Finite Group Databasebeta

← #27142896

Subgroups — #27142896

Order 12416.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261363C211yesmaximal, characteristic
2#20260745C211yesmaximal, characteristic
3#20260743C211yesmaximal, characteristic
4#20260641C211yesmaximal, characteristic
5#20260503C211yesmaximal, characteristic
6#20260495C211yesmaximal, characteristic
7#339×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095495C2211yescharacteristic
10#15690810—21no—
11#15690809—21no—
12#15690802—21no—
13#15690801—21no—
14#15690787—22no—
15#15690769C2211yescharacteristic
16C2×C4⋊Dic97C2211yescharacteristic
17#15690735—21no—
18#15690735—21no—
19#15690733—21no—
20#15690733—21no—
21#15690717—22no—
22C2×M16×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24C2×C42×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—22no—
31C2×Dic194Q812yescomplemented
32C22×Dic97—21no—
33C22×Dic97D412yes—
34#13095495—21no—
35#13095495—21no—
36C2×#11715895—21no—
37C4⋊Dic97—22no—
38C4⋊Dic97—21no—
39C4⋊Dic97—21no—
40C4⋊Dic97D411yescharacteristic
41C4⋊Dic97D411yescharacteristic
42#13095490—22no—
43#13095490—22no—
44C22⋊Dic97—21no—
45C4×Dic97—21no—
46C22×C4×C97—21no—
47C22×C4×C97C2311yescharacteristic
48M16×C97—21no—
49M16×C97—21no—
50C2×C8×C97—21no—
51C4⋊C4×C97D411yescharacteristic
52C4⋊C4×C97D411yescharacteristic
53C22⋊C4×C97—21no—
54C42×C97—21no—
55C42×C97—21no—
56C42×C97—21no—
57C23×C97C2×D411yescharacteristic
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×Q811yescharacteristic
66C2×C4×C97#4111yescharacteristic
67C2×C4×C97#4111yescharacteristic
68C2×C4×C97#4111yescharacteristic
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—41no—
74C2×Dic97—41no—
75C2×Dic97—42no—
76Dic194—22no—
77Dic194—22no—
78Dic194—41no—
79Dic194—41no—
80Dic194—82no—
81C8×C97—41no—
82#11715895—41no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97#12011yescharacteristic
86C22×C97C22⋊Q811yescharacteristic
87C22×C97#12411yescharacteristic
88C4×C97—42no—
89C4×C97—41no—
90C4×C97—41no—
91C4×C97C22⋊Q811yescharacteristic
92C4×C97C22⋊Q811yescharacteristic
93C4×C97#12411yescharacteristic
94C4×C97#12411yescharacteristic
95Dic97—42no—
96Dic97—41no—
97Dic97—42no—
98Dic97—41no—
99C2×C97—21no—
100C2×C97—21no—
101C2×C97#39311yescharacteristic
102C2×C97#45511yescharacteristic
103C2×C97#45611yescharacteristic
104#2028—971nomaximal, Sylow
105C97#202811yescharacteristic, complemented, Sylow
106#544—971no—
107#419—971no—
108#419—971no—
109#391—971no—
110#343—971no—
111#339—971no—
112#339D9711yescharacteristic
113C22×Q8—971no—
114C2×M16—971no—
115C2×M16C2×D9711yescharacteristic
116C4⋊Q8—1941no—
117C4⋊Q8—1941no—
118#125—1941no—
119#125—1941no—
120C22⋊Q8—1942no—
121#117—971no—
122#117C2×D9711yescharacteristic
123C2×C4⋊C4—971no—
124C2×C42C2×D9711yescharacteristic
125#103—1941no—
126#103—1941no—
127#103—1941no—
128#103—1941no—
129#95—1942no—
130C2×Q8—972no—
131C2×Q8—1942no—
132C2×Q8—1941no—
133C2×Q8—1941no—
134C2×Q8—1941no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C22×C4—21no—
138C22×C4—972no—
139C22×C4—1941no—
140C22×C4C22×D9711yescharacteristic
141M16—21no—
142M16—21no—
143M16—1941no—
144M16—1941no—
145C2×C8—21no—
146C2×C8—1941no—
147C4⋊C4—971no—
148C4⋊C4—971no—
149C4⋊C4—1942no—
150C4⋊C4—1942no—
151C4⋊C4—1942no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C4⋊C4D38811yescharacteristic
155C4⋊C4D38811yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—1941no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—1941no—
162C23D4×D9711yescharacteristic
163Q8—7762no—
164Q8—3881no—
165Q8—3881no—
166Q8—1942no—
167Q8—1942no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—3882no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1942no—
179C2×C4—1942no—
180C2×C4D4×D9711yescharacteristic
181C2×C4C2×D38811yescharacteristic
182C2×C4Q8×D9711yescharacteristic
183C2×C4#1309551011yescharacteristic
184C2×C4#1309551111yescharacteristic
185C2×C4#1309551111yescharacteristic
186C8—41no—
187C8—3881no—
188C22—21no—
189C22—21no—
190C22#1569078011yescharacteristic
191C22#1569078811yescharacteristic
192C22#1569079611yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—3882no—
197C4—3882no—
198C4—3881no—
199C4—3881no—
200C4#1569078811yescharacteristic
201C4#1569079011yescharacteristic
202C4#1569079611yescharacteristic
203C4#1569079811yescharacteristic
204C2—21no—
205C2—21no—
206C2#2026064811yescharacteristic
207C2#2026091411yescharacteristic
208C2#2026091911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97#120
prime radicalC23×C97C2×D4
hypercenter#339D97
hypocenterC97#2028
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
31. C2×Dic194Q8semidirect
105. C97#2028semidirect

Maximal subgroups

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

subgroupindex
#202613632
#202607452
#202607432
#202606412
#202605032
#202604952
#339×C972
#202897