Finite Group Databasebeta

← #480264

Subgroups — #480264

Order 512.

1618 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34044C211yesmaximal, characteristic
2C2×#1464C212yesmaximal, complemented
3#11510C212yesmaximal, complemented
4#11509C212yesmaximal, complemented
5#11507C214yesmaximal, complemented
6#11432C214yesmaximal, complemented
7C23×#41C411yescharacteristic
8C22×#117C411yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#432—22no—
11#2311—22no—
12#2311C412yes—
13#2311C2212yes—
14#2310C411yescharacteristic
15#2310C2211yescharacteristic
16#2309C411yescharacteristic
17#2309C2211yescharacteristic
18#2308—22no—
19#2297—22no—
20#2297C412yes—
21#2297C2212yes—
22#2280—22no—
23#1464—22no—
24#1464—22no—
25#1464C2214yes—
26#1464C2214yescomplemented
27C2×#324C2214yescomplemented
28C2×#323—22no—
29#1319C2218yescomplemented
30#1318C2218yescomplemented
31C22×#41—28no—
32C22×#41—21no—
33C22×#41—21no—
34C22×#41—42no—
35C22×#41—42no—
36C22×#41—44no—
37C22×#41C2×C412yes—
38C22×#41C2×C412yes—
39C23×Q8C2×C411yescharacteristic
40C23×D4—21no—
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C24×C4C2×C411yescharacteristic
44C2×#117—21no—
45C2×#117—21no—
46C2×#117—42no—
47C2×#117—42no—
48C2×#117—44no—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117C2×C411yescharacteristic
53C2×#117C2×C412yes—
54C2×#117D412yes—
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C2×#117C2311yescharacteristic
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C22⋊C4—21no—
62C22×C22⋊C4D412yes—
63C22×C42C2×C412yes—
64C22×C42D412yes—
65#432—24no—
66#432—24no—
67#432—42no—
68#432—42no—
69C2×C4⋊C8—42no—
70C2×C22⋊C8—24no—
71C2×#97—42no—
72#391—22no—
73#391—22no—
74#391C2×C412yes—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2312yes—
78#390—22no—
79#390—22no—
80#390C2×C412yes—
81#390C2×C412yes—
82#390C2×C412yes—
83#390C2312yes—
84#386—24no—
85#386—24no—
86#386—24no—
87#386—24no—
88#381—22no—
89#381—22no—
90#380—24no—
91#380—24no—
92#379—22no—
93#379—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96#324—24no—
97#324C2318yes—
98#323—24no—
99#323—24no—
100C25—21no—
101C25C22⋊C411yescharacteristic
102C2×#41—28no—
103C2×#41—28no—
104C2×#41—24no—
105C2×#41—24no—
106C2×#41—24no—
107C2×#41—432no—
108C2×#41—432no—
109C2×#41—44no—
110C2×#41—44no—
111C2×#41—44no—
112C2×#41—44no—
113C2×#41—44no—
114C2×#41—44no—
115C2×#41—42no—
116C2×#41—42no—
117C2×#41—42no—
118C2×#41—42no—
119C2×#41M1618yes—
120C22×Q8—24no—
121C22×Q8—21no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8C22×C412yes—
125C22×D4—28no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—24no—
129C22×D4—21no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—44no—
133C22×D4—44no—
134C22×D4C22×C412yes—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—28no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C2×D412yes—
155C22×C8—44no—
156#117—42no—
157#117—41no—
158#117—41no—
159#117—42no—
160#117—42no—
161#117—42no—
162#117—42no—
163#117—41no—
164#117—42no—
165#117—41no—
166#117—44no—
167#117—44no—
168#117—42no—
169#117—42no—
170#117—44no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C411yescharacteristic
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2411yescharacteristic
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×D412yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—44no—
203C2×C42C22⋊C412yes—
204C2×C42C22⋊C412yes—
205C2×C42C22×C412yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C4⋊C8—44no—
210C4⋊C8—44no—
211C22⋊C8—28no—
212C22⋊C8—44no—
213#97—44no—
214#97—44no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219C24—22no—
220C24—22no—
221C24—24no—
222C24—22no—
223C24—42no—
224C24—42no—
225C24—81no—
226C24—81no—
227C24—82no—
228C24—82no—
229C24#10014yes—
230C24C2×C22⋊C411yescharacteristic
231C24C2×C22⋊C411yescharacteristic
232C24C2×C22⋊C411yescharacteristic
233#41—48no—
234#41—48no—
235#41—48no—
236#41—48no—
237#41—432no—
238#41—416no—
239#41—416no—
240#41—416no—
241#41—416no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—416no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8C2×M1614yes—
250C2×D4—24no—
251C2×D4—28no—
252C2×D4—28no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—416no—
257C2×D4—42no—
258C2×D4—432no—
259C2×D4—44no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—42no—
264C2×D4C2×M1614yes—
265C22×C4—216no—
266C22×C4—216no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—28no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—416no—
296C22×C4—416no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4#9912yes—
310C22×C4#9912yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×M1614yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D412yes—
320C2×C8—48no—
321C2×C8—88no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—84no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—44no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—28no—
358C23—24no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23—48no—
368C23—42no—
369C23—42no—
370C23—82no—
371C23—82no—
372C23—84no—
373C23—84no—
374C23—88no—
375C23—88no—
376C23—84no—
377C23—84no—
378C23#40712yes—
379C23C2×#9912yes—
380C23C2×#10012yes—
381C23C2×#10012yes—
382C23C2×#10012yes—
383C23C22×C22⋊C411yescharacteristic
384Q8—44no—
385Q8—44no—
386Q8—48no—
387D4—44no—
388D4—44no—
389D4—48no—
390D4—416no—
391D4—416no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—28no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—416no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—88no—
435C2×C4#40712yes—
436C2×C4C2×#9911yescharacteristic
437C2×C4C2×#9911yescharacteristic
438C2×C4C2×#9912yes—
439C2×C4C22×M1612yes—
440C8—88no—
441C22—21no—
442C22—22no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—42no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—48no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—44no—
459C22—88no—
460C22—84no—
461C22—816no—
462C22—82no—
463C22—82no—
464C22#146214yes—
465C22C2×#40711yescharacteristic
466C22C22×#9911yescharacteristic
467C22C22×#10011yescharacteristic
468C4—22no—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—44no—
478C4—82no—
479C4—82no—
480C2—21no—
481C2—21no—
482C2—24no—
483C2—42no—
484C2—42no—
485C2—84no—
486C2C2×#146211yescharacteristic
487C2#3335811yescharacteristic
488C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC23×#41C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1464C2semidirect
3. #11510C2semidirect
4. #11509C2semidirect
5. #11507C2semidirect
6. #11432C2semidirect
26. #1464C22semidirect
27. C2×#324C22semidirect
29. #1319C22semidirect
30. #1318C22semidirect

Maximal subgroups

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

subgroupindex
#340442
C2×#14642
#115102
#115092
#115072
#114322