Finite Group Databasebeta

← #239654

Subgroups — #239654

Order 512.

1242 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#21804C211yesmaximal, characteristic, complemented
3#21170C211yesmaximal, characteristic, complemented
4#20732C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic
7#10413C214yesmaximal, complemented
8#10412C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#428—21no—
15C2×#423C2211yescharacteristic
16#2563—22no—
17#2563—22no—
18#2563—22no—
19#2563C2214yes—
20C2×#393—22no—
21C2×#390—21no—
22#2285C2211yescharacteristic
23C2×#382—21no—
24C2×#380—21no—
25C2×#380—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27#1981—22no—
28#1956C2212yescomplemented
29#1956C2212yescomplemented
30#1949—22no—
31#1948C2212yescomplemented
32#1947C2212yescomplemented
33#1823—21no—
34#1802C2212yes—
35#1802C2212yes—
36#1787—22no—
37#1786—21no—
38C4×#102—22no—
39#1736—21no—
40#1548—24no—
41C2×#333C2214yescomplemented
42#1486—24no—
43C2×D4⋊C8C2214yes—
44#1279C2214yescomplemented
45C23×Q8D412yes—
46C23×D4D411yescharacteristic
47C24×C4—21no—
48C22×M16—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8D412yes—
56C2×C4⋊Q8C2312yes—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4D412yes—
64C2×C4×D4C2312yes—
65C2×#117—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#428—44no—
81C2×#107—22no—
82C2×#106—22no—
83#423—22no—
84#423D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—24no—
90C2×#103—24no—
91C2×#102—22no—
92C2×#102—22no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97#393—28no—
98#393—28no—
99#390—24no—
100#390—24no—
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#382—24no—
104#382—24no—
105#382—21no—
106#380—24no—
107#380—24no—
108#380—24no—
109#380—24no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#333—28no—
126D4⋊C8—28no—
127C25—21no—
128C22×Q8—22no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—82no—
132C22×Q8—82no—
133C22×Q8C2×D414yes—
134C22×D4—24no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4C2×D412yes—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C2×M16—24no—
156C2×M16—42no—
157C2×M16—42no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—42no—
180C4×D4—42no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—84no—
217C2×C22⋊C4—84no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—44no—
221C2×C42—41no—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42#4112yes—
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#107—44no—
231#106—44no—
232C4⋊C8—24no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—42no—
237C4⋊C8C2×D414yes—
238#103—84no—
239#103—84no—
240#102—44no—
241#102—44no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C4×C8—22no—
245C4×C8—22no—
246C4×C8—42no—
247#95—28no—
248#95—28no—
249#95—22no—
250#95—22no—
251#95—41no—
252#95—41no—
253C24—24no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24#12011yescharacteristic
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×D4—24no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—48no—
278C2×D4—48no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4#13714yes—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—84no—
314C22×C4—84no—
315C22×C4—84no—
316C22×C4—84no—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4#12412yes—
326C22×C4#13512yes—
327C22×C4#13512yes—
328C22×C4AGL(1,ℤ/8)12yes—
329C22×C4#13712yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D411yescharacteristic
333M16—48no—
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—82no—
372C42—44no—
373C42—44no—
374C42—82no—
375C42#12112yes—
376C42#12112yes—
377C42#12112yes—
378C42#12312yes—
379C42#12312yes—
380C42#12412yes—
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C42C2×#4111yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—24no—
390C23—48no—
391C23—48no—
392C23—48no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—84no—
400C23—84no—
401C23—84no—
402C23—84no—
403C23#39412yes—
404C23#44912yes—
405C23#45212yes—
406C23C2×#12011yescharacteristic
407Q8—48no—
408Q8—84no—
409Q8—88no—
410D4—24no—
411D4—24no—
412D4—48no—
413D4—42no—
414D4—42no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—88no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—88no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#39412yes—
453C2×C4#39412yes—
454C2×C4#39412yes—
455C2×C4#44912yes—
456C2×C4#45212yes—
457C2×C4#46314yes—
458C2×C4#46414yes—
459C2×C4#48814yes—
460C2×C4#48914yes—
461C2×C4C2×#12111yescharacteristic
462C2×C4C2×#12111yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12311yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×#12411yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4C2×#13712yes—
471C2×C4C2×#13711yescharacteristic
472C2×C4#57512yes—
473C8—44no—
474C8—82no—
475C8—82no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—816no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22#204214yes—
491C22C2×#39411yescharacteristic
492C22C2×#44911yescharacteristic
493C22C2×#45211yescharacteristic
494C22#300411yescharacteristic
495C22#300511yescharacteristic
496C22#301911yescharacteristic
497C22#301911yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—82no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—161no—
507C4—161no—
508C4#204214yes—
509C4C2×#46311yescharacteristic
510C4C2×#46411yescharacteristic
511C4#305612yes—
512C4#309512yes—
513C4C2×#48811yescharacteristic
514C4C2×#48911yescharacteristic
515C2—24no—
516C2—84no—
517C2#1875211yescharacteristic
518C2#1875311yescharacteristic
519C2#1877211yescharacteristic
520C2#1877311yescharacteristic
521C2#2348111yescharacteristic
522C2#2349311yescharacteristic
523C2C2×#204211yescharacteristic
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. #21804C2semidirect
3. #21170C2semidirect
5. C2×#1548C2semidirect
7. #10413C2semidirect
8. #10412C2semidirect
9. #7892C2semidirect
28. #1956C22semidirect
29. #1956C22semidirect
31. #1948C22semidirect
32. #1947C22semidirect
41. C2×#333C22semidirect
44. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#218042
#211702
#207322
C2×#15482
C2×#14862
#104132
#104122
#78922