Finite Group Databasebeta

← #197620

Subgroups — #197620

Order 512.

1119 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3076C211yesmaximal, characteristic
2#60277C211yesmaximal, characteristic
3#20654C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic, complemented
5#17486C211yesmaximal, characteristic, complemented
6#9209C218yesmaximal, complemented
7C2×#1301C211yesmaximal, characteristic, complemented
8#7893C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#3076—22no—
12#3076—22no—
13#3076C414yescomplemented
14#3076C2214yes—
15C2×#479—21no—
16C2×#477C411yescharacteristic, complemented
17C2×#475C411yescharacteristic, complemented
18C2×#478—21no—
19C2×#476C411yescharacteristic, complemented
20C2×#474C411yescharacteristic, complemented
21#2989—22no—
22#2989—22no—
23#2983—21no—
24#2975—21no—
25#2973—22no—
26#2965—22no—
27#2965C2214yes—
28#2920—22no—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#427—21no—
32C2×#423C2211yescharacteristic
33C22×C4⋊C8C2211yescharacteristic
34C2×#421—21no—
35C2×#416—21no—
36#2285C2211yescharacteristic
37#1806—22no—
38#1806C2214yes—
39#1733—21no—
40#1520—22no—
41#1518—22no—
42#1503C2214yescomplemented
43#1493C2214yescomplemented
44#1487—22no—
45#1481—22no—
46#1301C2214yescomplemented
47#1279C2214yescomplemented
48C22×#41D411yescharacteristic
49C22×M16—21no—
50C23×C8—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533—21no—
57#533C2×C412yes—
58#533C2312yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C4×#41—21no—
65C4×#41—21no—
66C4×#41—21no—
67C4×#41—21no—
68C4×#41C2×C412yes—
69C4×#41D412yes—
70C4×#41D412yes—
71C4×#41C2312yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×#117—21no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#479—24no—
87#479—24no—
88#478—24no—
89#478—24no—
90#477—22no—
91#477C2×C414yes—
92#476—22no—
93#476C2×C414yes—
94#475—22no—
95#475C2×C414yes—
96#474—22no—
97#474C2×C414yes—
98#446—24no—
99#446—24no—
100#436—24no—
101#436—24no—
102#435—28no—
103#435—28no—
104#432—24no—
105#432—24no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—24no—
111#431D412yes—
112#431D412yes—
113#431D412yes—
114#431C2312yes—
115#427—24no—
116#427—24no—
117C2×#107—21no—
118C2×#107—21no—
119C2×#106—21no—
120C2×#106—21no—
121#423—24no—
122#423—21no—
123#423—21no—
124#423—21no—
125#423C2312yes—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8C2×C411yescharacteristic
134C2×C4⋊C8C2×C411yescharacteristic
135C2×C4⋊C8C2×C412yes—
136C2×C4⋊C8C2312yes—
137#421—44no—
138C2×#104—24no—
139#416—24no—
140#416—24no—
141C2×#103—21no—
142C2×#103—21no—
143C2×#102—21no—
144C2×#102—21no—
145#407—24no—
146C2×C22⋊C8—21no—
147C2×C22⋊C8D412yes—
148C4×M16—24no—
149C2×C4×C8—21no—
150C2×C4×C8C2312yes—
151#384—21no—
152#382—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2312yes—
157#339—24no—
158#328—28no—
159Q8⋊C8—28no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—22no—
163C2×#41—22no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41C22⋊C412yes—
167C2×#41C2×D412yes—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8C22⋊C411yescharacteristic
171C22×D4—21no—
172C22×D4C22⋊C411yescharacteristic
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4#4111yescharacteristic
179C2×M16—28no—
180C2×M16—22no—
181C2×M16—41no—
182C2×M16—42no—
183C2×M16—42no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—41no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8C22×C412yes—
200C4⋊Q8C22×C412yes—
201#125—22no—
202#125—22no—
203#125—41no—
204#125—41no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8C22⋊C412yes—
226C4×Q8C22⋊C412yes—
227C4×Q8C22×C412yes—
228C4×Q8C2×D412yes—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C22⋊C412yes—
242C4×D4C22⋊C412yes—
243C4×D4C22×C412yes—
244C4×D4C2×D412yes—
245#117—21no—
246#117—21no—
247#117—22no—
248#117—22no—
249#117—22no—
250#117—22no—
251#117—22no—
252#117—22no—
253#117—41no—
254#117—41no—
255#117—42no—
256#117—42no—
257#117C22⋊C412yes—
258#117C2×D412yes—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—41no—
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42C2411yescharacteristic
298#107—24no—
299#107—24no—
300#106—24no—
301#106—24no—
302C4⋊C8—28no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—41no—
314C4⋊C8—41no—
315C4⋊C8C22×C412yes—
316C4⋊C8C22×C412yes—
317#104—48no—
318#103—24no—
319#103—24no—
320#102—24no—
321#102—24no—
322C22⋊C8—28no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8C2×D414yes—
327#97—28no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8C2×D414yes—
332#95—22no—
333#95—41no—
334C24—21no—
335C24#12111yescharacteristic
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—24no—
341#41—24no—
342#41—42no—
343#41—42no—
344#41—42no—
345#41—42no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×Q8C2×C22⋊C412yes—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4C2×C22⋊C412yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—42no—
379C22×C4—82no—
380C22×C4—81no—
381C22×C4—82no—
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4#12011yescharacteristic
389C22×C4#12011yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12311yescharacteristic
396C22×C4#12311yescharacteristic
397C22×C4AGL(1,ℤ/8)12yes—
398C22×C4#13712yes—
399C22×C4#13712yes—
400C22×C4#13712yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4111yescharacteristic
404M16—42no—
405M16—42no—
406M16—88no—
407M16—82no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C2×C8—44no—
417C2×C8—42no—
418C2×C8—82no—
419C2×C8—82no—
420C2×C8—82no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4C2×C22⋊C412yes—
443C4⋊C4C2×C22⋊C412yes—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—82no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42C2×C22⋊C411yescharacteristic
464C42C2×C22⋊C411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42C4×D411yescharacteristic
469C42#12011yescharacteristic
470C42#12011yescharacteristic
471C42#12111yescharacteristic
472C42#12111yescharacteristic
473C42#12311yescharacteristic
474C42#12311yescharacteristic
475C42C23×C411yescharacteristic
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C42C2×#4111yescharacteristic
479C23—22no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23#38611yescharacteristic
486C23#38611yescharacteristic
487C23#46812yes—
488C23#47012yes—
489C23C2×#12111yescharacteristic
490Q8—22no—
491Q8—22no—
492Q8—42no—
493Q8—44no—
494Q8—82no—
495Q8—82no—
496Q8—82no—
497D4—22no—
498D4—22no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—42no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#38611yescharacteristic
536C2×C4#38611yescharacteristic
537C2×C4#38611yescharacteristic
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4#41712yes—
542C2×C4#41712yes—
543C2×C4#41912yes—
544C2×C4#41912yes—
545C2×C4#46812yes—
546C2×C4#46812yes—
547C2×C4#46812yes—
548C2×C4#47012yes—
549C2×C4#47012yes—
550C2×C4#47012yes—
551C2×C4C22×C22⋊C411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×#12011yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×#12311yescharacteristic
557C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
558C2×C4C2×#13711yescharacteristic
559C2×C4C2×#13711yescharacteristic
560C2×C4C2×#13711yescharacteristic
561C8—44no—
562C8—44no—
563C8—42no—
564C8—42no—
565C8—82no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—84no—
578C22#189412yes—
579C22#189412yes—
580C22C2×#38611yescharacteristic
581C22C2×#46811yescharacteristic
582C22C2×#47011yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—21no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—162no—
596C4#189412yes—
597C4#189412yes—
598C4C2×#41911yescharacteristic
599C4C2×#41711yescharacteristic
600C4#295712yes—
601C4C2×#46811yescharacteristic
602C4C2×#47011yescharacteristic
603C2—21no—
604C2—22no—
605C2—22no—
606C2—21no—
607C2—82no—
608C2#2091911yescharacteristic
609C2C2×#189411yescharacteristic
610C2#2100011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17575C2semidirect
5. #17486C2semidirect
6. #9209C2semidirect
7. C2×#1301C2semidirect
8. #7893C2semidirect
13. #3076C4semidirect
16. C2×#477C4semidirect
17. C2×#475C4semidirect
19. C2×#476C4semidirect
20. C2×#474C4semidirect
42. #1503C22semidirect
43. #1493C22semidirect
46. #1301C22semidirect
47. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#602772
#206542
#175752
#174862
#92092
C2×#13012
#78932