Finite Group Databasebeta

← #195867

Subgroups — #195867

Order 512.

1656 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#431C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4C2×#1487C212yesmaximal, complemented
5#9075C214yesmaximal, complemented
6#9071C214yesmaximal, complemented
7#7901C212yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#431—21no—
13C2×#431—28no—
14C2×#431C2212yes—
15C2×#431C2212yes—
16C2×#423C2212yes—
17C22×C4⋊C8—21no—
18C2×#407—21no—
19C22×C22⋊C8—21no—
20C22×C4×C8C2212yes—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8C412yes—
26C4×C22⋊Q8C412yes—
27C4×C22⋊Q8C2212yes—
28C4×C22⋊Q8C2212yes—
29C2×#391C411yescharacteristic
30C2×#389C411yescharacteristic
31C2×#387—21no—
32C2×#385—21no—
33C2×#380C411yescharacteristic
34C2×C4×C4⋊C4—21no—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4—21no—
37#1797—21no—
38#1797—21no—
39#1797C2212yes—
40#1797C2212yes—
41#1728—21no—
42#1487—28no—
43C2×Q8⋊C8C2218yescomplemented
44#1279C2218yescomplemented
45C23×Q8C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C22×M16—22no—
50C23×C8—22no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—28no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2312yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—21no—
73C22×C42—21no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#431—28no—
87#431—28no—
88#431—24no—
89#431—28no—
90#431D418yes—
91#423—24no—
92#423—24no—
93C2×C4⋊C8—28no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2314yes—
98#407—44no—
99C2×C22⋊C8—28no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—28no—
106C2×C4×C8C2314yes—
107#391—21no—
108#391—21no—
109#391C2×C412yes—
110#391C2×C412yes—
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#385—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#380—21no—
124#380—21no—
125#380C2×C412yes—
126#380C2×C412yes—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2×C412yes—
134C4×C4⋊C4C2312yes—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—22no—
138C4×C22⋊C4—22no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144#336—22no—
145#336—22no—
146Q8⋊C8—216no—
147C25C22⋊C411yescharacteristic
148C22×Q8—24no—
149C22×Q8—21no—
150C22×Q8—24no—
151C22×Q8—42no—
152C22×Q8C22×C412yes—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4SD1614yes—
176C23×C4Q1614yes—
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
186C2×M16—44no—
187C2×M16—44no—
188C2×M16—44no—
189C22×C8—28no—
190C22×C8—24no—
191C22×C8—24no—
192C22×C8—44no—
193C22×C8—44no—
194C22⋊Q8—28no—
195C22⋊Q8—28no—
196C22⋊Q8—24no—
197C22⋊Q8—24no—
198C22⋊Q8—24no—
199C22⋊Q8M1618yescomplemented
200C4×Q8—28no—
201C4×Q8—24no—
202C4×Q8—24no—
203C4×Q8—42no—
204C4×Q8—44no—
205C4×Q8—42no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—44no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—44no—
252C2×C42—44no—
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22⋊C411yescharacteristic
255C2×C42C22⋊C411yescharacteristic
256C2×C42C22⋊C411yescharacteristic
257C2×C42C22⋊C411yescharacteristic
258C2×C42C22⋊C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D414yes—
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42#4114yes—
270C2×C42C2411yescharacteristic
271C4⋊C8—28no—
272C4⋊C8—28no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—44no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—44no—
279C22⋊C8—44no—
280C4×C8—28no—
281C4×C8—24no—
282C4×C8C2×D418yes—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—41no—
298C24—41no—
299C24#10314yes—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C2×C22⋊C411yescharacteristic
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—416no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8C2×M1614yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—216no—
330C22×C4—216no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
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—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—416no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4#10314yes—
378C22×C4#10314yes—
379C22×C4#10314yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C4×D414yes—
390C22×C4#12012yes—
391C22×C4#12114yes—
392C22×C4#12312yes—
393C22×C4C2×M1614yes—
394C22×C4C2×SD1612yes—
395C22×C4C2×SD1612yes—
396C22×C4C2×SD1612yes—
397C22×C4C2×Q1612yes—
398C22×C4C2×Q1612yes—
399C22×C4C2×Q1612yes—
400C22×C4#13514yes—
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404M16—88no—
405C2×C8—28no—
406C2×C8—416no—
407C2×C8—44no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—22no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4C2×M1614yes—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C42—22no—
431C42—22no—
432C42—21no—
433C42—21no—
434C42—22no—
435C42—22no—
436C42—21no—
437C42—21no—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—44no—
445C42—44no—
446C42—44no—
447C42—44no—
448C42C4×D414yes—
449C42#12012yes—
450C42#12114yes—
451C42#12312yes—
452C42C22×D412yes—
453C42C2×#4112yes—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—28no—
468C23—28no—
469C23—21no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23#38612yes—
476C23#40712yes—
477C23C2×#10312yes—
478C23C2×#10312yes—
479C23C2×#10312yes—
480C23#41612yes—
481C23#42112yes—
482C23C22×C22⋊C411yescharacteristic
483Q8—28no—
484Q8—48no—
485Q8—44no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—28no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—84no—
543C2×C4—84no—
544C2×C4#38612yes—
545C2×C4#38612yes—
546C2×C4#38612yes—
547C2×C4#40712yes—
548C2×C4C2×#10312yes—
549C2×C4C2×#10312yes—
550C2×C4C2×#10312yes—
551C2×C4#41612yes—
552C2×C4#42112yes—
553C2×C4#46518yes—
554C2×C4#46718yes—
555C2×C4C2×C4×D412yes—
556C2×C4C2×#12011yescharacteristic
557C2×C4C2×#12112yes—
558C2×C4C2×#12311yescharacteristic
559C2×C4C22×M1612yes—
560C2×C4#56812yes—
561C2×C4C22×SD1611yescharacteristic
562C2×C4C22×Q1611yescharacteristic
563C2×C4C2×#13512yes—
564C8—48no—
565C8—84no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—416no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22#148714yescomplemented
587C22#189214yes—
588C22C2×#38611yescharacteristic
589C22C2×#40711yescharacteristic
590C22#287911yescharacteristic
591C22C22×#10311yescharacteristic
592C22C2×#41611yescharacteristic
593C22C2×#42111yescharacteristic
594C22C2×#42111yescharacteristic
595C4—22no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—22no—
602C4—22no—
603C4—44no—
604C4—41no—
605C4—44no—
606C4—41no—
607C4—82no—
608C4—82no—
609C4#189214yes—
610C4D4×M1614yes—
611C4C2×#46512yes—
612C4C2×#46712yes—
613C2—24no—
614C2—24no—
615C2—42no—
616C2C2×#148711yescharacteristic
617C2#1744311yescharacteristic
618C2#1748611yescharacteristic
619C2#1748711yescharacteristic
620C2#2089811yescharacteristic
621C2C2×#189211yescharacteristic
622C2#2097911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1487C2semidirect
5. #9075C2semidirect
6. #9071C2semidirect
7. #7901C2semidirect
43. C2×Q8⋊C8C22semidirect
44. #1279C22semidirect
199. C22⋊Q8M16semidirect
586. C22#1487semidirect

Maximal subgroups

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

subgroupindex
C22×#4312
C2×C4×C22⋊Q82
#206262
C2×#14872
#90752
#90712
#79012