Finite Group Databasebeta

← #194163

Subgroups — #194163

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3100C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic, complemented
3#21859C211yesmaximal, characteristic, complemented
4#21258C211yesmaximal, characteristic, complemented
5C2×#1486C211yesmaximal, characteristic, complemented
6#8984C214yesmaximal, complemented
7#8980C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7892C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13#3100—24no—
14#3100C418yescomplemented
15C2×#480—21no—
16C2×#474C2212yes—
17C2×#459C2212yes—
18C2×#447—21no—
19C2×C4×D8—22no—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C2×#417—21no—
24C2×#417—21no—
25#2354—22no—
26#2354—22no—
27#2354—22no—
28#2354C2214yes—
29C2×#390—21no—
30C2×#388—22no—
31C2×#384C2211yescharacteristic
32C2×#380—21no—
33C2×C4×C4⋊C4—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#1990C2212yescomplemented
36#1983—22no—
37#1982C2212yescomplemented
38#1981C2212yescomplemented
39#1957C2212yescomplemented
40#1874—22no—
41#1802—22no—
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C22×#41—21no—
47C23×D4D411yescharacteristic
48C24×C4—21no—
49C22×D8—22no—
50C22×M16—21no—
51#532—22no—
52#532—22no—
53#532—22no—
54#532C2×C414yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×#127C2311yescharacteristic
57C2×#124—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C414yes—
67C2×C4×D4D414yes—
68C2×C4×D4D412yes—
69C2×C4×D4C2312yes—
70C2×#117—24no—
71C2×#117—24no—
72C2×#117—44no—
73C2×#117—44no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—22no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#480—28no—
87#474—24no—
88#474C2×C418yescomplemented
89#459—24no—
90#459C2×C418yescomplemented
91#447—28no—
92C4×D8—216no—
93#431—22no—
94#431—22no—
95#431—22no—
96#431C2×C414yes—
97C2×#107—22no—
98#423—22no—
99#423—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104#417—44no—
105#417—44no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—24no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2312yes—
117#390—24no—
118#390—24no—
119#388—28no—
120#388—28no—
121#384—21no—
122#384—21no—
123#384D412yes—
124#384C2312yes—
125#380—24no—
126#380—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133C2×#95—22no—
134C2×#95—21no—
135#340—24no—
136D4⋊C8—28no—
137C25—21no—
138C2×#41—24no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C22×Q8—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—41no—
153C22×D4C22⋊C414yes—
154C22×D4C2×D412yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—21no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—41no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×D8—28no—
172C2×D8—44no—
173C2×D8—44no—
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8C22×C412yes—
184#127—21no—
185#127—21no—
186#127—21no—
187#127C22×C412yes—
188#124—24no—
189#124—42no—
190#121—24no—
191#121—24no—
192#121—42no—
193#121—42no—
194C4×D4—24no—
195C4×D4—24no—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4C22⋊C418yes—
206C4×D4C22×C414yes—
207C4×D4C2×D414yes—
208#117—48no—
209#117—416no—
210#117—48no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22⋊C414yes—
230C2×C4⋊C4D814yescomplemented
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—84no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—48no—
249C2×C42—44no—
250C2×C42C22×C412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D412yes—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4112yes—
256C2×C42#4112yes—
257C2×C42C2411yescharacteristic
258#107—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8C22×C414yes—
265#103—44no—
266#103—44no—
267#102—28no—
268#102—28no—
269#102—44no—
270#102—44no—
271#102—84no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274C4×C8—24no—
275C4×C8—22no—
276C4×C8—22no—
277#95—28no—
278#95—22no—
279#95—22no—
280C24—22no—
281C24—24no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—41no—
289C24#12011yescharacteristic
290#41—48no—
291#41—48no—
292#41—84no—
293#41—84no—
294C2×Q8—22no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—48no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—48no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4C2×C22⋊C414yes—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12312yes—
374C22×C4C2×D812yes—
375C22×C4#13512yes—
376C22×C4AGL(1,ℤ/8)12yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381D8—416no—
382M16—84no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—82no—
410C4⋊C4C2×C22⋊C414yes—
411C4⋊C4C2×D814yes—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—28no—
415C22⋊C4—416no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—88no—
419C42—44no—
420C42—416no—
421C42—44no—
422C42C2×C22⋊C412yes—
423C42C4×D412yes—
424C42C4×D412yes—
425C42#12112yes—
426C42#12112yes—
427C42#12312yes—
428C42C23×C411yescharacteristic
429C42C22×D411yescharacteristic
430C42C2×#4111yescharacteristic
431C42C2×#4111yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—24no—
437C23—48no—
438C23—48no—
439C23—48no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—84no—
448C23—84no—
449C23—84no—
450C23—84no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23#38612yes—
458C23#44712yes—
459C23#44912yes—
460C23C2×#12011yescharacteristic
461Q8—42no—
462Q8—42no—
463Q8—81no—
464Q8—81no—
465D4—24no—
466D4—24no—
467D4—48no—
468D4—44no—
469D4—42no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4—82no—
475D4—88no—
476D4—82no—
477D4—81no—
478D4—81no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—416no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—88no—
510C2×C4—84no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—88no—
514C2×C4#38612yes—
515C2×C4#38612yes—
516C2×C4#38612yes—
517C2×C4C4×D814yes—
518C2×C4#44214yes—
519C2×C4#44712yes—
520C2×C4#44912yes—
521C2×C4#45914yes—
522C2×C4#46314yes—
523C2×C4C22×C22⋊C411yescharacteristic
524C2×C4C2×C4×D411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×#12111yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4C22×D811yescharacteristic
530C2×C4C2×#13511yescharacteristic
531C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
532C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
533C2×C4#57512yes—
534C2×C4#57512yes—
535C8—44no—
536C8—82no—
537C8—82no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—816no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22#187914yes—
560C22C2×#38611yescharacteristic
561C22C2×#44711yescharacteristic
562C22C2×#44911yescharacteristic
563C22#300511yescharacteristic
564C22#300511yescharacteristic
565C22#301911yescharacteristic
566C22#302011yescharacteristic
567C4—22no—
568C4—21no—
569C4—21no—
570C4—22no—
571C4—44no—
572C4—44no—
573C4—84no—
574C4—82no—
575C4—82no—
576C4#187914yes—
577C4#289612yes—
578C4C2×C4×D811yescharacteristic
579C4C2×#44211yescharacteristic
580C4C2×#45911yescharacteristic
581C4C2×#46311yescharacteristic
582C4#309312yes—
583C2—24no—
584C2—84no—
585C2—162no—
586C2#1875611yescharacteristic
587C2#1876211yescharacteristic
588C2#1876611yescharacteristic
589C2#1877211yescharacteristic
590C2#2091011yescharacteristic
591C2C2×#187911yescharacteristic
592C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#3100C2
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. C2×#2354C2semidirect
3. #21859C2semidirect
4. #21258C2semidirect
5. C2×#1486C2semidirect
6. #8984C2semidirect
7. #8980C2semidirect
8. #8043C2semidirect
9. #7892C2semidirect
14. #3100C4semidirect
35. #1990C22semidirect
37. #1982C22semidirect
38. #1981C22semidirect
39. #1957C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
88. #474C2×C4semidirect
90. #459C2×C4semidirect
230. C2×C4⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#31002
C2×#23542
#218592
#212582
C2×#14862
#89842
#89802
#80432
#78922