Finite Group Databasebeta

← #27142043

Subgroups — #27142043

Order 12416.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15690733C212yesmaximal, complemented
3C2×C22⋊#11715895C211yesmaximal, characteristic
4#20260497C212yesmaximal, complemented
5C22×C4⋊C4×C97C211yesmaximal, characteristic
6C22×Dic194C2211yescharacteristic
7C22×#11715895C2212yes—
8#15690789—22no—
9#15690789C414yescomplemented
10C2×C4⋊Dic97C2211yescharacteristic
11C2×#13095490—21no—
12C2×C22⋊Dic97—21no—
13#15690733—24no—
14C22⋊#11715895—22no—
15C23×C4×C97—21no—
16C23×C4×C97C2211yescharacteristic
17C2×C4⋊C4×C97—21no—
18C2×C4⋊C4×C97—24no—
19C2×C4⋊C4×C97C2212yes—
20C2×Dic194—22no—
21C2×Dic194—22no—
22C2×Dic194—21no—
23C2×Dic194C2×C412yescomplemented
24C22×Dic97—21no—
25C22×Dic97—21no—
26C2×#11715895—22no—
27C2×#11715895—22no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97C2×C412yescomplemented
31#13095490—24no—
32C22⋊Dic97—24no—
33C24×C97D411yescharacteristic
34C22×C4×C97—22no—
35C22×C4×C97—22no—
36C22×C4×C97—24no—
37C22×C4×C97—22no—
38C22×C4×C97—42no—
39C22×C4×C97—42no—
40C22×C4×C97C2×C412yes—
41C22×C4×C97D412yes—
42C22×C4×C97D411yescharacteristic
43C22×C4×C97C2311yescharacteristic
44C4⋊C4×C97—24no—
45C4⋊C4×C97—22no—
46C4⋊C4×C97D414yes—
47C23×C97—21no—
48C23×C97—21no—
49C23×C97—21no—
50C23×C97—21no—
51C23×C97C22⋊C412yes—
52C23×C97SD1612yes—
53C23×C97Q1612yes—
54C23×C97C2×D411yescharacteristic
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—24no—
58C2×C4×C97—24no—
59C2×C4×C97—24no—
60C2×C4×C97—48no—
61C2×C4×C97—42no—
62C2×C4×C97—42no—
63C2×C4×C97C22⋊C412yes—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D412yes—
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97#4112yes—
68C2×Dic97—22no—
69C2×Dic97—22no—
70C2×Dic97—41no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97—41no—
74Dic194—24no—
75Dic194—42no—
76#11715895—42no—
77C22×C97—22no—
78C22×C97—22no—
79C22×C97—22no—
80C22×C97—22no—
81C22×C97—22no—
82C22×C97—22no—
83C22×C97#10314yescomplemented
84C22×C97C2×C22⋊C411yescharacteristic
85C22×C97#12011yescharacteristic
86C22×C97#12311yescharacteristic
87C22×C97C2×SD1611yescharacteristic
88C22×C97C2×Q1611yescharacteristic
89C22×C97AGL(1,ℤ/8)11yescharacteristic
90C22×C97#13711yescharacteristic
91C4×C97—22no—
92C4×C97—44no—
93C4×C97C4×D412yes—
94C4×C97#12112yes—
95Dic97—42no—
96Dic97—42no—
97C2×C97—24no—
98C2×C97#38611yescharacteristic
99C2×C97C2×#10311yescharacteristic
100C2×C97#41711yescharacteristic
101C2×C97#45011yescharacteristic
102C2×C97#45111yescharacteristic
103C2×C97#48311yescharacteristic
104C2×C97#48411yescharacteristic
105#1872—971nomaximal, Sylow
106C97#187211yescharacteristic, complemented, Sylow
107C2×C22⋊Q8—971no—
108C22×C4⋊C4D9711yescharacteristic
109C2×#103—972no—
110C2×C22⋊C8—971no—
111#340—972no—
112C22×Q8—971no—
113C23×C4—21no—
114C23×C4C2×D9711yescharacteristic
115C22×C8—972no—
116C22⋊Q8—974no—
117C22⋊Q8—1942no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4—1941no—
122C2×C4⋊C4C2×D9712yes—
123C2×C22⋊C4—1941no—
124#103—1944no—
125C22⋊C8—1942no—
126C24C22⋊D9711yescharacteristic
127C2×Q8—972no—
128C2×Q8—1942no—
129C2×Q8—1942no—
130C2×Q8—1941no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—24no—
134C22×C4—22no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—1941no—
138C22×C4—1941no—
139C22×C4C4×D9712yes—
140C22×C4D38812yes—
141C22×C4C22⋊D9711yescharacteristic
142C22×C4C22×D9711yescharacteristic
143C2×C8—1942no—
144C2×C8—1942no—
145C2×C8—1942no—
146C4⋊C4—24no—
147C4⋊C4—22no—
148C4⋊C4—972no—
149C4⋊C4—1944no—
150C4⋊C4—1941no—
151C4⋊C4C22⋊D9714yes—
152C22⋊C4—1944no—
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23#1309548812yes—
158C23#1309549912yes—
159C23#1309550112yes—
160C23C2×C22⋊D9711yescharacteristic
161Q8—3882no—
162Q8—1944no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—48no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—1942no—
176C2×C4—1942no—
177C2×C4#1309548812yes—
178C2×C4C2×C4×D9711yescharacteristic
179C2×C4C2×C22⋊D9712yes—
180C2×C4C2×D38811yescharacteristic
181C2×C4#1309551112yes—
182C8—3882no—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22#1569073314yescomplemented
190C22C2×#1309548811yescharacteristic
191C22#1569077911yescharacteristic
192C22#1569079311yescharacteristic
193C22C2×#1309549911yescharacteristic
194C22C2×#1309550111yescharacteristic
195C22#1569083611yescharacteristic
196C22#1569084111yescharacteristic
197C4—22no—
198C4—44no—
199C4—3882no—
200C4—3882no—
201C4C4×C22⋊D9712yes—
202C4#1569078412yes—
203C2—24no—
204C2#2026061411yescharacteristic
205C2C2×#1569073311yescharacteristic
206C2#2026073411yescharacteristic
207C2#2026088611yescharacteristic
208C2#2026089111yescharacteristic
209C2#2026106511yescharacteristic
210C2#2026107011yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×C4⋊C4×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
cosocleC22×C4×C97C23
hypercenterC22×C4⋊C4D97
hypocenterC97#1872
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15690733C2semidirect
4. #20260497C2semidirect
9. #15690789C4semidirect
23. C2×Dic194C2×C4semidirect
30. C4⋊Dic97C2×C4semidirect
83. C22×C97#103semidirect
106. C97#1872semidirect
189. C22#15690733semidirect

Maximal subgroups

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

subgroupindex
C2×#156907892
C2×#156907332
C2×C22⋊#117158952
#202604972
C22×C4⋊C4×C972
#187297