Finite Group Databasebeta

← #480433

Subgroups — #480433

Order 512.

1080 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#17370C212yesmaximal, complemented
3#11535C212yesmaximal, complemented
4#11529C212yesmaximal, complemented
5#11512C212yesmaximal, complemented
6#11508C212yesmaximal, complemented
7#11480C212yesmaximal, complemented
8#11478C212yesmaximal, complemented
9C2×#532C411yescharacteristic
10C2×#520C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2975—22no—
13#2361—22no—
14#2361—22no—
15#2354—22no—
16D4×C4⋊C4—22no—
17#2326C411yescharacteristic
18#2326C411yescharacteristic
19#2326C2211yescharacteristic
20#2326C2211yescharacteristic
21#2311C411yescharacteristic
22#2311C2211yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C411yescharacteristic
26C4×#127C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1465—22no—
30#1465C2214yes—
31#1463—22no—
32#1463C2214yes—
33#1461C2214yes—
34#1458—22no—
35#1325C2214yes—
36#1325C2214yes—
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—22no—
56C2×C4×D4—22no—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×#117—21no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C42C2×C411yescharacteristic
75#436—48no—
76#431—24no—
77#407—24no—
78#407—24no—
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#391C2311yescharacteristic
83#390—22no—
84#390—22no—
85#390C2×C411yescharacteristic
86#390C2×C411yescharacteristic
87#390C2×C411yescharacteristic
88#390C2311yescharacteristic
89#389—22no—
90#389—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#386—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#384—22no—
102#384C2×C411yescharacteristic
103#384C2×C411yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#380—22no—
107#380—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
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—22no—
133C22×D4—21no—
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—
148C2×M16—44no—
149C4⋊Q8—22no—
150C4⋊Q8M1614yescomplemented
151#127—22no—
152#127—22no—
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—
171C4×D4—84no—
172C4×D4—84no—
173#117—24no—
174#117—44no—
175#117—44no—
176#117—42no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2411yescharacteristic
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4C22⋊C411yescharacteristic
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C411yescharacteristic
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22⋊C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C4⋊C8—44no—
231C22⋊C8—28no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C4×C8—44no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24—21no—
244C24C2×C22⋊C412yes—
245C24C2×C22⋊C411yescharacteristic
246#41—48no—
247#41—48no—
248#41—48no—
249#41—48no—
250C2×Q8—21no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8C2×M1612yes—
256C2×Q8C2×M1612yes—
257C2×Q8C2×M1612yes—
258C2×D4—28no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—24no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4C2×M1612yes—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D412yes—
316M16—164no—
317M16—164no—
318C2×C8—48no—
319C4⋊C4—28no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—44no—
332C4⋊C4—44no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C22⋊C4—28no—
338C22⋊C4—22no—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C22⋊C4#9914yes—
350C42—24no—
351C42—21no—
352C42—21no—
353C42—24no—
354C42—22no—
355C42—22no—
356C42—42no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42C2×M1612yes—
366C42C2×M1612yes—
367C23—21no—
368C23—21no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23C2×#9912yes—
382C23C22×C22⋊C411yescharacteristic
383Q8—42no—
384Q8—42no—
385Q8—42no—
386Q8—42no—
387D4—48no—
388D4—42no—
389D4—416no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—48no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—48no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—162no—
437C2×C4—162no—
438C2×C4—162no—
439C2×C4—162no—
440C2×C4#40714yes—
441C2×C4C2×#9912yes—
442C2×C4C2×#9912yes—
443C2×C4#41312yes—
444C2×C4C22×M1611yescharacteristic
445C2×C4C22×M1611yescharacteristic
446C8—88no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—42no—
453C22—42no—
454C22—44no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—82no—
461C22—82no—
462C22—81no—
463C22—81no—
464C22C2×#40711yescharacteristic
465C22C22×#9911yescharacteristic
466C22C2×#41311yescharacteristic
467C4—24no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—21no—
472C4—21no—
473C4—24no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—81no—
484C2—21no—
485C2—21no—
486C2—42no—
487C2—82no—
488C2—82no—
489C2#1736211yescharacteristic
490C2#3335911yescharacteristic
491C2#3337211yescharacteristic
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. #17370C2semidirect
3. #11535C2semidirect
4. #11529C2semidirect
5. #11512C2semidirect
6. #11508C2semidirect
7. #11480C2semidirect
8. #11478C2semidirect
150. C4⋊Q8M16semidirect
156. #124M16semidirect

Maximal subgroups

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

subgroupindex
#342072
#173702
#115352
#115292
#115122
#115082
#114802
#114782