Finite Group Databasebeta

← #27140851

Subgroups — #27140851

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#20261109C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#476×C97C211yesmaximal, characteristic, complemented
8C22×D388—21no—
9C2×D4×D97—21no—
10C2×#13095498—22no—
11#15690807C2211yescharacteristic
12#15690785—21no—
13#15690784—21no—
14#15690780—22no—
15C4×D388C2211yescharacteristic
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic, complemented
18#15690730—21no—
19C4×#11715895C2211yescharacteristic, complemented
20C4⋊Q8×C97C2211yescharacteristic, complemented
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic, complemented
24#102×C97—21no—
25C23×D97—22no—
26C2×D388—22no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—42no—
30C2×D388—42no—
31C2×D388D411yescharacteristic
32C2×D388D412yes—
33C2×C22⋊D97—22no—
34D4×D97—42no—
35D4×D97—42no—
36C2×C4×D97—21no—
37#13095498—44no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C4⋊Dic97D411yescharacteristic
41#13095488—22no—
42#13095488—21no—
43C2×Q8×C97—21no—
44C2×D4×C97D411yescharacteristic
45C22×C4×C97—21no—
46C2×C8×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×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—
58Q8×C97—42no—
59D4×C97—21no—
60D4×C97SD1612yescomplemented
61C2×C4×C97—21no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C22⋊D97—44no—
69C2×Dic97—21no—
70D388—21no—
71D388—42no—
72D388—44no—
73D388—41no—
74D388—41no—
75D388—41no—
76D388—41no—
77D388SD1612yescomplemented
78C4×D97—41no—
79C4×D97—41no—
80C8×C97—41no—
81#11715895—22no—
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×C97—81no—
97C4×C97C2×SD1611yescharacteristic
98C4×C97C2×SD1611yescharacteristic
99C4×C97AGL(1,ℤ/8)11yescharacteristic
100C4×C97AGL(1,ℤ/8)11yescharacteristic
101Dic97—41no—
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#476D9711yescharacteristic, complemented
114#331—971no—
115D4⋊C8—971no—
116D4⋊C8—971no—
117C22×D4—1941no—
118C22×D4—1941no—
119C2×SD16—1942no—
120C4⋊Q8C2×D9711yescharacteristic, complemented
121#127—971no—
122#121—1941no—
123#121—1941no—
124#120—1942no—
125C4×D4—971no—
126C4×D4C2×D9711yescharacteristic
127#106—21no—
128#106—1941no—
129C4⋊C8—971no—
130C4⋊C8C2×D9711yescharacteristic, complemented
131#102—21no—
132#102—1941no—
133C4×C8—971no—
134C24—1942no—
135C2×Q8—21no—
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×D4C22⋊D9711yescharacteristic
147C22×C4—21no—
148C22×C4—1941no—
149SD16—3884no—
150C2×C8—21no—
151C2×C8—1941no—
152C2×C8—1941no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—971no—
156C4⋊C4C22⋊D9711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—1942no—
159C22⋊C4—1941no—
160C42C22×D9711yescharacteristic
161C23—21no—
162C23—7762no—
163C23—3884no—
164C23—3884no—
165C23—1942no—
166C23—1941no—
167Q8—42no—
168D4—21no—
169D4—972no—
170D4—3882no—
171D4—3884no—
172D4—3884no—
173D4—3881no—
174D4—3881no—
175D4—3881no—
176D4—3881no—
177D4—1941no—
178D4#1309549812yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—3881no—
184C2×C4—3881no—
185C2×C4—1941no—
186C2×C4D4×D9711yescharacteristic
187C2×C4D4×D9711yescharacteristic
188C2×C4C2×C22⋊D9711yescharacteristic
189C8—41no—
190C8—3881no—
191C8—1942no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—7768no—
196C22—3882no—
197C22—3882no—
198C22—3882no—
199C22—3881no—
200C22—3881no—
201C22—1942no—
202C22#1569077811yescharacteristic
203C4—21no—
204C4—41no—
205C4—81no—
206C4—3881no—
207C4SD16×D9711yescharacteristic
208C4C2×#1309549811yescharacteristic
209C4#1569083511yescharacteristic
210C4#1569083711yescharacteristic
211C2—42no—
212C2—7762no—
213C2—3882no—
214C2#2026088411yescharacteristic
215C2#2026088511yescharacteristic
216C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radicalD4×D388C2
hypercenter#476D97
hypocenterC97#1629
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20261109C2semidirect
3. #20261022C2semidirect
4. #20260478C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #476×C97C2semidirect
17. C4⋊#11715895C22semidirect
19. C4×#11715895C22semidirect
20. C4⋊Q8×C97C22semidirect
23. C4⋊C8×C97C22semidirect
60. D4×C97SD16semidirect
77. D388SD16semidirect
109. C97#1629semidirect
113. #476D97semidirect
120. C4⋊Q8C2×D97semidirect
130. C4⋊C8C2×D97semidirect
178. D4#13095498semidirect

Maximal subgroups

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

subgroupindex
D4×D3882
#202611092
#202610222
#202604782
D4⋊#117158952
#202604622
#476×C972
#162997