Finite Group Databasebeta

← #388071

Subgroups — #388071

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1870C211yesmaximal, characteristic
2#13905C211yesmaximal, characteristic
3#13871C211yesmaximal, characteristic
4#11268C212yesmaximal, complemented
5#11096C212yesmaximal, complemented
6#11096C212yesmaximal, complemented
7#10907C212yesmaximal, complemented
8#10731C212yesmaximal, complemented
9#10722C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#382C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1870C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31#1445—21no—
32#1440—21no—
33#1438—21no—
34#1438—21no—
35#1437—21no—
36#1437—21no—
37#1437—21no—
38#1436—21no—
39C4⋊#95C2212yes—
40C23×D4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—41no—
44C22×M16—41no—
45C22×M16—42no—
46C22×M16—42no—
47C22×M16C2×C414yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4Q814yes—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117—41no—
59C2×#117—41no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4D411yescharacteristic
71C22×C22⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—42no—
77C2×#107—42no—
78C2×C22⋊C8—42no—
79C2×C22⋊C8—42no—
80C2×C22⋊C8—41no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8C2×C418yes—
83C2×C22⋊C8C2×C414yes—
84#382—41no—
85#382—41no—
86#380—24no—
87#380—24no—
88#380—24no—
89#380—42no—
90#380—42no—
91#380—42no—
92#380—41no—
93#380—41no—
94C4×C4⋊C4—41no—
95C4×C4⋊C4—41no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105#340—24no—
106#340—42no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340—42no—
111#340—42no—
112#336C2×C418yes—
113C25C2×D411yescharacteristic
114C25C2×D411yescharacteristic
115C22×D4—41no—
116C22×D4—41no—
117C22×D4—42no—
118C22×D4—42no—
119C22×D4C4⋊C414yes—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×Q812yes—
147C23×C4C2×Q812yes—
148C23×C4C2411yescharacteristic
149C2×M16—48no—
150C2×M16—44no—
151C2×M16—44no—
152C2×M16—48no—
153C2×M16—44no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C22×C414yes—
167C22×C8C22×C414yes—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171C4×D4—44no—
172#117—44no—
173#117—48no—
174#117—44no—
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—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4C4⋊C414yes—
200C2×C4⋊C4#4114yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—82no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4C4⋊C414yes—
220C2×C22⋊C4C4⋊C414yes—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—44no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—82no—
229C2×C42C22×C412yes—
230C2×C42C2×Q812yes—
231#107—48no—
232C22⋊C8—44no—
233C22⋊C8—48no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—42no—
242#95—82no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24C2×C4⋊C412yes—
260C24C2×C4⋊C412yes—
261C24C22×D411yescharacteristic
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—48no—
266C2×D4—48no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
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—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
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—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4C2×C4⋊C412yes—
348C22×C4C2×C4⋊C412yes—
349C22×C4C2×C4⋊C412yes—
350C22×C4C2×C4⋊C412yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4112yes—
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361M16—416no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—48no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C22⋊C4—84no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—48no—
405C42—84no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23C22×C4⋊C411yescharacteristic
449C23#53411yescharacteristic
450C23#53411yescharacteristic
451C23#54612yes—
452C23#54612yes—
453C23#57512yes—
454C23#57512yes—
455C23#57612yes—
456C23#57812yes—
457D4—416no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4C4×#4112yes—
519C2×C4#51811yescharacteristic
520C2×C4#51812yes—
521C2×C4#51811yescharacteristic
522C2×C4#53812yes—
523C2×C4#53812yes—
524C2×C4#54012yes—
525C2×C4#54012yes—
526C2×C4#54311yescharacteristic
527C2×C4#54311yescharacteristic
528C2×C4#55111yescharacteristic
529C2×C4#55312yes—
530C2×C4#55412yes—
531C2×C4#55412yes—
532C2×C4#55911yescharacteristic
533C2×C4#56011yescharacteristic
534C2×C4#56012yes—
535C2×C4H(3,𝔽4)11yescharacteristic
536C8—44no—
537C8—44no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22#231511yescharacteristic
557C22#234912yes—
558C22#268411yescharacteristic
559C22#268812yes—
560C22#270911yescharacteristic
561C22#291111yescharacteristic
562C22#291111yescharacteristic
563C22#291312yes—
564C22#319712yes—
565C22#319712yes—
566C22#319911yescharacteristic
567C22#320211yescharacteristic
568C22#320312yes—
569C22#330912yes—
570C22#331312yes—
571C22#331512yes—
572C22#331912yes—
573C22#332212yes—
574C22#332212yes—
575C22#332712yes—
576C22#332712yes—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—84no—
582C4—84no—
583C4—162no—
584C4—162no—
585C4#233312yes—
586C4#237411yescharacteristic
587C4#237511yescharacteristic
588C4#274811yescharacteristic
589C4#278012yes—
590C4#280411yescharacteristic
591C2—82no—
592C2—82no—
593C2#1451311yescharacteristic
594C2#2194311yescharacteristic
595C2#2194711yescharacteristic
596C2#2271212yes—
597C2#2271612yes—
598C2#3127412yes—
599C2#3139912yes—
600C2#3140512yes—
601C2#3204012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11268C2semidirect
5. #11096C2semidirect
6. #11096C2semidirect
7. #10907C2semidirect
8. #10731C2semidirect
9. #10722C2semidirect
22. #1870C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18702
#139052
#138712
#112682
#110962
#109072
#107312
#107222