Finite Group Databasebeta

← #27157247

Subgroups — #27157247

Order 12416.

260 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261740C211yesmaximal, characteristic, complemented
2#20261736C211yesmaximal, characteristic, complemented
3#20261594C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261282C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×#13095519—21no—
8#15690877—21no—
9#15690877C2212yes—
10#15690877C2212yes—
11C4⋊#11715897—21no—
12C2×#13095511—21no—
13#15690809C2211yescharacteristic
14#15690806C2211yescharacteristic
15#15690796—21no—
16#15690785C411yescharacteristic
17#15690783—21no—
18#15690781C411yescharacteristic
19C4×D388C411yescharacteristic
20C4×C22⋊D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic
23#13095521—42no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27#13095511—42no—
28#13095511—42no—
29C2×Dic194—21no—
30C2×D388—21no—
31C2×D388C2×C411yescharacteristic
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C22⋊D97—21no—
35C2×C4×D97—21no—
36C2×C4×D97C2×C411yescharacteristic
37C2×C4×D97D411yescharacteristic
38C2×C4×D97D411yescharacteristic
39C4⋊Dic97C2×C411yescharacteristic
40#13095490—21no—
41C22⋊Dic97—21no—
42#13095488—21no—
43#13095488—21no—
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C2×D4×C97C2×C411yescharacteristic
48C22×C4×C97—21no—
49C4⋊C4×C97C2×C411yescharacteristic
50C22⋊C4×C97—21no—
51C42×C97C2×C411yescharacteristic
52C23×C97—21no—
53C22×D97—21no—
54C22×D97C22⋊C411yescharacteristic
55D4×C97—41no—
56C2×C4×C97—21no—
57C2×C4×C97—42no—
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C22⋊D97—42no—
62C22⋊D97—41no—
63C22⋊D97—41no—
64C22⋊D97—41no—
65C22⋊D97—41no—
66C2×Dic97—21no—
67C2×Dic97C22⋊C411yescharacteristic
68C2×Dic97C2×D411yescharacteristic
69C2×Dic97C2×D411yescharacteristic
70D388—41no—
71D388—42no—
72C4×D97—21no—
73C4×D97—21no—
74C4×D97—41no—
75C4×D97—42no—
76C4×D97D812yes—
77C4×D97SD1612yes—
78Dic194—41no—
79Dic194—41no—
80#11715897—42no—
81#11715897—42no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97C2×C22⋊C411yescharacteristic
86C2×D97—21no—
87C2×D97—21no—
88C2×D97—41no—
89C2×D97—41no—
90C2×D97—41no—
91C2×D97#10212yes—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—41no—
95C4×C97#10212yes—
96Dic97—21no—
97Dic97—41no—
98Dic97C2×D811yescharacteristic
99Dic97C2×SD1611yescharacteristic
100Dic97AGL(1,ℤ/8)11yescharacteristic
101Dic97#13711yescharacteristic
102C2×C97—81no—
103C2×C97#41311yescharacteristic
104C2×C97C2×#10211yescharacteristic
105C2×C97#41711yescharacteristic
106D97—22no—
107D97—81no—
108#1547—971nomaximal, Sylow
109C97#154711yescharacteristic, complemented, Sylow
110#532—971no—
111#423—971no—
112C2×C4⋊C8—971no—
113#332—972no—
114#331—972no—
115C2×#41—1941no—
116C2×M16—1941no—
117C22×C8—1941no—
118C4⋊Q8—971no—
119#127—971no—
120#124—1941no—
121#121—971no—
122#121—971no—
123#121—1941no—
124C4×D4—971no—
125C4×D4—1941no—
126C4×D4#8760411yescharacteristic
127C2×C42—971no—
128C4⋊C8—972no—
129C4⋊C8—972no—
130C4⋊C8—1941no—
131C4⋊C8—1941no—
132#41—3882no—
133#41—3882no—
134C2×Q8—1941no—
135C2×D4—971no—
136C2×D4—1941no—
137C2×D4—1941no—
138C2×D4—1941no—
139C2×D4—1941no—
140C2×D4C2×#8760411yescharacteristic
141C22×C4—21no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—971no—
145C22×C4—1941no—
146M16—3882no—
147C2×C8—3882no—
148C2×C8—1942no—
149C2×C8—1942no—
150C4⋊C4—971no—
151C4⋊C4—1941no—
152C4⋊C4C2×#8760411yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C42—971no—
158C42—971no—
159C42—971no—
160C42C2×#8760411yescharacteristic
161C23—21no—
162C23—971no—
163C23—1941no—
164Q8—3881no—
165Q8—3881no—
166D4—41no—
167D4—3881no—
168D4—3882no—
169D4—3881no—
170D4—3881no—
171D4—3881no—
172D4—3881no—
173D4—3882no—
174C2×C4—21no—
175C2×C4—42no—
176C2×C4—971no—
177C2×C4—971no—
178C2×C4—972no—
179C2×C4—972no—
180C2×C4—971no—
181C2×C4—3881no—
182C2×C4—3882no—
183C2×C4—1941no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4C22⋊#8760411yescharacteristic
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22×#8760411yescharacteristic
189C8—3882no—
190C8—3882no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—972no—
195C22—3881no—
196C22—3881no—
197C22—3881no—
198C22—1941no—
199C22—1941no—
200C22C2×C22⋊#8760411yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—971no—
205C4—971no—
206C4—971no—
207C4—971no—
208C4—3881no—
209C4—1941no—
210C4#1569087212yes—
211C2—81no—
212C2—7761no—
213C2—1942no—
214C2#2026172111yescharacteristic
215C2C2×#1569087211yescharacteristic
216C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC4×Dic97C23
hypercenterC4×D4#87604
hypocenterC97#1547
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261740C2semidirect
2. #20261736C2semidirect
3. #20261594C2semidirect
4. #20261593C2semidirect
109. C97#1547semidirect

Maximal subgroups

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

subgroupindex
#202617402
#202617362
#202615942
#202615932
#202612822
#154797