Finite Group Databasebeta

← #41011997

Subgroups — #41011997

Order 27008.

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

normal subgroups only

This group has 264 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
1C2×#20269965C211yesmaximal, characteristic
2C4×#16809946C211yesmaximal, characteristic, complemented
3#27212885C214yesmaximal, complemented
4#432×C211C211yesmaximal, characteristic, complemented
5C22×Dic422C411yescharacteristic
6C22×D844C411yescharacteristic
7C2×#16809946—21no—
8#20269965—21no—
9#20269965—21no—
10#20269965C412yes—
11#20269965C2212yes—
12C2×(C2×D211)⋊C4—21no—
13C2×(C2×D211)⋊C4—21no—
14#20269888C2212yescomplemented
15C4×C211⋊C8—21no—
16C2×C42×C211C2211yescharacteristic
17C4⋊C8×C211—21no—
18C22⋊C8×C211—21no—
19#97×C211C2212yescomplemented
20C23×D211—21no—
21C2×Dic422—21no—
22C2×Dic422—42no—
23C2×Dic422C2×C412yes—
24C2×D844—21no—
25C2×D844—42no—
26C2×D844C2×C412yes—
27C22×Dic211—21no—
28#16809946—24no—
29C2×C211⋊C8—22no—
30(C2×D211)⋊C4—22no—
31(C2×D211)⋊C4—22no—
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—
43C22×D211—42no—
44C22×D211—42no—
45C22×D211—42no—
46Dic422—42no—
47Dic422—42no—
48D844—42no—
49D844—42no—
50C2×Dic211—22no—
51C2×Dic211—42no—
52C211⋊C8—24no—
53C23×C211C22⋊C411yescharacteristic
54C2×C4×C211—22no—
55C2×C4×C211—22no—
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#10412yes—
67C22×C211C2×C22⋊C411yescharacteristic
68C2×D211—42no—
69C2×D211—42no—
70C2×D211—42no—
71C2×D211—84no—
72C4×C211—21no—
73C4×C211—21no—
74C4×C211—21no—
75C4×C211—21no—
76C4×C211—21no—
77C4×C211#10012yes—
78Dic211—42no—
79C2×C211—22no—
80C2×C211C2×#10011yescharacteristic
81C2×C211C2×#10411yescharacteristic
82C2×C211#42111yescharacteristic
83D211—82no—
84C211#152711yescharacteristic, complemented, Sylow
85#1527—2111nomaximal, Sylow
86C2×#124—2111no—
87#432D21111yescharacteristic, complemented
88C4×M16—2111no—
89#329—2114no—
90C22×Q8—2111no—
91C22×D4—2111no—
92C2×M16—4221no—
93#124—4221no—
94#124—4221no—
95#124—2112no—
96#124—2112no—
97C2×C22⋊C4—4221no—
98C2×C22⋊C4—4221no—
99C2×C42C2×D21111yescharacteristic
100C4⋊C8—21no—
101C22⋊C8—21no—
102#97—2112no—
103#97C2×D21112yescomplemented
104C4×C8—4221no—
105C24—4221no—
106C2×Q8—8442no—
107C2×Q8—4221no—
108C2×Q8—2112no—
109C2×D4—8442no—
110C2×D4—4221no—
111C2×D4—2112no—
112C22×C4—4221no—
113C22×C4C4×D21111yescharacteristic
114C22×C4C22⋊D21111yescharacteristic
115C22×C4D84411yescharacteristic
116M16—4224no—
117C2×C8—22no—
118C2×C8—4222no—
119C22⋊C4—4222no—
120C22⋊C4—4222no—
121C22⋊C4—4222no—
122C22⋊C4—4222no—
123C42C4×D21111yescharacteristic
124C42C22⋊D21111yescharacteristic
125C42D84411yescharacteristic
126C42C22×D21111yescharacteristic
127C23—8442no—
128C23—8442no—
129C23—8442no—
130C23—4222no—
131C23(C2×D211)⋊C411yescharacteristic
132Q8—8442no—
133Q8—8442no—
134D4—8442no—
135D4—8442no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—8442no—
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—4224no—
149C22—21no—
150C22—21no—
151C22—8442no—
152C22—8442no—
153C22—8442no—
154C22—16884no—
155C22#2026990312yes—
156C22C2×(C2×D211)⋊C411yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—8442no—
163C4#2026989312yes—
164C2—22no—
165C2—16882no—
166C2C2×#2026989311yescharacteristic
167C2C2×#2026990311yescharacteristic
168C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D844C4
cosocleC42×C211C23
hypercenter#432D211
hypocenterC211#1527
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. C4×#16809946C2semidirect
3. #27212885C2semidirect
4. #432×C211C2semidirect
14. #20269888C22semidirect
19. #97×C211C22semidirect
84. C211#1527semidirect
87. #432D211semidirect
103. #97C2×D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699652
C4×#168099462
#272128852
#432×C2112
#1527211