Finite Group Databasebeta

← #194138

Subgroups — #194138

Order 512.

1504 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic, complemented
4#21385C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8976C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117—21no—
13#3077—24no—
14#3077C418yescomplemented
15C2×#478—22no—
16C2×#476C2212yes—
17C2×#474C2212yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C2×#418—21no—
23#2354—22no—
24#2354—22no—
25#2354—22no—
26#2354C2214yes—
27C2×#390—21no—
28C2×#388—22no—
29C2×#384C2211yescharacteristic
30C2×#380—21no—
31C2×C4×C4⋊C4—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—
48C22×M16—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533C2×C414yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×C4×D4D414yes—
66C2×C4×D4D412yes—
67C2×C4×D4C2312yes—
68C2×#117—24no—
69C2×#117—24no—
70C2×#117—21no—
71C2×#117—48no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D412yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—22no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84#478—216no—
85#476—24no—
86#476C2×C418yescomplemented
87#474—24no—
88#474C2×C418yescomplemented
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#428—28no—
94C2×#107—22no—
95C2×#106—22no—
96#423—22no—
97#423C2×C414yes—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102#418—28no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—24no—
106C2×#102—22no—
107C2×#102—22no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2312yes—
112#390—24no—
113#390—24no—
114#388—28no—
115#388—28no—
116#384—21no—
117#384—21no—
118#384D412yes—
119#384C2312yes—
120#380—24no—
121#380—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2312yes—
128C2×#95—22no—
129C2×#95—21no—
130#340—24no—
131#340—22no—
132#340—22no—
133D4⋊C8—28no—
134C25—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22⋊C414yes—
144C22×D4C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172#125—24no—
173#125—42no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4C22⋊C418yes—
190C4×D4C22×C414yes—
191C4×D4C2×D414yes—
192#117—24no—
193#117—416no—
194#117—48no—
195#117—42no—
196#117—48no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C22⋊C414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D412yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—84no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—48no—
236C2×C42—41no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22×C412yes—
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42#4112yes—
245C2×C42#4112yes—
246C2×C42C2411yescharacteristic
247#107—28no—
248#106—28no—
249C4⋊C8—24no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—42no—
254C4⋊C8C22×C414yes—
255#103—44no—
256#103—44no—
257#103—84no—
258#102—28no—
259#102—28no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C4×C8—22no—
263C4×C8—22no—
264C4×C8—42no—
265#95—28no—
266#95—22no—
267#95—22no—
268C24—22no—
269C24—24no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24#12011yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—28no—
292C2×D4—28no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4C2×C22⋊C414yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—28no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—416no—
314C22×C4—416no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—44no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4C2×C22⋊C412yes—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12011yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12112yes—
336C22×C4#12112yes—
337C22×C4#12312yes—
338C22×C4#13512yes—
339C22×C4#13512yes—
340C22×C4#13712yes—
341C22×C4#13712yes—
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345M16—48no—
346C2×C8—24no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—44no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—44no—
374C4⋊C4—42no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—82no—
379C4⋊C4C2×C22⋊C414yes—
380C4⋊C4AGL(1,ℤ/8)14yes—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—28no—
384C22⋊C4—416no—
385C22⋊C4—44no—
386C22⋊C4—88no—
387C42—416no—
388C42—42no—
389C42—42no—
390C42—44no—
391C42—44no—
392C42C2×C22⋊C412yes—
393C42C4×D412yes—
394C42C4×D412yes—
395C42#12112yes—
396C42#12112yes—
397C42#12312yes—
398C42C23×C411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—24no—
407C23—48no—
408C23—48no—
409C23—48no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—84no—
417C23—84no—
418C23—84no—
419C23—84no—
420C23#38612yes—
421C23#45212yes—
422C23#45212yes—
423C23C2×#12011yescharacteristic
424Q8—44no—
425Q8—44no—
426Q8—82no—
427Q8—82no—
428D4—24no—
429D4—24no—
430D4—42no—
431D4—42no—
432D4—48no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—416no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—88no—
463C2×C4—84no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4#38612yes—
471C2×C4#41614yes—
472C2×C4#41914yes—
473C2×C4#45212yes—
474C2×C4#45212yes—
475C2×C4#48514yes—
476C2×C4#48514yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×#13511yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×#13711yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4#57512yes—
489C8—44no—
490C8—82no—
491C8—82no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—44no—
499C22—44no—
500C22—44no—
501C22—816no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22#187314yes—
507C22C2×#38611yescharacteristic
508C22C2×#45211yescharacteristic
509C22C2×#45211yescharacteristic
510C22#300311yescharacteristic
511C22#300511yescharacteristic
512C22#301911yescharacteristic
513C22#302011yescharacteristic
514C4—22no—
515C4—21no—
516C4—21no—
517C4—22no—
518C4—44no—
519C4—44no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4#187314yes—
525C4C2×#41611yescharacteristic
526C4C2×#41911yescharacteristic
527C4C4×AGL(1,ℤ/8)12yes—
528C4#303612yes—
529C4C2×#48511yescharacteristic
530C4C2×#48511yescharacteristic
531C2—24no—
532C2—84no—
533C2#1877111yescharacteristic
534C2#1877311yescharacteristic
535C2#1878111yescharacteristic
536C2#1878311yescharacteristic
537C2C2×#187311yescharacteristic
538C2#2092511yescharacteristic
539C2#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. #8984C2semidirect
7. #8976C2semidirect
8. #8043C2semidirect
9. #7901C2semidirect
14. #3077C4semidirect
33. #1990C22semidirect
34. #1987C22semidirect
35. #1981C22semidirect
36. #1956C22semidirect
44. #1296C22semidirect
45. #1279C22semidirect
86. #476C2×C4semidirect
88. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
C2×#23542
#218602
#213852
C2×#14782
#89842
#89762
#80432
#79012