Finite Group Databasebeta

← #27151084

Subgroups — #27151084

Order 12416.

724 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#20261571C211yesmaximal, characteristic, complemented
2#20261533C212yesmaximal, complemented
3#20260917C212yesmaximal, complemented
4#20260912C212yesmaximal, complemented
5#20260907C212yesmaximal, complemented
6#20260902C212yesmaximal, complemented
7C2×#15690738C212yesmaximal, complemented
8#20260719C211yesmaximal, characteristic, complemented
9#532×C97C211yesmaximal, characteristic, complemented
10#15690858C2212yes—
11#15690857—22no—
12#15690856—22no—
13#15690855—22no—
14#15690855C2212yes—
15C2×#13095511—22no—
16C2×#13095511C2212yes—
17#41×D97—24no—
18#15690841C2214yescomplemented
19#15690840—22no—
20#15690836C2214yescomplemented
21#15690835—22no—
22#15690830—24no—
23C2×#13095495C2212yescomplemented
24#15690738C2214yescomplemented
25#15690738C2214yescomplemented
26#15690729C2212yescomplemented
27#15690727C2212yescomplemented
28C2×#41×C97C2211yescharacteristic
29C2×#41×C97C2211yescharacteristic, complemented
30C4⋊Q8×C97C2211yescharacteristic, complemented
31#127×C97C2211yescharacteristic, complemented
32#124×C97C2211yescharacteristic, complemented
33#124×C97C2211yescharacteristic, complemented
34#121×C97—22no—
35C4×D4×C97—22no—
36C2×C42×C97C2211yescharacteristic, complemented
37#13095512—24no—
38#13095511—24no—
39#13095511—24no—
40#13095511—22no—
41#13095511—24no—
42#13095511—22no—
43#13095511D414yescomplemented
44#13095511C2314yes—
45#13095510—28no—
46#13095510—24no—
47C2×Dic194—22no—
48C2×Dic194D412yescomplemented
49Q8×D97—24no—
50C2×D388—22no—
51C2×D388D412yescomplemented
52C2×C22⋊D97—24no—
53C2×C22⋊D97—22no—
54D4×D97—28no—
55D4×D97—24no—
56C2×C4×D97—24no—
57C2×C4×D97—22no—
58#13095501—24no—
59#13095499—24no—
60#13095498—24no—
61#13095496—24no—
62#13095495—22no—
63#13095495C2314yes—
64C2×#11715895—22no—
65#41×C97—22no—
66#41×C97—22no—
67#41×C97—22no—
68#41×C97—22no—
69C2×Q8×C97C2311yescharacteristic
70C2×Q8×C97C2311yescharacteristic
71C2×D4×C97—22no—
72C2×D4×C97—21no—
73C2×D4×C97D412yes—
74C2×D4×C97C2311yescharacteristic
75C2×D4×C97C2311yescharacteristic
76C22×C4×C97—21no—
77C22×C4×C97—21no—
78C22×C4×C97D412yes—
79C22×C4×C97C2311yescharacteristic
80C4⋊C4×C97—22no—
81C22⋊C4×C97—22no—
82C22⋊C4×C97—22no—
83C42×C97—21no—
84C42×C97C2311yescharacteristic
85C42×C97C2311yescharacteristic
86C23×C97—21no—
87C23×C97C2×D412yes—
88C23×C97C2×D411yescharacteristic
89C22×D97—24no—
90C22×D97—22no—
91Q8×C97—22no—
92Q8×C97—22no—
93D4×C97—24no—
94D4×C97—22no—
95D4×C97—22no—
96D4×C97—44no—
97D4×C97—42no—
98C2×C4×C97—21no—
99C2×C4×C97—22no—
100C2×C4×C97—22no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—42no—
104C2×C4×C97—41no—
105C2×C4×C97—41no—
106C2×C4×C97C2×D412yes—
107C2×C4×C97C2×D412yes—
108C2×C4×C97C2×D412yes—
109C2×C4×C97C2×D411yescharacteristic
110C2×C4×C97C2411yescharacteristic
111C22⋊D97—28no—
112C22⋊D97—24no—
113C22⋊D97—24no—
114C2×Dic97—24no—
115C2×Dic97—22no—
116D388—24no—
117D388—22no—
118D388C2×D414yes—
119C4×D97—28no—
120C4×D97—24no—
121C4×D97—24no—
122Dic194—24no—
123Dic194—22no—
124Dic194C2×D414yes—
125#11715895—24no—
126C22×C97—22no—
127C22×C97—41no—
128C22×C97—42no—
129C22×C97—42no—
130C22×C97—41no—
131C22×C97#12012yes—
132C22×C97C22×D411yescharacteristic
133C2×D97—24no—
134C2×D97—82no—
135C2×D97—84no—
136C4×C97—22no—
137C4×C97—22no—
138C4×C97—41no—
139C4×C97—41no—
140C4×C97#12011yescharacteristic
141C4×C97#12011yescharacteristic
142C4×C97C22×D412yes—
143Dic97—24no—
144C2×C97—21no—
145C2×C97—22no—
146C2×C97—42no—
147C2×C97—81no—
148C2×C97C2×#12011yescharacteristic
149D97—44no—
150#3018—971nomaximal, Sylow
151C97#301811yescharacteristic, complemented, Sylow
152#585—971no—
153#575—972no—
154#532D9711yescharacteristic, complemented
155#456—972no—
156#455—972no—
157#454—972no—
158#453—972no—
159C2×#104—972no—
160#413—971no—
1612−1+4—972no—
1622−1+4—1942no—
163H(5,𝔽2)—972no—
164H(5,𝔽2)—1942no—
165H(5,𝔽2)—1942no—
166C2×#41—972no—
167C2×#41—1944no—
168C2×#41—1942no—
169C2×#41C2×D9711yescharacteristic
170C2×#41C2×D9711yescharacteristic, complemented
171#137—974no—
172#137—1942no—
173AGL(1,ℤ/8)—974no—
174AGL(1,ℤ/8)—1942no—
175#135—1944no—
176C2×M16—972no—
177C4⋊Q8C2×D9711yescharacteristic, complemented
178#127C2×D9711yescharacteristic, complemented
179#124C2×D9711yescharacteristic, complemented
180#124C2×D9711yescharacteristic, complemented
181#121—22no—
182C4×D4—22no—
183C2×C42C2×D9711yescharacteristic, complemented
184#104—974no—
185#104—974no—
186#101—972no—
187#100—972no—
188#41—22no—
189#41—22no—
190#41—22no—
191#41—22no—
192#41—974no—
193#41—974no—
194#41—1944no—
195#41—1944no—
196#41—1944no—
197#41—1948no—
198#41—1944no—
199#41—1942no—
200#41—1942no—
201#41—1944no—
202C2×Q8—972no—
203C2×Q8—1942no—
204C2×Q8—1944no—
205C2×Q8C22×D9711yescharacteristic
206C2×Q8C22×D9711yescharacteristic
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—972no—
210C2×D4—1942no—
211C2×D4—1944no—
212C2×D4—1948no—
213C2×D4—1942no—
214C2×D4—1944no—
215C2×D4C22⋊D9712yes—
216C2×D4C22×D9711yescharacteristic
217C2×D4C22×D9711yescharacteristic
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—1944no—
221C22×C4—1942no—
222C22×C4C22⋊D9712yes—
223C22×C4C22×D9711yescharacteristic
224Q16—1944no—
225SD16—1944no—
226SD16—1944no—
227D8—1944no—
228M16—974no—
229M16—1942no—
230C2×C8—1942no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42—21no—
235C42C22×D9711yescharacteristic
236C42C22×D9711yescharacteristic
237C23—21no—
238C23—1944no—
239C23—1942no—
240C23C2×C22⋊D9712yes—
241C23C2×C22⋊D9711yescharacteristic
242Q8—22no—
243Q8—22no—
244Q8—974no—
245Q8—1944no—
246Q8—1942no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—44no—
251D4—42no—
252D4—974no—
253D4—1944no—
254D4—1948no—
255D4—1944no—
256D4—1944no—
257D4—1942no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—1948no—
267C2×C4—1944no—
268C2×C4—1942no—
269C2×C4—1944no—
270C2×C4—1944no—
271C2×C4D4×D9712yes—
272C2×C4D4×D9712yes—
273C2×C4C2×C22⋊D9712yes—
274C2×C4C2×C22⋊D9711yescharacteristic
275C2×C4C23×D9711yescharacteristic
276C8—1944no—
277C22—22no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—7762no—
283C22—7764no—
284C22—1944no—
285C22#1569077812yes—
286C22C22×C22⋊D9711yescharacteristic
287C4—22no—
288C4—22no—
289C4—41no—
290C4—41no—
291C4—1944no—
292C4#1569077811yescharacteristic
293C4#1569077811yescharacteristic
294C4C2×D4×D9712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—81no—
299C2—3884no—
300C2C2×#1569077811yescharacteristic
subgroupG/H
centerC4#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#532×C97C2
socleC2×C97C2×#120
hypercenter#532D97
hypocenterC97#3018
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20261571C2semidirect
2. #20261533C2semidirect
3. #20260917C2semidirect
4. #20260912C2semidirect
5. #20260907C2semidirect
6. #20260902C2semidirect
7. C2×#15690738C2semidirect
8. #20260719C2semidirect
9. #532×C97C2semidirect
18. #15690841C22semidirect
20. #15690836C22semidirect
23. C2×#13095495C22semidirect
24. #15690738C22semidirect
25. #15690738C22semidirect
26. #15690729C22semidirect
27. #15690727C22semidirect
29. C2×#41×C97C22semidirect
30. C4⋊Q8×C97C22semidirect
31. #127×C97C22semidirect
32. #124×C97C22semidirect
33. #124×C97C22semidirect
36. C2×C42×C97C22semidirect
43. #13095511D4semidirect
48. C2×Dic194D4semidirect
51. C2×D388D4semidirect
151. C97#3018semidirect
154. #532D97semidirect
170. C2×#41C2×D97semidirect
177. C4⋊Q8C2×D97semidirect
178. #127C2×D97semidirect
179. #124C2×D97semidirect
180. #124C2×D97semidirect
183. C2×C42C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615712
#202615332
#202609172
#202609122
#202609072
#202609022
C2×#156907382
#202607192
#532×C972
#301897