Finite Group Databasebeta

← #27151012

Subgroups — #27151012

Order 12416.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690856C211yesmaximal, characteristic
2C22×#13095498C211yesmaximal, characteristic, complemented
3#20261309C211yesmaximal, characteristic, complemented
4#20260896C214yesmaximal, complemented
5#20260884C214yesmaximal, complemented
6#20260741C211yesmaximal, characteristic, complemented
7#20260731C211yesmaximal, characteristic, complemented
8#20260703C211yesmaximal, characteristic, complemented
9C2×#137×C97C211yesmaximal, characteristic, complemented
10#15690856—24no—
11#15690856—24no—
12C2×#13095512C2212yes—
13C2×#13095511—21no—
14C2×#13095511C2211yescharacteristic
15C22×D388—21no—
16C22×D388C2211yescharacteristic
17C2×D4×D97—22no—
18C2×D4×D97C2212yes—
19C2×#13095498—24no—
20C2×#13095498C2214yescomplemented
21C22×#11715895C2211yescharacteristic, complemented
22#15690808—21no—
23#15690799—21no—
24#15690793—22no—
25#15690786C2214yescomplemented
26#15690769C2211yescharacteristic, complemented
27#15690734C2212yescomplemented
28#15690732C2214yescomplemented
29#15690731C2212yescomplemented
30#15690722C2214yescomplemented
31C2×#41×C97C2211yescharacteristic
32C22×Q8×C97C2211yescharacteristic, complemented
33#137×C97—24no—
34C2×Q16×C97C2212yescomplemented
35C2×SD16×C97C2212yescomplemented
36C2×M16×C97C2211yescharacteristic, complemented
37C23×D97—21no—
38C23×D97—22no—
39#13095512—24no—
40#13095512—24no—
41#13095511—28no—
42#13095511—22no—
43#13095511—22no—
44C2×Dic194D411yescharacteristic
45C2×D388—28no—
46C2×D388—22no—
47C2×D388—21no—
48C2×D388—22no—
49C2×D388D411yescharacteristic
50C2×D388D414yes—
51C2×D388C2312yes—
52C2×C22⋊D97—22no—
53C2×C22⋊D97D412yes—
54D4×D97—216no—
55D4×D97—24no—
56D4×D97—24no—
57C2×C4×D97—22no—
58C2×C4×D97C2312yes—
59#13095498—28no—
60C2×#11715895—22no—
61C2×#11715895C2312yes—
62C4⋊Dic97C2312yes—
63#13095490—24no—
64C22⋊Dic97—21no—
65#13095488—24no—
66C4×Dic97—21no—
67#41×C97—22no—
68#41×C97D414yes—
69C2×Q8×C97—21no—
70C2×Q8×C97—22no—
71C2×Q8×C97—22no—
72C2×Q8×C97C2312yes—
73C2×Q8×C97C2311yescharacteristic
74C2×D4×C97—21no—
75C2×D4×C97C2311yescharacteristic
76C22×C4×C97—21no—
77C22×C4×C97—21no—
78C22×C4×C97C2311yescharacteristic
79Q16×C97—24no—
80SD16×C97—24no—
81M16×C97—22no—
82C2×C8×C97C2312yes—
83C23×C97—21no—
84C23×C97C2×D411yescharacteristic
85C22×D97—22no—
86C22×D97—28no—
87C22×D97—24no—
88C22×D97—84no—
89C22×D97—82no—
90C22×D97C2×D412yes—
91Q8×C97—21no—
92Q8×C97—22no—
93Q8×C97—22no—
94Q8×C97—42no—
95Q8×C97C2×D412yes—
96D4×C97—24no—
97D4×C97—21no—
98D4×C97C2×D412yes—
99C2×C4×C97—21no—
100C2×C4×C97—24no—
101C2×C4×C97—22no—
102C2×C4×C97—42no—
103C2×C4×C97C2×D411yescharacteristic
104C2×C4×C97C2×D412yes—
105C2×C4×C97C2×D412yes—
106C2×C4×C97C2411yescharacteristic
107C22⋊D97—28no—
108C22⋊D97—24no—
109C2×Dic97—22no—
110C2×Dic97C2×D412yes—
111D388—28no—
112D388—22no—
113D388—24no—
114D388—22no—
115D388C2×D414yes—
116C4×D97—28no—
117C4×D97—22no—
118C4×D97—22no—
119Dic194—22no—
120C8×C97—22no—
121#11715895—22no—
122C22×C97—22no—
123C22×C97—22no—
124C22×C97—42no—
125C22×C97#12012yes—
126C22×C97C22×D411yescharacteristic
127C2×D97—22no—
128C2×D97—24no—
129C2×D97—24no—
130C2×D97—44no—
131C2×D97—84no—
132C2×D97—88no—
133C4×C97—22no—
134C4×C97—42no—
135C4×C97#12012yes—
136C4×C97C22×D411yescharacteristic
137C4×C97C22×D411yescharacteristic
138Dic97—22no—
139Dic97—42no—
140C2×C97—22no—
141C2×C97—42no—
142C2×C97C2×#12011yescharacteristic
143C2×C97#57712yes—
144D97—42no—
145D97—44no—
146#3009—971nomaximal, Sylow
147C97#300911yescharacteristic, complemented, Sylow
148C2×H(5,𝔽2)—971no—
149C2×#137D9711yescharacteristic, complemented
150C22×SD16—971no—
151#536—971no—
152#452—974no—
153#450—974no—
154#418—971no—
155#416—971no—
156#408—971no—
157H(5,𝔽2)—1944no—
158H(5,𝔽2)—1944no—
159C2×#41—971no—
160C2×#41—972no—
161C2×#41—1941no—
162C2×#41C2×D9711yescharacteristic
163C22×Q8C2×D9711yescharacteristic, complemented
164C22×D4—972no—
165C22×D4—971no—
166C22×D4—1941no—
167C22×D4—1942no—
168#137—24no—
169C2×Q16C2×D9712yescomplemented
170C2×SD16—974no—
171C2×SD16—1944no—
172C2×SD16C2×D9712yescomplemented
173C2×M16C2×D9711yescharacteristic, complemented
174C22×C8—971no—
175C4⋊Q8—1941no—
176#124—1941no—
177#123—1942no—
178C22⋊Q8—974no—
179#117—971no—
180#103—972no—
181#102—972no—
182#102—974no—
183C22⋊C8—974no—
184C24—1942no—
185C24—1941no—
186#41—22no—
187#41—1948no—
188#41—1944no—
189#41—1944no—
190#41—1942no—
191#41—1942no—
192#41C22⋊D9714yes—
193C2×Q8—21no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—971no—
197C2×Q8C22×D9712yes—
198C2×Q8C22×D9711yescharacteristic
199C2×D4—21no—
200C2×D4—972no—
201C2×D4—974no—
202C2×D4—971no—
203C2×D4—972no—
204C2×D4—1942no—
205C2×D4—1948no—
206C2×D4—1942no—
207C2×D4—1944no—
208C2×D4—19416no—
209C2×D4—1944no—
210C2×D4—1941no—
211C2×D4—1942no—
212C2×D4C22×D9711yescharacteristic
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—972no—
216C22×C4—1942no—
217C22×C4C22×D9711yescharacteristic
218Q16—24no—
219SD16—24no—
220SD16—1948no—
221M16—22no—
222C2×C8—972no—
223C2×C8—1942no—
224C2×C8C22×D9712yes—
225C4⋊C4—972no—
226C4⋊C4—1944no—
227C22⋊C4—1944no—
228C22⋊C4—1941no—
229C42—1941no—
230C23—21no—
231C23—7762no—
232C23—7764no—
233C23—972no—
234C23—1948no—
235C23—1944no—
236C23—1942no—
237C23D4×D9711yescharacteristic
238Q8—21no—
239Q8—22no—
240Q8—22no—
241Q8—42no—
242Q8—1942no—
243Q8C2×C22⋊D9712yes—
244D4—24no—
245D4—21no—
246D4—974no—
247D4—1948no—
248D4—1948no—
249D4—1944no—
250D4—1944no—
251D4—1942no—
252D4—1942no—
253D4C2×C22⋊D9712yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—972no—
259C2×C4—1948no—
260C2×C4—1942no—
261C2×C4—1942no—
262C2×C4—1942no—
263C2×C4D4×D9711yescharacteristic
264C2×C4D4×D9712yes—
265C2×C4C2×C22⋊D9712yes—
266C2×C4C23×D9711yescharacteristic
267C8—22no—
268C8—1942no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—7764no—
273C22—7768no—
274C22—3884no—
275C22—1944no—
276C22—1942no—
277C22—1944no—
278C22#1569077812yes—
279C22C2×D4×D9711yescharacteristic
280C4—22no—
281C4—42no—
282C4—3882no—
283C4—1942no—
284C4#1569077812yes—
285C4C2×D4×D9711yescharacteristic
286C4C22×C22⋊D9711yescharacteristic
287C2—22no—
288C2—42no—
289C2—3884no—
290C2—3882no—
291C2C2×#1569077811yescharacteristic
292C2#2026154412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
prime radicalC2×#15690856C2
hypercenterC2×#137D97
hypocenterC97#3009
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C22×#13095498C2semidirect
3. #20261309C2semidirect
4. #20260896C2semidirect
5. #20260884C2semidirect
6. #20260741C2semidirect
7. #20260731C2semidirect
8. #20260703C2semidirect
9. C2×#137×C97C2semidirect
20. C2×#13095498C22semidirect
21. C22×#11715895C22semidirect
25. #15690786C22semidirect
26. #15690769C22semidirect
27. #15690734C22semidirect
28. #15690732C22semidirect
29. #15690731C22semidirect
30. #15690722C22semidirect
32. C22×Q8×C97C22semidirect
34. C2×Q16×C97C22semidirect
35. C2×SD16×C97C22semidirect
36. C2×M16×C97C22semidirect
147. C97#3009semidirect
149. C2×#137D97semidirect
163. C22×Q8C2×D97semidirect
169. C2×Q16C2×D97semidirect
172. C2×SD16C2×D97semidirect
173. C2×M16C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908562
C22×#130954982
#202613092
#202608962
#202608842
#202607412
#202607312
#202607032
C2×#137×C972
#300997