Finite Group Databasebeta

← #41012121

Subgroups — #41012121

Order 27008.

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

normal subgroups only

This group has 246 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#27213696C211yesmaximal, characteristic
2#27213170C211yesmaximal, characteristic, complemented
3#27212893C212yesmaximal, complemented
4#27212889C212yesmaximal, complemented
5#432×C211C211yesmaximal, characteristic, complemented
6C2×#16809962—21no—
7C2×#16809946—21no—
8#20269977C2211yescharacteristic
9#20269974C2211yescharacteristic
10#20269965—21no—
11#20269951—21no—
12#20269951C412yes—
13C4×D844—21no—
14C4×D844C412yes—
15C4⋊(C211⋊C8)—21no—
16C4⋊(C211⋊C8)C2212yes—
17C2×C42×C211C2211yescharacteristic
18C4⋊C8×C211C2212yes—
19C22⋊C8×C211—21no—
20#97×C211—21no—
21#16809962—44no—
22C2×Dic422—21no—
23C2×D844—21no—
24C2×D844—21no—
25C2×D844C2×C412yes—
26C2×C22⋊D211—22no—
27C2×C4×D211—22no—
28#16809946—42no—
29C2×C211⋊C8—22no—
30C4⋊Dic211—21no—
31C4⋊Dic211C2×C412yes—
32(C2×D211)⋊C4—22no—
33(C2×D211)⋊C4—22no—
34C22×C4×C211C2×C411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C2×C8×C211—22no—
38C42×C211C2×C411yescharacteristic
39C42×C211D411yescharacteristic
40C42×C211D411yescharacteristic
41C42×C211C2311yescharacteristic
42C22×D211—22no—
43Dic422—42no—
44D844—42no—
45D844—41no—
46D844—42no—
47D844—41no—
48C22⋊D211—44no—
49C2×Dic211—22no—
50C4×D211—44no—
51C211⋊C8—42no—
52C23×C211C22⋊C411yescharacteristic
53C2×C4×C211—21no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—41no—
57C2×C4×C211C22⋊C411yescharacteristic
58C2×C4×C211C22⋊C411yescharacteristic
59C2×C4×C211C22⋊C411yescharacteristic
60C2×C4×C211C22×C411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C8×C211—42no—
64C22×C211—21no—
65C22×C211—21no—
66C22×C211—21no—
67C22×C211C2×C22⋊C411yescharacteristic
68C2×D211—42no—
69C2×D211—42no—
70C2×D211—42no—
71C4×C211—21no—
72C4×C211—21no—
73C4×C211—21no—
74C4×C211—21no—
75C4×C211AGL(1,ℤ/8)12yes—
76C4×C211AGL(1,ℤ/8)11yescharacteristic
77C4×C211#13711yescharacteristic
78Dic211—42no—
79C2×C211—41no—
80C2×C211#41311yescharacteristic
81C2×C211#41711yescharacteristic
82C2×C211#41811yescharacteristic
83D211—82no—
84C211#155711yescharacteristic, complemented, Sylow
85#1557—2111nomaximal, Sylow
86#532—2111no—
87#432D21111yescharacteristic, complemented
88#423—2111no—
89#332—2112no—
90#331—2112no—
91C2×#41—4221no—
92C2×M16—4221no—
93C4⋊Q8—2111no—
94#127—2111no—
95#124—4221no—
96#121—4221no—
97#121—2112no—
98C4×D4—4221no—
99C4×D4—2112no—
100C2×C42C2×D21111yescharacteristic
101C4⋊C8—4221no—
102C4⋊C8—2112no—
103C4⋊C8C2×D21112yes—
104C22⋊C8—21no—
105#97—21no—
106#41—8444no—
107C2×Q8—4221no—
108C2×D4—4222no—
109C2×D4—4221no—
110C2×D4—4221no—
111C2×D4—2112no—
112C22×C4—4222no—
113C22×C4C4×D21111yescharacteristic
114C22×C4C22⋊D21111yescharacteristic
115C22×C4D84411yescharacteristic
116M16—8442no—
117C2×C8—22no—
118C2×C8—4222no—
119C4⋊C4—4221no—
120C4⋊C4—2112no—
121C22⋊C4—4222no—
122C22⋊C4—4222no—
123C42C4×D21111yescharacteristic
124C42C22⋊D21111yescharacteristic
125C42D84411yescharacteristic
126C42C22×D21111yescharacteristic
127C23—4222no—
128C23(C2×D211)⋊C411yescharacteristic
129Q8—8442no—
130D4—8442no—
131D4—8444no—
132D4—8442no—
133D4—8441no—
134D4—8441no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—8444no—
140C2×C4—4222no—
141C2×C4(C2×D211)⋊C411yescharacteristic
142C2×C4(C2×D211)⋊C411yescharacteristic
143C2×C4(C2×D211)⋊C411yescharacteristic
144C2×C4C2×C4×D21111yescharacteristic
145C2×C4C2×C22⋊D21111yescharacteristic
146C2×C4C2×D84411yescharacteristic
147C8—42no—
148C8—8442no—
149C22—21no—
150C22—21no—
151C22—21no—
152C22—8442no—
153C22—8442no—
154C22—8442no—
155C22C2×(C2×D211)⋊C411yescharacteristic
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—8442no—
161C4#2027000212yes—
162C4#2027000511yescharacteristic
163C4#2027001011yescharacteristic
164C2—41no—
165C2—16882no—
166C2#2721312811yescharacteristic
167C2#2721314811yescharacteristic
168C2#2721315011yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213696C2
cosocleC42×C211C23
hypercenter#432D211
hypocenterC211#1557
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #27213170C2semidirect
3. #27212893C2semidirect
4. #27212889C2semidirect
5. #432×C211C2semidirect
84. C211#1557semidirect
87. #432D211semidirect

Maximal subgroups

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

subgroupindex
#272136962
#272131702
#272128932
#272128892
#432×C2112
#1557211