Finite Group Databasebeta

← #27140914

Subgroups — #27140914

Order 12416.

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

normal subgroups only

This group has 308 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#20261083C211yesmaximal, characteristic, complemented
3#20260937C211yesmaximal, characteristic, complemented
4#20260478C211yesmaximal, characteristic, complemented
5D4⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#474×C97C211yesmaximal, characteristic, complemented
8C22×D388—21no—
9C2×D4×D97—21no—
10C2×#13095498—21no—
11#15690807C2211yescharacteristic
12#15690785—21no—
13#15690784—21no—
14#15690780—22no—
15C4×D388C2211yescharacteristic
16C4⋊#11715895C2211yescharacteristic, complemented
17#15690734—21no—
18#15690730—22no—
19C4×#11715895C2211yescharacteristic, complemented
20C4⋊Q8×C97C2211yescharacteristic, complemented
21C4×D4×C97C2211yescharacteristic
22#107×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—42no—
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—41no—
59Q8×C97—41no—
60D4×C97—21no—
61D4×C97SD1612yescomplemented
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C22⋊D97—44no—
70C2×Dic97—21no—
71D388—21no—
72D388—42no—
73D388—44no—
74D388—41no—
75D388—41no—
76D388—41no—
77D388—41no—
78D388D812yes—
79C4×D97—41no—
80C4×D97—41no—
81C8×C97—41no—
82#11715895—41no—
83#11715895—41no—
84C22×C97—22no—
85C22×C97—41no—
86C22×C97—41no—
87C22×C97#12011yescharacteristic
88C2×D97—22no—
89C2×D97—41no—
90C2×D97—41no—
91C2×D97—42no—
92C2×D97—42no—
93C2×D97—42no—
94C2×D97—88no—
95C4×C97—21no—
96C4×C97—41no—
97C4×C97—81no—
98C4×C97C2×D811yescharacteristic
99C4×C97C2×SD1611yescharacteristic
100C4×C97AGL(1,ℤ/8)11yescharacteristic
101C4×C97AGL(1,ℤ/8)11yescharacteristic
102Dic97—41no—
103C2×C97—42no—
104C2×C97#44711yescharacteristic
105C2×C97#45011yescharacteristic
106C2×C97#45511yescharacteristic
107D97—42no—
108D97—82no—
109#1639—971nomaximal, Sylow
110C97#163911yescharacteristic, complemented, Sylow
111D42—971no—
112#486—971no—
113#474D9711yescharacteristic, complemented
114#461—971no—
115#331—971no—
116D4⋊C8—971no—
117D4⋊C8—971no—
118C22×D4—1941no—
119C22×D4—1941no—
120C2×SD16—1941no—
121C4⋊Q8C2×D9711yescharacteristic, complemented
122#127—971no—
123#121—1941no—
124#121—1941no—
125#120—1942no—
126C4×D4—971no—
127C4×D4C2×D9711yescharacteristic
128#107—21no—
129C4⋊C8—971no—
130C4⋊C8C2×D9711yescharacteristic, complemented
131#103—1941no—
132#102—21no—
133#102—1942no—
134C4×C8—971no—
135C24—1942no—
136C2×Q8—21no—
137C2×D4—971no—
138C2×D4—972no—
139C2×D4—3882no—
140C2×D4—3882no—
141C2×D4—3882no—
142C2×D4—3882no—
143C2×D4—1942no—
144C2×D4—1942no—
145C2×D4—1941no—
146C2×D4—1941no—
147C2×D4C22⋊D9711yescharacteristic
148C22×C4—21no—
149C22×C4—1941no—
150SD16—3882no—
151C2×C8—21no—
152C2×C8—1941no—
153C2×C8—1941no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—971no—
157C4⋊C4C22⋊D9711yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—1942no—
160C22⋊C4—1941no—
161C42C22×D9711yescharacteristic
162C23—21no—
163C23—7762no—
164C23—3884no—
165C23—3884no—
166C23—1942no—
167C23—1941no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—972no—
172D4—3882no—
173D4—3884no—
174D4—3884no—
175D4—3881no—
176D4—3881no—
177D4—3881no—
178D4—3881no—
179D4—1941no—
180D4#1309549812yescomplemented
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—3881no—
186C2×C4—3881no—
187C2×C4—1941no—
188C2×C4D4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C8—41no—
192C8—3881no—
193C8—3881no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—7768no—
198C22—3882no—
199C22—3882no—
200C22—3882no—
201C22—3881no—
202C22—3881no—
203C22—1942no—
204C22#1569077811yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—3881no—
209C4D8×D9711yescharacteristic
210C4C2×#1309549811yescharacteristic
211C4#1569083411yescharacteristic
212C4#1569083511yescharacteristic
213C2—42no—
214C2—7762no—
215C2—3882no—
216C2#2026086711yescharacteristic
217C2#2026088511yescharacteristic
218C2#2026091111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#474×C97C2
socleC22×C97#120
prime radicalD4×D388C2
hypercenter#474D97
hypocenterC97#1639
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20261083C2semidirect
3. #20260937C2semidirect
4. #20260478C2semidirect
5. D4⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #474×C97C2semidirect
16. C4⋊#11715895C22semidirect
19. C4×#11715895C22semidirect
20. C4⋊Q8×C97C22semidirect
23. C4⋊C8×C97C22semidirect
61. D4×C97SD16semidirect
110. C97#1639semidirect
113. #474D97semidirect
121. C4⋊Q8C2×D97semidirect
130. C4⋊C8C2×D97semidirect
180. D4#13095498semidirect

Maximal subgroups

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

subgroupindex
D4×D3882
#202610832
#202609372
#202604782
D4⋊#117158952
#202604622
#474×C972
#163997