Finite Group Databasebeta

← #194279

Subgroups — #194279

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic
4#21257C211yesmaximal, characteristic
5C2×#1481C211yesmaximal, characteristic, complemented
6#8983C212yesmaximal, complemented
7#8982C212yesmaximal, complemented
8#8979C212yesmaximal, complemented
9#8978C212yesmaximal, complemented
10C2×#1296C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3246—24no—
15#3246C418yescomplemented
16C2×#475C2212yes—
17C2×#476—21no—
18C2×#465—22no—
19C2×#460C2212yes—
20C2×#461—21no—
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#416—21no—
25C2×#416—21no—
26#2360—22no—
27#2360—22no—
28#2355—21no—
29#2355C2212yes—
30Q8×C4⋊C4—21no—
31Q8×C4⋊C4C2212yes—
32#2330—22no—
33#2317—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8—21no—
36C2×#384C2211yescharacteristic
37#2274—21no—
38C4×#117C2211yescharacteristic
39#1990C2212yes—
40#1983—21no—
41#1983—21no—
42#1982C2211yescharacteristic
43#1982C2211yescharacteristic
44#1981C2211yescharacteristic
45#1981C2211yescharacteristic
46#1957C2212yes—
47#1874—22no—
48#1802—22no—
49#1481—24no—
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41—21no—
57C22×SD16—22no—
58C23×C8—21no—
59C23×C8—21no—
60#533—22no—
61#533—22no—
62#533C2×C414yes—
63#533D414yes—
64#532—22no—
65#532—22no—
66#532—22no—
67#532C2×C414yes—
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8D412yes—
76C2×#121—21no—
77C2×#121—21no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×C4×Q8—21no—
81C2×C4×Q8C2312yes—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×#117—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4D412yes—
90C22×C42C2311yescharacteristic
91#476—28no—
92#475—24no—
93#475C2×C418yescomplemented
94#465—216no—
95#461—28no—
96#460—24no—
97#460C2×C418yescomplemented
98#431—22no—
99#431—22no—
100C2×#106—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8C2×C414yes—
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2312yes—
111#416—44no—
112#416—44no—
113C2×#103—22no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#103—21no—
118C2×#102—22no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—21no—
123C2×#102—24no—
124C2×C22⋊C8—21no—
125C2×C4×C8C2312yes—
126#391—22no—
127#391—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—24no—
132#389—22no—
133#389—24no—
134#387—24no—
135#387—24no—
136#385—24no—
137#384—22no—
138#384—21no—
139#384—21no—
140#384D412yes—
141#384C2312yes—
142#382—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4D411yescharacteristic
150C4×C4⋊C4D411yescharacteristic
151C4×C4⋊C4C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—24no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156#376—21no—
157C43—21no—
158#340—24no—
159Q8⋊C8—24no—
160Q8⋊C8—24no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8C2×D412yes—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4—41no—
174C23×C4—22no—
175C23×C4—41no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C2×SD16—28no—
180C2×SD16—82no—
181C2×SD16—82no—
182C22×C8—22no—
183C22×C8—24no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—81no—
191C22×C8—81no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22⋊C414yes—
198C4⋊Q8C22⋊C414yes—
199C4⋊Q8C22×C412yes—
200#127—21no—
201#127—21no—
202#127—21no—
203#127C22×C412yes—
204#125—24no—
205#125—24no—
206#124—24no—
207#124—42no—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8C22⋊C418yes—
212#121—24no—
213#121—24no—
214#121—42no—
215#121—42no—
216C4×Q8—24no—
217C4×Q8—24no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—82no—
221C4×Q8—82no—
222C4×Q8—84no—
223C4×Q8C22×C414yes—
224C4×Q8C2×D414yes—
225C4×D4—24no—
226C4×D4—24no—
227C4×D4—42no—
228C4×D4—42no—
229#117—24no—
230#117—24no—
231#117—22no—
232#117—22no—
233#117—22no—
234#117SD1614yescomplemented
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282#106—28no—
283C4⋊C8—24no—
284C4⋊C8—22no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8C22×C414yes—
292#103—28no—
293#103—42no—
294#103—42no—
295#103—42no—
296#103—42no—
297#102—28no—
298#102—42no—
299#102—42no—
300#102—42no—
301#102—42no—
302#102—84no—
303C22⋊C8—42no—
304C4×C8—22no—
305C4×C8—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314C24—41no—
315C24#12011yescharacteristic
316#41—48no—
317#41—48no—
318#41—84no—
319#41—84no—
320C2×Q8—22no—
321C2×Q8—24no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8C2×C22⋊C414yes—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—82no—
353C2×D4—82no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4C2×C22⋊C412yes—
375C22×C4C4×D412yes—
376C22×C4C4×D412yes—
377C22×C4#12011yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12112yes—
381C22×C4#12112yes—
382C22×C4#12312yes—
383C22×C4C2×SD1612yes—
384C22×C4#13512yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4#13712yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390SD16—88no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—84no—
404C2×C8—84no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4C2×C22⋊C414yes—
433C4⋊C4C2×SD1612yes—
434C4⋊C4C2×SD1612yes—
435C22⋊C4—28no—
436C22⋊C4—24no—
437C22⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—42no—
444C42—42no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—84no—
451C42—84no—
452C42C2×C22⋊C412yes—
453C42C4×D412yes—
454C42C4×D412yes—
455C42#12112yes—
456C42#12112yes—
457C42#12312yes—
458C42C23×C411yescharacteristic
459C42C22×D411yescharacteristic
460C42C2×#4111yescharacteristic
461C42C2×#4111yescharacteristic
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—42no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23#38612yes—
474C23#44812yes—
475C23#44912yes—
476C23C2×#12011yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—42no—
483Q8—42no—
484Q8—81no—
485Q8—81no—
486D4—44no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—82no—
492D4—88no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—88no—
534C2×C4—164no—
535C2×C4#38612yes—
536C2×C4#38612yes—
537C2×C4#38612yes—
538C2×C4C4×SD1614yes—
539C2×C4#44014yes—
540C2×C4#44812yes—
541C2×C4#44912yes—
542C2×C4#46014yes—
543C2×C4#46414yes—
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C22×SD1611yescharacteristic
551C2×C4C2×#13511yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C2×#13711yescharacteristic
554C2×C4#57611yescharacteristic
555C2×C4#57711yescharacteristic
556C2×C4#57711yescharacteristic
557C2×C4#57811yescharacteristic
558C8—44no—
559C8—82no—
560C8—81no—
561C8—81no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—161no—
572C22—161no—
573C22—161no—
574C22—161no—
575C22#188014yes—
576C22C2×#38611yescharacteristic
577C22C2×#44811yescharacteristic
578C22C2×#44911yescharacteristic
579C22#301211yescharacteristic
580C22#301311yescharacteristic
581C22#301911yescharacteristic
582C22#302011yescharacteristic
583C4—22no—
584C4—21no—
585C4—21no—
586C4—22no—
587C4—44no—
588C4—42no—
589C4—42no—
590C4—44no—
591C4—81no—
592C4—84no—
593C4—82no—
594C4—81no—
595C4#188014yes—
596C4#289711yescharacteristic
597C4#289811yescharacteristic
598C4C2×C4×SD1611yescharacteristic
599C4C2×#44011yescharacteristic
600C4C2×#46011yescharacteristic
601C4C2×#46411yescharacteristic
602C4#309811yescharacteristic
603C4#309911yescharacteristic
604C2—24no—
605C2—162no—
606C2#1887211yescharacteristic
607C2#1887411yescharacteristic
608C2#1888411yescharacteristic
609C2#1888611yescharacteristic
610C2#2091511yescharacteristic
611C2C2×#188011yescharacteristic
612C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34214C2semidirect
5. C2×#1481C2semidirect
6. #8983C2semidirect
7. #8982C2semidirect
8. #8979C2semidirect
9. #8978C2semidirect
10. C2×#1296C2semidirect
11. #7891C2semidirect
15. #3246C4semidirect
93. #475C2×C4semidirect
97. #460C2×C4semidirect
234. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
#342142
#218592
#212572
C2×#14812
#89832
#89822
#89792
#89782
C2×#12962
#78912