Finite Group Databasebeta

← #27150194

Subgroups — #27150194

Order 12416.

800 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#15690855C211yesmaximal, characteristic
2#20261208C212yesmaximal, complemented
3#20260707C214yesmaximal, complemented
4C2×#15690725C211yesmaximal, characteristic, complemented
5C2×#15690724C212yesmaximal, complemented
6#99×D97C214yesmaximal, complemented
7C2×#99×C97C211yesmaximal, characteristic, complemented
8#15690855—24no—
9#15690855—24no—
10C2×#13095511C412yescomplemented
11C2×#13095510C412yescomplemented
12C2×#13095510C2212yes—
13C22×C22⋊D97C412yescomplemented
14C22×C22⋊D97C2211yescharacteristic
15C22×C22⋊D97C2211yescharacteristic
16C2×D4×D97C412yescomplemented
17C2×D4×D97C2212yes—
18C4×C22⋊D97—24no—
19#15690770—24no—
20#15690766C2214yescomplemented
21C2×C22⋊Dic97C2212yescomplemented
22C22⋊C4×D97C2214yescomplemented
23#15690725C2214yescomplemented
24#15690724C2218yescomplemented
25C22×D4×C97C2211yescharacteristic
26C2×C22⋊C4×C97C2212yescomplemented
27#99×C97C2214yescomplemented
28C23×D97C2×C411yescharacteristic, complemented
29C23×D97C2×C412yescomplemented
30C23×D97C2311yescharacteristic
31#13095511—24no—
32#13095511—24no—
33#13095510—28no—
34#13095510—28no—
35C2×Dic194C2×C412yescomplemented
36C2×D388C2×C412yescomplemented
37C2×C22⋊D97—22no—
38C2×C22⋊D97—22no—
39C2×C22⋊D97—24no—
40C2×C22⋊D97—24no—
41C2×C22⋊D97C2×C412yes—
42C2×C22⋊D97C2×C412yes—
43C2×C22⋊D97C2×C412yescomplemented
44C2×C22⋊D97C2×C412yescomplemented
45C2×C22⋊D97C2×C412yescomplemented
46C2×C22⋊D97D414yes—
47C2×C22⋊D97D414yes—
48C2×C22⋊D97C2312yes—
49D4×D97—28no—
50D4×D97—28no—
51C22×Dic97—22no—
52C22×Dic97C2×C412yescomplemented
53C22×Dic97C2×C411yescharacteristic, complemented
54C22×Dic97C2311yescharacteristic
55C2×C4×D97—24no—
56C2×C4×D97C2×C412yescomplemented
57C2×C4×D97C2×C412yescomplemented
58#13095490—24no—
59C22⋊Dic97—22no—
60C22⋊Dic97C2314yes—
61#13095488—24no—
62C4×Dic97—24no—
63C24×C97C2×C411yescharacteristic, complemented
64C24×C97C2311yescharacteristic
65C2×D4×C97—22no—
66C2×D4×C97—22no—
67C2×D4×C97C2×C412yescomplemented
68C2×D4×C97C2312yes—
69C22×C4×C97—22no—
70C22×C4×C97C2×C411yescharacteristic, complemented
71C22⋊C4×C97—22no—
72C22⋊C4×C97C2314yes—
73C23×C97—21no—
74C23×C97—21no—
75C23×C97—81no—
76C23×C97—81no—
77C23×C97C22×C412yes—
78C23×C97C22×C411yescharacteristic
79C23×C97C22×C411yescharacteristic
80C23×C97C2×D412yes—
81C23×C97C2×D412yes—
82C23×C97C2411yescharacteristic
83C22×D97—22no—
84C22×D97—24no—
85C22×D97—81no—
86C22×D97—82no—
87C22×D97—81no—
88C22×D97C22×C412yes—
89C22×D97C22×C412yes—
90C22×D97C2×D414yes—
91D4×C97—24no—
92D4×C97—24no—
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—24no—
96C2×C4×C97—82no—
97C2×C4×C97C22×C412yes—
98C22⋊D97—22no—
99C22⋊D97—22no—
100C22⋊D97—24no—
101C22⋊D97—24no—
102C22⋊D97—24no—
103C22⋊D97C22⋊C418yes—
104C22⋊D97C22⋊C414yes—
105C22⋊D97C22⋊C414yes—
106C2×Dic97—22no—
107C2×Dic97—24no—
108C2×Dic97—24no—
109C2×Dic97—82no—
110C2×Dic97C22×C412yes—
111C2×Dic97C22×C412yes—
112C2×Dic97C2×D414yes—
113D388—24no—
114C4×D97—24no—
115C4×D97—24no—
116C4×D97—84no—
117Dic194—24no—
118C22×C97—22no—
119C22×C97—42no—
120C22×C97—42no—
121C22×C97—41no—
122C22×C97—41no—
123C22×C97—41no—
124C22×C97—42no—
125C22×C97—81no—
126C22×C97—81no—
127C22×C97C2×C22⋊C412yes—
128C22×C97C2×C22⋊C411yescharacteristic
129C22×C97C2×C22⋊C411yescharacteristic
130C22×C97C23×C411yescharacteristic
131C22×C97C22×D412yes—
132C2×D97—22no—
133C2×D97—44no—
134C2×D97—42no—
135C2×D97—42no—
136C2×D97—84no—
137C2×D97—82no—
138C2×D97C2×C22⋊C414yes—
139C4×C97—22no—
140C4×C97—44no—
141Dic97—22no—
142Dic97—44no—
143Dic97C2×C22⋊C414yes—
144C2×C97—22no—
145C2×C97—21no—
146C2×C97—22no—
147C2×C97—21no—
148C2×C97—21no—
149C2×C97—42no—
150C2×C97C22×C22⋊C411yescharacteristic
151D97—24no—
152D97—42no—
153#2883—971nomaximal, Sylow
154C97#288311yescharacteristic, complemented, Sylow
155C2×H(5,𝔽2)—971no—
156#518—972no—
157#410—974no—
158C2×#99—971no—
159C2×#99—972no—
160C2×#99—974no—
161C2×#99D9711yescharacteristic, complemented
162H(5,𝔽2)—1944no—
163H(5,𝔽2)—1944no—
164C2×#41—972no—
165C2×#41—972no—
166C2×#41—972no—
167C22×D4—971no—
168C22×D4—972no—
169C22×D4—971no—
170C22×D4—972no—
171C22×D4—972no—
172C22×D4C2×D9711yescharacteristic
173C4×D4—1944no—
174C4×D4—1944no—
175#117—974no—
176C2×C22⋊C4—972no—
177C2×C22⋊C4—974no—
178C2×C22⋊C4C2×D9712yescomplemented
179#99—974no—
180#99—978no—
181#99C2×D9714yescomplemented
182C24—971no—
183C24—972no—
184C24—971no—
185C24C4×D9711yescharacteristic, complemented
186C24C22×D9711yescharacteristic
187#41—1944no—
188#41—1944no—
189#41—1948no—
190#41—1948no—
191C2×Q8—972no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—972no—
195C2×D4—972no—
196C2×D4—974no—
197C2×D4—974no—
198C2×D4—972no—
199C2×D4—972no—
200C2×D4—972no—
201C2×D4—972no—
202C2×D4—972no—
203C2×D4—1948no—
204C2×D4—1948no—
205C2×D4—1944no—
206C2×D4—1944no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4C4×D9712yescomplemented
210C2×D4C22×D9712yes—
211C22×C4—22no—
212C22×C4—971no—
213C22×C4—972no—
214C22×C4—971no—
215C22×C4—972no—
216C22×C4—972no—
217C22×C4—1944no—
218C22×C4—1942no—
219C22×C4C4×D9711yescharacteristic, complemented
220C4⋊C4—1944no—
221C22⋊C4—22no—
222C22⋊C4—974no—
223C22⋊C4—1944no—
224C22⋊C4—1942no—
225C22⋊C4C22×D9714yes—
226C42—1944no—
227C23—21no—
228C23—21no—
229C23—81no—
230C23—81no—
231C23—7761no—
232C23—7762no—
233C23—7761no—
234C23—972no—
235C23—974no—
236C23—972no—
237C23—1944no—
238C23—1942no—
239C23C2×C4×D9712yes—
240C23C2×C4×D9711yescharacteristic
241C23C2×C4×D9711yescharacteristic
242C23D4×D9712yes—
243C23D4×D9712yes—
244C23C23×D9711yescharacteristic
245Q8—1944no—
246D4—24no—
247D4—24no—
248D4—974no—
249D4—974no—
250D4—978no—
251D4—1944no—
252D4—1944no—
253D4—1944no—
254D4—1942no—
255D4—1942no—
256D4—1944no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—24no—
260C2×C4—82no—
261C2×C4—7764no—
262C2×C4—7762no—
263C2×C4—972no—
264C2×C4—974no—
265C2×C4—972no—
266C2×C4—1944no—
267C2×C4—1944no—
268C2×C4—1944no—
269C2×C4—1944no—
270C2×C4—1942no—
271C2×C4C2×C4×D9712yes—
272C22—22no—
273C22—42no—
274C22—42no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—81no—
281C22—7762no—
282C22—7764no—
283C22—974no—
284C22—3882no—
285C22—3884no—
286C22—3882no—
287C22—1942no—
288C22C22⋊C4×D9711yescharacteristic
289C22C22⋊C4×D9711yescharacteristic
290C22C22⋊C4×D9712yes—
291C22C22×C4×D9711yescharacteristic
292C22C2×D4×D9712yes—
293C4—22no—
294C4—44no—
295C4—974no—
296C4—3884no—
297C4—1942no—
298C2—22no—
299C2—21no—
300C2—21no—
301C2—21no—
302C2—22no—
303C2—42no—
304C2—3882no—
305C2—1944no—
306C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D97
commutatorC22×C97C23×C4
FrattiniC23C23×D97
FittingC2×#99×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690855C2
cosocleC23×C97C24
hypercenterC2×#99D97
hypocenterC97#2883
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. #20261208C2semidirect
3. #20260707C2semidirect
4. C2×#15690725C2semidirect
5. C2×#15690724C2semidirect
6. #99×D97C2semidirect
7. C2×#99×C97C2semidirect
10. C2×#13095511C4semidirect
11. C2×#13095510C4semidirect
13. C22×C22⋊D97C4semidirect
16. C2×D4×D97C4semidirect
20. #15690766C22semidirect
21. C2×C22⋊Dic97C22semidirect
22. C22⋊C4×D97C22semidirect
23. #15690725C22semidirect
24. #15690724C22semidirect
26. C2×C22⋊C4×C97C22semidirect
27. #99×C97C22semidirect
28. C23×D97C2×C4semidirect
29. C23×D97C2×C4semidirect
35. C2×Dic194C2×C4semidirect
36. C2×D388C2×C4semidirect
43. C2×C22⋊D97C2×C4semidirect
44. C2×C22⋊D97C2×C4semidirect
45. C2×C22⋊D97C2×C4semidirect
52. C22×Dic97C2×C4semidirect
53. C22×Dic97C2×C4semidirect
56. C2×C4×D97C2×C4semidirect
57. C2×C4×D97C2×C4semidirect
63. C24×C97C2×C4semidirect
67. C2×D4×C97C2×C4semidirect
70. C22×C4×C97C2×C4semidirect
154. C97#2883semidirect
161. C2×#99D97semidirect
178. C2×C22⋊C4C2×D97semidirect
181. #99C2×D97semidirect
185. C24C4×D97semidirect
209. C2×D4C4×D97semidirect
219. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908552
#202612082
#202607072
C2×#156907252
C2×#156907242
#99×D972
C2×#99×C972
#288397