Finite Group Databasebeta

← #27154384

Subgroups — #27154384

Order 12416.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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#20261446C211yesmaximal, characteristic, complemented
3#20261088C212yesmaximal, complemented
4#20261087C211yesmaximal, characteristic, complemented
5#20261045C212yesmaximal, complemented
6#20260949C212yesmaximal
7#20260941C211yesmaximal, characteristic, complemented
8#20260859C211yesmaximal, characteristic, complemented
9#20260822C212yesmaximal, complemented
10C4×#13095496C211yesmaximal, characteristic
11#476×C97C211yesmaximal, characteristic, complemented
12C2×#13095500C2212yes—
13C2×#13095496C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690803—22no—
16#15690801C2212yes—
17#15690797C2212yes—
18#15690790—21no—
19#15690789—21no—
20#15690786—22no—
21C4×Dic194C2211yescharacteristic
22#15690775C2212yes—
23C4×D388C2211yescharacteristic
24#15690772C2212yes—
25#15690769—21no—
26C4⋊C4×D97—21no—
27#15690743C2211yescharacteristic
28#15690739C2212yes—
29#15690735C2212yes—
30#15690733C2212yes—
31#15690733C2212yes—
32#15690731C2211yescharacteristic
33#15690730C2211yescharacteristic
34C4×#11715895C2211yescharacteristic
35C8×Dic97C2212yes—
36C4⋊Q8×C97C2211yescharacteristic
37C4×D4×C97C2211yescharacteristic
38#106×C97C2212yes—
39C4⋊C8×C97C2211yescharacteristic
40#102×C97C2212yes—
41C2×Dic194D411yescharacteristic
42C2×Dic194C2312yes—
43C2×D388C2311yescharacteristic
44C2×C4×D97—21no—
45C2×C4×D97—22no—
46#13095500—24no—
47#13095496—22no—
48C2×#11715895C2311yescharacteristic
49C2×#11715895C2311yescharacteristic
50C4⋊Dic97—21no—
51C4⋊Dic97—21no—
52C4⋊Dic97—22no—
53C4⋊Dic97D411yescharacteristic
54C4⋊Dic97C2311yescharacteristic
55#13095490—22no—
56#13095490—22no—
57#13095490D412yes—
58C22⋊Dic97—22no—
59C22⋊Dic97—22no—
60#13095488—21no—
61#13095488—22no—
62C4×Dic97—22no—
63C4×Dic97C2312yes—
64C2×Q8×C97—21no—
65C2×D4×C97C2311yescharacteristic
66C22×C4×C97—21no—
67C2×C8×C97C2312yes—
68C4⋊C4×C97—21no—
69C4⋊C4×C97C2312yes—
70C4⋊C4×C97C2311yescharacteristic
71C22⋊C4×C97—21no—
72C42×C97C2311yescharacteristic
73C23×C97—21no—
74C22×D97—21no—
75Q8×C97—81no—
76D4×C97—21no—
77D4×C97—21no—
78C2×C4×C97—21no—
79C2×C4×C97—22no—
80C2×C4×C97—42no—
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2411yescharacteristic
84C2×Dic97—22no—
85C2×Dic97—21no—
86C2×Dic97C2×D412yes—
87C2×Dic97C2×D412yes—
88D388—21no—
89D388#4112yes—
90C4×D97—42no—
91C4×D97—82no—
92Dic194—22no—
93Dic194—22no—
94Dic194—22no—
95C8×C97—22no—
96#11715895—21no—
97#11715895—21no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97—41no—
101C22×C97C22×D411yescharacteristic
102C2×D97—22no—
103C2×D97—41no—
104C2×D97—41no—
105C4×C97—21no—
106C4×C97—41no—
107C4×C97—42no—
108C4×C97#13512yes—
109C4×C97C2×#4111yescharacteristic
110C4×C972−1+411yescharacteristic
111Dic97—22no—
112Dic97—22no—
113Dic97—42no—
114Dic97—41no—
115C2×C97—81no—
116C2×C97#54811yescharacteristic
117C2×C97C2×#13511yescharacteristic
118C2×C97#57711yescharacteristic
119D97—42no—
120#3360—971nomaximal, Sylow
121C97#336011yescharacteristic, complemented, Sylow
122#556—971no—
123#554—971no—
124#488—972no—
125#487—971no—
126#479—972no—
127#476D9711yescharacteristic, complemented
128#463—972no—
129#461—971no—
130C8×Q8—971no—
131C4×SD16—972no—
132C4×D8—971no—
133C2×SD16—972no—
134C2×D8—971no—
135C4⋊Q8—971no—
136C4⋊Q8C2×D9711yescharacteristic
137#126—1942no—
138#125—972no—
139#124—972no—
140C22⋊Q8—1941no—
141C22⋊Q8—1942no—
142C22⋊Q8—1941no—
143C4×Q8—971no—
144C4×Q8—972no—
145C4×D4—972no—
146C4×D4—971no—
147C4×D4C2×D9711yescharacteristic
148#117—1941no—
149C2×C4⋊C4—1941no—
150#107—971no—
151#106C2×D9712yes—
152C4⋊C8—972no—
153C4⋊C8C2×D9711yescharacteristic
154#103—972no—
155#103—972no—
156#103—972no—
157#102—971no—
158#102—971no—
159#102C2×D9712yes—
160C4×C8—972no—
161C4×C8—971no—
162C2×Q8—21no—
163C2×Q8—971no—
164C2×Q8—972no—
165C2×D4—971no—
166C2×D4C22×D9711yescharacteristic
167C22×C4—21no—
168C22×C4—1942no—
169C22×C4—1941no—
170SD16—1944no—
171D8—1942no—
172C2×C8—971no—
173C2×C8—971no—
174C2×C8C22×D9712yes—
175C4⋊C4—21no—
176C4⋊C4—972no—
177C4⋊C4—971no—
178C4⋊C4—971no—
179C4⋊C4—1942no—
180C4⋊C4—1942no—
181C4⋊C4—1942no—
182C4⋊C4—1941no—
183C4⋊C4—1941no—
184C4⋊C4C22×D9711yescharacteristic
185C4⋊C4C22×D9712yes—
186C22⋊C4—21no—
187C22⋊C4—1942no—
188C22⋊C4—1942no—
189C22⋊C4—1942no—
190C22⋊C4—1941no—
191C42—972no—
192C42—1942no—
193C42C22×D9711yescharacteristic
194C23—21no—
195C23—1941no—
196Q8—81no—
197Q8—1942no—
198Q8—1942no—
199Q8—1942no—
200D4—21no—
201D4—21no—
202D4—972no—
203D4—1941no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—7762no—
208C2×C4—972no—
209C2×C4—972no—
210C2×C4—3882no—
211C2×C4—1942no—
212C2×C4—1941no—
213C2×C4D4×D9711yescharacteristic
214C2×C4D4×D9711yescharacteristic
215C2×C4C23×D9711yescharacteristic
216C8—22no—
217C8—1941no—
218C8—1941no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3881no—
223C22—3881no—
224C22—1942no—
225C22C2×D4×D9711yescharacteristic
226C4—21no—
227C4—41no—
228C4—42no—
229C4—3882no—
230C4—3881no—
231C4—1942no—
232C4—1942no—
233C4#1569083112yes—
234C4#41×D9711yescharacteristic
235C4#1569085811yescharacteristic
236C2—81no—
237C2—3882no—
238C2#2026139511yescharacteristic
239C2C2×#1569083111yescharacteristic
240C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#476×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095496C22
hypercenter#476D97
hypocenterC97#3360
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #20261455C2semidirect
2. #20261446C2semidirect
3. #20261088C2semidirect
4. #20261087C2semidirect
5. #20261045C2semidirect
7. #20260941C2semidirect
8. #20260859C2semidirect
9. #20260822C2semidirect
11. #476×C97C2semidirect
121. C97#3360semidirect
127. #476D97semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202614462
#202610882
#202610872
#202610452
#202609492
#202609412
#202608592
#202608222
C4×#130954962
#476×C972
#336097