Finite Group Databasebeta

← #480306

Subgroups — #480306

Order 512.

1070 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#34183C211yesmaximal, characteristic
2#17368C212yesmaximal
3#11516C212yesmaximal, complemented
4#11514C212yesmaximal, complemented
5#11513C212yesmaximal, complemented
6#11512C212yesmaximal, complemented
7#11448C212yesmaximal, complemented
8C2×#1325C212yesmaximal, complemented
9C2×#520C411yescharacteristic
10C2×#520C2211yescharacteristic
11C2×C4×#41C411yescharacteristic
12#2973—22no—
13Q8×C4⋊C4—22no—
14D4×C4⋊C4—22no—
15D4×C4⋊C4—22no—
16D4×C4⋊C4—22no—
17#2317C411yescharacteristic
18#2317C411yescharacteristic
19#2317C411yescharacteristic
20#2317C2211yescharacteristic
21#2317C2211yescharacteristic
22#2317C2211yescharacteristic
23#2285C411yescharacteristic
24#2285C411yescharacteristic
25#2285C2211yescharacteristic
26C2×#384C411yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29#1467—22no—
30#1465—22no—
31#1464C2214yes—
32#1463C2214yes—
33#1459—22no—
34C2×#324C2214yes—
35#1325C2214yes—
36#1325C2214yes—
37#1325C2214yes—
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⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C411yescharacteristic
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77#435—48no—
78#432—24no—
79#407—24no—
80C2×C22⋊C8—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389—22no—
87#385—22no—
88#385—22no—
89#385—22no—
90#385—22no—
91#385—22no—
92#385—22no—
93#384—22no—
94#384—22no—
95#384—22no—
96#384C2×C411yescharacteristic
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#384C2×C411yescharacteristic
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#384C2×C411yescharacteristic
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#382C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110#382C2×C411yescharacteristic
111#382C2311yescharacteristic
112#380—22no—
113#380—22no—
114#380—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—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—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4—84no—
192C2×C4⋊C4—84no—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C2×D412yes—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42C22⋊C412yes—
225C2×C42C22⋊C412yes—
226C2×C42M1614yescomplemented
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D412yes—
232C4⋊C8—44no—
233C22⋊C8—28no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236#97—44no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C24C2×C22⋊C411yescharacteristic
244C24C2×C22⋊C411yescharacteristic
245C24C2×C22⋊C411yescharacteristic
246#41—216no—
247#41—216no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4#9914yes—
259C2×D4#10114yes—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C412yes—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×M1612yes—
306C22×C4C2×M1612yes—
307C22×C4C23×C411yescharacteristic
308C22×C4C22×D412yes—
309M16—164no—
310C2×C8—48no—
311C2×C8—164no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C42—24no—
349C42—24no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—24no—
355C42—24no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42C2×M1612yes—
362C42C2×M1612yes—
363C42C2×M1612yes—
364C42C2×M1612yes—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23#40712yes—
376C23C2×#9912yes—
377C23C2×#9912yes—
378C23C2×#10112yes—
379C23C2×#10112yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—28no—
382D4—28no—
383D4—28no—
384D4—28no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—48no—
410C2×C4—48no—
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—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
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—162no—
433C2×C4—162no—
434C2×C4—162no—
435C2×C4—162no—
436C2×C4#40712yes—
437C2×C4C2×#9912yes—
438C2×C4C2×#10112yes—
439C2×C4C22×M1611yescharacteristic
440C2×C4C22×M1611yescharacteristic
441C8—88no—
442C22—21no—
443C22—22no—
444C22—21no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22#146314yes—
459C22C2×#40711yescharacteristic
460C22C22×#9911yescharacteristic
461C22C22×#10111yescharacteristic
462C4—24no—
463C4—21no—
464C4—21no—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C2—21no—
480C2—21no—
481C2—24no—
482C2—42no—
483C2—42no—
484C2C2×#146311yescharacteristic
485C2#3335911yescharacteristic
486C2#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 ◁ 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. #11516C2semidirect
4. #11514C2semidirect
5. #11513C2semidirect
6. #11512C2semidirect
7. #11448C2semidirect
8. C2×#1325C2semidirect
167. #117M16semidirect
226. C2×C42M16semidirect

Maximal subgroups

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

subgroupindex
#341832
#173682
#115162
#115142
#115132
#115122
#114482
C2×#13252