Finite Group Databasebeta

← #27150990

Subgroups — #27150990

Order 12416.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13095511C211yesmaximal, characteristic, complemented
2C2×#15690838C212yesmaximal, complemented
3C2×#15690784C211yesmaximal, characteristic, complemented
4#20260882C214yesmaximal, complemented
5#20260876C212yesmaximal, complemented
6#20260871C212yesmaximal, complemented
7#20260737C212yesmaximal, complemented
8#407×C97C211yesmaximal, characteristic, complemented
9C2×#13095511—28no—
10C2×#13095511—21no—
11C2×#13095511—21no—
12C2×#13095511C2212yes—
13C2×#13095511C2212yescomplemented
14C22×Dic194C2211yescharacteristic, complemented
15C22×D388C2211yescharacteristic, complemented
16C22×D388C2211yescharacteristic, complemented
17C22×C22⋊D97—21no—
18C22×C22⋊D97—21no—
19C22×C4×D97—21no—
20#15690838—28no—
21C2×#13095500C2214yescomplemented
22C2×D776C2214yescomplemented
23#15690784—21no—
24#15690784—21no—
25#15690784C2212yes—
26#15690784C2212yescomplemented
27C2×C4⋊Dic97C2211yescharacteristic, complemented
28C2×#13095488—21no—
29#15690735C2214yescomplemented
30#15690732C2214yescomplemented
31C23×C4×C97C2211yescharacteristic
32C2×M16×C97C2212yescomplemented
33C22⋊C8×C97C2212yescomplemented
34C22⋊C8×C97C2212yescomplemented
35C23×D97—21no—
36C23×D97—21no—
37#13095511—28no—
38#13095511—28no—
39#13095511—24no—
40#13095511—24no—
41#13095511—24no—
42#13095511D418yescomplemented
43C2×Dic194—21no—
44C2×Dic194—24no—
45C2×Dic194C2312yes—
46C2×D388—21no—
47C2×D388—24no—
48C2×D388—24no—
49C2×D388—21no—
50C2×D388C2312yes—
51C2×D388C2312yes—
52C2×C22⋊D97—28no—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—22no—
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C2×C22⋊D97—44no—
58C22×Dic97—21no—
59C22×Dic97—21no—
60C2×C4×D97—28no—
61C2×C4×D97—22no—
62C2×C4×D97—22no—
63#13095500—28no—
64D776—28no—
65C4⋊Dic97—21no—
66C4⋊Dic97C2312yes—
67#13095488—22no—
68#13095488—22no—
69C24×C97D411yescharacteristic
70C22×C4×C97—24no—
71C22×C4×C97D411yescharacteristic
72C22×C4×C97D411yescharacteristic
73C22×C4×C97D411yescharacteristic
74C22×C4×C97C2311yescharacteristic
75C22×C4×C97C2311yescharacteristic
76C22×C4×C97C2311yescharacteristic
77M16×C97—24no—
78C2×C8×C97C2314yes—
79C23×C97—22no—
80C23×C97—22no—
81C23×C97—41no—
82C23×C97C2×D411yescharacteristic
83C23×C97C2×D411yescharacteristic
84C23×C97C2×D411yescharacteristic
85C22×D97—24no—
86C22×D97—22no—
87C22×D97—22no—
88C22×D97—42no—
89C22×D97—42no—
90C22×D97—42no—
91C22×D97—42no—
92C22×D97—42no—
93C2×C4×C97—22no—
94C2×C4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97—22no—
98C2×C4×C97C2×D414yes—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97C2×D411yescharacteristic
101C2×C4×C97C2×D411yescharacteristic
102C2×C4×C97C2411yescharacteristic
103C22⋊D97—28no—
104C22⋊D97—28no—
105C22⋊D97—44no—
106C22⋊D97—44no—
107C2×Dic97—24no—
108C2×Dic97—22no—
109C2×Dic97—22no—
110C2×Dic97—42no—
111D388—24no—
112D388—22no—
113D388—24no—
114D388—42no—
115D388C2×D414yescomplemented
116C4×D97—28no—
117C4×D97—28no—
118Dic194—24no—
119Dic194—22no—
120Dic194C2×D414yescomplemented
121C8×C97—24no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—21no—
125C22×C97—21no—
126C22×C97—21no—
127C22×C97—21no—
128C22×C97—21no—
129C22×C97—44no—
130C22×C97#12012yes—
131C22×C97AGL(1,ℤ/8)12yescomplemented
132C22×C97C22×D411yescharacteristic
133C2×D97—24no—
134C2×D97—48no—
135C2×D97—42no—
136C2×D97—42no—
137C2×D97—42no—
138C2×D97—42no—
139C2×D97—42no—
140C2×D97—84no—
141C4×C97—21no—
142C4×C97—21no—
143C4×C97#12012yes—
144C4×C97C22×D412yes—
145Dic97—24no—
146Dic97—42no—
147C2×C97—22no—
148C2×C97—22no—
149C2×C97—41no—
150C2×C97C2×#12011yescharacteristic
151C2×C97C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C97#57511yescharacteristic
153D97—44no—
154D97—82no—
155#3006—971nomaximal, Sylow
156C97#300611yescharacteristic, complemented, Sylow
157C22×#41—971no—
158C2×AGL(1,ℤ/8)—972no—
159C2×#121—971no—
160#449—974no—
161#448—972no—
162#447—972no—
163#417—972no—
164#407D9711yescharacteristic, complemented
165C2×#41—972no—
166C2×#41—972no—
167C2×#41—1948no—
168C2×#41—1941no—
169C2×#41—1941no—
170C22×Q8—971no—
171C22×D4—971no—
172C22×D4—971no—
173C22×D4—1941no—
174C22×D4—1941no—
175C23×C4—1941no—
176C23×C4C2×D9711yescharacteristic
177AGL(1,ℤ/8)—1948no—
178C2×SD16—974no—
179C2×D8—974no—
180C2×M16C2×D9712yescomplemented
181#121—972no—
182#121—972no—
183#121—1941no—
184#121—1941no—
185C2×C4⋊C4—971no—
186C2×C22⋊C4—1941no—
187#103—974no—
188#102—974no—
189C22⋊C8C2×D9712yescomplemented
190C22⋊C8C2×D9712yescomplemented
191C24—1941no—
192C24—1941no—
193C24D38811yescharacteristic
194#41—978no—
195#41—1944no—
196#41—1944no—
197#41—1948no—
198#41—1948no—
199#41—1944no—
200C2×Q8—972no—
201C2×Q8—1944no—
202C2×Q8—1941no—
203C2×D4—972no—
204C2×D4—972no—
205C2×D4—3884no—
206C2×D4—1944no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4—1948no—
210C2×D4—1942no—
211C2×D4—1942no—
212C2×D4—1944no—
213C2×D4—1941no—
214C2×D4—1941no—
215C22×C4—24no—
216C22×C4—1948no—
217C22×C4—1941no—
218C22×C4—1941no—
219C22×C4—1942no—
220C22×C4—1942no—
221C22×C4D38811yescharacteristic
222C22×C4D38811yescharacteristic
223C22×C4D38811yescharacteristic
224C22×C4C22×D9711yescharacteristic
225C22×C4C22×D9711yescharacteristic
226C22×C4C22×D9711yescharacteristic
227SD16—1948no—
228D8—1948no—
229M16—24no—
230C2×C8C22×D9714yes—
231C4⋊C4—972no—
232C4⋊C4—1941no—
233C22⋊C4—1942no—
234C22⋊C4—1942no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—3882no—
239C23—3882no—
240C23—3882no—
241C23—3882no—
242C23—3882no—
243C23—1944no—
244C23—1942no—
245C23—1942no—
246C23C2×D38811yescharacteristic
247C23C2×D38811yescharacteristic
248C23C2×D38811yescharacteristic
249Q8—974no—
250Q8—1944no—
251Q8—1942no—
252D4—974no—
253D4—3884no—
254D4—3884no—
255D4—3882no—
256D4—1944no—
257D4—1948no—
258D4—1948no—
259D4—1944no—
260D4—1942no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—3882no—
267C2×C4—1948no—
268C2×C4—1942no—
269C2×C4—1944no—
270C2×C4—1942no—
271C2×C4—1948no—
272C2×C4D4×D9714yes—
273C2×C4C2×D38811yescharacteristic
274C2×C4C2×D38811yescharacteristic
275C2×C4C2×D38811yescharacteristic
276C2×C4C23×D9711yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—7764no—
287C22—3888no—
288C22—3882no—
289C22—3882no—
290C22—3882no—
291C22—3882no—
292C22—3882no—
293C22—1944no—
294C22#1569078012yes—
295C22#1569083812yescomplemented
296C22C22×D38811yescharacteristic
297C4—21no—
298C4—21no—
299C4—3882no—
300C4—1944no—
301C4#1569078012yes—
302C4C2×D4×D9712yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—7762no—
307C2—3884no—
308C2C2×#1569078011yescharacteristic
309C2C2×#1569083811yescharacteristic
310C2#2026153511yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#407×C97C2
socleC22×C97C22×D4
hypercenter#407D97
hypocenterC97#3006
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C2×#15690838C2semidirect
3. C2×#15690784C2semidirect
4. #20260882C2semidirect
5. #20260876C2semidirect
6. #20260871C2semidirect
7. #20260737C2semidirect
8. #407×C97C2semidirect
13. C2×#13095511C22semidirect
14. C22×Dic194C22semidirect
15. C22×D388C22semidirect
16. C22×D388C22semidirect
21. C2×#13095500C22semidirect
22. C2×D776C22semidirect
26. #15690784C22semidirect
27. C2×C4⋊Dic97C22semidirect
29. #15690735C22semidirect
30. #15690732C22semidirect
32. C2×M16×C97C22semidirect
33. C22⋊C8×C97C22semidirect
34. C22⋊C8×C97C22semidirect
42. #13095511D4semidirect
115. D388C2×D4semidirect
120. Dic194C2×D4semidirect
131. C22×C97AGL(1,ℤ/8)semidirect
156. C97#3006semidirect
164. #407D97semidirect
180. C2×M16C2×D97semidirect
189. C22⋊C8C2×D97semidirect
190. C22⋊C8C2×D97semidirect
295. C22#15690838semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156908382
C2×#156907842
#202608822
#202608762
#202608712
#202607372
#407×C972
#300697