Finite Group Databasebeta

← #27151414

Subgroups — #27151414

Order 12416.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15690836C211yesmaximal, characteristic, complemented
2C2×#15690816C211yesmaximal, characteristic, complemented
3#20261316C211yesmaximal, characteristic, complemented
4#20261302C211yesmaximal, characteristic, complemented
5#20260987C212yesmaximal, complemented
6#20260980C212yesmaximal, complemented
7#20260970C212yesmaximal, complemented
8#20260963C212yesmaximal, complemented
9#20260775C211yesmaximal, characteristic, complemented
10#20260732C211yesmaximal, characteristic, complemented
11C2×#102×C97C211yesmaximal, characteristic, complemented
12C2×#13095511C2211yescharacteristic
13C2×#13095511C2211yescharacteristic
14#15690836—24no—
15C2×#13095499C2212yes—
16C2×#13095496C2212yes—
17#15690816—24no—
18C2×#13095495C2211yescharacteristic
19C2×#13095494C2212yescomplemented
20C2×C8×D97C2212yescomplemented
21#15690806—21no—
22#15690797—21no—
23#15690790—21no—
24#15690790C2212yescomplemented
25#15690785—21no—
26#15690785C2212yescomplemented
27#15690784—22no—
28#15690781C2212yes—
29#15690781C2212yes—
30#15690779—22no—
31#15690769C2211yescharacteristic
32#15690743C2212yescomplemented
33#15690741C2212yescomplemented
34#15690735C2211yescharacteristic, complemented
35#15690735C2211yescharacteristic, complemented
36#15690732C2211yescharacteristic, complemented
37#15690732C2211yescharacteristic, complemented
38C22×D4×C97C2211yescharacteristic
39C22×C8×C97C2211yescharacteristic
40C2×C4⋊C4×C97C2211yescharacteristic
41#102×C97C2212yes—
42#102×C97C2212yes—
43#13095511—22no—
44#13095511—22no—
45#13095511—22no—
46#13095511—22no—
47C2×Dic194D411yescharacteristic
48C2×Dic194C2311yescharacteristic
49C2×D388—22no—
50C2×D388D411yescharacteristic
51C2×D388C2311yescharacteristic
52C2×C22⋊D97—21no—
53C2×C22⋊D97—22no—
54C2×C22⋊D97—22no—
55C2×C22⋊D97D412yes—
56C2×C4×D97—21no—
57C2×C4×D97C2312yes—
58#13095499—24no—
59#13095496—24no—
60#13095495D414yescomplemented
61#13095494—24no—
62C2×#11715895C2312yes—
63C8×D97—24no—
64C4⋊Dic97—22no—
65C4⋊Dic97C2311yescharacteristic
66C4⋊Dic97C2311yescharacteristic
67C22⋊Dic97—21no—
68C22⋊Dic97—22no—
69C22⋊Dic97—22no—
70#13095488—22no—
71#13095488—22no—
72C4×Dic97—21no—
73C24×C97—21no—
74C2×D4×C97—22no—
75C2×D4×C97—21no—
76C2×D4×C97—22no—
77C2×D4×C97C2312yes—
78C22×C4×C97—21no—
79C22×C4×C97C2311yescharacteristic
80C2×C8×C97—22no—
81C2×C8×C97C2311yescharacteristic
82C2×C8×C97C2311yescharacteristic
83C4⋊C4×C97—21no—
84C4⋊C4×C97C2312yes—
85C23×C97—22no—
86C23×C97—42no—
87C23×C97—81no—
88C23×C97—81no—
89C23×C97C2×D411yescharacteristic
90C22×D97—21no—
91C22×D97C2×D412yes—
92D4×C97—22no—
93D4×C97—22no—
94D4×C97—42no—
95C2×C4×C97—22no—
96C2×C4×C97—81no—
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×D412yes—
99C2×C4×C97#4112yes—
100C2×C4×C97C2411yescharacteristic
101C22⋊D97—24no—
102C22⋊D97—42no—
103C22⋊D97—82no—
104C22⋊D97—82no—
105C2×Dic97—21no—
106C2×Dic97—21no—
107C2×Dic97—21no—
108C2×Dic97C2×D412yes—
109D388—22no—
110D388—21no—
111D388—21no—
112D388—82no—
113C4×D97—22no—
114C4×D97—22no—
115C4×D97—42no—
116Dic194—22no—
117Dic194—21no—
118Dic194—21no—
119C8×C97—21no—
120C8×C97—21no—
121#11715895C2×D414yes—
122C22×C97—21no—
123C22×C97—21no—
124C22×C97—42no—
125C22×C97—42no—
126C22×C97—42no—
127C22×C97—84no—
128C22×C97#12112yes—
129C22×C97C22×D411yescharacteristic
130C2×D97—22no—
131C2×D97—22no—
132C2×D97—22no—
133C2×D97—41no—
134C2×D97—41no—
135C2×D97—41no—
136C4×C97—42no—
137C4×C97#12112yes—
138C4×C97C22×D411yescharacteristic
139C4×C97C2×#4111yescharacteristic
140Dic97—22no—
141Dic97—41no—
142Dic97—41no—
143Dic97—41no—
144C2×C97—22no—
145C2×C97—82no—
146C2×C97C2×#12111yescharacteristic
147C2×C97#57611yescharacteristic
148C2×C97#57711yescharacteristic
149D97—42no—
150D97—81no—
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#428—971no—
162#416—971no—
163C2×#102D9711yescharacteristic, complemented
164C2×#41—971no—
165C2×#41—971no—
166C22×D4C2×D9711yescharacteristic
167AGL(1,ℤ/8)—1944no—
168C2×SD16—972no—
169C2×D8—972no—
170#131—1944no—
171C2×M16—971no—
172C2×M16—972no—
173C22×C8—972no—
174C22×C8C2×D9711yescharacteristic
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⋊C4C2×D9711yescharacteristic
187#107—972no—
188#106—972no—
189#103—971no—
190#103—971no—
191#102—971no—
192#102—971no—
193#102C2×D9712yes—
194#102C2×D9712yes—
195C24—21no—
196#41—1942no—
197#41—1942no—
198#41—1942no—
199#41—1942no—
200C2×Q8—971no—
201C2×Q8—971no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—972no—
206C2×D4—971no—
207C2×D4—971no—
208C2×D4—1942no—
209C2×D4—1942no—
210C2×D4—1942no—
211C2×D4—1941no—
212C2×D4C22×D9712yes—
213C22×C4—21no—
214C22×C4—972no—
215C22×C4—1941no—
216C22×C4C22×D9711yescharacteristic
217SD16—1944no—
218D8—1944no—
219M16—974no—
220M16—1944no—
221C2×C8—22no—
222C2×C8—972no—
223C2×C8—1944no—
224C2×C8C22×D9711yescharacteristic
225C2×C8C22×D9711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—971no—
228C4⋊C4—971no—
229C4⋊C4—1942no—
230C4⋊C4C22×D9712yes—
231C22⋊C4—1942no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C22⋊C4—1942no—
235C22⋊C4—1941no—
236C42—1941no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—972no—
242C23—1941no—
243C23D4×D9711yescharacteristic
244Q8—1941no—
245Q8—1941no—
246Q8—1942no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—7762no—
251D4—7762no—
252D4—7762no—
253D4—3882no—
254D4—1944no—
255D4—1942no—
256D4—1941no—
257D4—1941no—
258C2×C4—22no—
259C2×C4—81no—
260C2×C4—972no—
261C2×C4—3882no—
262C2×C4—1941no—
263C2×C4—1941no—
264C2×C4—1941no—
265C2×C4—1942no—
266C2×C4—1942no—
267C2×C4D4×D9711yescharacteristic
268C2×C4D4×D9712yes—
269C2×C4#1309551112yes—
270C2×C4C23×D9711yescharacteristic
271C8—21no—
272C8—21no—
273C8—974no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—3881no—
281C22—3881no—
282C22—3881no—
283C22—1942no—
284C22—1942no—
285C22—1942no—
286C22#1569078312yes—
287C22C2×D4×D9711yescharacteristic
288C4—42no—
289C4—3881no—
290C4—3881no—
291C4—3881no—
292C4—1942no—
293C4#1569078312yes—
294C4C2×D4×D9711yescharacteristic
295C4C2×#1309551111yescharacteristic
296C2—22no—
297C2—82no—
298C2—7761no—
299C2—3882no—
300C2C2×#1569078311yescharacteristic
301C2#2026153611yescharacteristic
302C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#102×C97C2
socleC22×C97C22×D4
hypercenterC2×#102D97
hypocenterC97#3062
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#15690836C2semidirect
2. C2×#15690816C2semidirect
3. #20261316C2semidirect
4. #20261302C2semidirect
5. #20260987C2semidirect
6. #20260980C2semidirect
7. #20260970C2semidirect
8. #20260963C2semidirect
9. #20260775C2semidirect
10. #20260732C2semidirect
11. C2×#102×C97C2semidirect
19. C2×#13095494C22semidirect
20. C2×C8×D97C22semidirect
24. #15690790C22semidirect
26. #15690785C22semidirect
32. #15690743C22semidirect
33. #15690741C22semidirect
34. #15690735C22semidirect
35. #15690735C22semidirect
36. #15690732C22semidirect
37. #15690732C22semidirect
60. #13095495D4semidirect
152. C97#3062semidirect
163. C2×#102D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908362
C2×#156908162
#202613162
#202613022
#202609872
#202609802
#202609702
#202609632
#202607752
#202607322
C2×#102×C972
#306297