Finite Group Databasebeta

← #127141

Subgroups — #127141

Order 512.

923 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20898C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7204C214yesmaximal, complemented
5C23×M16C411yescharacteristic, complemented
6C22×C4×D4C2211yescharacteristic
7#2920—21no—
8C22×#99C412yes—
9C2×#407—21no—
10C2×#407C411yescharacteristic, complemented
11C2×C4×C22⋊C4—21no—
12#1870—21no—
13#1870—21no—
14#1869C412yescomplemented
15#1869C2212yes—
16C4×#99—21no—
17C4×#99—21no—
18C4×#99C412yes—
19C4×#99C2212yes—
20C2×#342—21no—
21#1728—21no—
22#1466—21no—
23#1466C412yescomplemented
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C2212yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—21no—
34C22×M16—44no—
35C22×M16—41no—
36C22×M16—41no—
37C23×C8C2×C412yescomplemented
38C2×C4×D4—41no—
39C2×C4×D4—41no—
40C2×C4×D4—41no—
41C2×C4×D4—41no—
42C2×C4×D4C2×C411yescharacteristic
43C2×C4×D4C2×C411yescharacteristic
44C2×C4×D4C2×C411yescharacteristic
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D411yescharacteristic
47C2×C4×D4D411yescharacteristic
48C2×C4×D4Q811yescharacteristic
49C2×C4×D4C2311yescharacteristic
50C22×C4⋊C4Q811yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—41no—
55C22×C42D411yescharacteristic
56#432—22no—
57#432—41no—
58#431—22no—
59#431—41no—
60#423—41no—
61C2×#99—22no—
62C2×#99—28no—
63C2×#99C2×C414yes—
64#407—22no—
65#407—44no—
66#407—41no—
67#407—41no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×C412yescomplemented
72C4×M16—41no—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—41no—
76C4×C22⋊C4—41no—
77C2×#95—41no—
78#342—22no—
79#342—22no—
80#336—22no—
81#336—22no—
82#323—22no—
83#323—22no—
84#323—22no—
85#323C2×C414yescomplemented
86C25C4211yescharacteristic
87C25C22⋊C411yescharacteristic
88C22×D4—24no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—42no—
92C22×D4C4211yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C22×D4C4⋊C411yescharacteristic
95C22×D4C22×C411yescharacteristic
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4C4211yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C2×M16—24no—
123C2×M16—44no—
124C2×M16—42no—
125C2×M16—42no—
126C2×M16—44no—
127C2×M16—48no—
128C2×M16—41no—
129C2×M16—41no—
130C2×M16—44no—
131C2×M16—42no—
132C2×M16—42no—
133C2×M16—42no—
134C2×M16—41no—
135C2×M16—41no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—44no—
139C22×C8—42no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C4×D4—41no—
155C4×D4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4C4⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C2×Q811yescharacteristic
162C2×C22⋊C4—28no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4C4211yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C22⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C22×C411yescharacteristic
177C2×C22⋊C4C2×D411yescharacteristic
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—82no—
185C2×C42—82no—
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C4⋊C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C4⋊C8—42no—
191C4⋊C8—42no—
192#99—28no—
193#99—24no—
194#99M1618yescomplemented
195C22⋊C8—24no—
196C22⋊C8—44no—
197C22⋊C8—44no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200C22⋊C8—42no—
201#97—42no—
202C4×C8—42no—
203#95—42no—
204#95—42no—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—41no—
213C24—42no—
214C24—42no—
215C24—81no—
216C24—81no—
217C24#9511yescharacteristic
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24C2×C4211yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4C2×M1614yes—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—48no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—81no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4#9511yescharacteristic
296C22×C4#9511yescharacteristic
297C22×C4#9511yescharacteristic
298C22×C4#9511yescharacteristic
299C22×C4#9511yescharacteristic
300C22×C4#9911yescharacteristic
301C22×C4#9911yescharacteristic
302C22×C4#9911yescharacteristic
303C22×C4#9911yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C4⋊C411yescharacteristic
307C22×C4C2×C4⋊C411yescharacteristic
308C22×C4C2×C4⋊C411yescharacteristic
309M16—44no—
310M16—48no—
311M16—44no—
312M16—82no—
313M16—82no—
314M16—84no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—44no—
318C2×C8—42no—
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C22⋊C4—28no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—48no—
338C22⋊C4—44no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—82no—
348C42—82no—
349C42—84no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—42no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—42no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—84no—
384C23—84no—
385C23—84no—
386C23—84no—
387C23#34211yescharacteristic
388C23#34211yescharacteristic
389C23#34412yes—
390C23C2×#9511yescharacteristic
391C23C4×M1612yes—
392C23#40712yes—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—44no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—48no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—84no—
451C2×C4—816no—
452C2×C4—84no—
453C2×C4#34211yescharacteristic
454C2×C4#34211yescharacteristic
455C2×C4AΣL(1,𝔽16)12yes—
456C2×C4#35212yes—
457C2×C4C2×#9911yescharacteristic
458C2×C4C2×#9911yescharacteristic
459C2×C4#42312yes—
460C8—44no—
461C8—82no—
462C8—84no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—22no—
470C22—21no—
471C22—22no—
472C22—22no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—48no—
479C22—42no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—81no—
490C22—81no—
491C22—84no—
492C22—81no—
493C22—81no—
494C22—84no—
495C22—84no—
496C22#139012yes—
497C22#172811yescharacteristic
498C22C2×#34211yescharacteristic
499C22C2×#34411yescharacteristic
500C4—21no—
501C4—21no—
502C4—22no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—88no—
511C4#139012yes—
512C4C2×AΣL(1,𝔽16)11yescharacteristic
513C4C2×#35211yescharacteristic
514C2—22no—
515C2—21no—
516C2—21no—
517C2—42no—
518C2—41no—
519C2—42no—
520C2—42no—
521C2—81no—
522C2#792911yescharacteristic
523C2C2×#139011yescharacteristic
524C2#1257711yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7204C2semidirect
5. C23×M16C4semidirect
10. C2×#407C4semidirect
14. #1869C4semidirect
23. #1466C4semidirect
25. #1462C4semidirect
37. C23×C8C2×C4semidirect
71. C2×C22⋊C8C2×C4semidirect
85. #323C2×C4semidirect
194. #99M16semidirect

Maximal subgroups

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

subgroupindex
#208982
C2×C4×#992
#173642
#72042