Finite Group Databasebeta

← #27140688

Subgroups — #27140688

Order 12416.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#20260873C212yesmaximal, complemented
2#20260647C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#534×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic, complemented
7C2×#13095499—22no—
8C2×C22⋊Dic97—21no—
9#15690733—22no—
10C22⋊#11715895C2212yescomplemented
11#15690717—21no—
12#15690717C2211yescharacteristic, complemented
13C22×D4×C97—21no—
14#123×C97—22no—
15#121×C97—21no—
16#121×C97C2212yes—
17#120×C97—22no—
18C2×C22⋊C4×C97C2211yescharacteristic
19C2×Dic194—22no—
20C2×Dic194—42no—
21C22×Dic97—21no—
22C22×Dic97—21no—
23#13095499—44no—
24C2×#11715895—22no—
25C22⋊Dic97—42no—
26C24×C97—21no—
27C24×C97D411yescharacteristic
28C2×D4×C97—22no—
29C2×D4×C97—22no—
30C2×D4×C97—21no—
31C2×D4×C97—42no—
32C2×D4×C97—42no—
33C2×D4×C97D412yes—
34C22×C4×C97—21no—
35C22×C4×C97D411yescharacteristic
36C22×C4×C97C2311yescharacteristic
37C4⋊C4×C97—21no—
38C4⋊C4×C97D412yes—
39C22⋊C4×C97—22no—
40C22⋊C4×C97—22no—
41C22⋊C4×C97—22no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97—22no—
45C23×C97—41no—
46C23×C97—41no—
47C23×C97—42no—
48C23×C97—81no—
49C23×C97—81no—
50C23×C97SD1612yescomplemented
51C23×C97C2×D411yescharacteristic
52D4×C97—42no—
53D4×C97—44no—
54D4×C97—42no—
55D4×C97—42no—
56D4×C97—81no—
57C2×C4×C97—21no—
58C2×C4×C97—22no—
59C2×C4×C97—41no—
60C2×C4×C97—42no—
61C2×C4×C97—41no—
62C2×C4×C97C2×D412yes—
63C2×Dic97—22no—
64C2×Dic97—41no—
65C2×Dic97—41no—
66C2×Dic97—41no—
67C2×Dic97—41no—
68C2×Dic97—41no—
69Dic194—24no—
70Dic194—42no—
71#11715895—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—22no—
75C22×C97—42no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97—42no—
81C22×C97—81no—
82C22×C97—84no—
83C22×C97#12011yescharacteristic
84C22×C97C2×SD1611yescharacteristic
85C22×C97AGL(1,ℤ/8)11yescharacteristic
86C4×C97—22no—
87C4×C97—42no—
88C4×C97—41no—
89Dic97—42no—
90Dic97—81no—
91C2×C97—21no—
92C2×C97—21no—
93C2×C97—42no—
94C2×C97—82no—
95C2×C97#45011yescharacteristic
96C2×C97#45511yescharacteristic
97C2×C97B(4,𝔽2)11yescharacteristic
98#1601—971nomaximal, Sylow
99C97#160111yescharacteristic, complemented, Sylow
100#534D9711yescharacteristic
101#448—972no—
102#393—971no—
103#327—972no—
104#323—971no—
105C22×Q8—971no—
106C22×D4—21no—
107C2×SD16—1942no—
108#123—22no—
109#121—21no—
110#121C2×D9712yes—
111#120—22no—
112C2×C22⋊C4—1941no—
113C2×C22⋊C4C2×D9711yescharacteristic
114#103—1942no—
115C22⋊C8—972no—
116#95—971no—
117#95—1941no—
118C24—21no—
119C24C22⋊D9711yescharacteristic
120C2×Q8—3882no—
121C2×Q8—1942no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4—42no—
127C2×D4C22⋊D9712yes—
128C22×C4—21no—
129C22×C4—1941no—
130C22×C4—1941no—
131C22×C4C22⋊D9711yescharacteristic
132C22×C4C22×D9711yescharacteristic
133SD16—3884no—
134C2×C8—1942no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D9712yes—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—3882no—
141C23—21no—
142C23—21no—
143C23—22no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#1309549912yescomplemented
150C23C2×C22⋊D9711yescharacteristic
151Q8—3882no—
152Q8—1944no—
153D4—42no—
154D4—44no—
155D4—42no—
156D4—42no—
157D4—81no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—3881no—
164C2×C4—3881no—
165C2×C4—3881no—
166C2×C4—3881no—
167C2×C4—3881no—
168C2×C4—1942no—
169C2×C4C2×C22⋊D9712yes—
170C8—3882no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—81no—
181C22—84no—
182C22#1569077911yescharacteristic
183C22C2×#1309549911yescharacteristic
184C22#1569083611yescharacteristic
185C4—22no—
186C4—42no—
187C4—41no—
188C4—7761no—
189C4—3882no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—82no—
194C2#2026088611yescharacteristic
195C2#2026091311yescharacteristic
196C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#534×C97C2
socleC22×C97#120
hypercenter#534D97
hypocenterC97#1601
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #20260873C2semidirect
2. #20260647C2semidirect
3. #20260469C2semidirect
4. #20260459C2semidirect
6. C22×Dic194C22semidirect
10. C22⋊#11715895C22semidirect
12. #15690717C22semidirect
50. C23×C97SD16semidirect
99. C97#1601semidirect
149. C23#13095499semidirect

Maximal subgroups

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

subgroupindex
#202608732
#202606472
#202604692
#202604592
#534×C972
#160197