Finite Group Databasebeta

← #215394

Subgroups — #215394

Order 512.

1578 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic, complemented
3C4×#380C211yesmaximal, characteristic, complemented
4#9713C214yesmaximal, complemented
5#9682C214yesmaximal, complemented
6#8329C212yesmaximal, complemented
7#8328C212yesmaximal
8C22×#124C411yescharacteristic, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11C23×C42C2211yescharacteristic
12C2×#432—21no—
13#2850—21no—
14#2850—21no—
15#2850—21no—
16#2850—21no—
17#2850C412yescomplemented
18#2850C412yescomplemented
19#2850C2212yes—
20#2850C2212yes—
21C2×#391C411yescharacteristic, complemented
22C2×#390C411yescharacteristic, complemented
23C2×#388—21no—
24C2×#388—21no—
25C2×#386—21no—
26C2×#386—21no—
27C2×#382C411yescharacteristic, complemented
28C2×#380—21no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C22⋊C4—21no—
31#1817—21no—
32#1797—21no—
33#1797C2212yescomplemented
34#1779C2212yes—
35#1749—21no—
36#1749C2212yescomplemented
37C2×#328C2218yescomplemented
38C2×#327C2218yes—
39C4⋊#95C2212yescomplemented
40#1434—21no—
41C22⋊#95C2212yes—
42C4×#95C2212yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C2×#124—21no—
50C2×#124—21no—
51C2×#124—44no—
52C2×#124C2×C412yes—
53C2×#124C2×C412yes—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—44no—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2312yes—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—44no—
62C2×#121C2×C412yes—
63C2×#121C2312yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C42—22no—
76C22×C42—44no—
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#432—44no—
90C2×C4⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2314yes—
94C2×#97—22no—
95#391—21no—
96#391—21no—
97#391C2×C412yes—
98#391C2×C412yes—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
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#386—22no—
118#386—22no—
119#382—21no—
120#382—21no—
121#382C2×C412yes—
122#382C2×C412yes—
123#380—28no—
124C4×C4⋊C4—44no—
125C4×C22⋊C4—44no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136#328—48no—
137#327—48no—
138C25—21no—
139C25C2×D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8C22×C412yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4C22×C412yes—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22⋊C411yescharacteristic
176C23×C4C22⋊C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C22×C8—24no—
189#124—44no—
190#124—44no—
191#124—44no—
192#124—44no—
193C22⋊Q8—28no—
194C22⋊Q8—44no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197#121—28no—
198#121—44no—
199#121—44no—
200#121—44no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—28no—
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×C22⋊C4—22no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—42no—
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—44no—
247C2×C42—44no—
248C2×C42—44no—
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×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C4⋊C8—44no—
266C22⋊C8—28no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269#97—44no—
270#95—28no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—28no—
280#95—44no—
281#95—44no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24AGL(1,ℤ/8)12yes—
299C24#13712yes—
300C24C22×D411yescharacteristic
301C2×Q8—24no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—88no—
308C2×D4—24no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—88no—
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—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
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—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—48no—
383C22×C4—416no—
384C22×C4—48no—
385C22×C4—416no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4#10414yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4H(5,𝔽2)14yes—
415C2×C8—44no—
416C2×C8—44no—
417C2×C8—44no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—416no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—88no—
454C42#9914yes—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—84no—
487C23—84no—
488C23—84no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#41012yes—
494C23#41712yes—
495C23#41712yes—
496C23C2×#10412yes—
497C23C22×C22⋊C411yescharacteristic
498C23#53512yes—
499C23C2×AGL(1,ℤ/8)11yescharacteristic
500C23C2×#13711yescharacteristic
501C23#57512yes—
502Q8—84no—
503Q8—84no—
504D4—84no—
505D4—84no—
506D4—88no—
507D4—84no—
508D4—84no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4C2×#9912yes—
577C2×C4C2×#9912yes—
578C2×C4C2×#9912yes—
579C2×C4C2×#10412yes—
580C2×C4C2×#10412yes—
581C2×C4#42112yes—
582C2×C4#51812yes—
583C2×C4#53512yes—
584C8—84no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—48no—
603C22—48no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—168no—
616C22#150914yes—
617C22#230911yescharacteristic
618C22C22×#9911yescharacteristic
619C22C2×#41011yescharacteristic
620C22C2×#41711yescharacteristic
621C22#289611yescharacteristic
622C22C22×#10411yescharacteristic
623C22C2×#42111yescharacteristic
624C22#321312yes—
625C22#321512yes—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—88no—
639C4—82no—
640C4—82no—
641C4—82no—
642C2—22no—
643C2—24no—
644C2—22no—
645C2—42no—
646C2—162no—
647C2C2×#150911yescharacteristic
648C2#1754211yescharacteristic
649C2#1755411yescharacteristic
650C2#1759211yescharacteristic
651C2#2179311yescharacteristic
652C2#2180811yescharacteristic
653C2#2181511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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. #20715C2semidirect
3. C4×#380C2semidirect
4. #9713C2semidirect
5. #9682C2semidirect
6. #8329C2semidirect
8. C22×#124C4semidirect
17. #2850C4semidirect
18. #2850C4semidirect
21. C2×#391C4semidirect
22. C2×#390C4semidirect
27. C2×#382C4semidirect
33. #1797C22semidirect
36. #1749C22semidirect
37. C2×#328C22semidirect
39. C4⋊#95C22semidirect
42. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
#207152
C4×#3802
#97132
#96822
#83292
#83282