Finite Group Databasebeta

← #27154383

Subgroups — #27154383

Order 12416.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#20261455C211yesmaximal, characteristic, complemented
2#20261442C211yesmaximal, characteristic, complemented
3#20261088C212yesmaximal, complemented
4#20261086C211yesmaximal, characteristic, complemented
5#20261041C212yesmaximal, complemented
6#20261025C211yesmaximal, characteristic, complemented
7#20260948C212yesmaximal
8#20260940C211yesmaximal, characteristic, complemented
9C8×Dic194C211yesmaximal, characteristic, complemented
10#20260826C212yesmaximal, complemented
11C4×D8×C97C211yesmaximal, characteristic
12C2×#13095499C2212yes—
13#15690810C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690803—22no—
16#15690801C2212yes—
17#15690797C2212yes—
18#15690789—21no—
19#15690789—21no—
20#15690787—22no—
21C4×Dic194C2211yescharacteristic
22#15690775C2212yes—
23D4×Dic97C2212yes—
24#15690769—21no—
25C2×C4⋊Dic97—21no—
26#15690741C2212yes—
27C4⋊#11715895C2211yescharacteristic
28#15690739C2212yes—
29#15690735C2212yes—
30#15690735C2212yes—
31#15690733C2212yes—
32#15690731C2212yes—
33C8×Dic97C2212yes—
34C2×D8×C97C2211yescharacteristic
35C4×D4×C97C2211yescharacteristic
36C4×D4×C97C2211yescharacteristic
37#107×C97C2211yescharacteristic
38#102×C97C2211yescharacteristic
39#102×C97C2211yescharacteristic
40C4×C8×C97C2211yescharacteristic
41C2×Dic194—21no—
42C2×Dic194D411yescharacteristic
43C2×Dic194C2312yes—
44C22×Dic97—22no—
45#13095499—24no—
46C2×#11715895C2312yes—
47C4⋊Dic97—21no—
48C4⋊Dic97—21no—
49C4⋊Dic97—21no—
50C4⋊Dic97—22no—
51C4⋊Dic97D411yescharacteristic
52C4⋊Dic97C2312yes—
53#13095490—22no—
54#13095490—22no—
55#13095490D412yes—
56C22⋊Dic97—22no—
57C22⋊Dic97—22no—
58C22⋊Dic97—22no—
59C4×Dic97—22no—
60C4×Dic97C2312yes—
61C2×D4×C97C2311yescharacteristic
62C2×D4×C97C2311yescharacteristic
63C22×C4×C97—21no—
64C22×C4×C97—21no—
65D8×C97—22no—
66C2×C8×C97C2311yescharacteristic
67C2×C8×C97C2311yescharacteristic
68C4⋊C4×C97C2311yescharacteristic
69C4⋊C4×C97C2311yescharacteristic
70C22⋊C4×C97—21no—
71C22⋊C4×C97—21no—
72C42×C97C2311yescharacteristic
73C23×C97—21no—
74C23×C97—21no—
75D4×C97—21no—
76D4×C97—21no—
77D4×C97—21no—
78D4×C97#4112yes—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—42no—
82C2×C4×C97—42no—
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2411yescharacteristic
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—82no—
89C2×Dic97C2×D412yes—
90C2×Dic97C2×D412yes—
91Dic194—22no—
92Dic194—22no—
93Dic194—22no—
94Dic194—81no—
95C8×C97—21no—
96C8×C97—21no—
97#11715895—22no—
98C22×C97—22no—
99C22×C97—41no—
100C22×C97—41no—
101C22×C97—41no—
102C22×C97—41no—
103C22×C97—41no—
104C22×C97C22×D411yescharacteristic
105C4×C97—21no—
106C4×C97—41no—
107C4×C97—41no—
108C4×C97#13512yes—
109C4×C97C2×#4111yescharacteristic
110C4×C972−1+411yescharacteristic
111Dic97—22no—
112Dic97—22no—
113Dic97—42no—
114Dic97—42no—
115C2×C97—42no—
116C2×C97—81no—
117C2×C97#54811yescharacteristic
118C2×C97C2×#13511yescharacteristic
119C2×C97#57711yescharacteristic
120#3360—971nomaximal, Sylow
121C97#336011yescharacteristic, complemented, Sylow
122#556—971no—
123#554—971no—
124#488—972no—
125#487—971no—
126#479—972no—
127#476—971no—
128#463—972no—
129#461—971no—
130C8×Q8—971no—
131C4×SD16—972no—
132C4×D8D9711yescharacteristic
133C2×SD16—972no—
134C2×D8C2×D9711yescharacteristic
135C4⋊Q8—971no—
136C4⋊Q8—971no—
137#126—1942no—
138#125—972no—
139#124—972no—
140C22⋊Q8—1942no—
141C22⋊Q8—1941no—
142C22⋊Q8—1941no—
143C4×Q8—971no—
144C4×Q8—972no—
145C4×D4—972no—
146C4×D4C2×D9711yescharacteristic
147C4×D4C2×D9711yescharacteristic
148#117—1941no—
149C2×C4⋊C4—1941no—
150#107C2×D9711yescharacteristic
151#106—972no—
152C4⋊C8—972no—
153C4⋊C8—971no—
154#103—972no—
155#103—972no—
156#103—972no—
157#102—972no—
158#102C2×D9711yescharacteristic
159#102C2×D9711yescharacteristic
160C4×C8—972no—
161C4×C8C2×D9711yescharacteristic
162C2×Q8—971no—
163C2×Q8—972no—
164C2×Q8—1941no—
165C2×D4C22×D9711yescharacteristic
166C2×D4C22×D9711yescharacteristic
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—1942no—
170SD16—1944no—
171D8—22no—
172C2×C8—972no—
173C2×C8C22×D9711yescharacteristic
174C2×C8C22×D9711yescharacteristic
175C4⋊C4—972no—
176C4⋊C4—971no—
177C4⋊C4—972no—
178C4⋊C4—1942no—
179C4⋊C4—1942no—
180C4⋊C4—1942no—
181C4⋊C4—1941no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4C22×D9711yescharacteristic
185C4⋊C4C22×D9711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—1942no—
189C22⋊C4—1942no—
190C22⋊C4—1942no—
191C42—972no—
192C42—1942no—
193C42C22×D9711yescharacteristic
194C23—21no—
195C23—21no—
196Q8—7761no—
197Q8—1942no—
198Q8—1942no—
199Q8—1942no—
200D4—21no—
201D4—21no—
202D4—21no—
203D4#1309551112yes—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—7762no—
209C2×C4—972no—
210C2×C4—972no—
211C2×C4—1942no—
212C2×C4—1942no—
213C2×C4D4×D9711yescharacteristic
214C2×C4D4×D9711yescharacteristic
215C2×C4C23×D9711yescharacteristic
216C8—21no—
217C8—21no—
218C8—1942no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22C2×D4×D9711yescharacteristic
226C4—21no—
227C4—41no—
228C4—41no—
229C4—3882no—
230C4—3882no—
231C4—1942no—
232C4—1942no—
233C4#1569082912yes—
234C4C2×#1309551111yescharacteristic
235C4#1569085711yescharacteristic
236C2—42no—
237C2—81no—
238C2#2026139411yescharacteristic
239C2C2×#1569082911yescharacteristic
240C2#2026154211yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×D8×C97C2
socleC22×C97C22×D4
prime radicalC2×D8×C97C22
hypercenterC4×D8D97
hypocenterC97#3360
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #20261455C2semidirect
2. #20261442C2semidirect
3. #20261088C2semidirect
4. #20261086C2semidirect
5. #20261041C2semidirect
6. #20261025C2semidirect
8. #20260940C2semidirect
9. C8×Dic194C2semidirect
10. #20260826C2semidirect
121. C97#3360semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202614422
#202610882
#202610862
#202610412
#202610252
#202609482
#202609402
C8×Dic1942
#202608262
C4×D8×C972
#336097