Finite Group Databasebeta

← #27151417

Subgroups — #27151417

Order 12416.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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
2C2×#15690818C211yesmaximal, characteristic, complemented
3#20261316C211yesmaximal, characteristic, complemented
4#20261304C211yesmaximal, characteristic, complemented
5#20260989C212yesmaximal, complemented
6#20260982C212yesmaximal, complemented
7#20260972C212yesmaximal, complemented
8#20260965C212yesmaximal, complemented
9#20260729C211yesmaximal, characteristic, complemented
10C2×#15690730C211yesmaximal, characteristic, complemented
11#428×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×#13095511C2211yescharacteristic
14C22×D388C2211yescharacteristic
15#15690838—24no—
16C2×#13095500C2212yes—
17C2×D776C2212yes—
18#15690818—24no—
19C2×#13095494C2212yescomplemented
20C22×#11715895C2211yescharacteristic
21C2×C8×D97C2212yescomplemented
22#15690807—21no—
23#15690798—21no—
24#15690790—21no—
25#15690790C2212yescomplemented
26#15690785—21no—
27#15690785C2212yes—
28#15690785C2212yes—
29#15690785C2212yescomplemented
30#15690784—22no—
31#15690780—22no—
32#15690733C2211yescharacteristic, complemented
33#15690733C2211yescharacteristic, complemented
34#15690730C2212yes—
35#15690730C2212yes—
36#15690730C2211yescharacteristic, complemented
37#15690730C2211yescharacteristic, complemented
38C2×M16×C97C2211yescharacteristic
39#117×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#107×C97C2212yescomplemented
42#106×C97C2212yescomplemented
43C23×D97—21no—
44#13095511—22no—
45#13095511—22no—
46#13095511—22no—
47#13095511—22no—
48C2×Dic194D411yescharacteristic
49C2×Dic194C2311yescharacteristic
50C2×D388—22no—
51C2×D388—22no—
52C2×D388—21no—
53C2×D388—22no—
54C2×D388—22no—
55C2×D388—22no—
56C2×D388D411yescharacteristic
57C2×D388C2312yes—
58C2×D388C2311yescharacteristic
59C2×C22⋊D97—21no—
60C2×C22⋊D97D412yes—
61C2×C4×D97—21no—
62C2×C4×D97C2312yes—
63#13095500—24no—
64D776—24no—
65#13095494—24no—
66C2×#11715895—22no—
67C2×#11715895C2311yescharacteristic
68C2×#11715895C2311yescharacteristic
69C8×D97—24no—
70C4⋊Dic97—22no—
71#13095488—22no—
72#13095488—22no—
73#13095488—22no—
74#13095488—22no—
75C22×C4×C97—21no—
76C22×C4×C97C2311yescharacteristic
77M16×C97D414yescomplemented
78C2×C8×C97C2312yes—
79C4⋊C4×C97—21no—
80C4⋊C4×C97C2311yescharacteristic
81C4⋊C4×C97C2311yescharacteristic
82C4⋊C4×C97C2312yes—
83C22⋊C4×C97—21no—
84C42×C97—21no—
85C23×C97C2×D411yescharacteristic
86C22×D97—21no—
87C22×D97—22no—
88C22×D97—42no—
89C22×D97—81no—
90C22×D97—81no—
91C22×D97C2×D412yes—
92C2×C4×C97—21no—
93C2×C4×C97—21no—
94C2×C4×C97—22no—
95C2×C4×C97—81no—
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97C2×D412yes—
98C2×C4×C97#4112yes—
99C2×C4×C97C2411yescharacteristic
100C22⋊D97—24no—
101C22⋊D97—42no—
102C2×Dic97—21no—
103C2×Dic97C2×D412yes—
104D388—22no—
105D388—21no—
106D388—22no—
107D388—22no—
108D388—21no—
109D388—42no—
110D388—82no—
111D388—82no—
112D388—82no—
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—42no—
116Dic194—22no—
117Dic194—21no—
118Dic194—21no—
119C8×C97C2×D414yes—
120#11715895—21no—
121#11715895—21no—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97#12112yes—
125C22×C97C22×D411yescharacteristic
126C2×D97—22no—
127C2×D97—22no—
128C2×D97—22no—
129C2×D97—41no—
130C2×D97—42no—
131C2×D97—41no—
132C2×D97—42no—
133C2×D97—42no—
134C2×D97—41no—
135C2×D97—84no—
136C4×C97—41no—
137C4×C97—42no—
138C4×C97—41no—
139C4×C97#12112yes—
140C4×C97C22×D411yescharacteristic
141C4×C97C2×#4111yescharacteristic
142Dic97—22no—
143Dic97—41no—
144C2×C97—22no—
145C2×C97C2×#12111yescharacteristic
146C2×C97#57611yescharacteristic
147C2×C97#57711yescharacteristic
148D97—42no—
149D97—81no—
150D97—82no—
151#3062—971nomaximal, Sylow
152C97#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—971no—
154C2×#131—971no—
155#537—971no—
156#535—971no—
157#469—972no—
158#468—972no—
159#466—972no—
160#465—972no—
161#428D9711yescharacteristic, complemented
162#416—971no—
163C2×#102—971no—
164C2×#41—971no—
165C2×#41—971no—
166C22×D4—971no—
167AGL(1,ℤ/8)—1944no—
168C2×SD16—972no—
169C2×D8—972no—
170#131—1944no—
171C2×M16—972no—
172C2×M16C2×D9711yescharacteristic
173C22×C8—971no—
174C22×C8—972no—
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#117C2×D9711yescharacteristic
186C2×C4⋊C4C2×D9711yescharacteristic
187#107C2×D9712yescomplemented
188#106C2×D9712yescomplemented
189#103—971no—
190#103—971no—
191#102—972no—
192#102—972no—
193#102—971no—
194#102—971no—
195C24—1941no—
196#41—1942no—
197#41—1942no—
198#41—1942no—
199#41—1942no—
200C2×Q8—971no—
201C2×Q8—971no—
202C2×D4—972no—
203C2×D4—971no—
204C2×D4—971no—
205C2×D4—972no—
206C2×D4—1942no—
207C2×D4—1942no—
208C2×D4—1942no—
209C2×D4—1941no—
210C2×D4—1941no—
211C2×D4—1942no—
212C2×D4—1942no—
213C22×C4—21no—
214C22×C4—972no—
215C22×C4—1941no—
216C22×C4C22×D9711yescharacteristic
217SD16—1944no—
218D8—1944no—
219M16—1944no—
220M16D38814yescomplemented
221C2×C8—971no—
222C2×C8—971no—
223C2×C8—1942no—
224C2×C8—1944no—
225C2×C8C22×D9712yes—
226C4⋊C4—21no—
227C4⋊C4—1942no—
228C4⋊C4C22×D9712yes—
229C4⋊C4C22×D9711yescharacteristic
230C4⋊C4C22×D9711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C22⋊C4—1942no—
235C22⋊C4—1942no—
236C42—21no—
237C23—7761no—
238C23—7761no—
239C23—972no—
240C23—3882no—
241C23—1941no—
242C23—1942no—
243C23D4×D9711yescharacteristic
244Q8—1941no—
245Q8—1941no—
246Q8—1942no—
247D4—7762no—
248D4—7762no—
249D4—7762no—
250D4—3882no—
251D4—3882no—
252D4—1944no—
253D4—1942no—
254D4—1941no—
255D4—1942no—
256D4—1942no—
257D4—1941no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—972no—
263C2×C4—3882no—
264C2×C4—1941no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4D4×D9711yescharacteristic
268C2×C4C2×D38812yes—
269C2×C4#1309551212yes—
270C2×C4C23×D9711yescharacteristic
271C8—1941no—
272C8—1941no—
273C8C2×D38814yes—
274C22—21no—
275C22—21no—
276C22—7764no—
277C22—3881no—
278C22—3882no—
279C22—3881no—
280C22—3882no—
281C22—3882no—
282C22—3881no—
283C22—1942no—
284C22—1942no—
285C22—1942no—
286C22#1569078512yes—
287C22C2×D4×D9711yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—3881no—
292C4—1942no—
293C4#1569078512yes—
294C4C22×D38811yescharacteristic
295C4C2×#1309551211yescharacteristic
296C2—22no—
297C2—7762no—
298C2—7761no—
299C2—3882no—
300C2C2×#1569078511yescharacteristic
301C2#2026153911yescharacteristic
302C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
hypercenter#428D97
hypocenterC97#3062
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#15690838C2semidirect
2. C2×#15690818C2semidirect
3. #20261316C2semidirect
4. #20261304C2semidirect
5. #20260989C2semidirect
6. #20260982C2semidirect
7. #20260972C2semidirect
8. #20260965C2semidirect
9. #20260729C2semidirect
10. C2×#15690730C2semidirect
11. #428×C97C2semidirect
19. C2×#13095494C22semidirect
21. C2×C8×D97C22semidirect
25. #15690790C22semidirect
29. #15690785C22semidirect
32. #15690733C22semidirect
33. #15690733C22semidirect
36. #15690730C22semidirect
37. #15690730C22semidirect
41. #107×C97C22semidirect
42. #106×C97C22semidirect
77. M16×C97D4semidirect
152. C97#3062semidirect
161. #428D97semidirect
187. #107C2×D97semidirect
188. #106C2×D97semidirect
220. M16D388semidirect

Maximal subgroups

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

subgroupindex
C2×#156908382
C2×#156908182
#202613162
#202613042
#202609892
#202609822
#202609722
#202609652
#202607292
C2×#156907302
#428×C972
#306297