Finite Group Databasebeta

← #194276

Subgroups — #194276

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3246C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21311C211yesmaximal, characteristic, complemented
5C2×#1480C211yesmaximal, characteristic
6#8988C212yesmaximal, complemented
7#8985C212yesmaximal, complemented
8#8976C212yesmaximal, complemented
9#8973C212yesmaximal, complemented
10C2×#1296C211yesmaximal, characteristic, complemented
11#7891C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14C2×#520—21no—
15#3246—24no—
16#3246C418yescomplemented
17C2×#475—21no—
18C2×#476C2212yes—
19C2×#465—22no—
20C2×#460—21no—
21C2×#461C2212yes—
22C2×#431C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25#2361—22no—
26#2361—22no—
27#2354—21no—
28#2354C2212yes—
29D4×C4⋊C4—21no—
30D4×C4⋊C4C2212yes—
31#2326—22no—
32#2311—21no—
33C4×C4⋊Q8—21no—
34C4×#127—21no—
35C2×#384C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1990C2212yes—
38#1989—21no—
39#1989—21no—
40#1988C2211yescharacteristic
41#1988C2211yescharacteristic
42#1987C2211yescharacteristic
43#1987C2211yescharacteristic
44#1956C2212yes—
45#1872—22no—
46#1803—22no—
47#1735—21no—
48#1735—21no—
49#1480—24no—
50C2×D4⋊C8C2212yes—
51C2×D4⋊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#533—22no—
63#533C2×C414yescomplemented
64#532—22no—
65#532—22no—
66#532C2×C414yes—
67#532D414yes—
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127D411yescharacteristic
72C2×#125—21no—
73C2×#124—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×#121—21no—
77C2×#121D412yes—
78#520—44no—
79#520—44no—
80C2×C4×Q8—22no—
81C2×C4×Q8—21no—
82C2×C4×Q8—21no—
83C2×C4×D4—22no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4—22no—
87C2×C4×D4—21no—
88C2×C4×D4C2312yes—
89C2×#117—22no—
90C2×#117—21no—
91C2×#117D411yescharacteristic
92C2×#117D411yescharacteristic
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4D412yes—
97C22×C42C2311yescharacteristic
98#476—24no—
99#476C2×C418yescomplemented
100#475—28no—
101#465—216no—
102#461—24no—
103#461C2×C418yescomplemented
104#460—28no—
105#431—22no—
106#431—22no—
107C2×#106—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—22no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2×C414yescomplemented
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2312yes—
118C2×#103—22no—
119C2×#103—24no—
120C2×#102—22no—
121C2×C22⋊C8—21no—
122C2×C4×C8C2312yes—
123#391—22no—
124#390—22no—
125#390—22no—
126#390—22no—
127#389—22no—
128#389—22no—
129#388—24no—
130#388—24no—
131#386—24no—
132#385—24no—
133#385—24no—
134#384—21no—
135#384—21no—
136#384D412yes—
137#384C2312yes—
138#380—22no—
139#380—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148#376—21no—
149C43—21no—
150#340—24no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#340—21no—
159D4⋊C8—24no—
160D4⋊C8—24no—
161C2×#41—24no—
162C2×#41—24no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—22no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×D412yes—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C2×SD16—28no—
181C2×SD16—82no—
182C2×SD16—82no—
183C22×C8—22no—
184C22×C8—24no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—22no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8—81no—
192C22×C8—81no—
193C4⋊Q8—22no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C22⋊C414yes—
201C4⋊Q8C22×C412yes—
202C4⋊Q8C22×C412yes—
203#127—22no—
204#127C22⋊C414yes—
205#125—24no—
206#125—42no—
207#124—24no—
208#124—24no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213#121—24no—
214#121—24no—
215#121—24no—
216#121C22⋊C418yes—
217C4×Q8—24no—
218C4×Q8—24no—
219C4×Q8—48no—
220C4×Q8—42no—
221C4×Q8—42no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4—48no—
227C4×D4—416no—
228C4×D4—82no—
229C4×D4—82no—
230C4×D4—84no—
231C4×D4C22×C414yes—
232C4×D4C2×D414yes—
233#117—22no—
234#117—24no—
235#117—22no—
236#117—22no—
237#117—44no—
238#117—42no—
239#117—44no—
240#117SD1614yescomplemented
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4—81no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D414yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—41no—
275C2×C42—21no—
276C2×C42—24no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—41no—
283C2×C42C22×C412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42#4112yes—
289C2×C42#4112yes—
290C2×C42C2411yescharacteristic
291#106—28no—
292C4⋊C8—24no—
293C4⋊C8—22no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8C22×C414yes—
301#103—28no—
302#103—84no—
303#102—28no—
304C22⋊C8—42no—
305C4×C8—22no—
306C4×C8—22no—
307#95—24no—
308#95—24no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313C24—22no—
314C24#12011yescharacteristic
315#41—48no—
316#41—48no—
317#41—48no—
318#41—48no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×D4—28no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4C2×C22⋊C414yes—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—44no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4C2×C22⋊C412yes—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12011yescharacteristic
377C22×C4#12011yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4#12312yes—
382C22×C4C2×SD1612yes—
383C22×C4#13512yes—
384C22×C4AGL(1,ℤ/8)12yes—
385C22×C4#13712yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389SD16—88no—
390C2×C8—24no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—84no—
403C2×C8—84no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4C2×C22⋊C414yes—
432C4⋊C4C2×SD1612yes—
433C4⋊C4C2×SD1612yes—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—24no—
437C22⋊C4—28no—
438C22⋊C4—48no—
439C22⋊C4—44no—
440C22⋊C4—42no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C42—24no—
444C42—42no—
445C42—48no—
446C42—42no—
447C42—41no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—44no—
452C42—42no—
453C42C2×C22⋊C412yes—
454C42C4×D412yes—
455C42C4×D412yes—
456C42#12112yes—
457C42#12112yes—
458C42#12312yes—
459C42C23×C411yescharacteristic
460C42C22×D411yescharacteristic
461C42C2×#4111yescharacteristic
462C42C2×#4111yescharacteristic
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—24no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23#38612yes—
475C23#45012yes—
476C23#45212yes—
477C23C2×#12011yescharacteristic
478Q8—44no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—82no—
483Q8—81no—
484Q8—81no—
485D4—24no—
486D4—24no—
487D4—48no—
488D4—416no—
489D4—42no—
490D4—42no—
491D4—48no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—168no—
533C2×C4—164no—
534C2×C4#38612yes—
535C2×C4#38612yes—
536C2×C4#38612yes—
537C2×C4C4×SD1614yes—
538C2×C4#44014yes—
539C2×C4#45012yes—
540C2×C4#45212yes—
541C2×C4#46114yes—
542C2×C4#46314yes—
543C2×C4C22×C22⋊C411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×#12111yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12311yescharacteristic
549C2×C4C22×SD1611yescharacteristic
550C2×C4C2×#13511yescharacteristic
551C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
552C2×C4C2×#13711yescharacteristic
553C2×C4#57611yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57711yescharacteristic
556C2×C4#57811yescharacteristic
557C8—44no—
558C8—82no—
559C8—81no—
560C8—81no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22#187714yes—
575C22C2×#38611yescharacteristic
576C22C2×#45011yescharacteristic
577C22C2×#45211yescharacteristic
578C22#301211yescharacteristic
579C22#301311yescharacteristic
580C22#301911yescharacteristic
581C22#302011yescharacteristic
582C4—22no—
583C4—21no—
584C4—21no—
585C4—22no—
586C4—42no—
587C4—44no—
588C4—42no—
589C4—82no—
590C4—84no—
591C4—81no—
592C4—81no—
593C4—82no—
594C4#187714yes—
595C4#289711yescharacteristic
596C4#289811yescharacteristic
597C4C2×C4×SD1611yescharacteristic
598C4C2×#44011yescharacteristic
599C4C2×#46111yescharacteristic
600C4C2×#46311yescharacteristic
601C4#309811yescharacteristic
602C4#309911yescharacteristic
603C2—24no—
604C2—84no—
605C2#1887111yescharacteristic
606C2#1887711yescharacteristic
607C2#1888311yescharacteristic
608C2#1888911yescharacteristic
609C2#2091511yescharacteristic
610C2C2×#187711yescharacteristic
611C2#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
3. #21860C2semidirect
4. #21311C2semidirect
6. #8988C2semidirect
7. #8985C2semidirect
8. #8976C2semidirect
9. #8973C2semidirect
10. C2×#1296C2semidirect
11. #7891C2semidirect
16. #3246C4semidirect
63. #533C2×C4semidirect
99. #476C2×C4semidirect
103. #461C2×C4semidirect
114. C2×C4⋊C8C2×C4semidirect
240. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32462
#342072
#218602
#213112
C2×#14802
#89882
#89852
#89762
#89732
C2×#12962
#78912