Finite Group Databasebeta

← #450×D97 #27150915

Subgroups — #450×D97 #27150915

Order 12416.

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×D97C211yesmaximal, characteristic
2C2×SD16×D97C212yesmaximal, complemented
3C22⋊Q8×D97C211yesmaximal, characteristic, complemented
4#20260888C211yesmaximal, characteristic, complemented
5#20260887C212yesmaximal, complemented
6#20260886C211yesmaximal, characteristic, complemented
7#20260885C211yesmaximal, characteristic, complemented
8#20260884C212yesmaximal, complemented
9#102×D97C212yesmaximal, complemented
10C22⋊C8×D97C211yesmaximal, characteristic, complemented
11#450×C97C211yesmaximal, characteristic, complemented
12C24×D97—21no—
13C2×Q8×D97C2211yescharacteristic, complemented
14C22×D388C2211yescharacteristic
15C22×C22⋊D97—21no—
16C2×D4×D97—28no—
17C2×D4×D97—21no—
18C2×D4×D97—21no—
19C2×D4×D97C2212yes—
20C2×D4×D97C2212yes—
21C22×C4×D97C2211yescharacteristic
22C2×#13095500C2212yescomplemented
23C2×#13095499C2212yescomplemented
24C2×#13095498C2212yescomplemented
25SD16×D97—28no—
26C2×C8×D97C2212yescomplemented
27#15690790C2211yescharacteristic, complemented
28#15690789C2211yescharacteristic, complemented
29#15690788—21no—
30#15690787—21no—
31#15690786C2211yescharacteristic, complemented
32C4⋊C4×D97—21no—
33C4⋊C4×D97C2211yescharacteristic, complemented
34C22⋊C4×D97—21no—
35#15690732C2212yescomplemented
36#15690731C2212yescomplemented
37#15690730C2212yescomplemented
38C22⋊#11715895C2211yescharacteristic, complemented
39#15690722C2212yescomplemented
40C22×D4×C97C2211yescharacteristic
41C2×SD16×C97C2212yescomplemented
42C22⋊Q8×C97C2211yescharacteristic, complemented
43#102×C97C2212yescomplemented
44C22⋊C8×C97C2211yescharacteristic, complemented
45C23×D97—28no—
46C23×D97—21no—
47C23×D97—22no—
48C23×D97—22no—
49C23×D97—44no—
50C23×D97—44no—
51C23×D97D411yescharacteristic
52C2×Dic194—21no—
53C2×Dic194C2311yescharacteristic, complemented
54Q8×D97—24no—
55C2×D388—21no—
56C2×D388—24no—
57C2×D388C2312yes—
58C2×C22⋊D97—28no—
59C2×C22⋊D97—22no—
60C2×C22⋊D97—22no—
61D4×D97—28no—
62D4×D97—28no—
63D4×D97—24no—
64D4×D97—24no—
65D4×D97—24no—
66D4×D97D418yes—
67C22×Dic97D411yescharacteristic
68C2×C4×D97—24no—
69C2×C4×D97—21no—
70C2×C4×D97—21no—
71C2×C4×D97D411yescharacteristic
72C2×C4×D97D411yescharacteristic
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75#13095500—24no—
76#13095499—24no—
77#13095498—24no—
78C2×#11715895C2312yescomplemented
79C8×D97—24no—
80C4⋊Dic97—21no—
81C4⋊Dic97C2311yescharacteristic, complemented
82#13095490—21no—
83#13095490—21no—
84#13095490—21no—
85C22⋊Dic97—21no—
86#13095488—21no—
87#13095488—21no—
88C24×C97—21no—
89C2×Q8×C97C2311yescharacteristic, complemented
90C2×D4×C97—24no—
91C2×D4×C97—21no—
92C2×D4×C97C2312yes—
93C22×C4×C97C2311yescharacteristic
94SD16×C97—24no—
95C2×C8×C97C2312yescomplemented
96C4⋊C4×C97—21no—
97C4⋊C4×C97C2311yescharacteristic, complemented
98C22⋊C4×C97—21no—
99C23×C97—24no—
100C23×C97—22no—
101C23×C97—42no—
102C23×C97—42no—
103C23×C97C2×D411yescharacteristic
104C22×D97—22no—
105C22×D97—22no—
106C22×D97—28no—
107C22×D97—24no—
108C22×D97—22no—
109C22×D97—28no—
110C22×D97—44no—
111C22×D97—432no—
112C22×D97—42no—
113C22×D97—42no—
114C22×D97—44no—
115C22×D97—44no—
116C22×D97—44no—
117C22×D97SD1614yescomplemented
118C22×D97C2×D411yescharacteristic
119C22×D97C2×D411yescharacteristic
120Q8×C97—22no—
121D4×C97—24no—
122D4×C97—22no—
123D4×C97C2×D414yes—
124C2×C4×C97—22no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—28no—
130C22⋊D97—28no—
131C2×Dic97—22no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97C2×D411yescharacteristic
135C2×Dic97C2×D411yescharacteristic
136D388—24no—
137D388—22no—
138D388C2×D414yes—
139C4×D97—22no—
140C4×D97—22no—
141C4×D97—22no—
142C4×D97—42no—
143C4×D97—42no—
144C4×D97C2×D414yes—
145Dic194—22no—
146Dic194—42no—
147C8×C97—22no—
148#11715895—22no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—24no—
152C22×C97—48no—
153C22×C97—42no—
154C22×C97—42no—
155C22×C97C2×SD1612yescomplemented
156C22×C97C22×D411yescharacteristic
157C2×D97—21no—
158C2×D97—21no—
159C2×D97—28no—
160C2×D97—24no—
161C2×D97—416no—
162C2×D97—48no—
163C2×D97—42no—
164C2×D97—42no—
165C2×D97—416no—
166C2×D97#12012yes—
167C2×D97C2×SD1612yes—
168C2×D97C2×SD1612yescomplemented
169C2×D97AGL(1,ℤ/8)12yes—
170C4×C97—21no—
171C4×C97—41no—
172C4×C97—41no—
173C4×C97C22×D412yes—
174Dic97—21no—
175Dic97—41no—
176Dic97—41no—
177Dic97#12012yes—
178C2×C97—22no—
179C2×C97—44no—
180C2×C97C2×#12011yescharacteristic
181C2×C97C22×SD1611yescharacteristic
182C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
183D97—22no—
184D97—44no—
185D97#45014yesdirect product factor
186C2×#450—971nomaximal, Sylow
187C97C2×#45011yescharacteristic, complemented, Sylow
188C23×D4—971no—
189C22×SD16—972no—
190C2×C22⋊Q8—971no—
191#450—971no—
192#450—972no—
193#450—971no—
194#450—972no—
195#450—971no—
196#450D9711yescharacteristic, direct product factor
197C2×#102—972no—
198C2×C22⋊C8—971no—
199C25—1941no—
200C22×Q8—971no—
201C22×D4—972no—
202C22×D4—972no—
203C22×D4—971no—
204C22×D4—1941no—
205C22×D4—1941no—
206C22×D4—1941no—
207C22×D4—1948no—
208C22×D4C2×D9711yescharacteristic
209C23×C4—971no—
210C2×SD16—972no—
211C2×SD16—972no—
212C2×SD16—972no—
213C2×SD16—1948no—
214C2×SD16C2×D9712yescomplemented
215C22×C8—972no—
216C22⋊Q8—971no—
217C22⋊Q8—971no—
218C22⋊Q8—971no—
219C22⋊Q8—1941no—
220C22⋊Q8—1941no—
221C22⋊Q8C2×D9711yescharacteristic, complemented
222C2×C4⋊C4—971no—
223C2×C4⋊C4—1941no—
224C2×C22⋊C4—1941no—
225#102—972no—
226#102—972no—
227#102—972no—
228#102C2×D9712yescomplemented
229C22⋊C8—971no—
230C22⋊C8—972no—
231C22⋊C8C2×D9711yescharacteristic, complemented
232C24—21no—
233C24—971no—
234C24—3884no—
235C24—3884no—
236C24—1942no—
237C24—1942no—
238C24—1948no—
239C24—1941no—
240C2×Q8—971no—
241C2×Q8—1941no—
242C2×Q8—1944no—
243C2×Q8C22×D9711yescharacteristic, complemented
244C2×D4—24no—
245C2×D4—21no—
246C2×D4—978no—
247C2×D4—972no—
248C2×D4—1944no—
249C2×D4—1944no—
250C2×D4—1942no—
251C2×D4—1942no—
252C2×D4—1944no—
253C2×D4—1948no—
254C2×D4—1948no—
255C2×D4—1948no—
256C2×D4—1944no—
257C2×D4—1941no—
258C2×D4C22×D9712yes—
259C22×C4—971no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—971no—
264C22×C4—1944no—
265C22×C4—1941no—
266C22×C4—1941no—
267C22×C4C22×D9711yescharacteristic
268SD16—24no—
269SD16—1944no—
270SD16—1944no—
271SD16—1944no—
272C2×C8—972no—
273C2×C8—1944no—
274C2×C8C22×D9712yescomplemented
275C4⋊C4—21no—
276C4⋊C4—971no—
277C4⋊C4—1941no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4C22×D9711yescharacteristic, complemented
282C22⋊C4—21no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C23—24no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—971no—
291C23—974no—
292C23—971no—
293C23—3882no—
294C23—3884no—
295C23—3884no—
296C23—3884no—
297C23—38832no—
298C23—3884no—
299C23—3882no—
300C23—1948no—
301C23—1942no—
302C23—1942no—
303C23—1944no—
304C23—1948no—
305C23—1942no—
306C23D4×D9711yescharacteristic
307Q8—22no—
308Q8—3882no—
309Q8—1942no—
310D4—24no—
311D4—22no—
312D4—974no—
313D4—1944no—
314D4—1948no—
315D4—1948no—
316D4—1942no—
317D4D4×D9714yes—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—971no—
322C2×C4—974no—
323C2×C4—971no—
324C2×C4—3882no—
325C2×C4—3882no—
326C2×C4—1942no—
327C2×C4—1942no—
328C2×C4—1942no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1942no—
332C2×C4D4×D9711yescharacteristic
333C2×C4C23×D9711yescharacteristic
334C8—22no—
335C8—1942no—
336C22—21no—
337C22—21no—
338C22—24no—
339C22—48no—
340C22—42no—
341C22—42no—
342C22—972no—
343C22—972no—
344C22—972no—
345C22—972no—
346C22—3888no—
347C22—38816no—
348C22—38816no—
349C22—3882no—
350C22—3882no—
351C22—1948no—
352C22—1944no—
353C22—1941no—
354C22—1941no—
355C22SD16×D9712yescomplemented
356C22C2×D4×D9711yescharacteristic
357C4—21no—
358C4—41no—
359C4—41no—
360C4—972no—
361C4—3881no—
362C4—3881no—
363C4—1941no—
364C4C2×D4×D9712yes—
365C2—22no—
366C2—44no—
367C2—974no—
368C2—3884no—
369C2—1942no—
370C2#120×D9711yescharacteristic
371C2C2×SD16×D9711yescharacteristic
372C2AGL(1,ℤ/8)×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
prime radicalC22×D4×D97C2
hypercenter#450D97
hypocenterC97C2×#450
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×SD16×D97C2semidirect
3. C22⋊Q8×D97C2semidirect
4. #20260888C2semidirect
5. #20260887C2semidirect
6. #20260886C2semidirect
7. #20260885C2semidirect
8. #20260884C2semidirect
9. #102×D97C2semidirect
10. C22⋊C8×D97C2semidirect
11. #450×C97C2semidirect
13. C2×Q8×D97C22semidirect
22. C2×#13095500C22semidirect
23. C2×#13095499C22semidirect
24. C2×#13095498C22semidirect
26. C2×C8×D97C22semidirect
27. #15690790C22semidirect
28. #15690789C22semidirect
31. #15690786C22semidirect
33. C4⋊C4×D97C22semidirect
35. #15690732C22semidirect
36. #15690731C22semidirect
37. #15690730C22semidirect
38. C22⋊#11715895C22semidirect
39. #15690722C22semidirect
41. C2×SD16×C97C22semidirect
42. C22⋊Q8×C97C22semidirect
43. #102×C97C22semidirect
44. C22⋊C8×C97C22semidirect
53. C2×Dic194C23semidirect
78. C2×#11715895C23semidirect
81. C4⋊Dic97C23semidirect
89. C2×Q8×C97C23semidirect
95. C2×C8×C97C23semidirect
97. C4⋊C4×C97C23semidirect
117. C22×D97SD16semidirect
155. C22×C97C2×SD16semidirect
168. C2×D97C2×SD16semidirect
185. D97#450direct (N × H)
187. C97C2×#450semidirect
196. #450D97direct (N × H)
214. C2×SD16C2×D97semidirect
221. C22⋊Q8C2×D97semidirect
228. #102C2×D97semidirect
231. C22⋊C8C2×D97semidirect
243. C2×Q8C22×D97semidirect
274. C2×C8C22×D97semidirect
281. C4⋊C4C22×D97semidirect
355. C22SD16×D97semidirect

Maximal subgroups

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

subgroupindex
C22×D4×D972
C2×SD16×D972
C22⋊Q8×D972
#202608882
#202608872
#202608862
#202608852
#202608842
#102×D972
C22⋊C8×D972
#450×C972
C2×#45097