Finite Group Databasebeta

← #27140854

Subgroups — #27140854

Order 12416.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D388C211yesmaximal, characteristic
2#20261108C211yesmaximal, characteristic, complemented
3#20261026C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic, complemented
5#20260478C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7D4⋊C8×C97C211yesmaximal, characteristic, complemented
8C22×D388—21no—
9C2×D4×D97—21no—
10C2×#13095500—22no—
11#15690810C2211yescharacteristic, complemented
12#15690807C2211yescharacteristic
13#15690785—21no—
14#15690784—21no—
15#15690780—22no—
16C4×D388C2211yescharacteristic
17#15690742—21no—
18#15690741—21no—
19C4⋊#11715895C2211yescharacteristic, complemented
20#15690732—21no—
21#15690731—21no—
22C4×D4×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic, complemented
24C4×C8×C97C2211yescharacteristic, complemented
25C23×D97—22no—
26C2×Dic194—21no—
27C2×D388—22no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—42no—
31C2×D388—42no—
32C2×D388D411yescharacteristic
33C2×D388D412yes—
34C2×C22⋊D97—22no—
35D4×D97—42no—
36D4×D97—42no—
37C2×C4×D97—21no—
38#13095500—44no—
39C2×#11715895—21no—
40C4⋊Dic97—21no—
41C4⋊Dic97—21no—
42C4⋊Dic97D411yescharacteristic
43#13095488—22no—
44#13095488—21no—
45C2×D4×C97D411yescharacteristic
46C22×C4×C97—21no—
47C2×C8×C97—21no—
48C2×C8×C97—21no—
49C4⋊C4×C97D411yescharacteristic
50C22⋊C4×C97—21no—
51C42×C97C2311yescharacteristic
52C23×C97—21no—
53C22×D97—21no—
54C22×D97—22no—
55C22×D97—44no—
56C22×D97—44no—
57C22×D97—82no—
58D4×C97—21no—
59D4×C97SD1612yescomplemented
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C22⋊D97—44no—
67C2×Dic97—21no—
68C2×Dic97—41no—
69D388—21no—
70D388—42no—
71D388—44no—
72D388—41no—
73D388—41no—
74D388—41no—
75D388—41no—
76D388SD1612yescomplemented
77C4×D97—41no—
78C4×D97—41no—
79Dic194—42no—
80C8×C97—22no—
81C8×C97—41no—
82#11715895—41no—
83C22×C97—22no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97#12011yescharacteristic
87C2×D97—22no—
88C2×D97—41no—
89C2×D97—41no—
90C2×D97—42no—
91C2×D97—42no—
92C2×D97—42no—
93C2×D97—88no—
94C4×C97—21no—
95C4×C97—41no—
96C4×C97C2×SD1611yescharacteristic
97C4×C97C2×SD1611yescharacteristic
98C4×C97AGL(1,ℤ/8)11yescharacteristic
99C4×C97AGL(1,ℤ/8)11yescharacteristic
100Dic97—41no—
101Dic97—81no—
102C2×C97—42no—
103C2×C97#45011yescharacteristic
104C2×C97#45011yescharacteristic
105C2×C97#45311yescharacteristic
106D97—42no—
107D97—82no—
108#1629—971nomaximal, Sylow
109C97#162911yescharacteristic, complemented, Sylow
110D42—971no—
111#492—971no—
112#476—971no—
113#476—971no—
114#331—971no—
115D4⋊C8—971no—
116D4⋊C8D9711yescharacteristic, complemented
117C22×D4—1941no—
118C22×D4—1941no—
119C2×SD16—1942no—
120C4⋊Q8—971no—
121#127—971no—
122#121—1941no—
123#121—1941no—
124#120—1942no—
125C4×D4—971no—
126C4×D4C2×D9711yescharacteristic
127#106—1941no—
128#106—1941no—
129C4⋊C8—971no—
130C4⋊C8C2×D9711yescharacteristic, complemented
131#102—1941no—
132#102—1941no—
133C4×C8C2×D9711yescharacteristic, complemented
134C24—1942no—
135C2×Q8—1941no—
136C2×D4—971no—
137C2×D4—972no—
138C2×D4—3882no—
139C2×D4—3882no—
140C2×D4—3882no—
141C2×D4—3882no—
142C2×D4—1942no—
143C2×D4—1942no—
144C2×D4—1941no—
145C2×D4—1941no—
146C2×D4D38811yescharacteristic
147C22×C4—21no—
148C22×C4—1941no—
149SD16—3884no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—1941no—
153C4⋊C4—971no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4D38811yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—1942no—
159C22⋊C4—1941no—
160C42C22×D9711yescharacteristic
161C23—21no—
162C23—7762no—
163C23—3884no—
164C23—3884no—
165C23—1942no—
166C23—1941no—
167Q8—3882no—
168D4—21no—
169D4—972no—
170D4—3882no—
171D4—3884no—
172D4—3884no—
173D4—3881no—
174D4—3881no—
175D4—3881no—
176D4—3881no—
177D4—1941no—
178D4#1309550012yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—1941no—
186C2×C4D4×D9711yescharacteristic
187C2×C4D4×D9711yescharacteristic
188C2×C4C2×D38811yescharacteristic
189C8—22no—
190C8—41no—
191C8—3881no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—7768no—
196C22—3882no—
197C22—3882no—
198C22—3882no—
199C22—3881no—
200C22—3881no—
201C22—1942no—
202C22#1569078011yescharacteristic
203C4—21no—
204C4—41no—
205C4—7761no—
206C4—3881no—
207C4SD16×D9711yescharacteristic
208C4C2×#1309550011yescharacteristic
209C4#1569083411yescharacteristic
210C4#1569083811yescharacteristic
211C2—42no—
212C2—7762no—
213C2—3882no—
214C2#2026088711yescharacteristic
215C2#2026088811yescharacteristic
216C2#2026090411yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingD4⋊C8×C97C2
socleC22×C97#120
prime radicalD4×D388C2
hypercenterD4⋊C8D97
hypocenterC97#1629
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #20261108C2semidirect
3. #20261026C2semidirect
4. #20261025C2semidirect
5. #20260478C2semidirect
6. #20260464C2semidirect
7. D4⋊C8×C97C2semidirect
11. #15690810C22semidirect
19. C4⋊#11715895C22semidirect
23. C4⋊C8×C97C22semidirect
24. C4×C8×C97C22semidirect
59. D4×C97SD16semidirect
76. D388SD16semidirect
109. C97#1629semidirect
116. D4⋊C8D97semidirect
130. C4⋊C8C2×D97semidirect
133. C4×C8C2×D97semidirect
178. D4#13095500semidirect

Maximal subgroups

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

subgroupindex
D4×D3882
#202611082
#202610262
#202610252
#202604782
#202604642
D4⋊C8×C972
#162997