Finite Group Databasebeta

← #216212

Subgroups — #216212

Order 512.

1362 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#9879C212yesmaximal, complemented
5#9650C212yesmaximal, complemented
6#9085C214yesmaximal
7#8560C214yesmaximal, complemented
8C22×#117C411yescharacteristic
9C22×#117C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#423—21no—
13C2×#407C2212yes—
14#2296—21no—
15#2296—21no—
16#2296—21no—
17#2296—21no—
18#2296—21no—
19#2296—21no—
20#2296—21no—
21#2296—21no—
22C2×#389C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#385C412yes—
27C2×#385C2212yes—
28C2×C4×C4⋊C4C412yes—
29C2×C4×C4⋊C4C2212yes—
30#1817—21no—
31#1797—21no—
32#1797—21no—
33#1797—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1797C2212yes—
37#1779C2212yes—
38C4×C22⋊C8—21no—
39C4×C22⋊C8C2212yes—
40#1728—21no—
41C2×#324C2218yes—
42#1280C2218yes—
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C2×#117—21no—
51C2×#117—21no—
52C2×#117—21no—
53C2×#117—21no—
54C2×#117—21no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—44no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2×C412yes—
68C22×C42—44no—
69C22×C42C2×C412yes—
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#431—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⋊C8C2314yes—
93C2×C22⋊C8C2314yes—
94C2×C4×C8—22no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#385—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2×C412yes—
130C2×#95C2312yes—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#324—48no—
136C25C22×C411yescharacteristic
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×M16—44no—
171C2×M16—44no—
172C2×M16—44no—
173C22×C8—24no—
174C22×C8—24no—
175#117—28no—
176#117—28no—
177#117—28no—
178#117—28no—
179#117—44no—
180#117—44no—
181#117—44no—
182#117—44no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4M1614yescomplemented
209C2×C4⋊C4M1614yescomplemented
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—44no—
217C2×C42—24no—
218C2×C42—24no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—42no—
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—44no—
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×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C4⋊C8—44no—
253C4⋊C8—44no—
254C4⋊C8—44no—
255C22⋊C8—24no—
256C22⋊C8—28no—
257C22⋊C8—24no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261C4×C8—44no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270C24—21no—
271C24—21no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C24C23×C411yescharacteristic
281C22×C4—216no—
282C22×C4—216no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
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—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
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—24no—
330C22×C4—24no—
331C22×C4—41no—
332C22×C4—41no—
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—48no—
346C22×C4—416no—
347C22×C4—48no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4#9914yes—
355C22×C4#10114yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×M1614yes—
367C22×C4C2×M1612yes—
368C22×C4C2×M1612yes—
369C22×C4C2×M1614yes—
370C22×C4#13114yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+412yes—
375M16—88no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C2×C8—44no—
380C2×C8—44no—
381C2×C8—44no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C42—28no—
404C42—28no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—44no—
422C42—44no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—24no—
428C23—24no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23C2×#9912yes—
450C23C2×#10112yes—
451C23C22×C22⋊C411yescharacteristic
452C23#52012yes—
453C23C22×M1611yescharacteristic
454C23C22×M1611yescharacteristic
455C23C2×#13112yes—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
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—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—28no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4#40712yes—
522C2×C4#40712yes—
523C2×C4#40812yes—
524C2×C4#40812yes—
525C2×C4C2×#9912yes—
526C2×C4C2×#9912yes—
527C2×C4#41012yes—
528C2×C4C2×#10112yes—
529C2×C4C2×#10112yes—
530C2×C4#41312yes—
531C2×C4#53511yescharacteristic
532C2×C4#53611yescharacteristic
533C2×C4#53712yes—
534C8—84no—
535C8—84no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—48no—
551C22—48no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22#231111yescharacteristic
561C22C2×#40711yescharacteristic
562C22C2×#40811yescharacteristic
563C22C22×#9911yescharacteristic
564C22C2×#41011yescharacteristic
565C22C22×#10111yescharacteristic
566C22C2×#41311yescharacteristic
567C22#299012yes—
568C22#299112yes—
569C4—21no—
570C4—22no—
571C4—21no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—44no—
580C4—41no—
581C4—44no—
582C4—41no—
583C4—82no—
584C4—82no—
585C4#146314yes—
586C2—22no—
587C2—22no—
588C2—42no—
589C2—42no—
590C2C2×#146311yescharacteristic
591C2#1736611yescharacteristic
592C2#1738111yescharacteristic
593C2#1738511yescharacteristic
594C2#2184911yescharacteristic
595C2#2187411yescharacteristic
596C2#2187911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #9879C2semidirect
5. #9650C2semidirect
7. #8560C2semidirect
208. C2×C4⋊C4M16semidirect
209. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
#207142
#206262
#98792
#96502
#90852
#85602