Finite Group Databasebeta

← #27142268

Subgroups — #27142268

Order 12416.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261302C211yesmaximal, characteristic
2#20260753C212yesmaximal, complemented
3C2×#15690727C211yesmaximal, characteristic, complemented
4#20260491C212yesmaximal, complemented
5#518×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2211yescharacteristic
7C2×#13095495C2212yes—
8#15690806C412yescomplemented
9#15690797C412yescomplemented
10#15690781—22no—
11#15690779—22no—
12#15690769C2211yescharacteristic
13#15690738—24no—
14#15690727—22no—
15C22×D4×C97C2211yescharacteristic
16C4×D4×C97—24no—
17#117×C97C2212yes—
18C2×C22⋊C4×C97—22no—
19#13095511—22no—
20#13095511—22no—
21C2×Dic194C2×C411yescharacteristic, complemented
22C2×D388C2×C411yescharacteristic, complemented
23C2×C22⋊D97—21no—
24C2×C22⋊D97—24no—
25C2×C4×D97—21no—
26#13095495—22no—
27#13095495—22no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C22⋊Dic97—24no—
31C22⋊Dic97—21no—
32C4×Dic97C2×C412yescomplemented
33C24×C97D412yes—
34C2×D4×C97—22no—
35C2×D4×C97—22no—
36C2×D4×C97C2×C412yes—
37C2×D4×C97D412yes—
38C22×C4×C97—24no—
39C22×C4×C97C2311yescharacteristic
40C4⋊C4×C97—22no—
41C22⋊C4×C97—24no—
42C22⋊C4×C97—22no—
43C42×C97D414yescomplemented
44C23×C97—22no—
45C23×C97—22no—
46C23×C97—82no—
47C23×C97C22⋊C414yes—
48C23×C97C2×D411yescharacteristic
49C22×D97—21no—
50D4×C97—24no—
51D4×C97—24no—
52C2×C4×C97—24no—
53C2×C4×C97—84no—
54C2×C4×C97C22×C411yescharacteristic
55C2×C4×C97C2×D412yes—
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97#4112yes—
58C22⋊D97—41no—
59C22⋊D97—41no—
60C22⋊D97—84no—
61C2×Dic97—22no—
62C2×Dic97—21no—
63D388—22no—
64C4×D97—42no—
65Dic194—22no—
66#11715895—42no—
67C22×C97—24no—
68C22×C97—41no—
69C22×C97—44no—
70C22×C97—44no—
71C22×C97—82no—
72C22×C97C2×C22⋊C411yescharacteristic
73C22×C97#12011yescharacteristic
74C22×C97#12311yescharacteristic
75C2×D97—42no—
76C2×D97—41no—
77C4×C97—44no—
78C4×C97C4×D412yes—
79C4×C97#12112yes—
80Dic97—42no—
81Dic97—41no—
82C2×C97—21no—
83C2×C97—21no—
84C2×C97—21no—
85C2×C97—44no—
86C2×C97#38611yescharacteristic
87D97—81no—
88#1911—971nomaximal, Sylow
89C97#191111yescharacteristic, complemented, Sylow
90#535—971no—
91#518D9711yescharacteristic, complemented
92#421—972no—
93C2×#100—971no—
94H(3,ℤ/4)—972no—
95C2×#41—971no—
96C22×D4C2×D9711yescharacteristic
97C2×M16—972no—
98#127—972no—
99#124—972no—
100#121—1942no—
101#120—1942no—
102C4×D4—24no—
103#117—971no—
104#117C2×D9712yes—
105C2×C22⋊C4—22no—
106#104—1944no—
107#100—1942no—
108C24C22⋊D9712yes—
109#41—1942no—
110#41—1942no—
111C2×Q8—971no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—971no—
115C2×D4—1944no—
116C2×D4—1941no—
117C2×D4C4×D9712yes—
118C2×D4D38812yes—
119C22×C4—24no—
120C22×C4—1941no—
121C22×C4C22×D9711yescharacteristic
122M16—1942no—
123M16—1942no—
124C2×C8—1942no—
125C4⋊C4—22no—
126C4⋊C4—1941no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—1944no—
130C22⋊C4—1941no—
131C42—972no—
132C42C22⋊D9714yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—1941no—
137C23#1309548814yes—
138C23C2×C22⋊D9711yescharacteristic
139Q8—1942no—
140D4—24no—
141D4—24no—
142D4—7764no—
143D4—3881no—
144D4—3881no—
145D4—1942no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—3882no—
149C2×C4—1942no—
150C2×C4—1941no—
151C2×C4C2×C4×D9711yescharacteristic
152C2×C4C2×C22⋊D9712yes—
153C2×C4C2×D38811yescharacteristic
154C2×C4#1309551112yes—
155C8—3882no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—3882no—
162C22—3881no—
163C22C2×#1309548811yescharacteristic
164C22#1569077911yescharacteristic
165C22#1569079311yescharacteristic
166C4—44no—
167C4—3882no—
168C4—3881no—
169C4C4×C22⋊D9712yes—
170C4#1569078412yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—7761no—
176C2#2026061411yescharacteristic
subgroupG/H
centerC2#20260614
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#518×C97C2
socleC2×C97#386
prime radical#20261302C2
cosocleC22×C4×C97C23
hypercenter#518D97
hypocenterC97#1911
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20260753C2semidirect
3. C2×#15690727C2semidirect
4. #20260491C2semidirect
5. #518×C97C2semidirect
8. #15690806C4semidirect
9. #15690797C4semidirect
21. C2×Dic194C2×C4semidirect
22. C2×D388C2×C4semidirect
32. C4×Dic97C2×C4semidirect
43. C42×C97D4semidirect
89. C97#1911semidirect
91. #518D97semidirect
132. C42C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202613022
#202607532
C2×#156907272
#202604912
#518×C972
#191197