Finite Group Databasebeta

← #196025

Subgroups — #196025

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1467C212yesmaximal, complemented
5#9085C214yesmaximal
6#9082C214yesmaximal, complemented
7#7912C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yes—
14C2×#432C2212yes—
15C2×#431—21no—
16C2×#431C2212yes—
17C2×#423—21no—
18C2×#407C2212yes—
19C2×C4×M16—21no—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27C4×C22⋊Q8—21no—
28C2×#391C411yescharacteristic
29C2×#389C411yescharacteristic
30C2×#387C2212yes—
31C2×#385C2212yes—
32C2×#380C411yescharacteristic
33C2×C4×C4⋊C4C412yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C412yes—
36#1817C2212yes—
37#1797—21no—
38#1797—21no—
39#1779C2212yes—
40#1749—21no—
41#1749—21no—
42#1467—28no—
43C2×#324C2218yes—
44#1280C2218yes—
45C23×Q8C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—28no—
55C2×C4×Q8—24no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—24no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4C2×C412yes—
68C22×C22⋊C4C2×C412yes—
69C22×C42—24no—
70C22×C42—24no—
71C22×C42C2×C412yes—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C412yes—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#432—28no—
86#432—24no—
87#432—24no—
88#432—24no—
89#432D418yes—
90#431—28no—
91#431—28no—
92#423—28no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96#407—28no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C4×M16—28no—
103C2×#97—22no—
104C2×#97—22no—
105C2×C4×C8—22no—
106#391—21no—
107#391—21no—
108#391—21no—
109#391—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C2×#95C2×C412yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2312yes—
143#336—22no—
144#336—22no—
145#324—216no—
146C25C22×C411yescharacteristic
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—42no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C22×C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2411yescharacteristic
184C2×M16—28no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—24no—
188C22×C8—24no—
189C22⋊Q8—28no—
190C22⋊Q8—28no—
191C22⋊Q8—28no—
192C22⋊Q8—28no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—44no—
196C4×Q8—44no—
197C4×Q8—42no—
198C4×Q8—42no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4M1618yescomplemented
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—44no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—28no—
247C2×C42—44no—
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D414yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42#4114yes—
266C4⋊C8—28no—
267C4⋊C8—28no—
268C4⋊C8—44no—
269C22⋊C8—28no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8C2×D418yes—
276#97—28no—
277#97—44no—
278C4×C8—28no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24C23×C411yescharacteristic
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—416no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—416no—
368C22×C4—416no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4#10114yes—
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C4×D414yes—
389C22×C4C4×D414yes—
390C22×C4C2×M1614yes—
391C22×C4C2×M1614yes—
392C22×C4C2×M1614yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C2×#4112yes—
397M16—416no—
398C2×C8—28no—
399C2×C8—44no—
400C2×C8—44no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—44no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—21no—
422C42—21no—
423C42—22no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—44no—
436C42—44no—
437C42—44no—
438C42—44no—
439C42#12012yes—
440C42#12012yes—
441C42#12114yes—
442C42#12114yes—
443C42#12312yes—
444C42#12312yes—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—28no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23#41012yes—
469C23C2×#10112yes—
470C23C22×C22⋊C411yescharacteristic
471C23C2×C4×D412yes—
472C23C22×M1612yes—
473C23#56812yes—
474Q8—48no—
475Q8—48no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—28no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4#38612yes—
540C2×C4#38612yes—
541C2×C4#38612yes—
542C2×C4#38612yes—
543C2×C4#40712yes—
544C2×C4#40712yes—
545C2×C4#41012yes—
546C2×C4C2×#10112yes—
547C2×C4C2×#10112yes—
548C2×C4#41312yes—
549C2×C4C2×#12011yescharacteristic
550C2×C4C2×#12112yes—
551C2×C4C2×#12311yescharacteristic
552C8—48no—
553C8—84no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—416no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#38611yescharacteristic
576C22C2×#40711yescharacteristic
577C22#287911yescharacteristic
578C22C2×#41011yescharacteristic
579C22C2×#41011yescharacteristic
580C22C22×#10111yescharacteristic
581C22C2×#41311yescharacteristic
582C22D4×M1614yes—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—21no—
588C4—22no—
589C4—21no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—44no—
594C4—44no—
595C4—41no—
596C4—41no—
597C4—82no—
598C4—82no—
599C4#190114yes—
600C2—24no—
601C2—42no—
602C2—42no—
603C2#1736111yescharacteristic
604C2#1736511yescharacteristic
605C2#1740211yescharacteristic
606C2#1740411yescharacteristic
607C2#2089811yescharacteristic
608C2#2098311yescharacteristic
609C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1467C2semidirect
6. #9082C2semidirect
7. #7912C2semidirect
218. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×C4×C22⋊Q82
#207152
C2×#14672
#90852
#90822
#79122