Finite Group Databasebeta

← #216508

Subgroups — #216508

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C211yesmaximal, characteristic
2C2×#1797C211yesmaximal, characteristic, complemented
3#14683C211yesmaximal, characteristic, complemented
4#9911C212yesmaximal, complemented
5#9715C212yesmaximal, complemented
6#9113C214yesmaximal, complemented
7#8573C214yesmaximal, complemented
8C2×#532C411yescharacteristic, complemented
9C2×#525—21no—
10C23×C42C2211yescharacteristic
11C22×C4⋊C8—21no—
12C22×C22⋊C8C2212yes—
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2856C411yescharacteristic, complemented
16#2856C2211yescharacteristic
17#2854—21no—
18#2854C412yescomplemented
19#2850—21no—
20#2850—21no—
21#2847—21no—
22#2847C2212yes—
23C4×#121—21no—
24C4×#121—21no—
25C4×#121C412yescomplemented
26C4×#121C2212yes—
27#2297—21no—
28#2285C411yescharacteristic, complemented
29C2×#383—21no—
30#1797—21no—
31#1797—21no—
32#1797C2212yes—
33#1797C2212yes—
34C2×#336—21no—
35C2×#327C2218yes—
36#1434—21no—
37#1434C2212yes—
38#1434C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C22⋊#95C2212yes—
42#1294C2218yes—
43C22×#41—21no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48#532—44no—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52#525—48no—
53C2×#123—22no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×#121C2×C412yes—
63C2×#121C2312yes—
64C2×#120—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117—21no—
70C22×C22⋊C4—22no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—44no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—44no—
89C2×C22⋊C8C2314yes—
90#391—21no—
91#391—21no—
92#390—21no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yes—
97#390C2×C412yes—
98#389—21no—
99#389—21no—
100#389C2×C412yes—
101#389C2×C412yes—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#384C2×C411yescharacteristic
113#384C2×C411yescharacteristic
114#383—44no—
115#382—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2312yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95C2314yes—
132#336—22no—
133#336—22no—
134#327—48no—
135C25C2×D411yescharacteristic
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—24no—
152C23×C4—42no—
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×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2411yescharacteristic
176C22×C8—24no—
177C22×C8—44no—
178C22×C8—44no—
179C22×C8—44no—
180C4⋊Q8—42no—
181#127—42no—
182#124—44no—
183#123—48no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186#121—24no—
187#121—24no—
188#121—44no—
189#121—44no—
190#121—44no—
191#121—44no—
192#121—44no—
193#121—44no—
194#121—44no—
195#120—48no—
196C4×D4—44no—
197C4×D4—44no—
198C4×D4—48no—
199C4×D4—48no—
200#117—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
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—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
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×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C4⋊C8—44no—
255C4⋊C8—44no—
256C22⋊C8—28no—
257C22⋊C8—44no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—41no—
277C24—41no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C24AGL(1,ℤ/8)14yes—
281C24C22×D411yescharacteristic
282#41—816no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×D4—24no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
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—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
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—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
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—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—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—48no—
362C22×C4—416no—
363C22×C4—48no—
364C22×C4—416no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4#13514yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+412yes—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—88no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—44no—
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—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—88no—
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—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—28no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23#41012yes—
463C23#41812yes—
464C23#41812yes—
465C23#42112yes—
466C23C22×C22⋊C411yescharacteristic
467C23#53712yes—
468C23C2×#13512yes—
469C23C2×AGL(1,ℤ/8)12yes—
470Q8—84no—
471D4—84no—
472D4—84no—
473D4—84no—
474D4—88no—
475D4—88no—
476D4—88no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—28no—
480C2×C4—28no—
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—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—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
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—42no—
535C2×C4—88no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—84no—
543C2×C4C2×#9912yes—
544C2×C4C2×#9912yes—
545C2×C4C2×#9912yes—
546C2×C4#41612yes—
547C2×C4#41612yes—
548C2×C4#42112yes—
549C2×C4#52012yes—
550C2×C4#53511yescharacteristic
551C2×C4#53611yescharacteristic
552C8—84no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—416no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22#151414yes—
581C22#231111yescharacteristic
582C22C22×#9911yescharacteristic
583C22C2×#41011yescharacteristic
584C22C2×#41611yescharacteristic
585C22C2×#41811yescharacteristic
586C22C2×#42111yescharacteristic
587C22C2×#42111yescharacteristic
588C22#324714yes—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—164no—
606C2—24no—
607C2—24no—
608C2—42no—
609C2—162no—
610C2C2×#151411yescharacteristic
611C2#1757111yescharacteristic
612C2#1757211yescharacteristic
613C2#1757711yescharacteristic
614C2#2185711yescharacteristic
615C2#2187411yescharacteristic
616C2#2188811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#36624C2
LCSG ▷ C22×C4 ▷ C22 ▷ 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
2. C2×#1797C2semidirect
3. #14683C2semidirect
4. #9911C2semidirect
5. #9715C2semidirect
6. #9113C2semidirect
7. #8573C2semidirect
8. C2×#532C4semidirect
13. #2857C4semidirect
15. #2856C4semidirect
18. #2854C4semidirect
25. C4×#121C4semidirect
28. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#366242
C2×#17972
#146832
#99112
#97152
#91132
#85732