Finite Group Databasebeta

← #41011958

Subgroups — #41011958

Order 27008.

264 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 93 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#27213199C211yesmaximal, characteristic, complemented
2#27213012C211yesmaximal, characteristic, complemented
3#27212880C214yesmaximal, complemented
4#532×C211C211yesmaximal, characteristic
5C2×C4⋊Dic211—21no—
6C4⋊(C211⋊C8)—21no—
7C22⋊(C211⋊C8)C2212yes—
8C4×C211⋊C8—21no—
9C211⋊#95—21no—
10C211⋊#95C2212yes—
11C2×#41×C211—21no—
12C4⋊Q8×C211C411yescharacteristic, complemented
13#127×C211C411yescharacteristic, complemented
14#124×C211—21no—
15#121×C211—21no—
16#121×C211C2212yes—
17C4×D4×C211—21no—
18C4×D4×C211C412yescomplemented
19C2×C42×C211C2211yescharacteristic
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C4⋊Dic211—42no—
23#41×C211—42no—
24#41×C211—42no—
25C2×Q8×C211—21no—
26C2×D4×C211—22no—
27C2×D4×C211—21no—
28C2×D4×C211—21no—
29C2×D4×C211C2×C412yes—
30C22×C4×C211—22no—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211D411yescharacteristic
33C22×C4×C211C2311yescharacteristic
34C4⋊C4×C211—21no—
35C4⋊C4×C211C2×C412yes—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C42×C211C2×C411yescharacteristic
39C42×C211C2×C411yescharacteristic
40C42×C211D411yescharacteristic
41C42×C211D411yescharacteristic
42C2×Dic211—42no—
43C2×Dic211—42no—
44C2×Dic211—42no—
45C211⋊C8—42no—
46C23×C211—22no—
47C23×C211C2×D411yescharacteristic
48Q8×C211—42no—
49D4×C211—42no—
50D4×C211—42no—
51D4×C211—42no—
52D4×C211—42no—
53D4×C211—42no—
54C2×C4×C211—22no—
55C2×C4×C211—41no—
56C2×C4×C211—41no—
57C2×C4×C211—44no—
58C2×C4×C211C22⋊C411yescharacteristic
59C2×C4×C211C22⋊C411yescharacteristic
60C2×C4×C211C22⋊C411yescharacteristic
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211D812yes—
63C2×C4×C211SD1612yes—
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C22×C211—21no—
67C22×C211—21no—
68C22×C211—42no—
69C22×C211—42no—
70C22×C211—42no—
71C22×C211C2×C22⋊C411yescharacteristic
72C22×C211C2×D811yescharacteristic
73C22×C211C2×SD1611yescharacteristic
74C4×C211—21no—
75C4×C211—21no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—42no—
79C4×C211#10212yes—
80C4×C211#10212yes—
81Dic211—82no—
82C2×C211—21no—
83C2×C211—21no—
84C2×C211—82no—
85C2×C211#41011yescharacteristic
86C2×C211C2×#10211yescharacteristic
87C2×C211#42111yescharacteristic
88C211#151711yescharacteristic, complemented, Sylow
89#1517—2111nomaximal, Sylow
90#532D21111yescharacteristic
91#431—2111no—
92#382—2111no—
93#327—2114no—
94C2×#41—21no—
95C4⋊Q8Dic21111yescharacteristic, complemented
96#127Dic21111yescharacteristic, complemented
97#124—21no—
98#121—21no—
99#121C2×D21112yes—
100C4×D4—21no—
101C4×D4Dic21112yescomplemented
102C2×C4⋊C4—4221no—
103C2×C42C2×D21111yescharacteristic
104C4⋊C8—4221no—
105C22⋊C8—2112no—
106C4×C8—4221no—
107#95—4221no—
108#95—2112no—
109#41—42no—
110#41—42no—
111C2×Q8—21no—
112C2×D4—22no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4C2×Dic21112yes—
116C22×C4—22no—
117C22×C4—4222no—
118C22×C4C22⋊D21111yescharacteristic
119C22×C4C22⋊D21111yescharacteristic
120C22×C4C22×D21111yescharacteristic
121C2×C8—4222no—
122C4⋊C4—21no—
123C4⋊C4—8442no—
124C4⋊C4C2×Dic21112yes—
125C22⋊C4—22no—
126C22⋊C4—22no—
127C42C2×Dic21111yescharacteristic
128C42C2×Dic21111yescharacteristic
129C42C22⋊D21111yescharacteristic
130C42C22⋊D21111yescharacteristic
131C23—22no—
132C23C2×C22⋊D21111yescharacteristic
133Q8—42no—
134D4—42no—
135D4—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—44no—
143C2×C4—8442no—
144C2×C4—8442no—
145C2×C4—8442no—
146C2×C4C22⋊Dic21111yescharacteristic
147C2×C4C22⋊Dic21111yescharacteristic
148C2×C4C22⋊Dic21111yescharacteristic
149C2×C4C22⋊Dic21111yescharacteristic
150C2×C4#1680994712yes—
151C2×C4#1680995012yes—
152C2×C4C22×Dic21111yescharacteristic
153C2×C4C2×C22⋊D21111yescharacteristic
154C8—8442no—
155C22—21no—
156C22—21no—
157C22—42no—
158C22—42no—
159C22—42no—
160C22C2×C22⋊Dic21111yescharacteristic
161C22C2×#1680994711yescharacteristic
162C22C2×#1680995011yescharacteristic
163C4—21no—
164C4—21no—
165C4—21no—
166C4—21no—
167C4—42no—
168C4—16882no—
169C4#2026989812yes—
170C4#2026989812yes—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#2721312011yescharacteristic
175C2C2×#2026989811yescharacteristic
176C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#532×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#532D211
hypocenterC211#1517
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #27213199C2semidirect
2. #27213012C2semidirect
3. #27212880C2semidirect
12. C4⋊Q8×C211C4semidirect
13. #127×C211C4semidirect
18. C4×D4×C211C4semidirect
88. C211#1517semidirect
95. C4⋊Q8Dic211semidirect
96. #127Dic211semidirect
101. C4×D4Dic211semidirect

Maximal subgroups

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

subgroupindex
#272131992
#272130122
#272128802
#532×C2112
#1517211