Finite Group Databasebeta

← #27142204

Subgroups — #27142204

Order 12416.

338 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#15690789C211yesmaximal, characteristic
2C2×#15690728C212yesmaximal, complemented
3#20260699C211yesmaximal, characteristic
4#20260503C212yesmaximal, complemented
5C22×M16×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C2×#13095495C2212yes—
8#15690789—21no—
9#15690789—21no—
10#15690789—21no—
11#15690789—21no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490C412yes—
14C2×C22⋊Dic97C412yes—
15#15690728—24no—
16C22⋊#11715895—21no—
17C22⋊#11715895—21no—
18C23×C4×C97C2211yescharacteristic
19C2×M16×C97—21no—
20C2×M16×C97—24no—
21C2×M16×C97C2212yes—
22C22×C8×C97—21no—
23C2×Dic194—21no—
24C2×Dic194—21no—
25C2×Dic194—22no—
26C2×Dic194—22no—
27C22×Dic97C2×C412yes—
28C22×Dic97C2×C412yes—
29#13095495—22no—
30#13095495—22no—
31C2×#11715895—22no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34#13095490—22no—
35#13095490—22no—
36C22⋊Dic97—22no—
37C22⋊Dic97—22no—
38C24×C97C2×C411yescharacteristic
39C22×C4×C97—22no—
40C22×C4×C97—22no—
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44C22×C4×C97D411yescharacteristic
45C22×C4×C97D411yescharacteristic
46C22×C4×C97C2311yescharacteristic
47M16×C97—24no—
48M16×C97—22no—
49M16×C97D414yes—
50C2×C8×C97—24no—
51C2×C8×C97—22no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97C22⋊C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C2×D412yes—
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97#4112yes—
71C2×Dic97—22no—
72C2×Dic97—22no—
73C2×Dic97—24no—
74C2×Dic97—42no—
75Dic194—41no—
76Dic194—42no—
77Dic194—41no—
78C8×C97—24no—
79#11715895—42no—
80C22×C97—21no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—44no—
88C22×C97#10112yescomplemented
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97C4×D412yes—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97#12011yescharacteristic
94C4×C97#12112yes—
95C4×C97#12311yescharacteristic
96Dic97—42no—
97Dic97—42no—
98C2×C97—22no—
99C2×C97—22no—
100C2×C97—41no—
101C2×C97#38611yescharacteristic
102C2×C97C2×#10111yescharacteristic
103C2×C97#41311yescharacteristic
104#1901—971nomaximal, Sylow
105C97#190111yescharacteristic, complemented, Sylow
106C22×M16D9711yescharacteristic
107C2×C22⋊Q8—971no—
108C2×#101—972no—
109#407—971no—
110#343—972no—
111C22×Q8—971no—
112C23×C4C2×D9711yescharacteristic
113C2×M16—21no—
114C2×M16—24no—
115C2×M16—972no—
116C2×M16C2×D9712yes—
117C22×C8—21no—
118C22⋊Q8—1941no—
119C22⋊Q8—1941no—
120C22⋊Q8—1941no—
121C22⋊Q8—1941no—
122C2×C4⋊C4—972no—
123C2×C4⋊C4—971no—
124C2×C22⋊C4—972no—
125#101—1944no—
126C22⋊C8—1941no—
127C22⋊C8—1941no—
128C24C4×D9711yescharacteristic
129C2×Q8—1942no—
130C2×Q8—1941no—
131C2×Q8—1941no—
132C2×Q8—1942no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—972no—
136C22×C4—972no—
137C22×C4C4×D9711yescharacteristic
138C22×C4D38811yescharacteristic
139C22×C4D38811yescharacteristic
140C22×C4C22⋊D9711yescharacteristic
141C22×C4C22⋊D9711yescharacteristic
142C22×C4C22×D9711yescharacteristic
143M16—24no—
144M16—22no—
145M16—1942no—
146M16—1942no—
147M16C22⋊D9714yes—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—1942no—
151C4⋊C4—1942no—
152C4⋊C4—1942no—
153C4⋊C4—1941no—
154C4⋊C4—1941no—
155C22⋊C4—1942no—
156C22⋊C4—1942no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23#1309548811yescharacteristic
161C23#1309548811yescharacteristic
162C23C2×C4×D9711yescharacteristic
163Q8—3881no—
164Q8—3882no—
165Q8—3881no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—3882no—
174C2×C4—1942no—
175C2×C4—1942no—
176C2×C4—1944no—
177C2×C4#1309548811yescharacteristic
178C2×C4#1309548811yescharacteristic
179C2×C4C2×C22⋊D9712yes—
180C2×C4C2×C22⋊D9711yescharacteristic
181C2×C4C2×D38811yescharacteristic
182C2×C4#1309551112yes—
183C8—24no—
184C8—3882no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1569072812yescomplemented
194C22C2×#1309548811yescharacteristic
195C22C4×C22⋊D9712yes—
196C4—21no—
197C4—21no—
198C4—3882no—
199C4—3882no—
200C4#1569077911yescharacteristic
201C4#1569078412yes—
202C4#1569079311yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2026061411yescharacteristic
207C2C2×#1569072811yescharacteristic
208C2#2026071711yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×M16×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenterC22×M16D97
hypocenterC97#1901
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15690728C2semidirect
4. #20260503C2semidirect
88. C22×C97#101semidirect
105. C97#1901semidirect
193. C22#15690728semidirect

Maximal subgroups

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

subgroupindex
C2×#156907892
C2×#156907282
#202606992
#202605032
C22×M16×C972
#190197