Finite Group Databasebeta

← #196424

Subgroups — #196424

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic, complemented
2C2×#2854C211yesmaximal, characteristic
3C2×#1510C212yesmaximal
4#14684C211yesmaximal, characteristic, complemented
5#9118C214yesmaximal, complemented
6#9114C214yesmaximal, complemented
7#8015C212yesmaximal, complemented
8C22×#123—21no—
9C22×#121C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2920—28no—
12C2×#432—21no—
13C2×#432C2212yescomplemented
14C2×#431—21no—
15C2×#431C2212yescomplemented
16C2×#423—21no—
17C2×#407C2212yes—
18C2×C4×M16—21no—
19#2854—21no—
20#2854—21no—
21#2854—21no—
22#2854—21no—
23#2854C412yescomplemented
24#2854C412yescomplemented
25#2854C2212yes—
26#2854C2212yes—
27C2×#390C411yescharacteristic, complemented
28C2×#390C411yescharacteristic, complemented
29C2×#389C411yescharacteristic, complemented
30C2×#389C411yescharacteristic, complemented
31C2×#388—21no—
32C2×#388—21no—
33C2×#386—21no—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36#1510—28no—
37C2×#327C2218yes—
38#1434—21no—
39#1434—21no—
40#1434C2212yescomplemented
41#1434C2212yescomplemented
42C22⋊#95—21no—
43C22⋊#95—21no—
44#1294C2218yescomplemented
45C23×D4—21no—
46C23×D4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C2×#123—22no—
52C2×#123—22no—
53C2×#123—22no—
54C2×#123—22no—
55C2×#123—48no—
56C2×#121—21no—
57C2×#121—21no—
58C2×#121—24no—
59C2×#121—24no—
60C2×#121C2×C412yes—
61C2×#121C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C42—24no—
75C22×C42—24no—
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84C22×C42D411yescharacteristic
85C22×C42D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88#432—28no—
89#432—28no—
90#431—28no—
91#431—28no—
92#423—28no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95#407—24no—
96#407D418yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8C2314yes—
99C4×M16—28no—
100C2×#97—22no—
101C2×C4×C8—22no—
102#390—21no—
103#390—21no—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#390C2×C412yes—
108#390C2×C412yes—
109#390C2×C412yes—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2×C412yes—
117#389C2×C412yes—
118#388—22no—
119#388—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#388—22no—
125#388—22no—
126#386—22no—
127#386—22no—
128#386—22no—
129#386—22no—
130#381—24no—
131#381—24no—
132#380—21no—
133#380—21no—
134#380C2×C412yes—
135#380C2312yes—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95—22no—
144C2×#95C2314yes—
145#327—216no—
146C25—21no—
147C25C2×D411yescharacteristic
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—21no—
151C22×D4—24no—
152C22×D4—42no—
153C22×D4—42no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4C22×C412yes—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—24no—
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×C4C2×D414yes—
188C23×C4#4114yes—
189C23×C4C2411yescharacteristic
190C2×M16—28no—
191C2×M16—44no—
192C2×M16—44no—
193C22×C8—24no—
194#123—48no—
195#123—48no—
196#123—48no—
197#123—48no—
198#121—28no—
199#121—24no—
200#121—24no—
201#121—42no—
202#121—44no—
203#121—42no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
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×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
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×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C4⋊C8—28no—
276C4⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—24no—
279C22⋊C8C2×D418yes—
280#97—28no—
281C4×C8—28no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—41no—
301C24—41no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24—42no—
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24#12114yes—
312C24C22×D411yescharacteristic
313C2×D4—24no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—88no—
326C2×D4—816no—
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—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—22no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
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—48no—
376C22×C4—48no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—416no—
380C22×C4—416no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C4×D414yes—
408C22×C4C4×D414yes—
409C22×C4#12012yes—
410C22×C4#12012yes—
411C22×C4#12114yes—
412C22×C4#12312yes—
413C22×C4#12312yes—
414C22×C4AGL(1,ℤ/8)14yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D412yes—
418C22×C4C2×#4112yes—
419M16—416no—
420C2×C8—28no—
421C2×C8—44no—
422C2×C8—44no—
423C4⋊C4—24no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—88no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—28no—
482C23—28no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—84no—
496C23—84no—
497C23—84no—
498C23—84no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23#38612yes—
503C23#38612yes—
504C23#41012yes—
505C23#41012yes—
506C23C22×C22⋊C411yescharacteristic
507C23C2×#12112yes—
508C23C2×AGL(1,ℤ/8)12yes—
509C23#57512yes—
510D4—84no—
511D4—84no—
512D4—88no—
513D4—88no—
514C2×C4—28no—
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—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—28no—
535C2×C4—21no—
536C2×C4—21no—
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—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
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—44no—
562C2×C4—44no—
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—84no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4#38612yes—
583C2×C4#38612yes—
584C2×C4#41812yes—
585C2×C4#41812yes—
586C2×C4#42112yes—
587C2×C4#42112yes—
588C2×C4C2×C4×D412yes—
589C2×C4C2×#12011yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C8—48no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—416no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—168no—
620C22#189914yes—
621C22C2×#38611yescharacteristic
622C22C2×#41011yescharacteristic
623C22C2×#41011yescharacteristic
624C22C2×#41811yescharacteristic
625C22#289611yescharacteristic
626C22C2×#42111yescharacteristic
627C22C2×#42111yescharacteristic
628C22#305514yes—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—22no—
633C4—21no—
634C4—22no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—164no—
646C2—24no—
647C2—24no—
648C2—42no—
649C2—162no—
650C2#1755411yescharacteristic
651C2#1756111yescharacteristic
652C2#1758011yescharacteristic
653C2#1758911yescharacteristic
654C2#2090211yescharacteristic
655C2#2097311yescharacteristic
656C2C2×#189911yescharacteristic
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
1. C2×#2920C2semidirect
4. #14684C2semidirect
5. #9118C2semidirect
6. #9114C2semidirect
7. #8015C2semidirect
13. C2×#432C22semidirect
15. C2×#431C22semidirect
23. #2854C4semidirect
24. #2854C4semidirect
27. C2×#390C4semidirect
28. C2×#390C4semidirect
29. C2×#389C4semidirect
30. C2×#389C4semidirect
40. #1434C22semidirect
41. #1434C22semidirect
44. #1294C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28542
C2×#15102
#146842
#91182
#91142
#80152