Finite Group Databasebeta

← #196549

Subgroups — #196549

Order 512.

1604 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×#2850C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic, complemented
4C2×#1533C212yesmaximal, complemented
5#9123C214yesmaximal, complemented
6#9120C214yesmaximal, complemented
7#8030C212yesmaximal, complemented
8C22×#124C2211yescharacteristic
9C22×C22⋊Q8C411yescharacteristic
10C22×#121C411yescharacteristic
11C23×C42C2211yescharacteristic
12#2920—28no—
13C2×#432C2212yes—
14C2×#432C2212yes—
15C2×#432C2212yescomplemented
16C2×#431—21no—
17C2×#431—21no—
18C2×#423—21no—
19C2×#407—21no—
20C2×C4×M16C2212yescomplemented
21#2850—21no—
22#2850—21no—
23#2850—21no—
24#2850—21no—
25#2850C412yes—
26#2850C412yes—
27#2850C2212yes—
28#2850C2212yes—
29C2×#391C411yescharacteristic
30C2×#390C411yescharacteristic
31C2×#388—21no—
32C2×#388—21no—
33C2×#386—21no—
34C2×#386—21no—
35C2×#382C2211yescharacteristic
36#1817—21no—
37#1797—21no—
38#1797—21no—
39#1779—21no—
40#1749C2212yescomplemented
41#1749C2212yescomplemented
42#1533—28no—
43C2×#329C2218yescomplemented
44#1295C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4C2×C411yescharacteristic
49C24×C4D411yescharacteristic
50C24×C4D411yescharacteristic
51C22×M16—22no—
52C2×#124—21no—
53C2×#124—21no—
54C2×#124—24no—
55C2×#124—24no—
56C2×#124C2×C412yes—
57C2×#124C2312yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—44no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—44no—
66C2×#121C2×C412yes—
67C2×#121C2×C412yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—24no—
92#432—24no—
93#432—24no—
94#432—28no—
95#432D418yescomplemented
96#431—28no—
97#431—28no—
98#423—28no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102#407—28no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C4×M16—28no—
109C2×#97C2314yes—
110C2×#97C2314yes—
111C2×C4×C8—22no—
112#391—21no—
113#391—21no—
114#391C2×C412yes—
115#391C2×C412yes—
116#390—21no—
117#390—21no—
118#390C2×C412yes—
119#390C2×C412yes—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#386—22no—
129#386—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#386—22no—
135#386—22no—
136#382—21no—
137#382—21no—
138#382C2×C412yes—
139#382C2312yes—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144#336—22no—
145#336—22no—
146#329—216no—
147C25—21no—
148C25C22⋊C411yescharacteristic
149C22×Q8—21no—
150C22×Q8—24no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8C22×C412yes—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—24no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4C22×C412yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C2×M16—28no—
196C2×M16—44no—
197C2×M16—44no—
198C22×C8—24no—
199C22×C8—24no—
200#124—28no—
201#124—24no—
202#124—24no—
203#124—42no—
204#124—44no—
205#124—42no—
206C22⋊Q8—44no—
207C22⋊Q8—44no—
208C22⋊Q8—44no—
209C22⋊Q8—44no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—44no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C22⋊C4—22no—
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×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22⋊C411yescharacteristic
260C2×C42C22⋊C411yescharacteristic
261C2×C42C22⋊C411yescharacteristic
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D414yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4114yes—
275C2×C42C2411yescharacteristic
276C4⋊C8—28no—
277C4⋊C8—28no—
278C4⋊C8—44no—
279C22⋊C8—28no—
280C22⋊C8—28no—
281C22⋊C8—44no—
282C22⋊C8—44no—
283C22⋊C8—44no—
284#97—24no—
285#97—24no—
286#97—24no—
287#97C2×D418yescomplemented
288C4×C8—28no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C2×C22⋊C411yescharacteristic
315C2×Q8—24no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—88no—
322C2×D4—24no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—88no—
335C2×D4—816no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
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—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
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—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—416no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4#10014yes—
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×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C4×D414yes—
415C22×C4#12012yes—
416C22×C4#12114yes—
417C22×C4#12312yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421M16—416no—
422C2×C8—28no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—44no—
427C2×C8—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C42—22no—
448C42—22no—
449C42—21no—
450C42—21no—
451C42—22no—
452C42—22no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42C4×D414yes—
462C42#12012yes—
463C42#12114yes—
464C42#12312yes—
465C42C22×D412yes—
466C42C2×#4112yes—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—28no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23#38612yes—
505C23C2×#10012yes—
506C23#41312yes—
507C23#42112yes—
508C23#42112yes—
509C23C22×C22⋊C411yescharacteristic
510Q8—84no—
511Q8—84no—
512D4—84no—
513D4—84no—
514D4—88no—
515D4—88no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
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—28no—
536C2×C4—28no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
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—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4#38612yes—
576C2×C4#38612yes—
577C2×C4#38612yes—
578C2×C4C2×#10012yes—
579C2×C4C2×#10012yes—
580C2×C4#42112yes—
581C2×C4#42112yes—
582C2×C4C2×C4×D412yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12112yes—
585C2×C4C2×#12311yescharacteristic
586C8—48no—
587C8—84no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—416no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—168no—
617C22#190014yes—
618C22C2×#38611yescharacteristic
619C22C22×#10011yescharacteristic
620C22C2×#41311yescharacteristic
621C22C2×#42111yescharacteristic
622C22C2×#42111yescharacteristic
623C22C2×#42111yescharacteristic
624C22C2×#42111yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—21no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4—82no—
640C4—82no—
641C2—24no—
642C2—42no—
643C2—42no—
644C2—162no—
645C2#1765911yescharacteristic
646C2#1766311yescharacteristic
647C2#1766611yescharacteristic
648C2#1766911yescharacteristic
649C2#2090211yescharacteristic
650C2#2097911yescharacteristic
651C2C2×#190011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#121C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2920C2semidirect
3. #20715C2semidirect
4. C2×#1533C2semidirect
5. #9123C2semidirect
6. #9120C2semidirect
7. #8030C2semidirect
15. C2×#432C22semidirect
20. C2×C4×M16C22semidirect
40. #1749C22semidirect
41. #1749C22semidirect
43. C2×#329C22semidirect
44. #1295C22semidirect
95. #432D4semidirect
287. #97C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28502
#207152
C2×#15332
#91232
#91202
#80302