Finite Group Databasebeta

← #27140670

Subgroups — #27140670

Order 12416.

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

normal subgroups only

This group has 310 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#20261316C211yesmaximal, characteristic, complemented
2#20260869C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#394×C97C211yesmaximal, characteristic, complemented
6C2×#13095511—21no—
7C2×#13095496—22no—
8#15690790—22no—
9#15690785—22no—
10#15690784—21no—
11#15690784C2212yes—
12#15690731—22no—
13C22⋊#11715895C2212yes—
14C22×D4×C97C2211yescharacteristic
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C22⋊C4×C97—21no—
17C2×C22⋊C4×C97—21no—
18#95×C97C2211yescharacteristic
19#13095511—42no—
20#13095511—42no—
21C2×Dic194—21no—
22C2×D388—21no—
23C2×D388—21no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C2×C4×D97—22no—
27#13095496—44no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97D412yes—
31#13095488—22no—
32#13095488—22no—
33C24×C97—21no—
34C2×D4×C97—22no—
35C2×D4×C97—42no—
36C22×C4×C97—21no—
37C22×C4×C97D411yescharacteristic
38C22×C4×C97D411yescharacteristic
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—22no—
41C22⋊C4×C97—42no—
42C22⋊C4×C97—42no—
43C23×C97—22no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97—41no—
48C23×C97—41no—
49C23×C97—41no—
50C23×C97C2×D411yescharacteristic
51C22×D97—22no—
52D4×C97—24no—
53D4×C97—42no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97D812yescomplemented
63C2×C4×C97C2×D412yes—
64C22⋊D97—42no—
65C22⋊D97—42no—
66C2×Dic97—22no—
67D388—42no—
68D388—42no—
69D388—81no—
70C4×D97—44no—
71Dic194—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97—81no—
79C22×C97—81no—
80C22×C97—81no—
81C22×C97—81no—
82C22×C97#12011yescharacteristic
83C22×C97C2×D811yescharacteristic
84C22×C97AGL(1,ℤ/8)11yescharacteristic
85C2×D97—42no—
86C2×D97—42no—
87C2×D97—42no—
88C4×C97—22no—
89C4×C97—22no—
90C4×C97—41no—
91C4×C97—81no—
92Dic97—42no—
93C2×C97—21no—
94C2×C97—21no—
95C2×C97—82no—
96C2×C97#44711yescharacteristic
97C2×C97#45411yescharacteristic
98C2×C97B(4,𝔽2)11yescharacteristic
99D97—82no—
100#1598—971nomaximal, Sylow
101C97#159811yescharacteristic, complemented, Sylow
102#537—971no—
103#447—972no—
104#394D9711yescharacteristic, complemented
105#327—972no—
106#324—971no—
107C2×#41—1941no—
108C22×D4C2×D9711yescharacteristic
109C2×D8—1942no—
110C22⋊Q8—1942no—
111#121—972no—
112#121—1942no—
113#121—1941no—
114C2×C4⋊C4C2×D9711yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117#102—1942no—
118C22⋊C8—972no—
119#95C2×D9711yescharacteristic
120C24—21no—
121#41—3882no—
122#41—3882no—
123C2×Q8—1941no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4—972no—
127C2×D4—1941no—
128C2×D4—1941no—
129C2×D4—1942no—
130C22×C4—21no—
131C22×C4—1942no—
132C22×C4C22⋊D9711yescharacteristic
133C22×C4C22⋊D9711yescharacteristic
134C22×C4C22×D9711yescharacteristic
135D8—3884no—
136C2×C8—1942no—
137C4⋊C4—22no—
138C4⋊C4—972no—
139C4⋊C4—1941no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—1942no—
143C22⋊C4—1942no—
144C23—22no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—1942no—
152C23C2×C22⋊D9711yescharacteristic
153Q8—3882no—
154D4—24no—
155D4—42no—
156D4—7761no—
157D4—3882no—
158D4—3882no—
159D4—3882no—
160D4—3882no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—3884no—
170C2×C4—1942no—
171C2×C4#1309549612yescomplemented
172C2×C4D4×D9712yes—
173C8—3882no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—81no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—3882no—
184C22—3882no—
185C22—3882no—
186C22#1569077811yescharacteristic
187C22C2×#1309549611yescharacteristic
188C22#1569083511yescharacteristic
189C4—22no—
190C4—22no—
191C4—41no—
192C4—81no—
193C4—3882no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—7762no—
198C2#2026086811yescharacteristic
199C2#2026090711yescharacteristic
200C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#394×C97C2
socleC22×C97#120
hypercenter#394D97
hypocenterC97#1598
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #20261316C2semidirect
2. #20260869C2semidirect
3. #20260468C2semidirect
4. #20260461C2semidirect
5. #394×C97C2semidirect
62. C2×C4×C97D8semidirect
101. C97#1598semidirect
104. #394D97semidirect
171. C2×C4#13095496semidirect

Maximal subgroups

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

subgroupindex
#202613162
#202608692
#202604682
#202604612
#394×C972
#159897