Finite Group Databasebeta

← #238682

Subgroups — #238682

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic, complemented
3#20734C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10436C214yesmaximal, complemented
8#10434C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8—21no—
11C22×C4×Q8C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16C2×#418—21no—
17#2538—22no—
18#2538—22no—
19#2538—22no—
20#2538C2214yes—
21C2×#396—22no—
22C2×#390—21no—
23C2×#389—21no—
24#2285C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#380—21no—
27#1947—22no—
28#1852—22no—
29#1845C2212yescomplemented
30#1834C2212yescomplemented
31#1824—21no—
32#1804—22no—
33#1803C2212yescomplemented
34#1803C2212yescomplemented
35#1802C2212yes—
36#1802C2212yes—
37#1787—21no—
38C4×#103—22no—
39#1736—21no—
40#1487—24no—
41#1478—24no—
42C2×Q8⋊C8C2214yescomplemented
43C2×D4⋊C8C2214yes—
44#1296C2214yescomplemented
45C23×Q8D411yescharacteristic
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×M16—21no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—48no—
52C2×C4×Q8—21no—
53C2×C4×Q8—21no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8D412yes—
59C2×C4×Q8C2312yes—
60C2×C4×D4—21no—
61C2×C4×D4—21no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4D412yes—
67C2×C4×D4C2312yes—
68C2×#117—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#431—22no—
82#431—22no—
83#431—22no—
84#431D414yes—
85C2×#107—24no—
86#423—22no—
87#423—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92#418—44no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#396—28no—
102#396—28no—
103#390—24no—
104#390—24no—
105#389—24no—
106#389—24no—
107#384—21no—
108#384—21no—
109#384D412yes—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2312yes—
113#382—21no—
114#380—24no—
115#380—24no—
116#380—21no—
117#380—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C2×#95—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—24no—
126Q8⋊C8—28no—
127D4⋊C8—28no—
128C25—21no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8C2×D412yes—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4Q1614yes—
142C22×D4C2×D412yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—42no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22⋊Q8—416no—
163C22⋊Q8—48no—
164C22⋊Q8—48no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—24no—
176C4×D4—24no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185#117—42no—
186#117—42no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—48no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—84no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—41no—
231C2×C42C2×D412yes—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42#4112yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42C2411yescharacteristic
239#107—84no—
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—42no—
246#103—44no—
247#103—44no—
248#102—44no—
249#102—44no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C4×C8—22no—
253C4×C8—42no—
254C4×C8C2×D414yes—
255#95—28no—
256#95—41no—
257#95—41no—
258C24—22no—
259C24—24no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24#12011yescharacteristic
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—48no—
274C2×Q8—48no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8AGL(1,ℤ/8)14yes—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—28no—
285C2×D4—28no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4C2×Q1614yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12112yes—
328C22×C4#12312yes—
329C22×C4#12312yes—
330C22×C4#12412yes—
331C22×C4C2×Q1612yes—
332C22×C4#13512yes—
333C22×C4AGL(1,ℤ/8)12yes—
334C22×C4#13712yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338M16—84no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—416no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—82no—
373C22⋊C4—416no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—88no—
378C22⋊C4—82no—
379C42—42no—
380C42—42no—
381C42—44no—
382C42—82no—
383C42#12112yes—
384C42#12112yes—
385C42#12112yes—
386C42#12312yes—
387C42#12312yes—
388C42#12412yes—
389C42C22×D411yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—24no—
398C23—48no—
399C23—48no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—48no—
407C23—84no—
408C23—84no—
409C23—84no—
410C23—84no—
411C23#39412yes—
412C23#44912yes—
413C23#45112yes—
414C23C2×#12011yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—48no—
418Q8—42no—
419Q8—42no—
420D4—24no—
421D4—24no—
422D4—42no—
423D4—42no—
424D4—48no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—88no—
449C2×C4—88no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4#39412yes—
461C2×C4#39412yes—
462C2×C4#39412yes—
463C2×C4#44912yes—
464C2×C4#45112yes—
465C2×C4#46714yes—
466C2×C4#46814yes—
467C2×C4#48414yes—
468C2×C4#48514yes—
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12111yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4C2×#12311yescharacteristic
474C2×C4C2×#12411yescharacteristic
475C2×C4C22×Q1611yescharacteristic
476C2×C4C2×#13511yescharacteristic
477C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
478C2×C4C2×AGL(1,ℤ/8)12yes—
479C2×C4C2×#13711yescharacteristic
480C2×C4#57512yes—
481C8—44no—
482C8—82no—
483C8—82no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—44no—
491C22—44no—
492C22—44no—
493C22—816no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22#203614yes—
499C22C2×#39411yescharacteristic
500C22C2×#45111yescharacteristic
501C22C2×#44911yescharacteristic
502C22#300311yescharacteristic
503C22#300511yescharacteristic
504C22#301911yescharacteristic
505C22#302011yescharacteristic
506C4—22no—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—44no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—161no—
515C4—161no—
516C4#203614yes—
517C4C2×#46711yescharacteristic
518C4C2×#46811yescharacteristic
519C4#305512yes—
520C4C2×#48511yescharacteristic
521C4C2×#48411yescharacteristic
522C4#309412yes—
523C2—24no—
524C2—84no—
525C2#1874911yescharacteristic
526C2#1875911yescharacteristic
527C2#1877011yescharacteristic
528C2#1878011yescharacteristic
529C2C2×#203611yescharacteristic
530C2#2348011yescharacteristic
531C2#2349911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×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. #21982C2semidirect
3. #20734C2semidirect
5. C2×#1487C2semidirect
7. #10436C2semidirect
8. #10434C2semidirect
9. #8043C2semidirect
29. #1845C22semidirect
30. #1834C22semidirect
33. #1803C22semidirect
34. #1803C22semidirect
42. C2×Q8⋊C8C22semidirect
44. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
#219822
#207342
#206462
C2×#14872
C2×#14782
#104362
#104342
#80432