Finite Group Databasebeta

← #27140277

Subgroups — #27140277

Order 12416.

286 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 73 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#20261246C211yesmaximal, characteristic
2#20260699C211yesmaximal, characteristic, complemented
3#20260471C212yesmaximal, complemented
4#20260468C212yesmaximal, complemented
5#380×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7C2×#13095495—21no—
8#15690793—21no—
9#15690789C411yescharacteristic, complemented
10#15690789C2211yescharacteristic
11#15690784C411yescharacteristic, complemented
12#15690784C2211yescharacteristic
13#15690779—21no—
14C4×C22⋊D97—21no—
15C4×C22⋊D97—21no—
16#15690769C411yescharacteristic, complemented
17C22⋊#11715895—21no—
18C22⋊#11715895C2212yes—
19C23×C4×C97C2211yescharacteristic
20C2×C4⋊C4×C97—21no—
21C2×C22⋊C4×C97—21no—
22#95×C97C2212yes—
23#13095511—42no—
24C2×Dic194C2×C411yescharacteristic
25C2×D388C2×C411yescharacteristic
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C2×C4×D97—21no—
30#13095495—42no—
31C2×#11715895—22no—
32C4⋊Dic97C2×C411yescharacteristic
33C4⋊Dic97C2×C411yescharacteristic
34#13095490—21no—
35#13095490—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C24×C97D411yescharacteristic
43C22×C4×C97—22no—
44C22×C4×C97—42no—
45C22×C4×C97C2×C411yescharacteristic
46C22×C4×C97C2×C411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97D411yescharacteristic
50C22×C4×C97C2311yescharacteristic
51C4⋊C4×C97—42no—
52C22⋊C4×C97—42no—
53C23×C97—21no—
54C23×C97—21no—
55C23×C97—41no—
56C23×C97—41no—
57C23×C97C22⋊C411yescharacteristic
58C23×C97C22⋊C411yescharacteristic
59C23×C97C2×D411yescharacteristic
60C22×D97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—22no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—42no—
68C2×C4×C97—42no—
69C2×C4×C97—42no—
70C2×C4×C97C22⋊C411yescharacteristic
71C2×C4×C97C22⋊C411yescharacteristic
72C2×C4×C97C22×C411yescharacteristic
73C2×C4×C97C2×D411yescharacteristic
74C22⋊D97—42no—
75C22⋊D97—42no—
76C22⋊D97—41no—
77C22⋊D97—41no—
78C2×Dic97—21no—
79C2×Dic97—21no—
80C2×Dic97—21no—
81D388—41no—
82C4×D97—42no—
83Dic194—41no—
84#11715895—42no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97—21no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—42no—
94C22×C97—42no—
95C22×C97C2×C22⋊C411yescharacteristic
96C22×C97AGL(1,ℤ/8)11yescharacteristic
97C22×C97#13711yescharacteristic
98C2×D97—41no—
99C2×D97—41no—
100C2×D97—41no—
101C4×C97—21no—
102C4×C97—21no—
103C4×C97—21no—
104C4×C97—82no—
105C4×C97#9912yes—
106Dic97—41no—
107Dic97—41no—
108Dic97—41no—
109C2×C97—21no—
110C2×C97—21no—
111C2×C97—41no—
112C2×C97—41no—
113C2×C97C2×#9911yescharacteristic
114C2×C97#41711yescharacteristic
115C2×C97#42111yescharacteristic
116D97—81no—
117#1513—971nomaximal, Sylow
118C97#151311yescharacteristic, complemented, Sylow
119#525—971no—
120#407—971no—
121#380D9711yescharacteristic, complemented
122#328—972no—
123#327—972no—
124C2×#41—971no—
125C23×C4C2×D9711yescharacteristic
126C2×M16—1941no—
127#123—1941no—
128C22⋊Q8—971no—
129C22⋊Q8—971no—
130#121—971no—
131#121—971no—
132#120—1941no—
133C4×D4—1941no—
134C4×D4—1941no—
135#117—971no—
136C2×C4⋊C4—21no—
137C2×C22⋊C4—21no—
138C22⋊C8—972no—
139C22⋊C8—1941no—
140#95C2×D9712yes—
141C24C22⋊D9711yescharacteristic
142#41—3882no—
143C2×Q8—971no—
144C2×D4—971no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4—1941no—
148C22×C4—22no—
149C22×C4—42no—
150C22×C4—1941no—
151C22×C4C4×D9711yescharacteristic
152C22×C4C4×D9711yescharacteristic
153C22×C4D38811yescharacteristic
154C22×C4D38811yescharacteristic
155C22×C4C22⋊D9711yescharacteristic
156C22×C4C22×D9711yescharacteristic
157M16—3882no—
158C2×C8—1942no—
159C4⋊C4—42no—
160C4⋊C4—971no—
161C4⋊C4—971no—
162C4⋊C4—1941no—
163C4⋊C4—1941no—
164C22⋊C4—42no—
165C22⋊C4—1941no—
166C22⋊C4—1941no—
167C22⋊C4—1941no—
168C22⋊C4—1941no—
169C22⋊C4—1941no—
170C42—1941no—
171C23—21no—
172C23—21no—
173C23—41no—
174C23—41no—
175C23—1941no—
176C23#1309548811yescharacteristic
177C23#1309548811yescharacteristic
178C23C2×C22⋊D9711yescharacteristic
179Q8—3881no—
180D4—3881no—
181D4—3882no—
182D4—3882no—
183D4—3881no—
184D4—3881no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—3882no—
195C2×C4—1941no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4#1309548811yescharacteristic
199C2×C4#1309548811yescharacteristic
200C2×C4C2×C4×D9711yescharacteristic
201C2×C4C2×D38811yescharacteristic
202C8—3882no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—3881no—
214C22—3881no—
215C22—3881no—
216C22C2×#1309548811yescharacteristic
217C22#1569083611yescharacteristic
218C22#1569084111yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—82no—
223C4—3881no—
224C4—3881no—
225C4—3881no—
226C4#1569072412yes—
227C2—21no—
228C2—21no—
229C2—41no—
230C2—41no—
231C2—7761no—
232C2C2×#1569072411yescharacteristic
233C2#2026073411yescharacteristic
234C2#2026075311yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#380×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenter#380D97
hypocenterC97#1513
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #20260699C2semidirect
3. #20260471C2semidirect
4. #20260468C2semidirect
5. #380×C97C2semidirect
6. C2×#13095511C4semidirect
9. #15690789C4semidirect
11. #15690784C4semidirect
16. #15690769C4semidirect
118. C97#1513semidirect
121. #380D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202606992
#202604712
#202604682
#380×C972
#151397