Finite Group Databasebeta

← #480303

Subgroups — #480303

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2#17368C212yesmaximal
3#11515C212yesmaximal, complemented
4#11514C212yesmaximal, complemented
5#11513C212yesmaximal, complemented
6#11511C212yesmaximal, complemented
7#11448C212yesmaximal, complemented
8C2×#1324C212yesmaximal, complemented
9C2×#520C411yescharacteristic
10C2×#520C2211yescharacteristic
11C2×C4×#41C411yescharacteristic
12#2973—22no—
13#2351—22no—
14#2349—22no—
15#2349—22no—
16#2349—22no—
17#2317C411yescharacteristic
18#2317C411yescharacteristic
19#2317C411yescharacteristic
20#2317C2211yescharacteristic
21#2317C2211yescharacteristic
22#2317C2211yescharacteristic
23#2285C411yescharacteristic
24#2285C411yescharacteristic
25#2285C2211yescharacteristic
26C2×#382C411yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29#1467—22no—
30#1465—22no—
31#1464C2214yes—
32#1463C2214yes—
33#1459—22no—
34C2×#324C2214yes—
35#1325C2214yes—
36#1324C2214yes—
37#1324C2214yes—
38#1324C2214yes—
39C22×#41C2×C411yescharacteristic
40#520—44no—
41#520—44no—
42C4×#41—24no—
43C4×#41—24no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8D412yes—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×C4×D4D412yes—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C2×#117C2×C411yescharacteristic
60C2×#117C2×C411yescharacteristic
61C2×#117C2×C411yescharacteristic
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2×C411yescharacteristic
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74#435—48no—
75#432—24no—
76#407—24no—
77C2×C22⋊C8—24no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—22no—
83#389—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#384C2×C411yescharacteristic
91#384C2×C411yescharacteristic
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#384C2×C411yescharacteristic
95#384C2311yescharacteristic
96#382—22no—
97#382—22no—
98#382—22no—
99#382C2×C411yescharacteristic
100#382C2×C411yescharacteristic
101#382C2×C411yescharacteristic
102#382C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106#382C2311yescharacteristic
107#382C2311yescharacteristic
108#382C2311yescharacteristic
109#380—22no—
110#380—22no—
111#380—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118#324—24no—
119#324—24no—
120#324C2318yes—
121#323—24no—
122C2×#41—24no—
123C2×#41—24no—
124C2×#41—22no—
125C2×#41—22no—
126C22×Q8C22⋊C411yescharacteristic
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22⋊C411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C22⋊C412yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C2×M16—44no—
143C22×C8—44no—
144C4×Q8—28no—
145C4×Q8—44no—
146C4×Q8—44no—
147C4×D4—28no—
148C4×D4—28no—
149C4×D4—28no—
150C4×D4—28no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4—44no—
156#117—22no—
157#117—22no—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—42no—
162#117—42no—
163#117—42no—
164#117—42no—
165#117—42no—
166#117—42no—
167#117M1614yescomplemented
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42C22⋊C412yes—
218C2×C42C22⋊C412yes—
219C2×C42M1614yescomplemented
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—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24C2×C22⋊C411yescharacteristic
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C411yescharacteristic
243#41—216no—
244#41—216no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4#9914yes—
256C2×D4#10114yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C412yes—
302C22×C4C2×M1612yes—
303C22×C4C2×M1612yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D412yes—
306M16—164no—
307C2×C8—48no—
308C2×C8—164no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—82no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C42—24no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—24no—
348C42—24no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—42no—
354C42C2×M1612yes—
355C42C2×M1612yes—
356C42C2×M1612yes—
357C42C2×M1612yes—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23#40712yes—
369C23C2×#9912yes—
370C23C2×#9912yes—
371C23C2×#10112yes—
372C23C2×#10112yes—
373C23C22×C22⋊C411yescharacteristic
374Q8—28no—
375D4—28no—
376D4—28no—
377D4—28no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—84no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—162no—
426C2×C4—162no—
427C2×C4—162no—
428C2×C4—162no—
429C2×C4#40712yes—
430C2×C4C2×#9912yes—
431C2×C4C2×#10112yes—
432C2×C4C22×M1611yescharacteristic
433C2×C4C22×M1611yescharacteristic
434C8—88no—
435C22—21no—
436C22—22no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22#146314yes—
452C22C2×#40711yescharacteristic
453C22C22×#9911yescharacteristic
454C22C22×#10111yescharacteristic
455C4—21no—
456C4—21no—
457C4—22no—
458C4—22no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C2—21no—
473C2—21no—
474C2—24no—
475C2—42no—
476C2—42no—
477C2C2×#146311yescharacteristic
478C2#3336111yescharacteristic
479C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ 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. #11515C2semidirect
4. #11514C2semidirect
5. #11513C2semidirect
6. #11511C2semidirect
7. #11448C2semidirect
8. C2×#1324C2semidirect
167. #117M16semidirect
219. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#341842
#173682
#115152
#115142
#115132
#115112
#114482
C2×#13242