Finite Group Databasebeta

← #161938

Subgroups — #161938

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2#20626C211yesmaximal, characteristic, complemented
3C2×#1797C211yesmaximal, characteristic, complemented
4#7710C214yesmaximal, complemented
5#7667C212yesmaximal, complemented
6#7664C212yesmaximal, complemented
7#7597C214yesmaximal, complemented
8C2×#532C411yescharacteristic
9C2×#525C412yes—
10C23×C42C2211yescharacteristic
11C2×#423—21no—
12C22×C4⋊C8—21no—
13C2×#407—21no—
14C22×C22⋊C8—21no—
15#2857C411yescharacteristic
16#2857C2211yescharacteristic
17#2856C411yescharacteristic
18#2856C2211yescharacteristic
19#2854—21no—
20#2854—21no—
21#2850C412yes—
22#2850C2212yes—
23#2847—21no—
24#2847—21no—
25C4×#121—21no—
26C4×#121—21no—
27C4×#121—21no—
28C4×#121—21no—
29#2297C2212yes—
30#2285C411yescharacteristic
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1728C2212yes—
40C2×#336C2212yes—
41#1272C2218yescomplemented
42#1270C2218yescomplemented
43C22×#41C2×C412yes—
44C24×C4C2×C411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4D411yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49#532—44no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#124C2×C412yes—
53#525—48no—
54C2×#123—22no—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2×C412yes—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C2×#121C2×C412yes—
65C2×#120—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×#117C2×C412yes—
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—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—44no—
90#407—44no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—44no—
96#391C2×C412yes—
97#391C2312yes—
98#390—21no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390C2×C412yes—
103#390C2312yes—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
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#382C2×C412yes—
121#380—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—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—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
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×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—44no—
179C22×C8—24no—
180C22×C8—24no—
181C22×C8—44no—
182C22×C8—44no—
183C22×C8—44no—
184C4⋊Q8—42no—
185#127—42no—
186#124—44no—
187#123—48no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—42no—
199#121—44no—
200#120—48no—
201C4×D4—44no—
202C4×D4—48no—
203C4×D4—48no—
204C4×D4—44no—
205#117—44no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
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×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—42no—
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×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2411yescharacteristic
256C4⋊C8—44no—
257C4⋊C8—44no—
258C4⋊C8—44no—
259C4⋊C8—44no—
260C22⋊C8—44no—
261C22⋊C8—44no—
262C22⋊C8—44no—
263C22⋊C8—44no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C24C2×C22⋊C411yescharacteristic
281#41—816no—
282C2×Q8—24no—
283C2×Q8—42no—
284C2×Q8—42no—
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—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
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—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
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—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—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—416no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4#10014yes—
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×C4#13114yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4AGL(1,ℤ/8)12yes—
381C22×C4#13712yes—
382C22×C4#13712yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386M16—88no—
387C2×C8—44no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—88no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—44no—
433C42#9914yes—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—28no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23#40812yes—
465C23C2×#10012yes—
466C23#41712yes—
467C23#41712yes—
468C23#42112yes—
469C23C22×C22⋊C411yescharacteristic
470Q8—84no—
471D4—84no—
472D4—84no—
473D4—88no—
474D4—88no—
475D4—88no—
476D4—84no—
477C2×C4—24no—
478C2×C4—24no—
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—28no—
493C2×C4—24no—
494C2×C4—24no—
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—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
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—88no—
535C2×C4#40812yes—
536C2×C4C2×#9912yes—
537C2×C4C2×#9912yes—
538C2×C4C2×#9912yes—
539C2×C4C2×#10012yes—
540C2×C4C2×#10012yes—
541C2×C4#41712yes—
542C2×C4#41712yes—
543C2×C4#42112yes—
544C2×C4C2×#13112yes—
545C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
546C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
547C2×C4C2×#13711yescharacteristic
548C2×C4C2×#13711yescharacteristic
549C8—84no—
550C8—84no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—416no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22#156014yes—
580C22C2×#40811yescharacteristic
581C22C22×#9911yescharacteristic
582C22C22×#10011yescharacteristic
583C22C2×#41711yescharacteristic
584C22C2×#41711yescharacteristic
585C22C2×#42111yescharacteristic
586C22C2×#42111yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4#147114yes—
603C4#151314yes—
604C2—24no—
605C2—42no—
606C2—42no—
607C2—162no—
608C2C2×#147111yescharacteristic
609C2#1746611yescharacteristic
610C2#1748111yescharacteristic
611C2C2×#151311yescharacteristic
612C2#1759211yescharacteristic
613C2#1767111yescharacteristic
614C2C2×#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×#1797C2semidirect
4. #7710C2semidirect
5. #7667C2semidirect
6. #7664C2semidirect
7. #7597C2semidirect
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×#17972
#77102
#76672
#76642
#75972