Finite Group Databasebeta

← #161044

Subgroups — #161044

Order 512.

1636 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C2×#1817C211yesmaximal, characteristic, complemented
3#20626C211yesmaximal, characteristic, complemented
4#7647C214yesmaximal, complemented
5#7610C212yesmaximal, complemented
6#7606C212yesmaximal, complemented
7#7586C214yesmaximal, complemented
8C22×#121C411yescharacteristic
9C22×C4×D4—21no—
10C22×C4×D4C411yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407—21no—
14C22×C22⋊C8—21no—
15C4×#121—21no—
16C4×#121—21no—
17C4×#121—21no—
18C4×#121—21no—
19C4×#121C412yes—
20C4×#121C412yes—
21C4×#121C2212yes—
22C4×#121C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#386—21no—
27C2×#380C411yescharacteristic
28C2×C4×C4⋊C4C411yescharacteristic
29C2×C4×C22⋊C4—21no—
30#1817—21no—
31#1817—21no—
32#1817C2212yes—
33#1817C2212yescomplemented
34#1797—21no—
35#1797—21no—
36#1797C2212yescomplemented
37#1797C2212yescomplemented
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C23×C8—22no—
51C2×#121—21no—
52C2×#121—21no—
53C2×#121—28no—
54C2×#121C2×C412yes—
55C2×#121C2×C412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—44no—
63C2×C4×D4—44no—
64C2×C4×D4—44no—
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C42—21no—
77C22×C42—44no—
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#423—44no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93#407—44no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—44no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2312yes—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2312yes—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#380—21no—
116#380—21no—
117#380C2×C412yes—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2314yes—
135#336C2314yes—
136C25—21no—
137C25C22⋊C411yescharacteristic
138C22×D4—28no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—21no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4C22×C412yes—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4D814yes—
174C23×C4SD1614yes—
175C23×C4C22×C411yescharacteristic
176C23×C4C22×C411yescharacteristic
177C23×C4C22×C411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—24no—
188C22×C8—24no—
189C22×C8—44no—
190C22×C8—44no—
191C22×C8—44no—
192#121—28no—
193#121—28no—
194#121—24no—
195#121—24no—
196#121—24no—
197#121M1618yescomplemented
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—44no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—44no—
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—44no—
269C4⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24#10214yes—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4—22no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—416no—
306C2×D4—432no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4C2×M1614yes—
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—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
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—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—416no—
366C22×C4—416no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—816no—
382C22×C4#9914yes—
383C22×C4#10214yes—
384C22×C4#10214yes—
385C22×C4#10214yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×M1614yes—
396C22×C4C2×D812yes—
397C22×C4C2×D812yes—
398C22×C4C2×D812yes—
399C22×C4C2×SD1612yes—
400C22×C4C2×SD1612yes—
401C22×C4C2×SD1612yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405M16—88no—
406C2×C8—44no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—88no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4C2×M1614yes—
423C22⋊C4—28no—
424C22⋊C4—28no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—44no—
451C42—44no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—48no—
475C23—48no—
476C23—48no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23#40712yes—
488C23C2×#9912yes—
489C23C2×#10212yes—
490C23C2×#10212yes—
491C23C2×#10212yes—
492C23#42112yes—
493C23#42112yes—
494C23C22×C22⋊C411yescharacteristic
495D4—44no—
496D4—44no—
497D4—48no—
498D4—416no—
499D4—416no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—48no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—88no—
558C2×C4—88no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4#40712yes—
562C2×C4C2×#9912yes—
563C2×C4C2×#9912yes—
564C2×C4#41312yes—
565C2×C4C2×#10212yes—
566C2×C4C2×#10212yes—
567C2×C4C2×#10212yes—
568C2×C4C22×M1612yes—
569C2×C4C22×D811yescharacteristic
570C2×C4C22×SD1611yescharacteristic
571C2×C4#57512yes—
572C8—84no—
573C8—84no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—416no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—816no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22#148614yes—
603C22#151914yes—
604C22C2×#40711yescharacteristic
605C22C22×#9911yescharacteristic
606C22C2×#41311yescharacteristic
607C22C22×#10211yescharacteristic
608C22#289611yescharacteristic
609C22C2×#42111yescharacteristic
610C22C2×#42111yescharacteristic
611C4—21no—
612C4—22no—
613C4—22no—
614C4—21no—
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—22no—
620C4—22no—
621C4—22no—
622C4—44no—
623C4—41no—
624C4—41no—
625C4—82no—
626C4—82no—
627C2—24no—
628C2—24no—
629C2—42no—
630C2—84no—
631C2#1736211yescharacteristic
632C2C2×#148611yescharacteristic
633C2#1745211yescharacteristic
634C2C2×#151911yescharacteristic
635C2#1758711yescharacteristic
636C2#1766811yescharacteristic
637C2#1770411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1817C2semidirect
3. #20626C2semidirect
4. #7647C2semidirect
5. #7610C2semidirect
6. #7606C2semidirect
7. #7586C2semidirect
33. #1817C22semidirect
36. #1797C22semidirect
37. #1797C22semidirect
40. C2×#336C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect
197. #121M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C2×#18172
#206262
#76472
#76102
#76062
#75862