Finite Group Databasebeta

← #480434

Subgroups — #480434

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2#17367C212yesmaximal
3#11535C212yesmaximal, complemented
4#11531C212yesmaximal, complemented
5#11513C212yesmaximal, complemented
6#11510C212yesmaximal, complemented
7#11481C212yesmaximal, complemented
8#11477C212yesmaximal, complemented
9C22×#124C411yescharacteristic
10C2×#519C411yescharacteristic
11C2×#518C2211yescharacteristic
12#2973—22no—
13#2367—22no—
14#2367—22no—
15#2342—22no—
16#2342—22no—
17#2326C411yescharacteristic
18#2326C411yescharacteristic
19#2326C2211yescharacteristic
20#2326C2211yescharacteristic
21#2310C2211yescharacteristic
22#2309C411yescharacteristic
23C4×#124C411yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26C4×#124C2211yescharacteristic
27#2285C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1466—22no—
30#1465C2214yes—
31#1464C2214yes—
32#1462—22no—
33#1459C2214yes—
34C2×#323—22no—
35#1325C2214yescomplemented
36#1324C2214yescomplemented
37#1319C2214yescomplemented
38#1319C2214yescomplemented
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C2×#124—22no—
42C2×#124—24no—
43C2×#124—24no—
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#124C2×C411yescharacteristic
48#519—44no—
49#518—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C2×C4×D4D412yes—
60C2×C4×D4C2311yescharacteristic
61C2×C4×D4C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C22⋊C4D412yes—
71C22×C42C2×C411yescharacteristic
72#435—48no—
73#432—24no—
74#407—24no—
75C2×C22⋊C8—24no—
76#391—22no—
77#391C2×C411yescharacteristic
78#391C2×C411yescharacteristic
79#391C2311yescharacteristic
80#391C2311yescharacteristic
81#390—22no—
82#390C2×C411yescharacteristic
83#390C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#390C2×C411yescharacteristic
86#388—22no—
87#388—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#384C2×C411yescharacteristic
99#384C2311yescharacteristic
100#382C2×C411yescharacteristic
101#382C2311yescharacteristic
102#381—22no—
103#380—22no—
104#379—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118#324C2318yes—
119#323—24no—
120#323—24no—
121#323—24no—
122C25C22⋊C412yes—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×Q8C22×C411yescharacteristic
129C22×Q8C22×C411yescharacteristic
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4C22⋊C412yes—
135C22×D4C22×C411yescharacteristic
136C22×D4C22×C411yescharacteristic
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C22⋊C412yes—
140C23×C4C22⋊C412yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C2×M16—44no—
145C22×C8—44no—
146#124—216no—
147#124—22no—
148#124—22no—
149#124—22no—
150#124—22no—
151#124—22no—
152#124—22no—
153#124M1614yescomplemented
154#124M1614yescomplemented
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—44no—
160C4×Q8—44no—
161C4×D4—28no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—44no—
165C4×D4—42no—
166C4×D4—42no—
167#117—24no—
168#117—42no—
169#117—44no—
170#117—44no—
171#117—42no—
172#117—42no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22×C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C2411yescharacteristic
186C2×C22⋊C4—216no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4C22⋊C411yescharacteristic
200C2×C22⋊C4C22⋊C411yescharacteristic
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C22⋊C4C2×D412yes—
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×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D412yes—
225C4⋊C8—44no—
226C22⋊C8—28no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229#97—44no—
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—24no—
241C24—24no—
242C24—42no—
243C24—82no—
244C24—82no—
245C24C2×C22⋊C412yes—
246C24C2×C22⋊C412yes—
247C24C2×C22⋊C411yescharacteristic
248C2×Q8—24no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—48no—
252C2×Q8—48no—
253C2×Q8—42no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8C2×M1612yes—
260C2×Q8C2×M1612yes—
261C2×D4—24no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—48no—
271C2×D4—48no—
272C2×D4C2×M1612yes—
273C2×D4C2×M1612yes—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D412yes—
308M16—164no—
309C2×C8—48no—
310C2×C8—164no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4#9914yes—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C22⋊C4—82no—
341C42—24no—
342C42—21no—
343C42—21no—
344C42—24no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42—41no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42—81no—
355C42—81no—
356C42—81no—
357C42—81no—
358C42C2×M1612yes—
359C42C2×M1612yes—
360C23—21no—
361C23—21no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—48no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—88no—
375C23—162no—
376C23—162no—
377C23—164no—
378C23#40714yes—
379C23#41312yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—42no—
384Q8—42no—
385Q8—48no—
386D4—42no—
387D4—42no—
388D4—48no—
389D4—42no—
390D4—42no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—48no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4C2×#9912yes—
436C2×C4C2×#9912yes—
437C2×C4C2×#9912yes—
438C2×C4C22×M1611yescharacteristic
439C2×C4C22×M1611yescharacteristic
440C8—88no—
441C22—21no—
442C22—21no—
443C22—22no—
444C22—24no—
445C22—44no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—42no—
450C22—41no—
451C22—42no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—82no—
457C22—82no—
458C22—168no—
459C22C2×#40711yescharacteristic
460C22C22×#9911yescharacteristic
461C22C2×#41311yescharacteristic
462C4—22no—
463C4—22no—
464C4—21no—
465C4—21no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—41no—
471C4—41no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C2—21no—
480C2—21no—
481C2—42no—
482C2—44no—
483C2—82no—
484C2#1736211yescharacteristic
485C2#3336111yescharacteristic
486C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11535C2semidirect
4. #11531C2semidirect
5. #11513C2semidirect
6. #11510C2semidirect
7. #11481C2semidirect
8. #11477C2semidirect
35. #1325C22semidirect
36. #1324C22semidirect
37. #1319C22semidirect
38. #1319C22semidirect
153. #124M16semidirect
154. #124M16semidirect

Maximal subgroups

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

subgroupindex
#342112
#173672
#115352
#115312
#115132
#115102
#114812
#114772