Finite Group Databasebeta

← #27158028

Subgroups — #27158028

Order 12416.

250 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 52 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#20261722C212yesmaximal, complemented
2#20261646C212yesmaximal, complemented
3#20261643C212yesmaximal, complemented
4#20261283C211yesmaximal, characteristic
5C2×#13095520—22no—
6#15690871C2214yes—
7#15690870—22no—
8C2×#13095510C411yescharacteristic
9C2×#13095510C2211yescharacteristic
10#15690810C2211yescharacteristic
11#15690797C411yescharacteristic
12#15690797C2211yescharacteristic
13#15690781—22no—
14D4×Dic97—22no—
15C42×D97C411yescharacteristic
16#127×C97C411yescharacteristic
17#13095520—24no—
18#13095520—42no—
19C2×#11715897—42no—
20#13095510—41no—
21#13095510—41no—
22#13095510—41no—
23#13095510—41no—
24C2×Dic194C2×C411yescharacteristic
25C2×Dic194C2311yescharacteristic
26C2×C22⋊D97—21no—
27C2×C22⋊D97D412yes—
28C22×Dic97—21no—
29C22×Dic97D412yes—
30C2×C4×D97—21no—
31C2×C4×D97C2×C411yescharacteristic
32C4⋊Dic97—22no—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C4×Dic97—21no—
36C4×Dic97C2×C411yescharacteristic
37C2×D4×C97—22no—
38C2×D4×C97C2×C411yescharacteristic
39C2×D4×C97C2×C411yescharacteristic
40C42×C97C2×C411yescharacteristic
41C23×C97—21no—
42C23×C97C22⋊C412yes—
43C22×D97C22⋊C411yescharacteristic
44D4×C97—41no—
45D4×C97—41no—
46D4×C97—44no—
47C2×C4×C97—21no—
48C2×C4×C97C22×C411yescharacteristic
49C22⋊D97—42no—
50C22⋊D97—42no—
51C2×Dic97—22no—
52C2×Dic97—22no—
53C2×Dic97—21no—
54C2×Dic97—21no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C2×D412yes—
59C4×D97—21no—
60C4×D97—21no—
61C4×D97—42no—
62Dic194—41no—
63Dic194—41no—
64#11715897—44no—
65C22×C97—22no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97—41no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—41no—
72C4×C97—22no—
73C4×C97—21no—
74Dic97—22no—
75Dic97—41no—
76Dic97—41no—
77Dic97#9911yescharacteristic
78Dic97#9911yescharacteristic
79C2×C97—21no—
80C2×C97—42no—
81C2×C97—81no—
82C2×C97C2×#9911yescharacteristic
83D97—41no—
84#2132—971nomaximal, Sylow
85C97#213211yescharacteristic, complemented, Sylow
86#532—971no—
87#413—972no—
88#356—972no—
89#355—972no—
90C2×#41—971no—
91C2×#41—971no—
92C2×M16—1942no—
93C4⋊Q8—971no—
94#127#8760411yescharacteristic
95#124—971no—
96#124—971no—
97#121—1942no—
98C4×D4—1942no—
99C2×C42—971no—
100#101—974no—
101#100—1942no—
102#41—3881no—
103#41—3881no—
104#41—3881no—
105#41—3881no—
106C2×Q8—971no—
107C2×Q8—971no—
108C2×D4—22no—
109C2×D4—972no—
110C2×D4—1941no—
111C2×D4C2×#8760411yescharacteristic
112C2×D4C2×#8760411yescharacteristic
113C22×C4—972no—
114C22×C4—971no—
115C22×C4—1941no—
116C22×C4—1941no—
117M16—3882no—
118M16—1944no—
119C2×C8—3882no—
120C4⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—971no—
124C42—1941no—
125C42C2×#8760411yescharacteristic
126C23—21no—
127C23—971no—
128C23C22⋊#8760412yes—
129Q8—3881no—
130Q8—3881no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3882no—
135D4—3882no—
136C2×C4—21no—
137C2×C4—971no—
138C2×C4—972no—
139C2×C4—3882no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—1941no—
143C2×C4—1941no—
144C2×C4—1942no—
145C2×C4—1942no—
146C2×C4—1941no—
147C2×C4—1941no—
148C2×C4C22×#8760411yescharacteristic
149C8—3884no—
150C22—22no—
151C22—42no—
152C22—42no—
153C22—41no—
154C22—3881no—
155C22—1941no—
156C22C2×C22⋊#8760411yescharacteristic
157C4—22no—
158C4—21no—
159C4—971no—
160C4—971no—
161C4—3881no—
162C4—3881no—
163C4—1942no—
164C2—21no—
165C2—42no—
166C2—81no—
167C2—3881no—
168C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#127×C97C4
socleC2×C97C2×#99
prime radical#20261283C2
cosocleC2×Dic194C23
hypercenter#127#87604
hypocenterC97#2132
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #20261722C2semidirect
2. #20261646C2semidirect
3. #20261643C2semidirect
85. C97#2132semidirect

Maximal subgroups

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

subgroupindex
#202617222
#202616462
#202616432
#202612832
#213297