Finite Group Databasebeta

← #27151416

Subgroups — #27151416

Order 12416.

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

normal subgroups only

This group has 504 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×#15690838C211yesmaximal, characteristic, complemented
2#20261313C211yesmaximal, characteristic, complemented
3#20261299C211yesmaximal, characteristic, complemented
4#20260988C212yesmaximal, complemented
5#20260981C212yesmaximal, complemented
6#20260971C212yesmaximal, complemented
7#20260964C212yesmaximal, complemented
8#20260777C211yesmaximal, characteristic, complemented
9#20260732C211yesmaximal, characteristic, complemented
10C2×#15690732C211yesmaximal, characteristic, complemented
11C2×#131×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C22×D388C2211yescharacteristic
14#15690838—24no—
15C2×#13095500C2212yes—
16C2×D776C2212yes—
17#15690806—21no—
18#15690799—21no—
19#15690789C2212yescomplemented
20#15690786—21no—
21#15690784C2212yes—
22#15690784C2212yes—
23#15690784C2212yescomplemented
24#15690782—22no—
25#15690781—21no—
26#15690778—22no—
27#15690769C2211yescharacteristic
28C2×C4⋊Dic97C2211yescharacteristic
29#15690744C2212yescomplemented
30#15690742C2212yescomplemented
31#15690735C2211yescharacteristic, complemented
32#15690735C2211yescharacteristic, complemented
33#15690732C2212yes—
34#15690732C2212yes—
35#15690732C2211yescharacteristic, complemented
36#15690732C2211yescharacteristic, complemented
37C2×#41×C97C2211yescharacteristic
38#131×C97—24no—
39C2×M16×C97C2211yescharacteristic
40C2×M16×C97C2212yescomplemented
41C22×C8×C97C2211yescharacteristic
42C22×C8×C97C2212yescomplemented
43C23×D97—21no—
44#13095511—22no—
45#13095511—22no—
46C2×Dic194C2311yescharacteristic
47C2×D388—22no—
48C2×D388—22no—
49C2×D388—21no—
50C2×D388C2312yes—
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—21no—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—22no—
55C2×C22⋊D97—22no—
56C22×Dic97—21no—
57C2×C4×D97—21no—
58#13095500—24no—
59D776—24no—
60C4⋊Dic97—21no—
61C4⋊Dic97C2312yes—
62C4⋊Dic97C2311yescharacteristic
63C4⋊Dic97C2311yescharacteristic
64#13095490—22no—
65C22⋊Dic97—21no—
66C22⋊Dic97—22no—
67#13095488—22no—
68#13095488—22no—
69#13095488—22no—
70C4×Dic97—21no—
71#41×C97—22no—
72#41×C97—22no—
73C2×Q8×C97D411yescharacteristic
74C2×D4×C97D412yes—
75C2×D4×C97D411yescharacteristic
76C22×C4×C97C2312yes—
77C22×C4×C97C2311yescharacteristic
78M16×C97—24no—
79M16×C97D414yescomplemented
80C2×C8×C97—22no—
81C2×C8×C97—24no—
82C2×C8×C97C2311yescharacteristic
83C2×C8×C97C2311yescharacteristic
84C2×C8×C97C2312yes—
85C23×C97C2×D412yes—
86C23×C97C2×D411yescharacteristic
87C22×D97—21no—
88C22×D97—22no—
89C22×D97—42no—
90C22×D97—81no—
91C22×D97—81no—
92Q8×C97—22no—
93D4×C97—24no—
94D4×C97—22no—
95C2×C4×C97—22no—
96C2×C4×C97—22no—
97C2×C4×C97C2×D412yes—
98C2×C4×C97C2×D412yes—
99C2×C4×C97C2×D411yescharacteristic
100C2×C4×C97#4112yes—
101C2×C4×C97C2411yescharacteristic
102C22⋊D97—42no—
103C22⋊D97—82no—
104C22⋊D97—82no—
105C22⋊D97—82no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108C2×Dic97—22no—
109C2×Dic97—21no—
110C2×Dic97—81no—
111D388—21no—
112D388—22no—
113D388—22no—
114D388—21no—
115D388—42no—
116C4×D97—42no—
117Dic194—21no—
118Dic194—21no—
119C8×C97—21no—
120C8×C97—21no—
121C8×C97C2×D414yes—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—22no—
125C22×C97—22no—
126C22×C97—22no—
127C22×C97#12112yes—
128C22×C97C22×D411yescharacteristic
129C2×D97—41no—
130C2×D97—42no—
131C2×D97—41no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—41no—
135C2×D97—84no—
136C4×C97—22no—
137C4×C97#12112yes—
138C4×C97C22×D411yescharacteristic
139C4×C97C2×#4111yescharacteristic
140Dic97—41no—
141Dic97—42no—
142Dic97—41no—
143Dic97—41no—
144C2×C97—22no—
145C2×C97—42no—
146C2×C97C2×#12111yescharacteristic
147C2×C97#57611yescharacteristic
148C2×C97#57711yescharacteristic
149D97—81no—
150D97—82no—
151#3062—971nomaximal, Sylow
152C97#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—971no—
154C2×#131D9711yescharacteristic, complemented
155#537—971no—
156#535—971no—
157#469—972no—
158#468—972no—
159#466—972no—
160#465—972no—
161#428—971no—
162#416—971no—
163C2×#102—971no—
164C2×#41—971no—
165C2×#41C2×D9711yescharacteristic
166C22×D4—971no—
167AGL(1,ℤ/8)—1944no—
168C2×SD16—972no—
169C2×D8—972no—
170#131—24no—
171C2×M16C2×D9711yescharacteristic
172C2×M16C2×D9712yescomplemented
173C22×C8C2×D9711yescharacteristic
174C22×C8C2×D9712yescomplemented
175#127—1941no—
176#124—1941no—
177C22⋊Q8—972no—
178C22⋊Q8—1941no—
179#121—972no—
180#121—972no—
181#121—972no—
182#121—1941no—
183#121—1942no—
184#120—1942no—
185#117—971no—
186C2×C4⋊C4—971no—
187#107—972no—
188#106—972no—
189#103—971no—
190#103—971no—
191#102—972no—
192#102—972no—
193#102—971no—
194#102—971no—
195C24—1941no—
196#41—22no—
197#41—22no—
198#41—1942no—
199#41—1942no—
200C2×Q8—971no—
201C2×Q8D38811yescharacteristic
202C2×D4—971no—
203C2×D4—972no—
204C2×D4—1942no—
205C2×D4—1942no—
206C2×D4—1942no—
207C2×D4—1941no—
208C2×D4—1942no—
209C2×D4—1941no—
210C2×D4—1942no—
211C2×D4D38812yes—
212C2×D4D38811yescharacteristic
213C22×C4—1941no—
214C22×C4—1941no—
215C22×C4C22×D9712yes—
216C22×C4C22×D9711yescharacteristic
217SD16—1944no—
218D8—1944no—
219M16—24no—
220M16C22⋊D9714yescomplemented
221C2×C8—22no—
222C2×C8—24no—
223C2×C8C22×D9711yescharacteristic
224C2×C8C22×D9711yescharacteristic
225C2×C8C22×D9712yes—
226C4⋊C4—972no—
227C4⋊C4—971no—
228C4⋊C4—971no—
229C4⋊C4—1942no—
230C4⋊C4—1941no—
231C22⋊C4—1941no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C42—1941no—
237C23—7761no—
238C23—7761no—
239C23—3882no—
240C23—1941no—
241C23—1942no—
242C23C2×D38812yes—
243C23C2×D38811yescharacteristic
244Q8—22no—
245Q8—1941no—
246Q8—1941no—
247D4—24no—
248D4—22no—
249D4—7762no—
250D4—7762no—
251D4—7762no—
252D4—3882no—
253D4—3882no—
254D4—1941no—
255D4—1942no—
256D4—1942no—
257D4—1941no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—7761no—
261C2×C4—3882no—
262C2×C4—1941no—
263C2×C4—1941no—
264C2×C4—1942no—
265C2×C4—1941no—
266C2×C4C2×C22⋊D9712yes—
267C2×C4C2×D38812yes—
268C2×C4C2×D38811yescharacteristic
269C2×C4#1309551112yes—
270C2×C4C23×D9711yescharacteristic
271C8—21no—
272C8—21no—
273C8C2×C22⋊D9714yes—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—7764no—
280C22—3881no—
281C22—3882no—
282C22—3881no—
283C22—3882no—
284C22—3882no—
285C22—3881no—
286C22#1569078412yes—
287C22C22×D38811yescharacteristic
288C4—22no—
289C4—3881no—
290C4—3882no—
291C4—3881no—
292C4—3881no—
293C4#1569078412yes—
294C4C22×C22⋊D9711yescharacteristic
295C4C2×#1309551111yescharacteristic
296C2—22no—
297C2—42no—
298C2—7762no—
299C2—7761no—
300C2C2×#1569078411yescharacteristic
301C2#2026154011yescharacteristic
302C2#2026154711yescharacteristic
subgroupG/H
centerC22C22×D388
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#131×C97C2
socleC22×C97C22×D4
hypercenterC2×#131D97
hypocenterC97#3062
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. #20261313C2semidirect
3. #20261299C2semidirect
4. #20260988C2semidirect
5. #20260981C2semidirect
6. #20260971C2semidirect
7. #20260964C2semidirect
8. #20260777C2semidirect
9. #20260732C2semidirect
10. C2×#15690732C2semidirect
11. C2×#131×C97C2semidirect
19. #15690789C22semidirect
23. #15690784C22semidirect
29. #15690744C22semidirect
30. #15690742C22semidirect
31. #15690735C22semidirect
32. #15690735C22semidirect
35. #15690732C22semidirect
36. #15690732C22semidirect
40. C2×M16×C97C22semidirect
42. C22×C8×C97C22semidirect
79. M16×C97D4semidirect
152. C97#3062semidirect
154. C2×#131D97semidirect
172. C2×M16C2×D97semidirect
174. C22×C8C2×D97semidirect
220. M16C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
#202613132
#202612992
#202609882
#202609812
#202609712
#202609642
#202607772
#202607322
C2×#156907322
C2×#131×C972
#306297