Finite Group Databasebeta

← #27151400

Subgroups — #27151400

Order 12416.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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×#15690833C211yesmaximal, characteristic, complemented
2C2×#15690818C211yesmaximal, characteristic, complemented
3#20261315C212yesmaximal, complemented
4#20260990C211yesmaximal, characteristic, complemented
5#20260989C211yesmaximal, characteristic, complemented
6#20260988C211yesmaximal, characteristic, complemented
7#20260967C212yesmaximal, complemented
8#20260966C212yesmaximal, complemented
9#106×D97C211yesmaximal, characteristic, complemented
10#20260735C212yesmaximal, complemented
11#470×C97C211yesmaximal, characteristic, complemented
12C2×#13095512C2212yes—
13C2×#13095511C2211yescharacteristic
14#15690833—24no—
15C2×#13095498C2212yes—
16C2×D776C2211yescharacteristic
17#15690818—24no—
18C2×#13095494C2211yescharacteristic
19C2×#13095494C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21C2×C8×D97C2211yescharacteristic
22C2×C8×D97C2211yescharacteristic
23#15690788—22no—
24#15690786C2212yes—
25#15690785—22no—
26#15690785C2212yes—
27#15690784C2212yes—
28#15690783—22no—
29#15690783—22no—
30C4⋊C4×D97C2212yes—
31#15690742C2211yescharacteristic
32#15690741C2212yes—
33#15690734C2212yes—
34#15690732C2212yes—
35#15690730C2212yes—
36C2×Q16×C97C2211yescharacteristic
37C2×M16×C97C2211yescharacteristic
38C22⋊Q8×C97C2212yes—
39#106×C97C2211yescharacteristic
40#103×C97C2212yes—
41#13095512—24no—
42#13095512—24no—
43#13095511—22no—
44#13095511—22no—
45C2×Dic194D411yescharacteristic
46C2×D388—22no—
47C2×D388—22no—
48C2×D388D411yescharacteristic
49C2×D388C2312yes—
50C2×C22⋊D97—22no—
51C2×C22⋊D97D411yescharacteristic
52C2×C22⋊D97D411yescharacteristic
53C2×C4×D97—22no—
54C2×C4×D97—22no—
55C2×C4×D97C2311yescharacteristic
56C2×C4×D97C2311yescharacteristic
57#13095498—24no—
58D776—22no—
59#13095494—22no—
60#13095494—22no—
61C2×#11715895—22no—
62C2×#11715895C2311yescharacteristic
63C2×#11715895C2311yescharacteristic
64C8×D97—22no—
65C8×D97D414yescomplemented
66C4⋊Dic97C2312yes—
67#13095490—22no—
68#13095490—22no—
69#13095488—22no—
70#13095488—22no—
71#13095488—22no—
72C2×Q8×C97C2312yes—
73C22×C4×C97C2311yescharacteristic
74Q16×C97—22no—
75M16×C97—22no—
76C2×C8×C97C2311yescharacteristic
77C2×C8×C97C2311yescharacteristic
78C4⋊C4×C97—22no—
79C4⋊C4×C97C2312yes—
80C22⋊C4×C97—22no—
81C23×C97C2×D411yescharacteristic
82C22×D97—22no—
83C22×D97C2×D411yescharacteristic
84C22×D97C2×D411yescharacteristic
85Q8×C97—22no—
86Q8×C97—22no—
87C2×C4×C97—21no—
88C2×C4×C97—21no—
89C2×C4×C97—22no—
90C2×C4×C97—22no—
91C2×C4×C97C2×D411yescharacteristic
92C2×C4×C97C2411yescharacteristic
93C22⋊D97—22no—
94C22⋊D97—22no—
95C22⋊D97—82no—
96C2×Dic97—22no—
97C2×Dic97C2×D411yescharacteristic
98C2×Dic97C2×D411yescharacteristic
99D388—22no—
100D388—22no—
101D388—22no—
102D388—44no—
103D388—82no—
104C4×D97—21no—
105C4×D97—21no—
106C4×D97—44no—
107C4×D97—82no—
108C4×D97C2×D412yes—
109C4×D97#4112yes—
110Dic194—22no—
111C8×C97—21no—
112C8×C97C2×D412yes—
113#11715895—21no—
114#11715895C2×D412yes—
115C22×C97—21no—
116C22×C97—22no—
117C22×C97C22×D411yescharacteristic
118C2×D97—22no—
119C2×D97—21no—
120C2×D97—22no—
121C2×D97—42no—
122C2×D97—42no—
123C2×D97—42no—
124C2×D97#12112yes—
125C4×C97—21no—
126C4×C97—42no—
127C4×C97—42no—
128C4×C97C22×D411yescharacteristic
129C4×C97C2×#4111yescharacteristic
130Dic97—21no—
131Dic97—42no—
132Dic97#12112yes—
133C2×C97—41no—
134C2×C97C2×#12111yescharacteristic
135C2×C97#57712yes—
136D97—22no—
137D97—41no—
138D97—82no—
139#3060—971nomaximal, Sylow
140C97#306011yescharacteristic, complemented, Sylow
141C2×#135—971no—
142C2×#131—971no—
143#537—972no—
144#470—971no—
145#470D9711yescharacteristic, complemented
146#469—971no—
147#469—971no—
148#465—972no—
149#465—972no—
150C2×#106—971no—
151#417—972no—
152C2×#41—971no—
153C2×#41—972no—
154#135—1944no—
155C2×Q16C2×D9711yescharacteristic
156C2×SD16—972no—
157C2×D8—971no—
158#131—1944no—
159C2×M16—971no—
160C2×M16—971no—
161C2×M16C2×D9711yescharacteristic
162C22×C8—971no—
163C22×C8—971no—
164C22×C8—971no—
165C22⋊Q8—972no—
166C22⋊Q8—1942no—
167C22⋊Q8C2×D9712yes—
168#121—972no—
169#121—972no—
170#121—1942no—
171#121—1942no—
172#121—1942no—
173C2×C4⋊C4—972no—
174#106—971no—
175#106—972no—
176#106C2×D9711yescharacteristic
177#103—972no—
178#103C2×D9712yes—
179#102—972no—
180#102—972no—
181#41—1942no—
182#41—1942no—
183#41—1944no—
184#41—1944no—
185C2×Q8—971no—
186C2×Q8C22×D9712yes—
187C2×D4—971no—
188C2×D4—971no—
189C2×D4—971no—
190C2×D4—972no—
191C2×D4—1942no—
192C2×D4—1942no—
193C2×D4—1942no—
194C22×C4—971no—
195C22×C4—971no—
196C22×C4—1942no—
197C22×C4—1942no—
198C22×C4C22×D9711yescharacteristic
199Q16—22no—
200SD16—1944no—
201D8—1942no—
202M16—22no—
203M16—1942no—
204M16—1942no—
205C2×C8—971no—
206C2×C8—974no—
207C2×C8—971no—
208C2×C8—1942no—
209C2×C8—1942no—
210C2×C8C22×D9711yescharacteristic
211C2×C8C22×D9711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—972no—
214C4⋊C4—1942no—
215C4⋊C4—1942no—
216C4⋊C4C22×D9712yes—
217C22⋊C4—22no—
218C22⋊C4—1942no—
219C22⋊C4—1942no—
220C22⋊C4—1942no—
221C23—971no—
222C23—971no—
223C23—1942no—
224C23D4×D9711yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—1942no—
228D4—7762no—
229D4—7762no—
230D4—3884no—
231D4—1942no—
232D4—1942no—
233D4—1942no—
234D4—1942no—
235D4—1942no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—7762no—
241C2×C4—971no—
242C2×C4—972no—
243C2×C4—972no—
244C2×C4—971no—
245C2×C4—3884no—
246C2×C4—1942no—
247C2×C4—1941no—
248C2×C4—1941no—
249C2×C4D4×D9711yescharacteristic
250C2×C4C23×D9711yescharacteristic
251C8—21no—
252C8—972no—
253C8—1941no—
254C8D4×D9712yes—
255C22—21no—
256C22—22no—
257C22—972no—
258C22—3882no—
259C22—3882no—
260C22—3882no—
261C22—1941no—
262C22—1942no—
263C22—1942no—
264C22C2×D4×D9711yescharacteristic
265C4—21no—
266C4—42no—
267C4—42no—
268C4—972no—
269C4—3882no—
270C4—1941no—
271C4C2×D4×D9711yescharacteristic
272C4#41×D9711yescharacteristic
273C2—41no—
274C2—7762no—
275C2—3881no—
276C2—1942no—
277C2#121×D9711yescharacteristic
278C2#2026154412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#470×C97C2
socleC22×C97C22×D4
hypercenter#470D97
hypocenterC97#3060
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#15690833C2semidirect
2. C2×#15690818C2semidirect
3. #20261315C2semidirect
4. #20260990C2semidirect
5. #20260989C2semidirect
6. #20260988C2semidirect
7. #20260967C2semidirect
8. #20260966C2semidirect
9. #106×D97C2semidirect
10. #20260735C2semidirect
11. #470×C97C2semidirect
65. C8×D97D4semidirect
140. C97#3060semidirect
145. #470D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908332
C2×#156908182
#202613152
#202609902
#202609892
#202609882
#202609672
#202609662
#106×D972
#202607352
#470×C972
#306097