Finite Group Databasebeta

← #162818

Subgroups — #162818

Order 512.

1520 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1728C211yesmaximal, characteristic, complemented
4#7710C214yesmaximal, complemented
5#7667C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7631C214yesmaximal, complemented
8C23×M16—21no—
9C2×#532C411yescharacteristic
10C2×#525C412yes—
11C23×C42C2211yescharacteristic
12C2×#423—21no—
13C2×#407—21no—
14#2857C411yescharacteristic
15#2857C2211yescharacteristic
16#2856C411yescharacteristic
17#2856C2211yescharacteristic
18#2854—21no—
19#2854—21no—
20#2850C412yes—
21#2850C2212yes—
22#2847—21no—
23#2847—21no—
24C4×#121—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28#2297C2212yes—
29#2285C411yescharacteristic
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34#1728—21no—
35#1728—21no—
36#1728C2212yes—
37#1728C2212yes—
38#1728C2212yes—
39C2×#336—21no—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—28no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—44no—
52C22×M16—44no—
53C23×C8—22no—
54#532—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#127C2×C411yescharacteristic
57C2×#124C2×C412yes—
58#525—48no—
59C2×#123—22no—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8C2×C412yes—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×#121C2×C412yes—
70C2×#120—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×#117C2×C412yes—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#423—44no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92#407—44no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95#391C2×C412yes—
96#391C2312yes—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2312yes—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
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#386—22no—
116#386—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#382C2×C412yes—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336C2314yes—
135#336C2314yes—
136C25C22⋊C411yescharacteristic
137C2×#41—44no—
138C2×#41—44no—
139C2×#41—44no—
140C22×Q8C22×C412yes—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C22×C412yes—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
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—28no—
178C2×M16—28no—
179C2×M16—432no—
180C2×M16—44no—
181C2×M16—44no—
182C2×M16—44no—
183C2×M16—44no—
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—28no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—44no—
191C22×C8—44no—
192C4⋊Q8—42no—
193#127—42no—
194#124—44no—
195#123—48no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198#121—44no—
199#121—44no—
200#121—44no—
201#121—44no—
202#121—44no—
203#121—42no—
204#121—42no—
205#121—42no—
206#121—42no—
207#121—44no—
208#120—48no—
209C4×D4—44no—
210C4×D4—44no—
211C4×D4—48no—
212C4×D4—48no—
213#117—44no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
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—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
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×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—44no—
265C4⋊C8—44no—
266C22⋊C8—44no—
267C22⋊C8—44no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—41no—
281C24—41no—
282C24C2×C22⋊C411yescharacteristic
283C24C2×C22⋊C411yescharacteristic
284C24C2×C22⋊C411yescharacteristic
285#41—816no—
286C2×Q8—24no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—44no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C22×C4—216no—
308C22×C4—216no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
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—21no—
340C22×C4—21no—
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—41no—
352C22×C4—41no—
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—48no—
366C22×C4—48no—
367C22×C4—416no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4#9914yes—
372C22×C4#10014yes—
373C22×C4#10414yes—
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4AGL(1,ℤ/8)12yes—
385C22×C4#13712yes—
386C22×C4#13712yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390M16—416no—
391M16—416no—
392M16—88no—
393C2×C8—28no—
394C2×C8—44no—
395C2×C8—416no—
396C2×C8—44no—
397C2×C8—44no—
398C2×C8—44no—
399C2×C8—44no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—22no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—44no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—28no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23C2×#9912yes—
471C23C2×#10012yes—
472C23#41712yes—
473C23#41712yes—
474C23C2×#10412yes—
475C23C22×C22⋊C411yescharacteristic
476Q8—84no—
477D4—84no—
478D4—84no—
479D4—88no—
480D4—88no—
481D4—88no—
482D4—84no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—24no—
490C2×C4—28no—
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—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—88no—
541C2×C4C2×#9912yes—
542C2×C4C2×#9912yes—
543C2×C4C2×#10012yes—
544C2×C4C2×#10012yes—
545C2×C4#41712yes—
546C2×C4#41712yes—
547C2×C4C2×#10412yes—
548C2×C4C2×#10412yes—
549C2×C4#42112yes—
550C2×C4#56812yes—
551C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C2×C4C2×#13711yescharacteristic
555C8—48no—
556C8—84no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—416no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22#156014yes—
586C22#287911yescharacteristic
587C22C22×#9911yescharacteristic
588C22C22×#10011yescharacteristic
589C22C2×#41711yescharacteristic
590C22C2×#41711yescharacteristic
591C22C22×#10411yescharacteristic
592C22C2×#42111yescharacteristic
593C4—21no—
594C4—21no—
595C4—22no—
596C4—22no—
597C4—21no—
598C4—21no—
599C4—22no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4#150414yes—
610C2—24no—
611C2—42no—
612C2—42no—
613C2—162no—
614C2#1739611yescharacteristic
615C2#1749811yescharacteristic
616C2#1750411yescharacteristic
617C2C2×#150411yescharacteristic
618C2#1759211yescharacteristic
619C2#1764711yescharacteristic
620C2C2×#156011yescharacteristic
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×#1728C2semidirect
4. #7710C2semidirect
5. #7667C2semidirect
6. #7664C2semidirect
7. #7631C2semidirect
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
#206262
C2×#17282
#77102
#76672
#76642
#76312