23C22×C22⋊C411yescharacteristic
465C23C2×#11711yescharacteristic
466C23C2×#11711yescharacteristic
467C23C4×#4111yescharacteristic
468C23C4×#4111yescharacteristic
469C23#52011yescharacteristic
470C23#52011yescharacteristic
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—416no—
499C2×C4—416no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4C2×#10012yes—
533C2×C4C2×#10012yes—
534C2×C4#52512yes—
535C2×C4#52512yes—
536C2×C4#52912yes—
537C2×C4#52912yes—
538C2×C4#54011yescharacteristic
539C2×C4#54111yescharacteristic
540C2×C4#54211yescharacteristic
541C2×C4#54211yescharacteristic
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54712yes—
545C2×C4#54712yes—
546C2×C4#55212yes—
547C2×C4#55312yes—
548C2×C4#55512yes—
549C2×C4#55612yes—
550C8—44no—
551C8—44no—
552C8—82no—
553C8—82no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22#228011yescharacteristic
585C22#229811yescharacteristic
586C22#231411yescharacteristic
587C22#233911yescharacteristic
588C22#233911yescharacteristic
589C22#234611yescharacteristic
590C22#234711yescharacteristic
591C22C22×#10011yescharacteristic
592C22C2×#41311yescharacteristic
593C22#288911yescharacteristic
594C22#288911yescharacteristic
595C4—22no—
596C4—22no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—44no—
603C4—44no—
604C4—82no—
605C4—82no—
606C4#247012yes—
607C4#247412yes—
608C4#247612yes—
609C4#247812yes—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—42no—
615C2—42no—
616C2#1394311yescharacteristic
617C2#2072611yescharacteristic
618C2#2073011yescharacteristic
619C2#2247911yescharacteristic
620C2#2247911yescharacteristic
621C2#2248011yescharacteristic
622C2#2248011yescharacteristic
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×#23442
#210492
#112042
#111982
#109132
#108472
#108022
#107272
C4×#3432