Finite Group Databasebeta

← #27140860

Subgroups — #27140860

Order 12416.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261405C211yesmaximal, characteristic
2#20261027C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic, complemented
4#20260479C211yesmaximal, characteristic, complemented
5#20260467C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7#492×C97C211yesmaximal, characteristic, complemented
8C2×#13095510—21no—
9#15690806—21no—
10#15690782—22no—
11#15690781—21no—
12C4×Dic194C2211yescharacteristic
13C4×D388C2211yescharacteristic
14D4×Dic97—21no—
15#15690743—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690733—21no—
20#15690730—21no—
21C2×SD16×C97—22no—
22C4⋊Q8×C97C2211yescharacteristic
23#127×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25#13095510—42no—
26#13095510—42no—
27C2×Dic194D411yescharacteristic
28C2×D388D411yescharacteristic
29C2×C22⋊D97—22no—
30C2×C22⋊D97—22no—
31C22×Dic97—22no—
32C2×C4×D97—21no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97D411yescharacteristic
36C4⋊Dic97D411yescharacteristic
37#13095490—21no—
38C22⋊Dic97—22no—
39#13095488—21no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97D412yes—
45SD16×C97—44no—
46C2×C8×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C42×C97C2311yescharacteristic
50C23×C97—22no—
51C22×D97—21no—
52Q8×C97—42no—
53D4×C97—42no—
54D4×C97—41no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C2×C4×C97C2×D411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C22⋊D97—44no—
63C22⋊D97—82no—
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—44no—
68D388—21no—
69D388—21no—
70C4×D97—41no—
71C4×D97—41no—
72Dic194—21no—
73Dic194SD1612yescomplemented
74C8×C97—22no—
75#11715895—41no—
76#11715895—41no—
77C22×C97—42no—
78C22×C97—42no—
79C22×C97—42no—
80C22×C97#12011yescharacteristic
81C2×D97—41no—
82C2×D97—41no—
83C2×D97—41no—
84C4×C97—21no—
85C4×C97—81no—
86C4×C97C2×SD1611yescharacteristic
87C4×C97#13511yescharacteristic
88C4×C97AGL(1,ℤ/8)11yescharacteristic
89C4×C97#13711yescharacteristic
90Dic97—22no—
91Dic97—41no—
92Dic97—41no—
93C2×C97—82no—
94C2×C97#45011yescharacteristic
95C2×C97#45211yescharacteristic
96C2×C97#45311yescharacteristic
97D97—81no—
98#1631—971nomaximal, Sylow
99C97#163111yescharacteristic, complemented, Sylow
100#550—971no—
101#492D9711yescharacteristic, complemented
102#477—971no—
103#476—971no—
104#331—971no—
105Q8⋊C8—971no—
106D4⋊C8—971no—
107C2×#41—1941no—
108C2×SD16—22no—
109C4⋊Q8C2×D9711yescharacteristic
110#127—1941no—
111#127C2×D9711yescharacteristic
112#121—1941no—
113#121—1942no—
114C4×Q8—971no—
115C4×D4—971no—
116C4×D4—1941no—
117#107—1941no—
118#106—1941no—
119C4⋊C8—971no—
120C4⋊C8—971no—
121#103—1941no—
122#102—1941no—
123C4×C8C2×D9711yescharacteristic
124#41—3882no—
125#41—3882no—
126C2×Q8—21no—
127C2×Q8—971no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—971no—
131C2×D4—1942no—
132C2×D4—1942no—
133C2×D4C22⋊D9712yes—
134C22×C4—1942no—
135C22×C4—1941no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—1941no—
139C2×C8—1941no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—971no—
144C4⋊C4—1941no—
145C22⋊C4—1942no—
146C22⋊C4—1941no—
147C42—1941no—
148C42C22×D9711yescharacteristic
149C23—22no—
150C23—1941no—
151Q8—42no—
152Q8—972no—
153Q8—1941no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—7762no—
160D4—3884no—
161D4—1941no—
162D4—1941no—
163C2×C4—41no—
164C2×C4—3884no—
165C2×C4—3881no—
166C2×C4—3881no—
167C2×C4—1941no—
168C2×C4—1941no—
169C2×C4—1941no—
170C2×C4D4×D9711yescharacteristic
171C2×C4D4×D9711yescharacteristic
172C2×C4C2×C22⋊D9711yescharacteristic
173C8—22no—
174C8—3881no—
175C8—3881no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—3881no—
180C22—3881no—
181C22—3881no—
182C22#1569077811yescharacteristic
183C4—21no—
184C4—81no—
185C4—3881no—
186C4—3881no—
187C4—1942no—
188C4SD16×D9711yescharacteristic
189C4#1569083111yescharacteristic
190C4#1569083711yescharacteristic
191C4#1569083911yescharacteristic
192C2—82no—
193C2—7761no—
194C2#2026088411yescharacteristic
195C2#2026089411yescharacteristic
196C2#2026090211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#492×C97C2
socleC22×C97#120
prime radical#20261405C2
hypercenter#492D97
hypocenterC97#1631
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
2. #20261027C2semidirect
3. #20261022C2semidirect
4. #20260479C2semidirect
5. #20260467C2semidirect
6. #20260464C2semidirect
7. #492×C97C2semidirect
73. Dic194SD16semidirect
99. C97#1631semidirect
101. #492D97semidirect

Maximal subgroups

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

subgroupindex
#202614052
#202610272
#202610222
#202604792
#202604672
#202604642
#492×C972
#163197