Finite Group Databasebeta

← #27157166

Subgroups — #27157166

Order 12416.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261754C211yesmaximal, characteristic, complemented
2#20261679C211yesmaximal, characteristic, complemented
3#20261587C212yesmaximal, complemented
4#20261586C212yesmaximal, complemented
5#20261282C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9C4×#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12C2×#13095511—21no—
13#15690809C411yescharacteristic, complemented
14#15690806C411yescharacteristic, complemented
15#15690796—21no—
16#15690785C2211yescharacteristic
17#15690783—21no—
18#15690781C2211yescharacteristic
19C4×D388C411yescharacteristic, complemented
20C4×C22⋊D97—21no—
21C42×D97C2211yescharacteristic
22C4×D4×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—42no—
26#13095511—42no—
27#13095511—42no—
28C2×Dic194—21no—
29C2×D388—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C2×C4×D97—21no—
35C2×C4×D97D411yescharacteristic
36C2×C4×D97D411yescharacteristic
37C2×C4×D97C2311yescharacteristic
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40C22⋊Dic97—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C2×D4×C97C2×C411yescharacteristic
47C22×C4×C97—21no—
48C4⋊C4×C97C2×C411yescharacteristic
49C22⋊C4×C97—21no—
50C42×C97C2×C411yescharacteristic
51C23×C97—21no—
52C22×D97—21no—
53C22×D97C2×D411yescharacteristic
54C2×#87604—42no—
55C2×#87604—42no—
56C2×#87604—42no—
57D4×C97—41no—
58C2×C4×C97—21no—
59C2×C4×C97—42no—
60C2×C4×C97C22⋊C411yescharacteristic
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22×C411yescharacteristic
63C22⋊D97—42no—
64C22⋊D97—41no—
65C22⋊D97—41no—
66C22⋊D97—42no—
67C2×Dic97—21no—
68C2×Dic97C22⋊C411yescharacteristic
69C2×Dic97C22⋊C411yescharacteristic
70C2×Dic97C2×D411yescharacteristic
71D388—41no—
72D388—41no—
73D388—41no—
74C4×D97—41no—
75C4×D97—41no—
76C4×D97—42no—
77C4×D97D812yes—
78C4×D97SD1612yes—
79Dic194—42no—
80#11715897—42no—
81C22×C97—41no—
82C22×C97—41no—
83C22×C97—41no—
84C22×C97C2×C22⋊C411yescharacteristic
85C2×D97—21no—
86C2×D97—21no—
87C2×D97—41no—
88C2×D97—41no—
89C2×D97—41no—
90C2×D97C2×D811yescharacteristic
91C2×D97C2×SD1611yescharacteristic
92#87604—82no—
93C4×C97—21no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97#10212yes—
97Dic97—21no—
98Dic97—21no—
99Dic97—41no—
100Dic97#10212yes—
101C2×C97—81no—
102C2×C97#41011yescharacteristic
103C2×C97C2×#10211yescharacteristic
104C2×C97#42111yescharacteristic
105D97—21no—
106D97—21no—
107D97—81no—
108#1517—971nomaximal, Sylow
109C97#151711yescharacteristic, complemented, Sylow
110#532—971no—
111#431—971no—
112#382—971no—
113#327—972no—
114#327—972no—
115C2×#41—1941no—
116C4⋊Q8—971no—
117#127—971no—
118#124—1941no—
119#121—971no—
120#121—971no—
121#121—1941no—
122C4×D4—971no—
123C4×D4—1941no—
124C4×D4#8760411yescharacteristic, complemented
125C2×C4⋊C4—1941no—
126C2×C42—971no—
127C4⋊C8—1941no—
128C22⋊C8—972no—
129C4×C8—1941no—
130#95—972no—
131#95—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—1942no—
146C22×C4—1941no—
147C2×C8—1942no—
148C4⋊C4—971no—
149C4⋊C4—3882no—
150C4⋊C4—1941no—
151C4⋊C4C2×#8760411yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C42—971no—
157C42—971no—
158C42—971no—
159C42C2×#8760411yescharacteristic
160C23—21no—
161C23—971no—
162C23—1941no—
163Q8—3882no—
164D4—41no—
165D4—3881no—
166D4—3881no—
167D4—3881no—
168D4—3882no—
169D4—3881no—
170D4—3881no—
171D4—3882no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—971no—
175C2×C4—972no—
176C2×C4—972no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—3881no—
180C2×C4—3881no—
181C2×C4—3882no—
182C2×C4—3882no—
183C2×C4—3882no—
184C2×C4—3882no—
185C2×C4—1941no—
186C2×C4C22⋊#8760411yescharacteristic
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22×#8760411yescharacteristic
189C8—3882no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—971no—
194C22—971no—
195C22—3881no—
196C22—3881no—
197C22—3881no—
198C22—1941no—
199C22—1941no—
200C22C2×C22⋊#8760411yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—7762no—
205C4—972no—
206C4—3881no—
207C4—1941no—
208C4—1941no—
209C4#1569087212yes—
210C2—81no—
211C2—7761no—
212C2—1941no—
213C2—1941no—
214C2#2026171411yescharacteristic
215C2C2×#1569087211yescharacteristic
216C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×D4×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261282C2
cosocleC2×C4×D97C23
hypercenterC4×D4#87604
hypocenterC97#1517
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. #20261754C2semidirect
2. #20261679C2semidirect
3. #20261587C2semidirect
4. #20261586C2semidirect
13. #15690809C4semidirect
14. #15690806C4semidirect
19. C4×D388C4semidirect
22. C4×D4×C97C4semidirect
109. C97#1517semidirect
124. C4×D4#87604semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202616792
#202615872
#202615862
#202612822
#151797