Finite Group Databasebeta

← #27141004

Subgroups — #27141004

Order 12416.

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

normal subgroups only

This group has 200 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
1#20261458C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic, complemented
3#20261012C211yesmaximal, characteristic
4#20260480C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic
7#462×C97C211yesmaximal, characteristic, complemented
8#15690810C2211yescharacteristic
9#15690805—22no—
10#15690798—21no—
11#15690786—21no—
12C4×Dic194—21no—
13C4×D388C2211yescharacteristic
14#15690768—21no—
15#15690743—21no—
16#15690741—22no—
17C4⋊#11715895C2211yescharacteristic
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C2×Q16×C97—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194—21no—
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97D411yescharacteristic
32C4⋊Dic97D412yes—
33#13095490—22no—
34#13095488—22no—
35#13095488—22no—
36#13095488—21no—
37C4×Dic97—22no—
38C2×Q8×C97—21no—
39C2×Q8×C97D411yescharacteristic
40Q16×C97—42no—
41C2×C8×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—21no—
49Q8×C97—21no—
50Q8×C97—21no—
51Q8×C97—41no—
52Q8×C97—41no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97—41no—
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—22no—
61D388—21no—
62D388—21no—
63C4×D97—41no—
64C4×D97—41no—
65Dic194—41no—
66Dic194—41no—
67C8×C97—41no—
68#11715895—22no—
69#11715895—41no—
70C22×C97#12011yescharacteristic
71C2×D97—41no—
72C2×D97—41no—
73C2×D97—41no—
74C4×C97—21no—
75C4×C97—41no—
76C4×C97—41no—
77C4×C97—81no—
78C4×C97#13511yescharacteristic
79C4×C97#13511yescharacteristic
80C4×C97AGL(1,ℤ/8)11yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—41no—
83Dic97—42no—
84C2×C97#44911yescharacteristic
85C2×C97#45211yescharacteristic
86C2×C97#45611yescharacteristic
87D97—81no—
88#1652—971nomaximal, Sylow
89C97#165211yescharacteristic, complemented, Sylow
90#556—971no—
91#498—971no—
92#474—971no—
93#462D9711yescharacteristic, complemented
94#332—971no—
95Q8⋊C8—971no—
96D4⋊C8—971no—
97C2×Q16—21no—
98C4⋊Q8—971no—
99C4⋊Q8C2×D9711yescharacteristic
100#126—1942no—
101#124—1941no—
102C22⋊Q8—1941no—
103C4×Q8—1941no—
104C4×Q8C2×D9711yescharacteristic
105C4×D4—971no—
106#117—1941no—
107#107—1941no—
108#106—1942no—
109C4⋊C8—971no—
110C4⋊C8C2×D9711yescharacteristic
111#103—21no—
112#102—1941no—
113C4×C8—971no—
114C2×Q8—21no—
115C2×Q8—1941no—
116C2×Q8C22⋊D9711yescharacteristic
117C2×D4—971no—
118C22×C4—1941no—
119Q16—42no—
120C2×C8—21no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—972no—
127C4⋊C4—971no—
128C4⋊C4—1942no—
129C4⋊C4—1941no—
130C4⋊C4C22⋊D9711yescharacteristic
131C22⋊C4—1942no—
132C22⋊C4—1942no—
133C22⋊C4—1941no—
134C42—21no—
135C42—1942no—
136C42C22×D9711yescharacteristic
137C23—1941no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—3881no—
143Q8—3881no—
144D4—1941no—
145D4—1941no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—3881no—
150C2×C4—3881no—
151C2×C4—1941no—
152C2×C4—1942no—
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C2×C4C2×C22⋊D9711yescharacteristic
156C8—41no—
157C8—3881no—
158C8—1942no—
159C22—3881no—
160C22—3881no—
161C22—3881no—
162C22#1569077811yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—3881no—
168C4—3882no—
169C4#1569083011yescharacteristic
170C4#1569083311yescharacteristic
171C4#1569083611yescharacteristic
172C4#1569084211yescharacteristic
173C2—7761no—
174C2#2026087811yescharacteristic
175C2#2026089611yescharacteristic
176C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#462×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#462D97
hypocenterC97#1652
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
2. #20261131C2semidirect
4. #20260480C2semidirect
5. Q8⋊#11715895C2semidirect
7. #462×C97C2semidirect
89. C97#1652semidirect
93. #462D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202611312
#202610122
#202604802
Q8⋊#117158952
#202604622
#462×C972
#165297