Finite Group Databasebeta

← #27142960

Subgroups — #27142960

Order 12416.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#20261316C211yesmaximal, characteristic
2C2×#15690738C212yesmaximal, complemented
3#20260734C212yesmaximal, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#384×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C2212yes—
7C2×#13095495C2212yes—
8#15690790—22no—
9#15690785—22no—
10#15690784—22no—
11#15690738—24no—
12#15690733—22no—
13#15690730—22no—
14C2×C4⋊C4×C97—22no—
15C2×C4⋊C4×C97C2211yescharacteristic
16C2×C4⋊C4×C97C2211yescharacteristic
17C2×C42×C97C2211yescharacteristic
18#13095511—22no—
19#13095511—22no—
20#13095511—22no—
21#13095511—22no—
22C2×Dic194D412yes—
23C2×D388—22no—
24C2×D388D412yes—
25C2×C22⋊D97—22no—
26C2×C4×D97—22no—
27#13095495—22no—
28#13095495—22no—
29C2×#11715895—22no—
30C4⋊Dic97—22no—
31#13095488—24no—
32C22×C4×C97—21no—
33C22×C4×C97—21no—
34C22×C4×C97D412yes—
35C22×C4×C97C2311yescharacteristic
36C4⋊C4×C97—22no—
37C4⋊C4×C97—22no—
38C4⋊C4×C97—44no—
39C42×C97—21no—
40C42×C97D412yescomplemented
41C23×C97C2×D411yescharacteristic
42C22×D97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—22no—
45C2×C4×C97—22no—
46C2×C4×C97—22no—
47C2×C4×C97—42no—
48C2×C4×C97—42no—
49C2×C4×C97—41no—
50C2×C4×C97C2×D412yes—
51C2×C4×C97C2×D411yescharacteristic
52C2×C4×C97#4111yescharacteristic
53C2×C4×C97#4112yes—
54C22⋊D97—42no—
55C22⋊D97—42no—
56C2×Dic97—22no—
57D388—22no—
58D388—22no—
59D388—82no—
60C4×D97—42no—
61C4×D97—42no—
62Dic194—22no—
63Dic194—22no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97#12011yescharacteristic
68C22×C97#12111yescharacteristic
69C22×C97#12411yescharacteristic
70C2×D97—42no—
71C2×D97—42no—
72C2×D97—42no—
73C4×C97—24no—
74C4×C97—42no—
75C4×C97—81no—
76C4×C97#12112yes—
77C4×C97#12312yes—
78Dic97—42no—
79C2×C97—21no—
80C2×C97—21no—
81C2×C97#39411yescharacteristic
82C2×C97#45311yescharacteristic
83C2×C97#45611yescharacteristic
84D97—82no—
85#2039—971nomaximal, Sylow
86C97#203911yescharacteristic, complemented, Sylow
87#537—971no—
88C2×#104—972no—
89#417—972no—
90#384D9711yescharacteristic, complemented
91#343—971no—
92C2×#41—972no—
93C2×M16—972no—
94C22⋊Q8—1942no—
95#121—1942no—
96#121—1942no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D9711yescharacteristic
99C2×C4⋊C4C2×D9711yescharacteristic
100C2×C42C2×D9711yescharacteristic
101#104—1944no—
102#103—1942no—
103#102—1942no—
104#41—1942no—
105#41—1942no—
106#41—1942no—
107#41—1942no—
108C2×Q8—972no—
109C2×D4—972no—
110C2×D4—1942no—
111C2×D4—1942no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—1942no—
115C22×C4C22⋊D9712yes—
116C22×C4C22×D9711yescharacteristic
117M16—1942no—
118M16—1942no—
119C2×C8—1942no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—1942no—
124C22⋊C4—1944no—
125C42—21no—
126C42D38812yescomplemented
127C23—1942no—
128C23C2×C22⋊D9711yescharacteristic
129Q8—1942no—
130Q8—1942no—
131D4—7762no—
132D4—3882no—
133D4—3882no—
134D4—1942no—
135D4—1942no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—3882no—
144C2×C4—3882no—
145C2×C4—1942no—
146C2×C4D4×D9712yes—
147C2×C4C2×D38811yescharacteristic
148C2×C4#1309551111yescharacteristic
149C2×C4#1309551212yes—
150C8—3882no—
151C22—21no—
152C22—21no—
153C22—3882no—
154C22—3882no—
155C22—3882no—
156C22#1569077811yescharacteristic
157C22#1569078411yescharacteristic
158C22#1569079911yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—3882no—
163C4#1569078512yes—
164C4#1569079512yes—
165C2—21no—
166C2—21no—
167C2—7762no—
168C2#2026065211yescharacteristic
169C2#2026090211yescharacteristic
170C2#2026091711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#384×C97C2
socleC22×C97#120
prime radical#20261316C2
hypercenter#384D97
hypocenterC97#2039
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#15690738C2semidirect
3. #20260734C2semidirect
4. #20260504C2semidirect
5. #384×C97C2semidirect
40. C42×C97D4semidirect
86. C97#2039semidirect
90. #384D97semidirect
126. C42D388semidirect

Maximal subgroups

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

subgroupindex
#202613162
C2×#156907382
#202607342
#202605042
#384×C972
#203997