Finite Group Databasebeta

← #27157248

Subgroups — #27157248

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#20261738C211yesmaximal, characteristic, complemented
2C2×#15690877C211yesmaximal, characteristic, complemented
3#20261594C212yesmaximal, complemented
4#20261593C212yesmaximal, complemented
5#20261280C211yesmaximal, characteristic
6C2×#13095520—21no—
7C22×#11715897—21no—
8#15690877—21no—
9#15690877C2212yes—
10#15690877C2212yes—
11C4⋊#11715897—21no—
12C2×#13095510—21no—
13#15690809C2211yescharacteristic
14#15690806C2211yescharacteristic
15#15690796—21no—
16#15690784C411yescharacteristic
17#15690782—21no—
18D4×Dic97C411yescharacteristic
19#15690772C411yescharacteristic
20#15690770—21no—
21C2×C4×Dic97C2211yescharacteristic
22#121×C97C411yescharacteristic
23#13095520—42no—
24C2×#11715897—22no—
25C2×#11715897—22no—
26C2×#11715897—42no—
27#13095510—42no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×D388C2×C411yescharacteristic
31C2×C22⋊D97—21no—
32C2×C22⋊D97—21no—
33C2×C22⋊D97—21no—
34C22×Dic97—21no—
35C22×Dic97D411yescharacteristic
36C22×Dic97D411yescharacteristic
37C2×C4×D97—21no—
38C4⋊Dic97C2×C411yescharacteristic
39#13095490—21no—
40C22⋊Dic97—21no—
41#13095488—21no—
42#13095488—21no—
43C4×Dic97C2×C411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97C2311yescharacteristic
47C2×D4×C97—21no—
48C2×D4×C97C2×C411yescharacteristic
49C22×C4×C97C2×C411yescharacteristic
50C4⋊C4×C97C2×C411yescharacteristic
51C22⋊C4×C97—21no—
52C23×C97—21no—
53C23×C97C22⋊C411yescharacteristic
54C22×D97—21no—
55D4×C97—41no—
56D4×C97—42no—
57C2×C4×C97—21no—
58C2×C4×C97—41no—
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×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—42no—
70C2×Dic97C22⋊C411yescharacteristic
71C2×Dic97C22⋊C411yescharacteristic
72C2×Dic97D812yes—
73C2×Dic97SD1612yes—
74C2×Dic97C2×D411yescharacteristic
75C2×Dic97C2×D411yescharacteristic
76D388—41no—
77C4×D97—42no—
78Dic194—41no—
79Dic194—41no—
80#11715897—42no—
81#11715897—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97#10212yes—
88C22×C97C2×C22⋊C411yescharacteristic
89C2×D97—41no—
90C2×D97—41no—
91C2×D97—41no—
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—41no—
95Dic97—21no—
96Dic97—41no—
97Dic97#10212yes—
98Dic97C2×D811yescharacteristic
99Dic97C2×SD1611yescharacteristic
100Dic97AGL(1,ℤ/8)11yescharacteristic
101Dic97#13711yescharacteristic
102C2×C97—22no—
103C2×C97—81no—
104C2×C97#41311yescharacteristic
105C2×C97C2×#10211yescharacteristic
106C2×C97#41711yescharacteristic
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—1941no—
123#121#8760411yescharacteristic
124C4×D4—971no—
125C4×D4—971no—
126C4×D4—1941no—
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—21no—
136C2×D4—971no—
137C2×D4—1941no—
138C2×D4—1941no—
139C2×D4—1941no—
140C2×D4C2×#8760411yescharacteristic
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4C2×#8760411yescharacteristic
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—
160C42—971no—
161C23—21no—
162C23—1941no—
163C23C22⋊#8760411yescharacteristic
164Q8—3881no—
165Q8—3881no—
166D4—41no—
167D4—42no—
168D4—3881no—
169D4—3881no—
170D4—3881no—
171D4—3881no—
172D4—3881no—
173D4—3882no—
174C2×C4—21no—
175C2×C4—41no—
176C2×C4—972no—
177C2×C4—972no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—971no—
182C2×C4—3882no—
183C2×C4—3882no—
184C2×C4—1941no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4C22⋊#8760411yescharacteristic
188C2×C4C22×#8760411yescharacteristic
189C8—3882no—
190C8—3882no—
191C22—21no—
192C22—21no—
193C22—41no—
194C22—41no—
195C22—41no—
196C22—3881no—
197C22—3881no—
198C22—3881no—
199C22#1569087212yes—
200C22C2×C22⋊#8760411yescharacteristic
201C4—21no—
202C4—21no—
203C4—41no—
204C4—971no—
205C4—972no—
206C4—971no—
207C4—971no—
208C4—971no—
209C4—3881no—
210C4—1941no—
211C2—22no—
212C2—81no—
213C2—7761no—
214C2#2026172211yescharacteristic
215C2C2×#1569087211yescharacteristic
216C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#121×C97C4
socleC22×C97C2×C22⋊C4
prime radical#20261280C2
cosocleC4×Dic97C23
hypercenter#121#87604
hypocenterC97#1547
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #20261738C2semidirect
2. C2×#15690877C2semidirect
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
#202617382
C2×#156908772
#202615942
#202615932
#202612802
#154797