Finite Group Databasebeta

← #480435

Subgroups — #480435

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34209C211yesmaximal, characteristic
2#17371C212yesmaximal
3#11536C212yesmaximal, complemented
4#11532C212yesmaximal, complemented
5#11513C212yesmaximal
6#11510C212yesmaximal
7#11482C212yesmaximal, complemented
8#11478C212yesmaximal, complemented
9C22×C4⋊Q8C411yescharacteristic
10C2×#519C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2983—22no—
13#2365—22no—
14#2365—22no—
15#2344—22no—
16#2344—22no—
17#2327C411yescharacteristic
18#2327C411yescharacteristic
19#2327C2211yescharacteristic
20#2327C2211yescharacteristic
21#2310C411yescharacteristic
22#2310C2211yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C411yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27#2285C2211yescharacteristic
28C4×#117C411yescharacteristic
29#1467—22no—
30#1467C2214yes—
31#1464—22no—
32#1464C2214yes—
33#1461C2214yes—
34#1458—22no—
35#1325C2214yes—
36#1324C2214yes—
37#1319C2214yes—
38#1319C2214yes—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48#519—44no—
49#519—44no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×#117—21no—
63C2×#117—21no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117D412yes—
68C2×#117D412yes—
69C2×#117C2311yescharacteristic
70C22×C4⋊C4—22no—
71C22×C42C2×C411yescharacteristic
72#446—48no—
73#432—24no—
74#432—24no—
75#431—24no—
76#391—22no—
77#391—22no—
78#391C2×C411yescharacteristic
79#391C2×C411yescharacteristic
80#391C2×C411yescharacteristic
81#391C2×C411yescharacteristic
82#391C2×C411yescharacteristic
83#391C2×C411yescharacteristic
84#391C2311yescharacteristic
85#391C2311yescharacteristic
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#384C2×C411yescharacteristic
99#384C2311yescharacteristic
100#382C2×C411yescharacteristic
101#382C2311yescharacteristic
102#381—22no—
103#381—22no—
104#380—22no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118#324—24no—
119#324—24no—
120#324C2318yes—
121#323—24no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—21no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8C22⋊C412yes—
132C22×Q8C22×C411yescharacteristic
133C22×Q8C22×C411yescharacteristic
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C23×C4—21no—
137C23×C4—21no—
138C23×C4C22⋊C412yes—
139C23×C4C22×C411yescharacteristic
140C4⋊Q8—216no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8M1614yes—
148C4⋊Q8M1614yes—
149C4×Q8—28no—
150C4×Q8—42no—
151C4×Q8—42no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—44no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8—44no—
160C4×Q8—44no—
161#117—24no—
162#117—42no—
163#117—44no—
164#117—44no—
165#117—44no—
166#117—44no—
167#117—42no—
168#117—42no—
169C2×C4⋊C4—216no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C22⋊C411yescharacteristic
194C2×C22⋊C4C22⋊C411yescharacteristic
195C2×C22⋊C4C2×D412yes—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22⋊C412yes—
209C2×C42C22⋊C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C4⋊C8—44no—
220C4⋊C8—44no—
221C4⋊C8—44no—
222C22⋊C8—28no—
223#97—44no—
224#97—44no—
225C4×C8—44no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
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
241C2×Q8—24no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—48no—
254C2×Q8—48no—
255C2×Q8—48no—
256C2×Q8—48no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8C2×M1612yes—
264C2×Q8C2×M1612yes—
265C2×Q8C2×M1612yes—
266C2×Q8C2×M1612yes—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
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×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C2×C8—48no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—28no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4#9914yes—
320C22⋊C4—22no—
321C22⋊C4—22no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C42—24no—
326C42—21no—
327C42—21no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C42—42no—
345C42—42no—
346C42C2×M1612yes—
347C42C2×M1612yes—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23#41312yes—
353C23C22×C22⋊C411yescharacteristic
354Q8—42no—
355Q8—42no—
356Q8—48no—
357Q8—42no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—48no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—48no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4#40714yes—
411C2×C4C2×#9912yes—
412C2×C4C2×#9912yes—
413C2×C4C2×#9912yes—
414C2×C4C22×M1611yescharacteristic
415C2×C4C22×M1611yescharacteristic
416C8—88no—
417C22—21no—
418C22—21no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22C2×#40711yescharacteristic
425C22C22×#9911yescharacteristic
426C22C2×#41311yescharacteristic
427C4—24no—
428C4—22no—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—24no—
434C4—42no—
435C4—42no—
436C4—41no—
437C4—41no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—81no—
446C2—21no—
447C2—21no—
448C2—42no—
449C2#1736211yescharacteristic
450C2#3336011yescharacteristic
451C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC24C2×C22⋊C4
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
3. #11536C2semidirect
4. #11532C2semidirect
7. #11482C2semidirect
8. #11478C2semidirect

Maximal subgroups

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

subgroupindex
#342092
#173712
#115362
#115322
#115132
#115102
#114822
#114782