Finite Group Databasebeta

← #480432

Subgroups — #480432

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34208C211yesmaximal, characteristic
2#17368C212yesmaximal, complemented
3#11536C212yesmaximal, complemented
4#11530C212yesmaximal, complemented
5#11511C212yesmaximal, complemented
6#11508C212yesmaximal, complemented
7#11479C212yesmaximal, complemented
8#11477C212yesmaximal, complemented
9C2×#532C411yescharacteristic
10C2×#520C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2973—22no—
13#2364—22no—
14#2364—22no—
15#2352—22no—
16#2352—22no—
17#2328C411yescharacteristic
18#2328C2211yescharacteristic
19#2327C411yescharacteristic
20#2327C2211yescharacteristic
21#2311C411yescharacteristic
22#2311C2211yescharacteristic
23C4×#124C411yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1467C2214yes—
30#1465—22no—
31#1464—22no—
32#1463C2214yes—
33#1459C2214yes—
34C2×#324—22no—
35#1324C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1318C2214yes—
39C22×#41C2×C411yescharacteristic
40C22×#41C2×C411yescharacteristic
41#532—24no—
42#532—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#121—22no—
48#520—44no—
49#520—44no—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2311yescharacteristic
54C2×C4×D4—22no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117D412yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72#435—48no—
73#432—24no—
74#407—24no—
75C2×C22⋊C8—24no—
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#391C2311yescharacteristic
80#390C2×C411yescharacteristic
81#390C2×C411yescharacteristic
82#390C2×C411yescharacteristic
83#390C2311yescharacteristic
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#386—22no—
95#386—22no—
96#386—22no—
97#386—22no—
98#382—22no—
99#382C2×C411yescharacteristic
100#382C2×C411yescharacteristic
101#382C2311yescharacteristic
102#382C2311yescharacteristic
103#380—22no—
104#380—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113#376C2×C411yescharacteristic
114#376C2×C411yescharacteristic
115C2×#95—22no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118#324—24no—
119#324—24no—
120#324C2318yes—
121#323—24no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—42no—
127C2×#41—42no—
128C2×#41—42no—
129C2×#41—42no—
130C22×Q8C22×C411yescharacteristic
131C22×Q8C22×C411yescharacteristic
132C22×D4—21no—
133C22×D4—22no—
134C22×D4C22⋊C412yes—
135C22×D4C22×C411yescharacteristic
136C22×D4C22×C411yescharacteristic
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C2×M16—44no—
148C22×C8—44no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151#127—22no—
152#127M1614yescomplemented
153#124—22no—
154#124—22no—
155#124—22no—
156#124M1614yescomplemented
157#121—216no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—216no—
163C4×D4—28no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—44no—
169C4×D4—44no—
170C4×D4—44no—
171#117—24no—
172#117—44no—
173#117—44no—
174#117—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2411yescharacteristic
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C411yescharacteristic
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C411yescharacteristic
205C2×C22⋊C4C2×D412yes—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C22⋊C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D412yes—
223C4⋊C8—44no—
224C22⋊C8—28no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227#97—44no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24—21no—
241C24C2×C22⋊C412yes—
242C24C2×C22⋊C411yescharacteristic
243#41—48no—
244#41—48no—
245#41—48no—
246#41—48no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8C2×M1612yes—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—28no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4C2×M1612yes—
270C2×D4C2×M1612yes—
271C2×D4C2×M1612yes—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D412yes—
313M16—164no—
314C2×C8—48no—
315C2×C8—164no—
316C4⋊C4—28no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—22no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C22⋊C4#9914yes—
343C42—24no—
344C42—21no—
345C42—21no—
346C42—24no—
347C42—22no—
348C42—22no—
349C42—42no—
350C42—41no—
351C42—41no—
352C42—42no—
353C42—42no—
354C42—81no—
355C42—81no—
356C42—81no—
357C42—81no—
358C42C2×M1612yes—
359C42C2×M1612yes—
360C23—21no—
361C23—22no—
362C23—21no—
363C23—22no—
364C23—22no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23#40714yes—
374C23C2×#9912yes—
375C23C22×C22⋊C411yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—42no—
379Q8—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—42no—
384D4—48no—
385D4—48no—
386D4—88no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—162no—
431C2×C4—162no—
432C2×C4—162no—
433C2×C4—162no—
434C2×C4C2×#9912yes—
435C2×C4C2×#9912yes—
436C2×C4#41312yes—
437C2×C4C22×M1611yescharacteristic
438C2×C4C22×M1611yescharacteristic
439C8—88no—
440C22—21no—
441C22—21no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—24no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—82no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22C2×#40711yescharacteristic
458C22C22×#9911yescharacteristic
459C22C2×#41311yescharacteristic
460C4—22no—
461C4—22no—
462C4—22no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C2—21no—
478C2—21no—
479C2—42no—
480C2—44no—
481C2—82no—
482C2#1736211yescharacteristic
483C2#3336111yescharacteristic
484C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17368C2semidirect
3. #11536C2semidirect
4. #11530C2semidirect
5. #11511C2semidirect
6. #11508C2semidirect
7. #11479C2semidirect
8. #11477C2semidirect
152. #127M16semidirect
156. #124M16semidirect

Maximal subgroups

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

subgroupindex
#342082
#173682
#115362
#115302
#115112
#115082
#114792
#114772