Finite Group Databasebeta

← #480363

Subgroups — #480363

Order 512.

1001 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#17368C212yesmaximal
3#11536C212yesmaximal, complemented
4#11535C212yesmaximal, complemented
5#11532C212yesmaximal, complemented
6#11531C212yesmaximal, complemented
7#11448C212yesmaximal, complemented
8C2×#1319C212yesmaximal, complemented
9C2×#536C411yescharacteristic
10C2×#520C2211yescharacteristic
11C2×#519C411yescharacteristic
12#2973—22no—
13#2362—22no—
14#2360—22no—
15#2359—22no—
16#2359—22no—
17#2327C411yescharacteristic
18#2327C2211yescharacteristic
19#2326C411yescharacteristic
20#2326C2211yescharacteristic
21#2317C2211yescharacteristic
22C4×C4⋊Q8C411yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24C4×#124C411yescharacteristic
25C4×#124C2211yescharacteristic
26#2297C411yescharacteristic
27C2×#391C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1467C2214yes—
30#1465C2214yes—
31#1464—22no—
32#1463—22no—
33#1459—22no—
34C2×#324C2214yes—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1319C2214yes—
39C22×#41C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41#536—24no—
42#536—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#124C2×C411yescharacteristic
46C2×#124C2×C411yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49#520—44no—
50#519—44no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C2×C4×D4D412yes—
62C2×C4×D4C2311yescharacteristic
63C2×#117—21no—
64C2×#117—21no—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117D412yes—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
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#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2311yescharacteristic
82#391C2311yescharacteristic
83#390C2×C411yescharacteristic
84#390C2×C411yescharacteristic
85#389—22no—
86#389—22no—
87#389—22no—
88#388—22no—
89#388—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#384C2×C411yescharacteristic
101#384C2311yescharacteristic
102#382C2×C411yescharacteristic
103#382C2311yescharacteristic
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
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—42no—
125C2×#41—42no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8M1614yescomplemented
131C22×Q8C22×C411yescharacteristic
132C22×Q8C22×C411yescharacteristic
133C22×Q8C22×C411yescharacteristic
134C22×D4C22⋊C412yes—
135C22×D4C22×C411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C22⋊C412yes—
139C23×C4C22⋊C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C2×M16—44no—
144C22×C8—44no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149#124—22no—
150#124—22no—
151#124—22no—
152#124—22no—
153#123—216no—
154C22⋊Q8—216no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×Q8—42no—
161C4×Q8—44no—
162C4×Q8—44no—
163C4×D4—28no—
164C4×D4—42no—
165C4×D4—42no—
166C4×D4—44no—
167#117—22no—
168#117—24no—
169#117—44no—
170#117—44no—
171#117—44no—
172#117—42no—
173#117M1614yescomplemented
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊C411yescharacteristic
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C411yescharacteristic
203C2×C22⋊C4C2×D412yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
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—
240C24C2×C22⋊C412yes—
241C24C2×C22⋊C411yescharacteristic
242#41—48no—
243#41—48no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—24no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—48no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—48no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8C2×M1612yes—
260C2×Q8C2×M1612yes—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×M1612yes—
301C22×C4C2×M1612yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D412yes—
304M16—164no—
305C2×C8—48no—
306C2×C8—164no—
307C4⋊C4—28no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—82no—
334C22⋊C4#9914yes—
335C42—21no—
336C42—21no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—42no—
351C42—81no—
352C42—81no—
353C42C2×M1612yes—
354C42C2×M1612yes—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23#40712yes—
363C23C2×#9912yes—
364C23#41312yes—
365C23C22×C22⋊C411yescharacteristic
366Q8—42no—
367Q8—42no—
368Q8—48no—
369Q8—42no—
370Q8—42no—
371Q8—42no—
372Q8—42no—
373D4—42no—
374D4—42no—
375D4—48no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—48no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—48no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—162no—
423C2×C4#40712yes—
424C2×C4C2×#9912yes—
425C2×C4C2×#9912yes—
426C2×C4C22×M1611yescharacteristic
427C2×C4C22×M1611yescharacteristic
428C8—88no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—42no—
434C22—42no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—82no—
440C22#146514yes—
441C22C2×#40711yescharacteristic
442C22C22×#9911yescharacteristic
443C22C2×#41311yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C2—21no—
463C2—21no—
464C2—24no—
465C2—82no—
466C2C2×#146511yescharacteristic
467C2#3336111yescharacteristic
468C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
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
3. #11536C2semidirect
4. #11535C2semidirect
5. #11532C2semidirect
6. #11531C2semidirect
7. #11448C2semidirect
8. C2×#1319C2semidirect
130. C22×Q8M16semidirect
173. #117M16semidirect

Maximal subgroups

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

subgroupindex
#342762
#173682
#115362
#115352
#115322
#115312
#114482
C2×#13192