Finite Group Databasebeta

← #450×D101 #27180311

Subgroups — #450×D101 #27180311

Order 12928.

996 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×D101C211yesmaximal, characteristic
2C2×SD16×D101C212yesmaximal, complemented
3C22⋊Q8×D101C211yesmaximal, characteristic, complemented
4#20265330C211yesmaximal, characteristic, complemented
5#20265329C212yesmaximal, complemented
6#20265328C211yesmaximal, characteristic, complemented
7#20265327C211yesmaximal, characteristic, complemented
8#20265326C212yesmaximal, complemented
9#102×D101C212yesmaximal, complemented
10C22⋊C8×D101C211yesmaximal, characteristic, complemented
11#450×C101C211yesmaximal, characteristic, complemented
12C24×D101—21no—
13C2×Q8×D101C2211yescharacteristic, complemented
14C22×D404C2211yescharacteristic
15C22×C22⋊D101—21no—
16C2×D4×D101—28no—
17C2×D4×D101—21no—
18C2×D4×D101—21no—
19C2×D4×D101C2212yes—
20C2×D4×D101C2212yes—
21C22×C4×D101C2211yescharacteristic
22C2×#13108063C2212yescomplemented
23C2×#13108062C2212yescomplemented
24C2×#13108061C2212yescomplemented
25SD16×D101—28no—
26C2×C8×D101C2212yescomplemented
27#15692235C2211yescharacteristic, complemented
28#15692234C2211yescharacteristic, complemented
29#15692233—21no—
30#15692232—21no—
31#15692231C2211yescharacteristic, complemented
32C4⋊C4×D101—21no—
33C4⋊C4×D101C2211yescharacteristic, complemented
34C22⋊C4×D101—21no—
35#15692177C2212yescomplemented
36#15692176C2212yescomplemented
37#15692175C2212yescomplemented
38C22⋊#11717579C2211yescharacteristic, complemented
39#15692167C2212yescomplemented
40C22×D4×C101C2211yescharacteristic
41C2×SD16×C101C2212yescomplemented
42C22⋊Q8×C101C2211yescharacteristic, complemented
43#102×C101C2212yescomplemented
44C22⋊C8×C101C2211yescharacteristic, complemented
45C23×D101—28no—
46C23×D101—21no—
47C23×D101—22no—
48C23×D101—22no—
49C23×D101—44no—
50C23×D101—44no—
51C23×D101D411yescharacteristic
52C2×Dic202—21no—
53C2×Dic202C2311yescharacteristic, complemented
54Q8×D101—24no—
55C2×D404—21no—
56C2×D404—24no—
57C2×D404C2312yes—
58C2×C22⋊D101—28no—
59C2×C22⋊D101—22no—
60C2×C22⋊D101—22no—
61D4×D101—28no—
62D4×D101—28no—
63D4×D101—24no—
64D4×D101—24no—
65D4×D101—24no—
66D4×D101D418yes—
67C22×Dic101D411yescharacteristic
68C2×C4×D101—24no—
69C2×C4×D101—21no—
70C2×C4×D101—21no—
71C2×C4×D101D411yescharacteristic
72C2×C4×D101D411yescharacteristic
73C2×C4×D101C2311yescharacteristic
74C2×C4×D101C2311yescharacteristic
75#13108063—24no—
76#13108062—24no—
77#13108061—24no—
78C2×#11717579C2312yescomplemented
79C8×D101—24no—
80C4⋊Dic101—21no—
81C4⋊Dic101C2311yescharacteristic, complemented
82#13108053—21no—
83#13108053—21no—
84#13108053—21no—
85C22⋊Dic101—21no—
86#13108051—21no—
87#13108051—21no—
88C24×C101—21no—
89C2×Q8×C101C2311yescharacteristic, complemented
90C2×D4×C101—24no—
91C2×D4×C101—21no—
92C2×D4×C101C2312yes—
93C22×C4×C101C2311yescharacteristic
94SD16×C101—24no—
95C2×C8×C101C2312yescomplemented
96C4⋊C4×C101—21no—
97C4⋊C4×C101C2311yescharacteristic, complemented
98C22⋊C4×C101—21no—
99C23×C101—24no—
100C23×C101—22no—
101C23×C101—42no—
102C23×C101—42no—
103C23×C101C2×D411yescharacteristic
104C22×D101—22no—
105C22×D101—22no—
106C22×D101—28no—
107C22×D101—24no—
108C22×D101—22no—
109C22×D101—28no—
110C22×D101—44no—
111C22×D101—432no—
112C22×D101—42no—
113C22×D101—42no—
114C22×D101—44no—
115C22×D101—44no—
116C22×D101—44no—
117C22×D101SD1614yescomplemented
118C22×D101C2×D411yescharacteristic
119C22×D101C2×D411yescharacteristic
120Q8×C101—22no—
121D4×C101—24no—
122D4×C101—22no—
123D4×C101C2×D414yes—
124C2×C4×C101—22no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2411yescharacteristic
129C22⋊D101—28no—
130C22⋊D101—28no—
131C2×Dic101—22no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101C2×D411yescharacteristic
135C2×Dic101C2×D411yescharacteristic
136D404—24no—
137D404—22no—
138D404C2×D414yes—
139C4×D101—22no—
140C4×D101—22no—
141C4×D101—22no—
142C4×D101—42no—
143C4×D101—42no—
144C4×D101C2×D414yes—
145Dic202—22no—
146Dic202—42no—
147C8×C101—22no—
148#11717579—22no—
149C22×C101—21no—
150C22×C101—21no—
151C22×C101—24no—
152C22×C101—48no—
153C22×C101—42no—
154C22×C101—42no—
155C22×C101C2×SD1612yescomplemented
156C22×C101C22×D411yescharacteristic
157C2×D101—21no—
158C2×D101—21no—
159C2×D101—28no—
160C2×D101—24no—
161C2×D101—416no—
162C2×D101—48no—
163C2×D101—42no—
164C2×D101—42no—
165C2×D101—416no—
166C2×D101#12012yes—
167C2×D101C2×SD1612yes—
168C2×D101C2×SD1612yescomplemented
169C2×D101AGL(1,ℤ/8)12yes—
170C4×C101—21no—
171C4×C101—41no—
172C4×C101—41no—
173C4×C101C22×D412yes—
174Dic101—21no—
175Dic101—41no—
176Dic101—41no—
177Dic101#12012yes—
178C2×C101—22no—
179C2×C101—44no—
180C2×C101C2×#12011yescharacteristic
181C2×C101C22×SD1611yescharacteristic
182C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
183D101—22no—
184D101—44no—
185D101#45014yesdirect product factor
186C2×#450—1011nomaximal, Sylow
187C101C2×#45011yescharacteristic, complemented, Sylow
188C23×D4—1011no—
189C22×SD16—1012no—
190C2×C22⋊Q8—1011no—
191#450—1011no—
192#450—1012no—
193#450—1011no—
194#450—1012no—
195#450—1011no—
196#450D10111yescharacteristic, direct product factor
197C2×#102—1012no—
198C2×C22⋊C8—1011no—
199C25—2021no—
200C22×Q8—1011no—
201C22×D4—1012no—
202C22×D4—1012no—
203C22×D4—1011no—
204C22×D4—2021no—
205C22×D4—2021no—
206C22×D4—2021no—
207C22×D4—2028no—
208C22×D4C2×D10111yescharacteristic
209C23×C4—1011no—
210C2×SD16—1012no—
211C2×SD16—1012no—
212C2×SD16—1012no—
213C2×SD16—2028no—
214C2×SD16C2×D10112yescomplemented
215C22×C8—1012no—
216C22⋊Q8—1011no—
217C22⋊Q8—1011no—
218C22⋊Q8—1011no—
219C22⋊Q8—2021no—
220C22⋊Q8—2021no—
221C22⋊Q8C2×D10111yescharacteristic, complemented
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—2021no—
224C2×C22⋊C4—2021no—
225#102—1012no—
226#102—1012no—
227#102—1012no—
228#102C2×D10112yescomplemented
229C22⋊C8—1011no—
230C22⋊C8—1012no—
231C22⋊C8C2×D10111yescharacteristic, complemented
232C24—21no—
233C24—1011no—
234C24—4044no—
235C24—4044no—
236C24—2022no—
237C24—2022no—
238C24—2028no—
239C24—2021no—
240C2×Q8—1011no—
241C2×Q8—2021no—
242C2×Q8—2024no—
243C2×Q8C22×D10111yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—1018no—
247C2×D4—1012no—
248C2×D4—2024no—
249C2×D4—2024no—
250C2×D4—2022no—
251C2×D4—2022no—
252C2×D4—2024no—
253C2×D4—2028no—
254C2×D4—2028no—
255C2×D4—2028no—
256C2×D4—2024no—
257C2×D4—2021no—
258C2×D4C22×D10112yes—
259C22×C4—1011no—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1011no—
263C22×C4—1011no—
264C22×C4—2024no—
265C22×C4—2021no—
266C22×C4—2021no—
267C22×C4C22×D10111yescharacteristic
268SD16—24no—
269SD16—2024no—
270SD16—2024no—
271SD16—2024no—
272C2×C8—1012no—
273C2×C8—2024no—
274C2×C8C22×D10112yescomplemented
275C4⋊C4—21no—
276C4⋊C4—1011no—
277C4⋊C4—2021no—
278C4⋊C4—2021no—
279C4⋊C4—2021no—
280C4⋊C4—2021no—
281C4⋊C4C22×D10111yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C22⋊C4—2021no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—1011no—
291C23—1011no—
292C23—1014no—
293C23—4042no—
294C23—4044no—
295C23—4044no—
296C23—4044no—
297C23—40432no—
298C23—4044no—
299C23—4042no—
300C23—2028no—
301C23—2022no—
302C23—2022no—
303C23—2024no—
304C23—2028no—
305C23—2022no—
306C23D4×D10111yescharacteristic
307Q8—22no—
308Q8—4042no—
309Q8—2022no—
310D4—24no—
311D4—22no—
312D4—1014no—
313D4—2024no—
314D4—2028no—
315D4—2028no—
316D4—2022no—
317D4D4×D10114yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—1011no—
322C2×C4—1014no—
323C2×C4—1011no—
324C2×C4—4042no—
325C2×C4—4042no—
326C2×C4—2022no—
327C2×C4—2022no—
328C2×C4—2022no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2022no—
332C2×C4D4×D10111yescharacteristic
333C2×C4C23×D10111yescharacteristic
334C8—22no—
335C8—2022no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—1012no—
343C22—1012no—
344C22—1012no—
345C22—1012no—
346C22—4048no—
347C22—40416no—
348C22—40416no—
349C22—4042no—
350C22—4042no—
351C22—2028no—
352C22—2024no—
353C22—2021no—
354C22—2021no—
355C22SD16×D10112yescomplemented
356C22C2×D4×D10111yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—1012no—
361C4—4041no—
362C4—4041no—
363C4—2021no—
364C4C2×D4×D10112yes—
365C2—22no—
366C2—44no—
367C2—1014no—
368C2—4044no—
369C2—2022no—
370C2#120×D10111yescharacteristic
371C2C2×SD16×D10111yescharacteristic
372C2AGL(1,ℤ/8)×D10111yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#450×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×D101C2
hypercenter#450D101
hypocenterC101C2×#450
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D101C2semidirect
3. C22⋊Q8×D101C2semidirect
4. #20265330C2semidirect
5. #20265329C2semidirect
6. #20265328C2semidirect
7. #20265327C2semidirect
8. #20265326C2semidirect
9. #102×D101C2semidirect
10. C22⋊C8×D101C2semidirect
11. #450×C101C2semidirect
13. C2×Q8×D101C22semidirect
22. C2×#13108063C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108061C22semidirect
26. C2×C8×D101C22semidirect
27. #15692235C22semidirect
28. #15692234C22semidirect
31. #15692231C22semidirect
33. C4⋊C4×D101C22semidirect
35. #15692177C22semidirect
36. #15692176C22semidirect
37. #15692175C22semidirect
38. C22⋊#11717579C22semidirect
39. #15692167C22semidirect
41. C2×SD16×C101C22semidirect
42. C22⋊Q8×C101C22semidirect
43. #102×C101C22semidirect
44. C22⋊C8×C101C22semidirect
53. C2×Dic202C23semidirect
78. C2×#11717579C23semidirect
81. C4⋊Dic101C23semidirect
89. C2×Q8×C101C23semidirect
95. C2×C8×C101C23semidirect
97. C4⋊C4×C101C23semidirect
117. C22×D101SD16semidirect
155. C22×C101C2×SD16semidirect
168. C2×D101C2×SD16semidirect
185. D101#450direct (N × H)
187. C101C2×#450semidirect
196. #450D101direct (N × H)
214. C2×SD16C2×D101semidirect
221. C22⋊Q8C2×D101semidirect
228. #102C2×D101semidirect
231. C22⋊C8C2×D101semidirect
243. C2×Q8C22×D101semidirect
274. C2×C8C22×D101semidirect
281. C4⋊C4C22×D101semidirect
355. C22SD16×D101semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D1012
C2×SD16×D1012
C22⋊Q8×D1012
#202653302
#202653292
#202653282
#202653272
#202653262
#102×D1012
C22⋊C8×D1012
#450×C1012
C2×#450101