Finite Group Databasebeta

← #238462

Subgroups — #238462

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2656C211yesmaximal, characteristic
2#21758C212yesmaximal, complemented
3C2×#1519C212yesmaximal, complemented
4#14275C211yesmaximal, characteristic, complemented
5#10403C214yesmaximal, complemented
6#10402C214yesmaximal, complemented
7C2×#1280C211yesmaximal, characteristic, complemented
8C22×#121—21no—
9C22×#121—21no—
10C22×#121C2212yes—
11C22×C22⋊C8C2212yes—
12#2656—22no—
13#2656—24no—
14#2656C2214yes—
15C2×#399—22no—
16C2×#392—22no—
17C2×#385—21no—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C22×#95C2211yescharacteristic
21#1893—24no—
22#1874C2214yescomplemented
23#1874C2214yescomplemented
24#1823—22no—
25#1786—22no—
26#1519—28no—
27C2×#327C2218yescomplemented
28#1444C2212yescomplemented
29#1442C2212yescomplemented
30#1441—22no—
31C22⋊#95—21no—
32#1280C2214yescomplemented
33C23×D4—22no—
34C23×D4—21no—
35C23×D4—21no—
36C23×D4D412yes—
37C24×C4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4C2311yescharacteristic
40C23×C8—22no—
41C2×#121—24no—
42C2×#121—24no—
43C2×#121—22no—
44C2×#121—22no—
45C2×#121—24no—
46C2×#121—24no—
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—44no—
52C2×#121—44no—
53C2×#121—44no—
54C2×#121—44no—
55C2×#121D414yes—
56C2×#121C2314yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C2×#102—24no—
70C2×#102—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8C2314yes—
75#399—28no—
76#399—28no—
77#392—28no—
78#392—28no—
79#385—44no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380D414yescomplemented
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95D412yes—
91C2×#95C2312yes—
92C2×#95C2312yes—
93#340—24no—
94#340—24no—
95#327—216no—
96C25—22no—
97C25C2×D411yescharacteristic
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—24no—
101C22×D4—24no—
102C22×D4—22no—
103C22×D4—24no—
104C22×D4—24no—
105C22×D4—44no—
106C22×D4—44no—
107C22×D4—44no—
108C22×D4—44no—
109C22×D4—44no—
110C22×D4—44no—
111C22×D4—44no—
112C22×D4—44no—
113C22×D4—82no—
114C22×D4—82no—
115C22×D4—82no—
116C22×D4—82no—
117C22×D4—82no—
118C22×D4—82no—
119C22×D4C2×D414yes—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—81no—
131C23×C4—81no—
132C23×C4—81no—
133C23×C4—81no—
134C23×C4—81no—
135C23×C4—81no—
136C23×C4D814yes—
137C23×C4SD1614yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4#4114yes—
145C23×C4#4112yes—
146C23×C4C2411yescharacteristic
147C22×C8—24no—
148C22×C8—44no—
149C22×C8—82no—
150C22×C8—82no—
151#121—28no—
152#121—28no—
153#121—24no—
154#121—24no—
155#121—24no—
156#121—24no—
157#121—48no—
158#121—48no—
159#121—416no—
160#121—416no—
161#121—44no—
162#121—44no—
163#121—44no—
164#121—44no—
165#121—48no—
166#121—48no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—81no—
179C2×C4⋊C4—81no—
180C2×C4⋊C4—81no—
181C2×C4⋊C4—81no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4C2×D414yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200#102—84no—
201#102—84no—
202C22⋊C8—24no—
203C22⋊C8—24no—
204C22⋊C8—44no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—42no—
209#95—44no—
210#95—82no—
211#95—82no—
212#95C2×D414yescomplemented
213C24—24no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—44no—
219C24—44no—
220C24—44no—
221C24—44no—
222C24—44no—
223C24—44no—
224C24—44no—
225C24#12011yescharacteristic
226C24#12011yescharacteristic
227C24#12112yes—
228C24#12412yes—
229C24C2×D812yes—
230C24C2×SD1612yes—
231C24AGL(1,ℤ/8)12yes—
232C24AGL(1,ℤ/8)12yes—
233C24C22×D411yescharacteristic
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4—48no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—48no—
247C2×D4—48no—
248C2×D4—48no—
249C2×D4—44no—
250C2×D4—82no—
251C2×D4—82no—
252C2×D4—88no—
253C2×D4—84no—
254C2×D4—84no—
255C2×D4—84no—
256C2×D4—84no—
257C2×D4—816no—
258C2×D4—816no—
259C2×D4—88no—
260C2×D4—88no—
261C2×D4—88no—
262C2×D4—88no—
263C2×D4—88no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4#12012yes—
321C22×C4#12114yes—
322C22×C4#12114yes—
323C22×C4#12112yes—
324C22×C4#12314yes—
325C22×C4#12314yes—
326C22×C4#12412yes—
327C22×C4C2×D814yes—
328C22×C4C2×SD1614yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—88no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—84no—
345C4⋊C4—88no—
346C22⋊C4—416no—
347C22⋊C4—416no—
348C22⋊C4—48no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—48no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—88no—
386C23—88no—
387C23—88no—
388C23—88no—
389C23—88no—
390C23—88no—
391C23—88no—
392C23#39412yes—
393C23#39412yes—
394C23#44712yes—
395C23#44712yes—
396C23#45012yes—
397C23#45012yes—
398C23#45914yes—
399C23#46014yes—
400C23#48614yes—
401C23#49014yes—
402C23C2×#12011yescharacteristic
403C23C2×#12111yescharacteristic
404C23C2×#12411yescharacteristic
405C23C22×D811yescharacteristic
406C23C22×SD1611yescharacteristic
407C23C2×AGL(1,ℤ/8)11yescharacteristic
408C23C2×AGL(1,ℤ/8)11yescharacteristic
409D4—48no—
410D4—84no—
411D4—88no—
412D4—816no—
413D4—816no—
414D4—84no—
415D4—84no—
416D4—84no—
417D4—84no—
418D4—816no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—88no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4#39414yes—
468C2×C4C2×#12112yes—
469C2×C4C2×#12312yes—
470C8—84no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—22no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—22no—
483C22—21no—
484C22—22no—
485C22—21no—
486C22—22no—
487C22—21no—
488C22—22no—
489C22—21no—
490C22—22no—
491C22—21no—
492C22—22no—
493C22—21no—
494C22—22no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—84no—
500C22—84no—
501C22—84no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—1616no—
507C22#204114yes—
508C22#204114yes—
509C22C2×#39411yescharacteristic
510C22C2×#44711yescharacteristic
511C22C2×#45011yescharacteristic
512C22#301911yescharacteristic
513C22#302011yescharacteristic
514C22#302611yescharacteristic
515C22#302611yescharacteristic
516C22C2×#45911yescharacteristic
517C22C2×#46011yescharacteristic
518C22C2×#48611yescharacteristic
519C22C2×#49011yescharacteristic
520C4—22no—
521C4—22no—
522C4—24no—
523C4—44no—
524C4—44no—
525C4—84no—
526C4—84no—
527C4—161no—
528C4—161no—
529C2—21no—
530C2—24no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—24no—
535C2—164no—
536C2#1828411yescharacteristic
537C2#1828511yescharacteristic
538C2#1829211yescharacteristic
539C2#1829311yescharacteristic
540C2#2349911yescharacteristic
541C2C2×#204111yescharacteristic
542C2#2351511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2656C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21758C2semidirect
3. C2×#1519C2semidirect
4. #14275C2semidirect
5. #10403C2semidirect
6. #10402C2semidirect
7. C2×#1280C2semidirect
22. #1874C22semidirect
23. #1874C22semidirect
27. C2×#327C22semidirect
28. #1444C22semidirect
29. #1442C22semidirect
32. #1280C22semidirect
83. #380D4semidirect
212. #95C2×D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26562
#217582
C2×#15192
#142752
#104032
#104022
C2×#12802