Finite Group Databasebeta

← #27151658

Subgroups — #27151658

Order 12416.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#20261245C211yesmaximal, characteristic
2C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
3#20261071C211yesmaximal, characteristic, complemented
4#20261070C211yesmaximal, characteristic, complemented
5#20261067C212yesmaximal, complemented
6#20261051C211yesmaximal, characteristic, complemented
7#20261047C212yesmaximal, complemented
8#20260774C212yesmaximal, complemented
9#20260733C212yesmaximal, complemented
10#20260697C211yesmaximal, characteristic, complemented
11#481×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2211yescharacteristic
13C22×C4×D97—21no—
14C22×C4×D97C2211yescharacteristic
15C2×#13095494C2212yes—
16#15690792—21no—
17#15690790C2211yescharacteristic
18#15690789C2211yescharacteristic
19#15690781C2211yescharacteristic
20#15690778—21no—
21D4×Dic97—21no—
22#15690770—21no—
23#15690768C2211yescharacteristic
24C2×C4⋊Dic97C2211yescharacteristic
25C2×#13095490—21no—
26C4⋊C4×D97—21no—
27C4⋊C4×D97—21no—
28C4⋊C4×D97C2212yes—
29C4⋊C4×D97C2212yes—
30#15690744C2212yescomplemented
31#15690743C2212yescomplemented
32#15690741C2212yescomplemented
33#15690735C2212yescomplemented
34#15690733C2212yescomplemented
35#15690731C2212yescomplemented
36C22⋊#11715895C2211yescharacteristic
37#15690722C2212yes—
38#121×C97C2211yescharacteristic
39C2×C4⋊C4×C97C2211yescharacteristic
40#106×C97C2212yescomplemented
41#102×C97C2212yescomplemented
42C22⋊C8×C97C2211yescharacteristic
43C23×D97D411yescharacteristic
44#13095510—24no—
45C2×Dic194C2311yescharacteristic
46C2×C22⋊D97—21no—
47C2×C22⋊D97—21no—
48C22×Dic97—21no—
49C22×Dic97—21no—
50C22×Dic97D411yescharacteristic
51C2×C4×D97—24no—
52C2×C4×D97—22no—
53C2×C4×D97—22no—
54C2×C4×D97—44no—
55C2×C4×D97D411yescharacteristic
56C2×C4×D97D411yescharacteristic
57C2×C4×D97C2311yescharacteristic
58C2×C4×D97C2311yescharacteristic
59#13095494—24no—
60C2×#11715895C2312yes—
61C4⋊Dic97—21no—
62C4⋊Dic97—21no—
63C4⋊Dic97C2312yes—
64C4⋊Dic97C2311yescharacteristic
65#13095490—21no—
66#13095490—22no—
67#13095490—22no—
68C22⋊Dic97—21no—
69C22⋊Dic97—21no—
70#13095488—21no—
71#13095488—21no—
72C4×Dic97—21no—
73C2×D4×C97—21no—
74C2×D4×C97C2311yescharacteristic
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77C2×C8×C97C2312yes—
78C4⋊C4×C97—21no—
79C4⋊C4×C97C2312yes—
80C4⋊C4×C97C2311yescharacteristic
81C22⋊C4×C97—21no—
82C23×C97—21no—
83C23×C97C2×D411yescharacteristic
84C22×D97—22no—
85C22×D97—22no—
86C22×D97—22no—
87C22×D97C2×D411yescharacteristic
88C22×D97C2×D411yescharacteristic
89D4×C97—22no—
90D4×C97—42no—
91C2×C4×C97—22no—
92C2×C4×C97—21no—
93C2×C4×C97—22no—
94C2×C4×C97—42no—
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2411yescharacteristic
97C22⋊D97—42no—
98C22⋊D97—42no—
99C2×Dic97—22no—
100C2×Dic97—21no—
101C2×Dic97—21no—
102C2×Dic97—22no—
103C2×Dic97—42no—
104C2×Dic97—42no—
105C2×Dic97C2×D411yescharacteristic
106C2×Dic97C2×D411yescharacteristic
107C4×D97—22no—
108C4×D97—22no—
109C4×D97—22no—
110C4×D97—44no—
111C4×D97—44no—
112C4×D97#4114yes—
113Dic194—22no—
114C8×C97—22no—
115#11715895—22no—
116C22×C97—21no—
117C22×C97—21no—
118C22×C97—41no—
119C22×C97—41no—
120C22×C97—41no—
121C22×C97#13712yes—
122C22×C97C22×D411yescharacteristic
123C2×D97—28no—
124C2×D97—21no—
125C2×D97—21no—
126C2×D97—21no—
127C2×D97—21no—
128C2×D97#12312yes—
129C2×D97AGL(1,ℤ/8)12yes—
130C4×C97—21no—
131C4×C97—42no—
132C4×C97—41no—
133C4×C97C2×#4112yes—
134Dic97—21no—
135Dic97—41no—
136Dic97—42no—
137Dic97—41no—
138Dic97#12312yes—
139C2×C97—22no—
140C2×C97—81no—
141C2×C97C2×#12311yescharacteristic
142C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C97C2×#13711yescharacteristic
144D97—22no—
145D97—22no—
146#3092—971nomaximal, Sylow
147C97#309211yescharacteristic, complemented, Sylow
148#525—971no—
149C22×C4⋊C4—971no—
150#484—971no—
151#484—971no—
152#483—972no—
153#481—971no—
154#481D9711yescharacteristic, complemented
155#480—972no—
156#428—972no—
157#417—972no—
158#407—971no—
159C2×#41—971no—
160C23×C4—971no—
161C23×C4—1941no—
162C2×M16—972no—
163#123—1941no—
164C22⋊Q8—971no—
165C22⋊Q8—971no—
166#121—971no—
167#121C2×D9711yescharacteristic
168#120—1941no—
169C4×D4—1941no—
170C4×D4—1941no—
171#117—971no—
172C2×C4⋊C4—971no—
173C2×C4⋊C4—972no—
174C2×C4⋊C4—972no—
175C2×C4⋊C4—1941no—
176C2×C4⋊C4—1941no—
177C2×C4⋊C4—1941no—
178C2×C4⋊C4C2×D9711yescharacteristic
179#107—972no—
180#107—972no—
181#106—972no—
182#106C2×D9712yescomplemented
183#103—972no—
184#103—972no—
185#102—972no—
186#102C2×D9712yescomplemented
187C22⋊C8—971no—
188C22⋊C8—972no—
189C22⋊C8C2×D9711yescharacteristic
190C24—971no—
191#41—1944no—
192C2×Q8—971no—
193C2×D4—21no—
194C2×D4—1941no—
195C2×D4—1941no—
196C2×D4C22×D9711yescharacteristic
197C22×C4—21no—
198C22×C4—971no—
199C22×C4—971no—
200C22×C4—971no—
201C22×C4—971no—
202C22×C4—971no—
203C22×C4—3884no—
204C22×C4—1944no—
205C22×C4—1942no—
206C22×C4—1942no—
207C22×C4—1941no—
208C22×C4—1941no—
209C22×C4C22×D9711yescharacteristic
210M16—1944no—
211C2×C8—972no—
212C2×C8C22×D9712yes—
213C4⋊C4—21no—
214C4⋊C4—972no—
215C4⋊C4—971no—
216C4⋊C4—1942no—
217C4⋊C4—1942no—
218C4⋊C4—1941no—
219C4⋊C4—1941no—
220C4⋊C4—1941no—
221C4⋊C4C22×D9712yes—
222C4⋊C4C22×D9711yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—1941no—
225C22⋊C4—1941no—
226C22⋊C4—1941no—
227C22⋊C4—1941no—
228C42—1941no—
229C23—21no—
230C23—971no—
231C23—971no—
232C23—1942no—
233C23—1942no—
234C23—1942no—
235C23D4×D9711yescharacteristic
236Q8—1942no—
237D4—22no—
238D4—42no—
239D4—3882no—
240D4—3882no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—971no—
246C2×C4—974no—
247C2×C4—971no—
248C2×C4—3884no—
249C2×C4—3884no—
250C2×C4—3882no—
251C2×C4—3882no—
252C2×C4—1942no—
253C2×C4—1942no—
254C2×C4—1942no—
255C2×C4—1941no—
256C2×C4—1941no—
257C2×C4—1942no—
258C2×C4—1942no—
259C2×C4D4×D9711yescharacteristic
260C2×C4C23×D9711yescharacteristic
261C8—22no—
262C8—1942no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—972no—
269C22—972no—
270C22—1948no—
271C22—1941no—
272C22—1941no—
273C22—1941no—
274C22—1941no—
275C22#1569083912yes—
276C22C2×D4×D9711yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—972no—
281C4—3881no—
282C4—3882no—
283C4—3881no—
284C4—1941no—
285C4#41×D9712yes—
286C2—22no—
287C2—81no—
288C2—1942no—
289C2—1942no—
290C2#123×D9711yescharacteristic
291C2AGL(1,ℤ/8)×D9711yescharacteristic
292C2C2×#1569083911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#481×C97C2
socleC22×C97C22×D4
prime radical#20261245C2
hypercenter#481D97
hypocenterC97#3092
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D97C2semidirect
3. #20261071C2semidirect
4. #20261070C2semidirect
5. #20261067C2semidirect
6. #20261051C2semidirect
7. #20261047C2semidirect
8. #20260774C2semidirect
9. #20260733C2semidirect
10. #20260697C2semidirect
11. #481×C97C2semidirect
30. #15690744C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690735C22semidirect
34. #15690733C22semidirect
35. #15690731C22semidirect
40. #106×C97C22semidirect
41. #102×C97C22semidirect
147. C97#3092semidirect
154. #481D97semidirect
182. #106C2×D97semidirect
186. #102C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612452
C2×C4⋊C4×D972
#202610712
#202610702
#202610672
#202610512
#202610472
#202607742
#202607332
#202606972
#481×C972
#309297