Finite Group Databasebeta

← #27143399

Subgroups — #27143399

Order 12416.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261281C211yesmaximal, characteristic
2#20260709C211yesmaximal, characteristic, complemented
3#20260529C212yesmaximal, complemented
4#20260527C212yesmaximal, complemented
5#410×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7#15690808C411yescharacteristic, complemented
8#15690806C2211yescharacteristic
9#15690799C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690782—21no—
12#15690782—21no—
13C4×C22⋊D97—21no—
14C4×C22⋊D97—21no—
15#15690769—21no—
16C2×C4×Dic97C411yescharacteristic, complemented
17#15690725—21no—
18#15690725C2212yes—
19C2×#41×C97C2211yescharacteristic
20#117×C97—21no—
21#99×C97—21no—
22#99×C97C2212yes—
23#13095511—41no—
24#13095511—41no—
25C2×Dic194C2×C411yescharacteristic
26C2×D388C2×C411yescharacteristic
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C2×C22⋊D97—21no—
30C22×Dic97—21no—
31C2×C4×D97—21no—
32C4⋊Dic97—41no—
33#13095490—21no—
34#13095490—21no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—22no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97—41no—
42C4×Dic97C2×C411yescharacteristic
43C4×Dic97C2×C411yescharacteristic
44#41×C97—41no—
45#41×C97—41no—
46C2×Q8×C97C2×C411yescharacteristic
47C2×D4×C97D411yescharacteristic
48C2×D4×C97D411yescharacteristic
49C2×D4×C97C2311yescharacteristic
50C22×C4×C97C2×C411yescharacteristic
51C22×C4×C97D411yescharacteristic
52C22×C4×C97D411yescharacteristic
53C4⋊C4×C97—41no—
54C22⋊C4×C97—22no—
55C42×C97—41no—
56C23×C97C22⋊C411yescharacteristic
57C23×C97C2×D411yescharacteristic
58C23×C97C2×D411yescharacteristic
59C22×D97—21no—
60Q8×C97—41no—
61D4×C97—41no—
62D4×C97—41no—
63D4×C97—41no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—42no—
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C22⋊C411yescharacteristic
73C2×C4×C97C22⋊C411yescharacteristic
74C2×C4×C97C22×C411yescharacteristic
75C22⋊D97—42no—
76C22⋊D97—42no—
77C22⋊D97—42no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81C2×Dic97—42no—
82C2×Dic97—42no—
83D388—41no—
84C4×D97—41no—
85C4×D97—41no—
86Dic194—41no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—41no—
91C22×C97—41no—
92C22×C97—41no—
93C22×C97C2×C22⋊C411yescharacteristic
94C2×D97—41no—
95C2×D97—41no—
96C2×D97—41no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100C4×C97—82no—
101C4×C97#9911yescharacteristic
102C4×C97#9911yescharacteristic
103Dic97—22no—
104Dic97—41no—
105Dic97—41no—
106Dic97—82no—
107C2×C97—21no—
108C2×C97—41no—
109C2×C97—41no—
110C2×C97—41no—
111C2×C97C2×#9911yescharacteristic
112D97—81no—
113#2126—971nomaximal, Sylow
114C97#212611yescharacteristic, complemented, Sylow
115#532—971no—
116#410—971no—
117#410D9711yescharacteristic, complemented
118#354—972no—
119#353—972no—
120C2×#41—971no—
121C2×#41C2×D9711yescharacteristic
122C4⋊Q8—971no—
123#127—971no—
124#124—971no—
125#124—971no—
126#121—1941no—
127#121—1941no—
128C4×D4—1941no—
129C4×D4—1941no—
130#117—21no—
131#117—1941no—
132C2×C42—971no—
133#99—21no—
134#99—972no—
135#99—1941no—
136#99C2×D9712yes—
137#41—41no—
138#41—41no—
139#41—3881no—
140#41—3881no—
141C2×Q8—971no—
142C2×Q8C4×D9711yescharacteristic
143C2×D4—971no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4D38811yescharacteristic
148C2×D4C22⋊D9711yescharacteristic
149C2×D4C22×D9711yescharacteristic
150C22×C4—1941no—
151C22×C4—1941no—
152C22×C4C4×D9711yescharacteristic
153C22×C4D38811yescharacteristic
154C22×C4C22⋊D9711yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—3881no—
157C4⋊C4—1941no—
158C4⋊C4—1941no—
159C22⋊C4—22no—
160C22⋊C4—1942no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C22⋊C4—1941no—
164C22⋊C4—1941no—
165C42—41no—
166C42—971no—
167C42—971no—
168C42—3881no—
169C42—1941no—
170C23—1941no—
171C23#1309548811yescharacteristic
172C23C2×C22⋊D9711yescharacteristic
173C23C2×D38811yescharacteristic
174Q8—41no—
175Q8—3881no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—3881no—
180D4—3882no—
181D4—3882no—
182D4—3882no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—3882no—
191C2×C4—3882no—
192C2×C4—3881no—
193C2×C4—3881no—
194C2×C4—1941no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4#1309548811yescharacteristic
198C2×C4#1309548811yescharacteristic
199C2×C4#1309548811yescharacteristic
200C2×C4C2×C4×D9711yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—3881no—
208C22—3881no—
209C22—3881no—
210C22C2×#1309548811yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—7762no—
216C4—3881no—
217C4—3881no—
218C4—1942no—
219C4#1569072411yescharacteristic
220C4#1569072411yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—7761no—
226C2C2×#1569072411yescharacteristic
subgroupG/H
centerC4#15690724
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#410×C97C2
socleC2×C97C2×#99
prime radical#20261281C2
cosocleC2×D4×C97C23
hypercenter#410D97
hypocenterC97#2126
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20260709C2semidirect
3. #20260529C2semidirect
4. #20260527C2semidirect
5. #410×C97C2semidirect
6. C2×#13095511C4semidirect
7. #15690808C4semidirect
9. #15690799C4semidirect
16. C2×C4×Dic97C4semidirect
114. C97#2126semidirect
117. #410D97semidirect

Maximal subgroups

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

subgroupindex
#202612812
#202607092
#202605292
#202605272
#410×C972
#212697