Finite Group Databasebeta

← #163018

Subgroups — #163018

Order 512.

1360 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3#20626C211yesmaximal, characteristic, complemented
4#7714C214yesmaximal, complemented
5#7698C214yesmaximal, complemented
6#7607C212yesmaximal, complemented
7#7606C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#423—21no—
13C2×#407—21no—
14#2854—21no—
15#2854—21no—
16#2854—21no—
17#2854—21no—
18#2854C412yes—
19#2854C412yes—
20#2854C2212yes—
21#2854C2212yes—
22C2×#390C411yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#388—21no—
27C2×#388—21no—
28C2×#386—21no—
29C2×#380C411yescharacteristic
30#1817C2212yes—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yescomplemented
35#1797C2212yescomplemented
36#1797C2212yescomplemented
37#1779—21no—
38#1749—21no—
39#1749C2212yescomplemented
40#1728C2212yes—
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C23×D4—21no—
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C22×M16—22no—
49C2×#123—22no—
50C2×#123—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—48no—
54C2×#121—21no—
55C2×#121—21no—
56C2×#121—44no—
57C2×#121C2×C412yes—
58C2×#121C2×C412yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—44no—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#432—44no—
84#423—44no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88#407—44no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—22no—
95#390—21no—
96#390—21no—
97#390—21no—
98#390—21no—
99#390C2×C412yes—
100#390C2×C412yes—
101#390C2×C412yes—
102#390C2312yes—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389C2×C412yes—
108#389C2×C412yes—
109#389C2×C412yes—
110#389C2312yes—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#388—22no—
119#386—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#380—21no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C412yes—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130#336—22no—
131#336—22no—
132#336C2314yes—
133#336C2314yes—
134C25—21no—
135C25C22⋊C411yescharacteristic
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4C22×C412yes—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
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×C411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C2×M16—44no—
178C2×M16—44no—
179C2×M16—44no—
180C22×C8—24no—
181C22×C8—24no—
182#123—48no—
183#123—48no—
184#123—48no—
185#123—48no—
186#121—44no—
187#121—44no—
188#121—44no—
189#121—44no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C22⋊C411yescharacteristic
237C2×C42C22⋊C411yescharacteristic
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2411yescharacteristic
252C4⋊C8—44no—
253C4⋊C8—44no—
254C4⋊C8—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257C22⋊C8—44no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260#97—44no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—41no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—88no—
298C2×D4—816no—
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—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
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—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
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—416no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—416no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4#9914yes—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4AGL(1,ℤ/8)12yes—
379C22×C4AGL(1,ℤ/8)14yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—88no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—44no—
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—28no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—84no—
461C23—84no—
462C23—84no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23C2×#9912yes—
468C23#41712yes—
469C23#41812yes—
470C23#42112yes—
471C23#42112yes—
472C23C22×C22⋊C411yescharacteristic
473D4—84no—
474D4—84no—
475D4—88no—
476D4—88no—
477C2×C4—28no—
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—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4C2×#9912yes—
539C2×C4C2×#9912yes—
540C2×C4#41312yes—
541C2×C4#41712yes—
542C2×C4#41812yes—
543C2×C4#56812yes—
544C2×C4C2×AGL(1,ℤ/8)12yes—
545C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
546C2×C4C2×#13711yescharacteristic
547C8—84no—
548C8—84no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—416no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—168no—
578C22#151314yes—
579C22#287911yescharacteristic
580C22C22×#9911yescharacteristic
581C22C2×#41311yescharacteristic
582C22C2×#41811yescharacteristic
583C22C2×#41711yescharacteristic
584C22C2×#42111yescharacteristic
585C22C2×#42111yescharacteristic
586C4—21no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—82no—
600C4—82no—
601C4—82no—
602C2—24no—
603C2—42no—
604C2—42no—
605C2—162no—
606C2#1739211yescharacteristic
607C2#1749211yescharacteristic
608C2#1749511yescharacteristic
609C2C2×#151311yescharacteristic
610C2#1757711yescharacteristic
611C2#1766811yescharacteristic
612C2#1774511yescharacteristic
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. #20715C2semidirect
3. #20626C2semidirect
4. #7714C2semidirect
5. #7698C2semidirect
6. #7607C2semidirect
7. #7606C2semidirect
34. #1797C22semidirect
35. #1797C22semidirect
36. #1797C22semidirect
39. #1749C22semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
#207152
#206262
#77142
#76982
#76072
#76062