Finite Group Databasebeta

← #27143697

Subgroups — #27143697

Order 12416.

210 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 47 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#20261549C211yesmaximal, characteristic, complemented
2C2×#15690754C211yesmaximal, characteristic, complemented
3#20261002C211yesmaximal, characteristic, complemented
4#20260552C211yesmaximal, characteristic, complemented
5#20260542C211yesmaximal, characteristic, complemented
6#20260520C211yesmaximal, characteristic, complemented
7#509×C97C211yesmaximal, characteristic, complemented
8#15690857—21no—
9#15690840—21no—
10#15690839—22no—
11#15690836—21no—
12#15690829—21no—
13C2×Dic388C2211yescharacteristic
14Q16×D97—21no—
15#15690816C2211yescharacteristic
16#15690754—22no—
17C2×#13095486C2211yescharacteristic
18#15690746C2211yescharacteristic
19#15690727—21no—
20#135×C97C2211yescharacteristic
21C2×D8×C97C2211yescharacteristic
22SD32×C97—21no—
23D16×C97—21no—
24M32×C97C2211yescharacteristic
25#13095511—41no—
26#13095511D411yescharacteristic
27#13095510—22no—
28#13095510—42no—
29C2×Dic194—21no—
30C2×Dic194—41no—
31Q8×D97—22no—
32Dic388—21no—
33Dic388D412yes—
34#13095501—22no—
35#13095499—22no—
36#13095499—41no—
37#13095496—41no—
38#13095495—21no—
39#13095495D411yescharacteristic
40#13095494—21no—
41C8×D97—21no—
42#13095486—21no—
43#41×C97—21no—
44C2×D4×C97—21no—
45Q16×C97D411yescharacteristic
46SD16×C97—21no—
47D8×C97—21no—
48D8×C97—21no—
49D8×C97D411yescharacteristic
50C2×C8×C97C2311yescharacteristic
51C16×C97—21no—
52C23×C97—41no—
53Q8×C97—21no—
54D4×C97—21no—
55D4×C97—41no—
56D4×C97—41no—
57D4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C22⋊D97—21no—
61C22⋊D97—42no—
62C2×Dic97—42no—
63C2×Dic97—41no—
64D388D811yescharacteristic
65C4×D97—21no—
66C4×D97—42no—
67Dic194—22no—
68Dic194—42no—
69Dic194—41no—
70Dic194D811yescharacteristic
71C8×C97C2×D411yescharacteristic
72C8×C97C2×D411yescharacteristic
73#11715895—21no—
74#11715895—41no—
75C22×C97—41no—
76C22×C97—81no—
77C22×C97—161no—
78C22×C97AGL(1,ℤ/8)11yescharacteristic
79C2×D97—21no—
80C4×C97—41no—
81C4×C97#12011yescharacteristic
82C4×C97C2×D811yescharacteristic
83Dic97—21no—
84Dic97—42no—
85C2×C97—21no—
86C2×C97—81no—
87C2×C97—161no—
88C2×C97#44711yescharacteristic
89D97—41no—
90#2193—971nomaximal, Sylow
91C97#219311yescharacteristic, complemented, Sylow
92#578—971no—
93#509D9711yescharacteristic, complemented
94C2×SD32—971no—
95#472—971no—
96#362—971no—
97#359—971no—
98#350—971no—
992−1+4—1941no—
100#137—1941no—
101#137—1942no—
102AGL(1,ℤ/8)—1941no—
103#135—1941no—
104#135C2×D9711yescharacteristic
105C2×Q16—971no—
106C2×Q16—1941no—
107C2×D8C2×D9711yescharacteristic
108#131—971no—
109SD32—21no—
110SD32—1942no—
111D16—21no—
112M32C2×D9711yescharacteristic
113C2×C16—971no—
114#108—971no—
115#100—1941no—
116#41—21no—
117#41—971no—
118#41—3881no—
119#41—3882no—
120#41—1942no—
121C2×Q8—3881no—
122C2×Q8—1942no—
123C2×Q8—1941no—
124C2×D4—21no—
125Q16—972no—
126Q16—1942no—
127Q16—1941no—
128Q16C22⋊D9711yescharacteristic
129SD16—21no—
130SD16—3881no—
131SD16—1942no—
132D8—21no—
133D8—21no—
134D8—3881no—
135D8C22⋊D9711yescharacteristic
136M16—971no—
137M16—1941no—
138M16—1941no—
139C2×C8—1941no—
140C2×C8C22×D9711yescharacteristic
141C16—21no—
142C16—1941no—
143C23—41no—
144Q8—21no—
145Q8—971no—
146Q8—3882no—
147Q8—3881no—
148Q8—1942no—
149D4—21no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—971no—
154D4—3882no—
155D4—1941no—
156C2×C4—41no—
157C2×C4—3882no—
158C2×C4—3881no—
159C2×C4—3882no—
160C2×C4—1941no—
161C2×C4D4×D9711yescharacteristic
162C8—3881no—
163C8—1941no—
164C8D4×D9711yescharacteristic
165C8C2×C22⋊D9711yescharacteristic
166C22—41no—
167C22—81no—
168C22—161no—
169C22—1941no—
170C22#1569083411yescharacteristic
171C4—41no—
172C4—3882no—
173C4—1941no—
174C4#1569077811yescharacteristic
175C4D8×D9711yescharacteristic
176C2—21no—
177C2—81no—
178C2—161no—
179C2—3881no—
180C2#2026086711yescharacteristic
subgroupG/H
centerC2#20260867
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#509×C97C2
socleC2×C97#447
hypercenter#509D97
hypocenterC97#2193
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #20261549C2semidirect
2. C2×#15690754C2semidirect
3. #20261002C2semidirect
4. #20260552C2semidirect
5. #20260542C2semidirect
6. #20260520C2semidirect
7. #509×C97C2semidirect
91. C97#2193semidirect
93. #509D97semidirect

Maximal subgroups

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

subgroupindex
#202615492
C2×#156907542
#202610022
#202605522
#202605422
#202605202
#509×C972
#219397