Finite Group Databasebeta

← #194130

Subgroups — #194130

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21385C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#8981C214yesmaximal, complemented
7#8973C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4—21no—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4—21no—
14#3077—24no—
15#3077C418yescomplemented
16C2×#478—22no—
17C2×#476C2212yes—
18C2×#474C2212yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#428—21no—
22C2×#423C2211yescharacteristic
23C2×#418—21no—
24D4×C4⋊C4—22no—
25D4×C4⋊C4—22no—
26D4×C4⋊C4—22no—
27D4×C4⋊C4C2214yes—
28C2×#389—21no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33#1990C2212yescomplemented
34#1987C2212yescomplemented
35#1981C2212yescomplemented
36#1956C2212yescomplemented
37#1872—22no—
38#1872—21no—
39#1872—21no—
40#1803—22no—
41C4×#103—22no—
42#1478—24no—
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45#1279C2214yescomplemented
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C24×C4—22no—
49C22×M16—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533C2×C414yescomplemented
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×#125—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—44no—
68C2×C4×D4—44no—
69C2×C4×D4C2×C414yes—
70C2×C4×D4D414yes—
71C2×C4×D4D412yes—
72C2×C4×D4C2312yes—
73C2×#117—21no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—48no—
80C22×C4⋊C4D411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—22no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#478—216no—
89#476—24no—
90#476C2×C418yescomplemented
91#474—24no—
92#474C2×C418yescomplemented
93#431—22no—
94#431—22no—
95#431—22no—
96#431—22no—
97#428—28no—
98C2×#107—22no—
99C2×#106—22no—
100#423—22no—
101#423C2×C414yescomplemented
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106#418—28no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×#102—22no—
111C2×#102—22no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2312yes—
116#389—24no—
117#389—24no—
118#385—28no—
119#385—28no—
120#384—21no—
121#384—21no—
122#384D412yes—
123#384C2312yes—
124#380—24no—
125#380—24no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4C2312yes—
130C2×#95—22no—
131C2×#95—21no—
132#340—24no—
133#340—22no—
134#340—22no—
135D4⋊C8—28no—
136C25—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×D4—21no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22⋊C414yes—
144C22×D4SD1614yes—
145C22×D4Q1614yes—
146C22×D4C2×D412yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—28no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—48no—
159C23×C4—48no—
160C23×C4—84no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8C22×C412yescomplemented
177C4⋊Q8C22×C412yescomplemented
178#125—24no—
179#125—42no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C4×Q8—24no—
185C4×Q8—24no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×D4—24no—
189C4×D4—24no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—48no—
196C4×D4—416no—
197C4×D4—48no—
198C4×D4C22⋊C418yes—
199C4×D4C22×C414yes—
200C4×D4C2×D414yes—
201#117—24no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—48no—
221C2×C4⋊C4—48no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22⋊C414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4—41no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—41no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42C22×C412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42#4112yes—
252C2×C42#4112yes—
253C2×C42C2411yescharacteristic
254#107—28no—
255#106—28no—
256C4⋊C8—24no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—42no—
261C4⋊C8C22×C414yescomplemented
262#103—44no—
263#103—44no—
264#103—84no—
265#102—28no—
266#102—28no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8—42no—
272#95—28no—
273#95—22no—
274#95—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24#12011yescharacteristic
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—28no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—28no—
300C2×D4—28no—
301C2×D4#103116yescomplemented
302C2×D4C2×C22⋊C414yes—
303C2×D4C2×SD1614yes—
304C2×D4C2×Q1614yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—416no—
321C22×C4—416no—
322C22×C4—416no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—416no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—88no—
339C22×C4—832no—
340C22×C4C2×C22⋊C412yes—
341C22×C4C4×D412yes—
342C22×C4C4×D412yes—
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12011yescharacteristic
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4C2×SD1612yes—
350C22×C4C2×Q1612yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356M16—48no—
357C2×C8—24no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—416no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4C2×C22⋊C414yes—
392C4⋊C4AGL(1,ℤ/8)14yes—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—28no—
396C22⋊C4—416no—
397C22⋊C4—44no—
398C42—42no—
399C42—42no—
400C42—44no—
401C42—44no—
402C42C2×C22⋊C412yes—
403C42C4×D412yes—
404C42C4×D412yes—
405C42#12112yes—
406C42#12112yes—
407C42#12312yes—
408C42C23×C411yescharacteristic
409C42C22×D411yescharacteristic
410C42C2×#4111yescharacteristic
411C42C2×#4111yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—216no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—48no—
421C23—48no—
422C23—48no—
423C23—48no—
424C23—48no—
425C23—48no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23#38612yes—
431C23#45012yes—
432C23#45112yes—
433C23C2×#12011yescharacteristic
434Q8—44no—
435Q8—44no—
436Q8—82no—
437Q8—82no—
438D4—24no—
439D4—24no—
440D4—24no—
441D4—216no—
442D4C2×#10318yes—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—416no—
456C2×C4—48no—
457C2×C4—48no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—832no—
473C2×C4—84no—
474C2×C4—88no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4#38612yes—
480C2×C4#38612yes—
481C2×C4#38612yes—
482C2×C4C2×#10314yes—
483C2×C4#41714yes—
484C2×C4#45012yes—
485C2×C4#45112yes—
486C2×C4#48314yes—
487C2×C4#48414yes—
488C2×C4C22×C22⋊C411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C2×#12111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12311yescharacteristic
494C2×C4C22×SD1611yescharacteristic
495C2×C4C22×Q1611yescharacteristic
496C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
497C2×C4C2×AGL(1,ℤ/8)12yes—
498C2×C4C2×#13711yescharacteristic
499C2×C4#57512yes—
500C8—44no—
501C8—82no—
502C8—82no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—28no—
510C22—432no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22#187214yes—
518C22C2×#38611yescharacteristic
519C22C2×#45011yescharacteristic
520C22C2×#45111yescharacteristic
521C22#300311yescharacteristic
522C22#300511yescharacteristic
523C22#301911yescharacteristic
524C22#302011yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—44no—
530C4—44no—
531C4—84no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4#187214yes—
536C4C22×#10311yescharacteristic
537C4C2×#41711yescharacteristic
538C4C4×AGL(1,ℤ/8)12yes—
539C4#303612yes—
540C4C2×#48311yescharacteristic
541C4C2×#48411yescharacteristic
542C2—24no—
543C2—48no—
544C2#1874711yescharacteristic
545C2#1874911yescharacteristic
546C2#1875711yescharacteristic
547C2#1875911yescharacteristic
548C2C2×#187211yescharacteristic
549C2#2092511yescharacteristic
550C2#2094311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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. #21385C2semidirect
6. #8981C2semidirect
7. #8973C2semidirect
8. #8043C2semidirect
9. #7901C2semidirect
15. #3077C4semidirect
33. #1990C22semidirect
34. #1987C22semidirect
35. #1981C22semidirect
36. #1956C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
53. #533C2×C4semidirect
90. #476C2×C4semidirect
92. #474C2×C4semidirect
101. #423C2×C4semidirect
176. C4⋊Q8C22×C4semidirect
177. C4⋊Q8C22×C4semidirect
261. C4⋊C8C22×C4semidirect
301. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×D4×C4⋊C42
#218602
#213852
C2×#14782
#89812
#89732
#80432
#79012