Finite Group Databasebeta

← #162858

Subgroups — #162858

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic, complemented
4#7698C214yesmaximal, complemented
5#7668C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7633C214yesmaximal, complemented
8C2×#532C411yescharacteristic
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C2×#407—21no—
13C22×C22⋊C8—21no—
14C22×#97—21no—
15#2857C411yescharacteristic
16#2857C2211yescharacteristic
17#2856C411yescharacteristic
18#2856C2211yescharacteristic
19#2854—21no—
20#2854C2212yes—
21#2850—21no—
22#2850—21no—
23#2847—21no—
24#2847C412yes—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121C412yes—
28C4×#121C2212yes—
29#2297—21no—
30#2285C411yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797C2212yes—
34#1797C2212yes—
35#1749—21no—
36#1749—21no—
37#1749C2212yes—
38#1749C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1271C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41—21no—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49#532—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124—21no—
53#525—48no—
54C2×#123—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C412yes—
63C2×#121C2×C412yes—
64C2×#121C2×C412yes—
65C2×#120—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C412yes—
70C2×#117—21no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#423—44no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87#407—44no—
88C2×C22⋊C8—28no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—44no—
96#391—21no—
97#391—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389—21no—
106#389C2×C412yes—
107#389C2312yes—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#382—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380C2×C412yes—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C412yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—22no—
131#336—22no—
132#336—22no—
133#336C2314yes—
134#336C2314yes—
135C25C22⋊C411yescharacteristic
136C2×#41—44no—
137C2×#41—44no—
138C2×#41—44no—
139C22×Q8—21no—
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—22no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
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
175C2×M16—44no—
176C2×M16—44no—
177C2×M16—44no—
178C22×C8—28no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C4⋊Q8—42no—
184#127—42no—
185#124—44no—
186#123—48no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189#121—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—42no—
195#121—44no—
196#121—42no—
197#121—44no—
198#120—48no—
199C4×D4—44no—
200C4×D4—44no—
201C4×D4—48no—
202C4×D4—48no—
203#117—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
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×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22⋊C411yescharacteristic
239C2×C42C22⋊C411yescharacteristic
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2411yescharacteristic
254C4⋊C8—44no—
255C4⋊C8—44no—
256C22⋊C8—28no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260#97—44no—
261#97—44no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278C24C2×C22⋊C411yescharacteristic
279#41—816no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×D4—24no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
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—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
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—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
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—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—41no—
356C22×C4—41no—
357C22×C4—416no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—416no—
363C22×C4—44no—
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×C4AGL(1,ℤ/8)14yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4#13712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380M16—88no—
381C2×C8—28no—
382C2×C8—44no—
383C2×C8—416no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C4⋊C4—24no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—44no—
426C42#9914yes—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—28no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23#41312yes—
458C23#41712yes—
459C23#41812yes—
460C23C2×#10412yes—
461C23#42112yes—
462C23C22×C22⋊C411yescharacteristic
463Q8—84no—
464D4—84no—
465D4—84no—
466D4—88no—
467D4—88no—
468D4—88no—
469D4—84no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
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—28no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—88no—
527C2×C4C2×#9912yes—
528C2×C4C2×#9912yes—
529C2×C4C2×#9912yes—
530C2×C4#41712yes—
531C2×C4#41812yes—
532C2×C4C2×#10412yes—
533C2×C4C2×#10412yes—
534C2×C4#56812yes—
535C2×C4C2×AGL(1,ℤ/8)12yes—
536C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
537C2×C4C2×#13711yescharacteristic
538C8—48no—
539C8—84no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—416no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22#155714yes—
569C22#287911yescharacteristic
570C22C22×#9911yescharacteristic
571C22C2×#41311yescharacteristic
572C22C2×#41811yescharacteristic
573C22C2×#41711yescharacteristic
574C22C22×#10411yescharacteristic
575C22C2×#42111yescharacteristic
576C4—22no—
577C4—22no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4#150714yes—
592C2—24no—
593C2—42no—
594C2—42no—
595C2—162no—
596C2#1740011yescharacteristic
597C2#1749511yescharacteristic
598C2#1750411yescharacteristic
599C2C2×#150711yescharacteristic
600C2#1759211yescharacteristic
601C2#1764611yescharacteristic
602C2C2×#155711yescharacteristic
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. #20626C2semidirect
3. C2×#1749C2semidirect
4. #7698C2semidirect
5. #7668C2semidirect
6. #7664C2semidirect
7. #7633C2semidirect
41. #1271C22semidirect
42. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#206262
C2×#17492
#76982
#76682
#76642
#76332