Finite Group Databasebeta

← #27140731

Subgroups — #27140731

Order 12416.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261332C211yesmaximal, characteristic
2#20261065C211yesmaximal, characteristic, complemented
3#20261048C211yesmaximal, characteristic
4#20260471C211yesmaximal, characteristic, complemented
5#20260468C211yesmaximal, characteristic
6#20260461C211yesmaximal, characteristic, complemented
7#398×C97C211yesmaximal, characteristic, complemented
8#15690805—21no—
9#15690800—21no—
10#15690795—21no—
11#15690793—21no—
12#15690789C2211yescharacteristic
13#15690788—21no—
14#15690784C2211yescharacteristic
15C4×C22⋊D97—21no—
16#15690743—21no—
17#15690741—21no—
18#15690733—21no—
19#15690730—21no—
20C22⋊#11715895C2211yescharacteristic
21C22⋊#11715895C2211yescharacteristic
22C2×C4⋊C4×C97—21no—
23C2×C4⋊C4×C97C2211yescharacteristic
24C2×C4⋊C4×C97C2211yescharacteristic
25#95×C97—21no—
26#95×C97C2211yescharacteristic
27C2×Dic194D411yescharacteristic
28C2×D388D411yescharacteristic
29C2×C22⋊D97—21no—
30C2×C4×D97—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97D411yescharacteristic
34C4⋊Dic97D411yescharacteristic
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39C22⋊Dic97—21no—
40#13095488—21no—
41#13095488—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97—21no—
45C22×C4×C97—21no—
46C22×C4×C97—21no—
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97—21no—
54C4⋊C4×C97—42no—
55C23×C97C2×D411yescharacteristic
56C22×D97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—41no—
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C22⋊D97—41no—
72C22⋊D97—41no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76D388—41no—
77C4×D97—81no—
78Dic194—41no—
79#11715895—41no—
80#11715895—41no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97#12011yescharacteristic
84C22×C97#13511yescharacteristic
85C22×C97AGL(1,ℤ/8)11yescharacteristic
86C2×D97—41no—
87C2×D97—41no—
88C2×D97—41no—
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—42no—
92C4×C97—41no—
93C4×C97—41no—
94C4×C97—81no—
95Dic97—41no—
96Dic97—41no—
97Dic97—41no—
98C2×C97—21no—
99C2×C97—21no—
100C2×C97#44911yescharacteristic
101C2×C97#45611yescharacteristic
102C2×C97#45811yescharacteristic
103D97—81no—
104#1609—971nomaximal, Sylow
105C97#160911yescharacteristic, complemented, Sylow
106#539—971no—
107#483—971no—
108#480—971no—
109#398D9711yescharacteristic, complemented
110#328—971no—
111#327—971no—
112#324—971no—
113#126—1941no—
114#125—1941no—
115#123—1941no—
116#123—1941no—
117C22⋊Q8—971no—
118C22⋊Q8—1941no—
119#121—971no—
120C4×D4—1941no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×D9711yescharacteristic
123C2×C4⋊C4C2×D9711yescharacteristic
124#107—1941no—
125#106—1941no—
126#103—1941no—
127#102—1941no—
128C22⋊C8—971no—
129C22⋊C8—971no—
130#95—21no—
131#95C2×D9711yescharacteristic
132C2×Q8—971no—
133C2×D4—971no—
134C2×D4—1941no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—1941no—
138C22×C4C22⋊D9711yescharacteristic
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22×D9711yescharacteristic
141C2×C8—1941no—
142C2×C8—1941no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—971no—
149C4⋊C4—971no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C22⋊C4—1941no—
159C42—1941no—
160C23—1941no—
161C23C2×C22⋊D9711yescharacteristic
162Q8—3881no—
163D4—3881no—
164D4—3881no—
165D4—3881no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—7761no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4D4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C8—3881no—
185C8—3881no—
186C22—21no—
187C22—21no—
188C22—3881no—
189C22—3881no—
190C22—3881no—
191C22#1569077811yescharacteristic
192C22#1569083011yescharacteristic
193C22#1569083611yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—41no—
199C4—81no—
200C4—3881no—
201C4—3881no—
202C4—3881no—
203C2—21no—
204C2—21no—
205C2—7761no—
206C2#2026087811yescharacteristic
207C2#2026091711yescharacteristic
208C2#2026092411yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#398×C97C2
socleC22×C97#120
prime radical#15690784C22
hypercenter#398D97
hypocenterC97#1609
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #20261065C2semidirect
4. #20260471C2semidirect
6. #20260461C2semidirect
7. #398×C97C2semidirect
105. C97#1609semidirect
109. #398D97semidirect

Maximal subgroups

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

subgroupindex
#202613322
#202610652
#202610482
#202604712
#202604682
#202604612
#398×C972
#160997