Finite Group Databasebeta

← #27143421

Subgroups — #27143421

Order 12416.

250 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 73 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#20260719C212yesmaximal, complemented
2#20260534C212yesmaximal, complemented
3#20260532C212yesmaximal, complemented
4#532×C97C211yesmaximal, characteristic
5C2×#13095495—22no—
6#15690729C2214yes—
7#15690727—22no—
8C2×#41×C97C411yescharacteristic
9C2×#41×C97C2211yescharacteristic
10C4⋊Q8×C97C2211yescharacteristic
11#127×C97C411yescharacteristic
12#124×C97C411yescharacteristic
13#124×C97C2211yescharacteristic
14#121×C97—22no—
15C4×D4×C97—22no—
16C2×C42×C97C411yescharacteristic
17#13095495—24no—
18#13095495—42no—
19C2×#11715895—42no—
20#41×C97—41no—
21#41×C97—41no—
22#41×C97—41no—
23#41×C97—41no—
24C2×Q8×C97C2×C411yescharacteristic
25C2×Q8×C97C2311yescharacteristic
26C2×D4×C97—22no—
27C2×D4×C97—21no—
28C2×D4×C97C2×C411yescharacteristic
29C2×D4×C97C2×C411yescharacteristic
30C2×D4×C97D412yes—
31C22×C4×C97—21no—
32C22×C4×C97—21no—
33C22×C4×C97C2×C411yescharacteristic
34C22×C4×C97D412yes—
35C4⋊C4×C97—22no—
36C22⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C42×C97—21no—
39C42×C97C2×C411yescharacteristic
40C42×C97C2×C411yescharacteristic
41C23×C97—21no—
42C23×C97C22⋊C412yes—
43C23×C97C22⋊C411yescharacteristic
44Q8×C97—41no—
45Q8×C97—41no—
46D4×C97—41no—
47D4×C97—42no—
48D4×C97—41no—
49D4×C97—44no—
50D4×C97—42no—
51C2×C4×C97—21no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—22no—
55C2×C4×C97—22no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—42no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22×C411yescharacteristic
63C2×C4×C97C2×D412yes—
64#11715895—44no—
65C22×C97—21no—
66C22×C97—22no—
67C22×C97—41no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97—41no—
71C22×C97C2×C22⋊C411yescharacteristic
72C4×C97—22no—
73C4×C97—22no—
74C4×C97—21no—
75C4×C97—41no—
76C4×C97—41no—
77C4×C97#9911yescharacteristic
78C4×C97#9911yescharacteristic
79C2×C97—21no—
80C2×C97—41no—
81C2×C97—42no—
82C2×C97—81no—
83C2×C97C2×#9911yescharacteristic
84#2132—971nomaximal, Sylow
85C97#213211yescharacteristic, complemented, Sylow
86#532D9711yescharacteristic
87#413—972no—
88#356—972no—
89#355—972no—
90C2×#41Dic9711yescharacteristic
91C2×#41C2×D9711yescharacteristic
92C2×M16—1942no—
93C4⋊Q8C2×D9711yescharacteristic
94#127Dic9711yescharacteristic
95#124Dic9711yescharacteristic
96#124C2×D9711yescharacteristic
97#121—22no—
98C4×D4—22no—
99C2×C42Dic9711yescharacteristic
100#101—974no—
101#100—1942no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—41no—
106C2×Q8C2×Dic9711yescharacteristic
107C2×Q8C22×D9711yescharacteristic
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C2×Dic9711yescharacteristic
111C2×D4C2×Dic9711yescharacteristic
112C2×D4C22⋊D9712yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4C2×Dic9711yescharacteristic
116C22×C4C22⋊D9712yes—
117M16—3882no—
118M16—1944no—
119C2×C8—3882no—
120C4⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42—21no—
124C42C2×Dic9711yescharacteristic
125C42C2×Dic9711yescharacteristic
126C23—21no—
127C23C22⋊Dic9712yes—
128C23C22⋊Dic9711yescharacteristic
129Q8—41no—
130Q8—41no—
131D4—41no—
132D4—42no—
133D4—41no—
134D4—44no—
135D4—42no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4C22⋊Dic9711yescharacteristic
147C2×C4C22×Dic9711yescharacteristic
148C2×C4C2×C22⋊D9712yes—
149C8—3884no—
150C22—21no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—41no—
156C22C2×C22⋊Dic9711yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4#1569072511yescharacteristic
163C4#1569072511yescharacteristic
164C2—21no—
165C2—41no—
166C2—42no—
167C2—81no—
168C2C2×#1569072511yescharacteristic
subgroupG/H
centerC4#15690725
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting, prime radical#532×C97C2
socleC2×C97C2×#99
cosocleC2×Q8×C97C23
hypercenter#532D97
hypocenterC97#2132
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #20260719C2semidirect
2. #20260534C2semidirect
3. #20260532C2semidirect
85. C97#2132semidirect

Maximal subgroups

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

subgroupindex
#202607192
#202605342
#202605322
#532×C972
#213297