Finite Group Databasebeta

← #480537

Subgroups — #480537

Order 512.

1039 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35677C211yesmaximal, characteristic
2#17371C211yesmaximal, characteristic, complemented
3#17352C211yesmaximal, characteristic, complemented
4#11567C214yesmaximal, complemented
5#11565C214yesmaximal, complemented
6#11433C212yesmaximal, complemented
7#11431C212yesmaximal, complemented
8C2×#536C411yescharacteristic
9C2×#535C411yescharacteristic
10C22×#124C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2982—21no—
14#2548C412yes—
15#2548C2212yes—
16#2545C412yes—
17#2545C2212yes—
18#2426—21no—
19#2426—21no—
20#2426—21no—
21#2426—21no—
22#2411—22no—
23#2411—22no—
24#2309C2211yescharacteristic
25#2297C2211yescharacteristic
26#2296C411yescharacteristic
27#1467C2214yescomplemented
28#1464—22no—
29#1461—21no—
30#1461—21no—
31#1461C2214yescomplemented
32#1458—22no—
33#1458C2212yes—
34#1458C2212yes—
35#1319C2214yescomplemented
36#1318C2218yes—
37#1318C2214yescomplemented
38C22×#41C2×C411yescharacteristic
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41#536—44no—
42#535—44no—
43C2×C4⋊Q8C2×C412yes—
44C2×#127C2×C412yes—
45C2×#124—21no—
46C2×#124—21no—
47C2×#124—44no—
48C2×#124C2×C412yes—
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#123—22no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×#121—21no—
56C2×#121—21no—
57C2×#120—22no—
58#519—44no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8C2312yes—
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C42C2×C411yescharacteristic
72#446—44no—
73C8×Q8—44no—
74#432—24no—
75#431—24no—
76#431—22no—
77#431—22no—
78#396—22no—
79#396—22no—
80#396—22no—
81#396—22no—
82#394—22no—
83#394—22no—
84#394—22no—
85#394—22no—
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#390—22no—
89#390—22no—
90#390—21no—
91#390—21no—
92#390—22no—
93#390—22no—
94#390C2×C412yes—
95#390C2×C411yescharacteristic
96#390C2312yes—
97#390C2311yescharacteristic
98#389—21no—
99#389—21no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#386—21no—
105#386—21no—
106#385—22no—
107#379—21no—
108#379—21no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#324—24no—
114#324—24no—
115#324C2314yes—
116#324C2314yes—
117#323—22no—
118#323—22no—
119C25—21no—
120C2×#41—24no—
121C2×#41—42no—
122C2×#41—42no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—42no—
126C22×Q8C22⋊C411yescharacteristic
127C22×Q8C22⋊C411yescharacteristic
128C22×Q8C22×C411yescharacteristic
129C22×Q8C22×C412yes—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—24no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C412yes—
138C22×D4C22×C411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C22⋊C412yes—
142C23×C4C22×C411yescharacteristic
143C4⋊Q8—44no—
144#127—44no—
145#124—44no—
146#124—44no—
147#124—44no—
148#124—44no—
149#124—44no—
150#124—44no—
151#123—48no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154#121—44no—
155#121—44no—
156#120—48no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—44no—
160C4×Q8—44no—
161C4×Q8—42no—
162C4×Q8—42no—
163#117—24no—
164#117—42no—
165#117—44no—
166#117—44no—
167#117—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C2×D412yes—
179C2×C4⋊C4C2×D412yes—
180C2×C4⋊C4C2411yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C2×D411yescharacteristic
201C2×C22⋊C4C2×D411yescharacteristic
202C2×C42—21no—
203C2×C42—42no—
204C2×C42C22⋊C412yes—
205C2×C42C22⋊C412yes—
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C4⋊C8—44no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217C4⋊C8—44no—
218C22⋊C8—24no—
219C22⋊C8—24no—
220#97—44no—
221C4×C8—44no—
222C4×C8—42no—
223C4×C8—42no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230C24—22no—
231C24—21no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—82no—
236C24—82no—
237C24C2×C22⋊C411yescharacteristic
238#41—88no—
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—42no—
248C2×Q8—416no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8#10014yes—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—44no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—42no—
266C2×D4—88no—
267C2×D4—82no—
268C2×D4—82no—
269C2×D4—82no—
270C2×D4—82no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4#13114yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C2×C8—44no—
298C2×C8—44no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C4⋊C4#9914yes—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C22⋊C4—81no—
331C22⋊C4—81no—
332C42—24no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—41no—
340C42—41no—
341C42—42no—
342C42—42no—
343C23—21no—
344C23—21no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—84no—
362C23—88no—
363C23—164no—
364C23—162no—
365C23—162no—
366C23#40812yes—
367C23C22×C22⋊C411yescharacteristic
368Q8—28no—
369Q8—48no—
370Q8—44no—
371Q8—82no—
372D4—84no—
373D4—82no—
374D4—84no—
375D4—84no—
376D4—164no—
377D4—164no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—28no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
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—41no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4#40812yes—
413C2×C4C2×#9914yes—
414C2×C4C2×#9912yes—
415C2×C4C2×#10014yes—
416C2×C4C2×#10012yes—
417C2×C4C2×#13112yes—
418C8—84no—
419C8—84no—
420C22—21no—
421C22—21no—
422C22—22no—
423C22—42no—
424C22—42no—
425C22—41no—
426C22—41no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22—81no—
436C22—82no—
437C22—81no—
438C22—82no—
439C22—81no—
440C22—82no—
441C22—168no—
442C22C2×#40811yescharacteristic
443C22C22×#9911yescharacteristic
444C22C22×#10011yescharacteristic
445C4—24no—
446C4—22no—
447C4—24no—
448C4—41no—
449C4—41no—
450C4—42no—
451C4—41no—
452C4—41no—
453C4—42no—
454C4—82no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4#147114yes—
459C2—21no—
460C2—21no—
461C2—42no—
462C2—161no—
463C2—162no—
464C2C2×#147111yescharacteristic
465C2#3336211yescharacteristic
466C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#535C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17371C2semidirect
3. #17352C2semidirect
4. #11567C2semidirect
5. #11565C2semidirect
6. #11433C2semidirect
7. #11431C2semidirect
27. #1467C22semidirect
31. #1461C22semidirect
35. #1319C22semidirect
37. #1318C22semidirect

Maximal subgroups

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

subgroupindex
#356772
#173712
#173522
#115672
#115652
#114332
#114312