Finite Group Databasebeta

← #27143657

Subgroups — #27143657

Order 12416.

230 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×#15690832C211yesmaximal, characteristic, complemented
2C2×#15690754C211yesmaximal, characteristic, complemented
3C2×#15690751C211yesmaximal, characteristic, complemented
4#20260538C211yesmaximal, characteristic, complemented
5#20260535C211yesmaximal, characteristic, complemented
6C22⋊#13095486C211yesmaximal, characteristic, complemented
7#466×C97C211yesmaximal, characteristic, complemented
8C2×#13095511—21no—
9#15690832—22no—
10#15690832—22no—
11C2×Dic388C2211yescharacteristic
12C2×#13095500—21no—
13C2×D776C2211yescharacteristic
14#15690754—22no—
15#15690751—22no—
16C2×#13095486C2211yescharacteristic
17C2×#13095486C2211yescharacteristic
18C2×D8×C97C2211yescharacteristic
19C22×C8×C97C2211yescharacteristic
20#121×C97—21no—
21#107×C97C2211yescharacteristic
22#102×C97—21no—
23#13095511—24no—
24#13095511—42no—
25C2×Dic194—21no—
26C2×D388—21no—
27C2×C22⋊D97—41no—
28C2×C4×D97—41no—
29Dic388—21no—
30Dic388D412yescomplemented
31#13095500—24no—
32D776—21no—
33D776D412yescomplemented
34#13095486—21no—
35#13095486—21no—
36C2×D4×C97—21no—
37C2×D4×C97—41no—
38C22×C4×C97D411yescharacteristic
39D8×C97—21no—
40D8×C97—21no—
41C2×C8×C97—21no—
42C2×C8×C97—21no—
43C2×C8×C97D411yescharacteristic
44C2×C8×C97C2311yescharacteristic
45C4⋊C4×C97—21no—
46C22⋊C4×C97—41no—
47C23×C97—41no—
48C23×C97D811yescharacteristic
49C22×D97—41no—
50D4×C97—41no—
51D4×C97—41no—
52D4×C97—82no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97D811yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C22⋊D97—44no—
59C2×Dic97—41no—
60D388—22no—
61D388—41no—
62C4×D97—44no—
63Dic194—22no—
64Dic194—41no—
65C8×C97—21no—
66C8×C97C2×D411yescharacteristic
67C8×C97C2×D411yescharacteristic
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—21no—
71C22×C97—81no—
72C22×C97—81no—
73C22×C97—81no—
74C22×C97C2×D811yescharacteristic
75C2×D97—42no—
76C2×D97—81no—
77C2×D97—81no—
78C4×C97—21no—
79C4×C97—81no—
80C4×C97#12011yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82Dic97—42no—
83C2×C97—41no—
84C2×C97—161no—
85C2×C97#44711yescharacteristic
86C2×C97#50811yescharacteristic
87C2×C97#50911yescharacteristic
88D97—82no—
89#2186—971nomaximal, Sylow
90C97#218611yescharacteristic, complemented, Sylow
91C2×#135—971no—
92C2×SD32—971no—
93C2×D16—971no—
94#466D9711yescharacteristic, complemented
95#358—971no—
96#357—971no—
97C22⋊C16—971no—
98C2×#41—1941no—
99#135—1942no—
100#135—1942no—
101C2×Q16—971no—
102C2×SD16—1941no—
103C2×D8—971no—
104C2×D8C2×D9711yescharacteristic
105C22×C8C2×D9711yescharacteristic
106#121—21no—
107SD32—1942no—
108D16—1942no—
109C2×C16—971no—
110C2×C16—971no—
111#107C2×D9711yescharacteristic
112#102—21no—
113#41—3882no—
114#41—1944no—
115C2×Q8—1941no—
116C2×D4—21no—
117C2×D4—41no—
118C2×D4—3881no—
119C2×D4—1941no—
120C22×C4—3881no—
121C22×C4C22⋊D9711yescharacteristic
122Q16—972no—
123Q16—1941no—
124SD16—1944no—
125D8—21no—
126D8—21no—
127D8—972no—
128D8—1941no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8C22⋊D9711yescharacteristic
132C2×C8C22×D9711yescharacteristic
133C4⋊C4—21no—
134C22⋊C4—41no—
135C16—1941no—
136C16—1941no—
137C23—41no—
138C23—3881no—
139C23#1309549611yescharacteristic
140Q8—3881no—
141Q8—1942no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—3884no—
146D4—3881no—
147D4—1942no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—41no—
151C2×C4—3884no—
152C2×C4—3881no—
153C2×C4#1309549611yescharacteristic
154C2×C4C2×C22⋊D9711yescharacteristic
155C8—21no—
156C8D4×D9711yescharacteristic
157C8D4×D9711yescharacteristic
158C22—21no—
159C22—21no—
160C22—21no—
161C22—81no—
162C22—81no—
163C22—81no—
164C22—7761no—
165C22—7761no—
166C22—3882no—
167C22C2×#1309549611yescharacteristic
168C4—21no—
169C4—81no—
170C4—3882no—
171C4#1569077811yescharacteristic
172C4#1569083511yescharacteristic
173C2—41no—
174C2—7762no—
175C2—161no—
176C2#2026086811yescharacteristic
177C2#2026116511yescharacteristic
178C2#2026117111yescharacteristic
subgroupG/H
centerC22C2×#13095496
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#466×C97C2
socleC22×C97C2×D8
hypercenter#466D97
hypocenterC97#2186
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#15690832C2semidirect
2. C2×#15690754C2semidirect
3. C2×#15690751C2semidirect
4. #20260538C2semidirect
5. #20260535C2semidirect
6. C22⋊#13095486C2semidirect
7. #466×C97C2semidirect
30. Dic388D4semidirect
33. D776D4semidirect
90. C97#2186semidirect
94. #466D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908322
C2×#156907542
C2×#156907512
#202605382
#202605352
C22⋊#130954862
#466×C972
#218697