Finite Group Databasebeta

← #27157999

Subgroups — #27157999

Order 12416.

234 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261722C211yesmaximal, characteristic
2#20261715C211yesmaximal, characteristic
3#20261638C212yesmaximal, complemented
4#20261637C212yesmaximal, complemented
5#20261307C211yesmaximal, characteristic
6C2×#13095520—21no—
7#15690889—21no—
8#15690871—21no—
9#15690870C2212yes—
10#15690868—21no—
11#15690868C2212yes—
12C2×#13095510C2211yescharacteristic
13C22×Dic194C411yescharacteristic, complemented
14#15690809—21no—
15#15690796—21no—
16#15690794C2211yescharacteristic
17#15690793C2211yescharacteristic
18#15690791C411yescharacteristic, complemented
19#15690788—21no—
20#15690787—21no—
21#15690766C411yescharacteristic, complemented
22#123×C97C411yescharacteristic, complemented
23#13095520—22no—
24#13095520—41no—
25C2×#11715897—41no—
26C4⋊#87604—41no—
27C22⋊#87604—22no—
28C4×#87604—41no—
29#13095510—41no—
30#13095510—41no—
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194—41no—
34C2×Dic194—41no—
35C2×Dic194D411yescharacteristic
36C2×C22⋊D97D411yescharacteristic
37C2×C22⋊D97C2311yescharacteristic
38C22×Dic97C2×C411yescharacteristic
39C22×Dic97C2×C411yescharacteristic
40C22×Dic97D411yescharacteristic
41C2×C4×D97D411yescharacteristic
42C4⋊Dic97—21no—
43#13095490—21no—
44#13095490—21no—
45#13095490—21no—
46C22⋊Dic97—21no—
47C22⋊Dic97C2×C411yescharacteristic
48#13095488—21no—
49#13095488—21no—
50C4×Dic97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic
53C4⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97C2×C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C22×D97C2×D411yescharacteristic
59C2×#87604—42no—
60D4×C97—41no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97C22⋊C411yescharacteristic
65C22⋊D97—41no—
66C22⋊D97—41no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—21no—
72C2×Dic97—21no—
73C2×Dic97—41no—
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C411yescharacteristic
76C2×Dic97C2×D411yescharacteristic
77C4×D97—21no—
78C4×D97—21no—
79Dic194—41no—
80Dic194—42no—
81Dic194—41no—
82Dic194—41no—
83#11715897—42no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97C2×C22⋊C411yescharacteristic
89C2×D97—21no—
90C2×D97—41no—
91#87604—82no—
92C4×C97—21no—
93C4×C97—41no—
94C4×C97—41no—
95Dic97—21no—
96Dic97—21no—
97Dic97—41no—
98Dic97—41no—
99Dic97#9911yescharacteristic
100Dic97#9911yescharacteristic
101C2×C97—21no—
102C2×C97—41no—
103C2×C97—41no—
104C2×C97C2×#9911yescharacteristic
105D97—41no—
106#2123—971nomaximal, Sylow
107C97#212311yescharacteristic, complemented, Sylow
108#536—971no—
109#413—971no—
110#410—971no—
111#352—972no—
112#352—972no—
113C2×#41—971no—
114C22×Q8—971no—
115C2×M16—1941no—
116C4⋊Q8—1941no—
117#124—1941no—
118#123—971no—
119#123—971no—
120#123—971no—
121#123#8760411yescharacteristic, complemented
122C22⋊Q8—1941no—
123C22⋊Q8—1941no—
124#117—971no—
125#117—1941no—
126#101—1941no—
127#100—972no—
128#99—972no—
129#99—1941no—
130#41—3881no—
131#41—3881no—
132C2×Q8—971no—
133C2×Q8—3881no—
134C2×Q8—3881no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×D4—971no—
138C2×D4—971no—
139C2×D4C2×#8760411yescharacteristic
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4C2×#8760411yescharacteristic
145M16—3881no—
146M16—1942no—
147C2×C8—3881no—
148C4⋊C4—21no—
149C4⋊C4—3881no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C22⋊C4—21no—
155C22⋊C4—971no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C22⋊C4—1942no—
159C22⋊C4—1941no—
160C22⋊C4C2×#8760411yescharacteristic
161C42—3881no—
162C42—1941no—
163C23—971no—
164C23C22⋊#8760411yescharacteristic
165C23C22×#8760411yescharacteristic
166Q8—3881no—
167Q8—3881no—
168Q8—3881no—
169Q8—3882no—
170D4—41no—
171D4—3881no—
172D4—3881no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—971no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—3882no—
180C2×C4—3881no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4—1941no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4C22⋊#8760411yescharacteristic
190C8—3882no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—3881no—
196C22—1941no—
197C22C2×C22⋊#8760411yescharacteristic
198C4—21no—
199C4—41no—
200C4—41no—
201C4—7762no—
202C4—971no—
203C4—971no—
204C4—3881no—
205C4—3881no—
206C4—1941no—
207C4—1941no—
208C2—21no—
209C2—41no—
210C2—41no—
211C2—3881no—
212C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#123×C97C4
socleC2×C97C2×#99
prime radicalC2×#13095510C22
cosocleC2×C22⋊D97C23
hypercenter#123#87604
hypocenterC97#2123
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #20261638C2semidirect
4. #20261637C2semidirect
13. C22×Dic194C4semidirect
18. #15690791C4semidirect
21. #15690766C4semidirect
22. #123×C97C4semidirect
107. C97#2123semidirect
121. #123#87604semidirect

Maximal subgroups

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

subgroupindex
#202617222
#202617152
#202616382
#202616372
#202613072
#212397