Finite Group Databasebeta

← #161502

Subgroups — #161502

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C22×#336C211yesmaximal, characteristic, complemented
4#7690C214yesmaximal, complemented
5#7667C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7578C214yesmaximal, complemented
8C24×C8—21no—
9C2×#532C411yescharacteristic
10C2×#525C412yes—
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13#2857C411yescharacteristic
14#2857C2211yescharacteristic
15#2856C411yescharacteristic
16#2856C2211yescharacteristic
17#2854—21no—
18#2854—21no—
19#2850C412yes—
20#2850C2212yes—
21#2847—21no—
22#2847—21no—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27#2297C2212yes—
28#2285C411yescharacteristic
29#1817C2212yes—
30#1797—21no—
31#1797C2212yes—
32#1779—21no—
33#1749—21no—
34#1749C2212yes—
35C2×#336—21no—
36C2×#336—21no—
37C2×#336—21no—
38C2×#336C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C23×C8—28no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—44no—
52C23×C8—44no—
53#532—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#124C2×C412yes—
57#525—48no—
58C2×#123—22no—
59C2×C22⋊Q8C2×C412yes—
60C2×C22⋊Q8C2×C412yes—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×#121C2×C412yes—
69C2×#120—22no—
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×#117C2×C412yes—
75C22×C42—44no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—44no—
89C2×C4⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94#391C2×C412yes—
95#391C2312yes—
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2312yes—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382C2×C412yes—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25C22⋊C411yescharacteristic
136C2×#41—28no—
137C2×#41—44no—
138C2×#41—44no—
139C22×Q8C22×C412yes—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4C22×C412yes—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
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×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C22×C8—28no—
176C22×C8—28no—
177C22×C8—28no—
178C22×C8—24no—
179C22×C8—24no—
180C22×C8—432no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8—44no—
184C22×C8—44no—
185C22×C8—44no—
186C22×C8—44no—
187C4⋊Q8—42no—
188#127—42no—
189#124—44no—
190#123—48no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193#121—44no—
194#121—44no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—44no—
199#121—44no—
200#121—42no—
201#121—42no—
202#120—48no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—48no—
206C4×D4—48no—
207#117—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
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×C42—24no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2411yescharacteristic
258C4⋊C8—44no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262#97—44no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24#10014yes—
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280#41—816no—
281C2×Q8—24no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×D4—24no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C22×C4—216no—
303C22×C4—216no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
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—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—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—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—41no—
358C22×C4—416no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—416no—
362C22×C4—44no—
363C22×C4#9914yes—
364C22×C4#10414yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4#13114yes—
375C22×C4#13514yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4#13712yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C2×C8—28no—
382C2×C8—416no—
383C2×C8—416no—
384C2×C8—44no—
385C2×C8—416no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C2×C8—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—44no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—28no—
447C23—28no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23#40812yes—
456C23C2×#9912yes—
457C23C2×#10012yes—
458C23C2×#10012yes—
459C23C2×#10012yes—
460C23#41612yes—
461C23#41712yes—
462C23C2×#10412yes—
463C23#42112yes—
464C23C22×C22⋊C411yescharacteristic
465Q8—84no—
466D4—84no—
467D4—84no—
468D4—88no—
469D4—88no—
470D4—88no—
471D4—84no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—21no—
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—28no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—28no—
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—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—88no—
529C2×C4#40812yes—
530C2×C4C2×#9912yes—
531C2×C4C2×#9912yes—
532C2×C4#41612yes—
533C2×C4#41712yes—
534C2×C4C2×#10412yes—
535C2×C4C2×#10412yes—
536C2×C4C2×#13112yes—
537C2×C4C2×#13512yes—
538C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
539C2×C4C2×#13711yescharacteristic
540C8—48no—
541C8—84no—
542C22—232no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
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—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22#147114yes—
569C22#153014yes—
570C22#155114yes—
571C22C2×#40811yescharacteristic
572C22C22×#9911yescharacteristic
573C22C22×#10011yescharacteristic
574C22C2×#41611yescharacteristic
575C22C2×#41711yescharacteristic
576C22C22×#10411yescharacteristic
577C22C2×#42111yescharacteristic
578C4—21no—
579C4—21no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4#150414yes—
594C2—24no—
595C2—24no—
596C2—24no—
597C2—162no—
598C2C2×#147111yescharacteristic
599C2#1746611yescharacteristic
600C2#1747311yescharacteristic
601C2C2×#150411yescharacteristic
602C2#1757211yescharacteristic
603C2C2×#153011yescharacteristic
604C2C2×#155111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36624C2
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. C22×#336C2semidirect
4. #7690C2semidirect
5. #7667C2semidirect
6. #7664C2semidirect
7. #7578C2semidirect
41. #1272C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#207152
C22×#3362
#76902
#76672
#76642
#75782