Finite Group Databasebeta

← #369329

Subgroups — #369329

Order 512.

1830 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3C2×#1982C211yesmaximal, characteristic
4C2×#1961C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic, complemented
6#11076C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#10927C212yesmaximal, complemented
9#10925C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#124—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#446—24no—
16C2×#446—24no—
17C2×#446—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#102C2212yes—
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2548—24no—
29#2548C418yescomplemented
30C2×#394—21no—
31C2×#394—21no—
32C2×#390—21no—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1982—24no—
39#1961—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1446—21no—
45#1445—21no—
46#1436—21no—
47#1436—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C23×Q8D411yescharacteristic
51C23×D4—21no—
52C23×D4—21no—
53C23×D4C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C2×#127—22no—
59C2×#127—22no—
60C2×#127—22no—
61C2×#127—22no—
62C2×#127—22no—
63C2×#127C2×C414yes—
64C2×#124—28no—
65C2×#124—48no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8C2×C414yes—
71C2×C4×Q8D414yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#446—232no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8D414yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—24no—
102C2×#102—24no—
103C2×#102—24no—
104C2×#102—24no—
105C2×#102—42no—
106C2×#102—42no—
107C2×#102—42no—
108C2×#102—42no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97D414yes—
115C2×C4×C8—22no—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118#394—28no—
119#394—28no—
120#390—28no—
121#390—24no—
122#390—24no—
123#390C2×C418yes—
124#390C2×C418yes—
125#382—22no—
126#382—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2×C414yes—
131C2×#95—21no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—22no—
139#340—24no—
140#340—24no—
141#340—42no—
142#340—42no—
143C25—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8C22⋊C414yes—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—82no—
171C22×D4—82no—
172C22×D4C22×C414yes—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2411yescharacteristic
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8C2×D412yes—
199C22×C8C2×D412yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8#4114yes—
203#127—28no—
204#127—28no—
205#127—44no—
206#127—44no—
207#124—432no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—28no—
211C4×Q8—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22⋊C414yes—
236C2×C4⋊C4C22⋊C414yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4#4114yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—24no—
260C2×C42—44no—
261C2×C42—42no—
262C2×C42—82no—
263C2×C42C22×C412yes—
264C2×C42C22×C412yes—
265C2×C42C22×C412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C4⋊C8—28no—
274C4⋊C8—28no—
275C4⋊C8—28no—
276#102—48no—
277#102—48no—
278#102—48no—
279#102—48no—
280#97—28no—
281#97—28no—
282C4×C8—28no—
283#95—24no—
284#95—44no—
285#95—42no—
286#95—42no—
287#95—42no—
288#95—44no—
289#95—82no—
290#95—82no—
291C24—24no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—42no—
306C24C22×D411yescharacteristic
307C2×Q8—28no—
308C2×Q8—28no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—48no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4—88no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
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×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C412yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C412yes—
395C22×C4C2×C22⋊C412yes—
396C22×C4#12112yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C42−1+412yes—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8#12414yescomplemented
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—48no—
459C42—84no—
460C42C4×D414yes—
461C42C4×D414yes—
462C42#12014yes—
463C42#12314yes—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23—82no—
497C23—82no—
498C23—82no—
499C23C22×C22⋊C411yescharacteristic
500C23C2×#12111yescharacteristic
501C23C2×#12111yescharacteristic
502C23#54812yes—
503C23#55012yes—
504C23C2×AGL(1,ℤ/8)12yes—
505C23#57512yes—
506C23#57612yes—
507C23#57712yes—
508Q8—416no—
509D4—44no—
510D4—44no—
511D4—44no—
512D4—44no—
513D4—816no—
514D4—816no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4#38614yes—
568C2×C4#38614yes—
569C2×C4#41814yes—
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C4×#4112yes—
573C2×C4#52012yes—
574C2×C4C2×#12011yescharacteristic
575C2×C4C2×#12311yescharacteristic
576C2×C4C2×#12411yescharacteristic
577C2×C4#52912yes—
578C2×C4#53211yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53611yescharacteristic
581C2×C4#54212yes—
582C2×C4#54612yes—
583C2×C4#54812yes—
584C2×C4#54812yes—
585C2×C4#55012yes—
586C2×C4#55612yes—
587C2×C4#55912yes—
588C8—24no—
589C8—42no—
590C8—42no—
591C8—42no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—162no—
600C22—162no—
601C22—162no—
602C22—162no—
603C22—162no—
604C22—162no—
605C22—162no—
606C22—162no—
607C22C2×#38611yescharacteristic
608C22#234112yes—
609C22#242612yes—
610C22#254511yescharacteristic
611C22#254811yescharacteristic
612C22C2×#41811yescharacteristic
613C22#289611yescharacteristic
614C22#289712yes—
615C22#305512yes—
616C22#305512yes—
617C22#305711yescharacteristic
618C22#305811yescharacteristic
619C22#306212yes—
620C22#336612yes—
621C22#336812yes—
622C22#337012yes—
623C22#337112yes—
624C22#339712yes—
625C22#339912yes—
626C22#340112yes—
627C22#340312yes—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—84no—
636C4—162no—
637C4C4×#12411yescharacteristic
638C4#232611yescharacteristic
639C4#236712yes—
640C4#243111yescharacteristic
641C4#243611yescharacteristic
642C4#244812yes—
643C2—164no—
644C2#1402611yescharacteristic
645C2#2097311yescharacteristic
646C2#2097511yescharacteristic
647C2#2254512yes—
648C2#2254912yes—
649C2#2530412yes—
650C2#2531012yes—
651C2#2865012yes—
652C2#2874612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
5. #13925C2semidirect
6. #11076C2semidirect
7. #11076C2semidirect
8. #10927C2semidirect
9. #10925C2semidirect
10. #10850C2semidirect
11. #10703C2semidirect
29. #2548C4semidirect
430. C2×C8#124semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#25482
C2×#19822
C2×#19612
#139252
#110762
#109272
#109252
#108502
#107032