Finite Group Databasebeta

← #27140755

Subgroups — #27140755

Order 12416.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261297C211yesmaximal, characteristic
2#20261053C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#399×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97—21no—
7#15690794—22no—
8#15690784—21no—
9#15690784C2212yes—
10#15690780—22no—
11#15690741—22no—
12#15690730—22no—
13C22⋊#11715895C2212yescomplemented
14C2×C4⋊C4×C97—21no—
15C2×C4⋊C4×C97C2211yescharacteristic, complemented
16C2×C22⋊C4×C97—21no—
17C2×C22⋊C4×C97C2211yescharacteristic
18#95×C97C2211yescharacteristic, complemented
19C23×D97—21no—
20C2×D388—21no—
21C2×D388D412yes—
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97—42no—
25C2×C22⋊D97—42no—
26C22×Dic97—21no—
27C2×#11715895—22no—
28C4⋊Dic97—21no—
29C4⋊Dic97D412yes—
30#13095488—22no—
31#13095488—22no—
32C24×C97D411yescharacteristic
33C22×C4×C97—21no—
34C22×C4×C97—21no—
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C4⋊C4×C97—22no—
38C4⋊C4×C97—42no—
39C22⋊C4×C97—22no—
40C22⋊C4×C97—42no—
41C23×C97—21no—
42C23×C97—21no—
43C23×C97—41no—
44C23×C97—41no—
45C23×C97SD1612yes—
46C23×C97C2×D411yescharacteristic
47C22×D97—22no—
48C22×D97—42no—
49C22×D97—81no—
50C22×D97—81no—
51C2×C4×C97—22no—
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97C2×D412yes—
61C22⋊D97—44no—
62C22⋊D97—42no—
63C22⋊D97—42no—
64C2×Dic97—22no—
65C2×Dic97—42no—
66D388—42no—
67D388—81no—
68#11715895—42no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—22no—
72C22×C97—42no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—81no—
76C22×C97#12011yescharacteristic
77C22×C97C2×SD1611yescharacteristic
78C22×C97AGL(1,ℤ/8)11yescharacteristic
79C2×D97—42no—
80C2×D97—42no—
81C2×D97—42no—
82C2×D97—84no—
83C4×C97—22no—
84C4×C97—42no—
85C4×C97—41no—
86C4×C97—81no—
87Dic97—42no—
88C2×C97—21no—
89C2×C97—21no—
90C2×C97—42no—
91C2×C97#45011yescharacteristic
92C2×C97#45311yescharacteristic
93C2×C97#45811yescharacteristic
94D97—82no—
95#1613—971nomaximal, Sylow
96C97#161311yescharacteristic, complemented, Sylow
97#534—971no—
98#481—972no—
99#399D9711yescharacteristic, complemented
100#327—972no—
101#323—971no—
102C22×D4—1941no—
103#123—1942no—
104#121—972no—
105#121—1941no—
106#120—1942no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D9711yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D9711yescharacteristic
111#106—1942no—
112#102—1942no—
113C22⋊C8—972no—
114#95C2×D9711yescharacteristic, complemented
115C24—1941no—
116C24C22⋊D9711yescharacteristic
117C2×D4—972no—
118C2×D4—3882no—
119C2×D4—3882no—
120C2×D4—1941no—
121C2×D4—1942no—
122C2×D4—1942no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—1941no—
126C22×C4C22⋊D9711yescharacteristic
127C22×C4C22×D9711yescharacteristic
128C2×C8—1942no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—972no—
132C4⋊C4—1941no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—1942no—
136C22⋊C4—1942no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—7761no—
142C23—7761no—
143C23—3882no—
144C23—1942no—
145C23#1309549812yes—
146C23C2×C22⋊D9711yescharacteristic
147D4—7761no—
148D4—3882no—
149D4—3884no—
150D4—3882no—
151D4—3882no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—3882no—
162C2×C4—1942no—
163C2×C4D4×D9712yes—
164C8—3882no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—7764no—
173C22—3882no—
174C22—3882no—
175C22—3882no—
176C22#1569077811yescharacteristic
177C22C2×#1309549811yescharacteristic
178C22#1569083511yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—3882no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—7762no—
188C2#2026088511yescharacteristic
189C2#2026090211yescharacteristic
190C2#2026092411yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#399×C97C2
socleC22×C97#120
prime radical#20261297C2
hypercenter#399D97
hypocenterC97#1613
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #20261053C2semidirect
3. #20260468C2semidirect
4. #20260459C2semidirect
5. #399×C97C2semidirect
13. C22⋊#11715895C22semidirect
15. C2×C4⋊C4×C97C22semidirect
18. #95×C97C22semidirect
96. C97#1613semidirect
99. #399D97semidirect
108. C2×C4⋊C4C2×D97semidirect
114. #95C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612972
#202610532
#202604682
#202604592
#399×C972
#161397