Finite Group Databasebeta

← #41012866

Subgroups — #41012866

Order 27008.

182 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 51 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#27213520C211yesmaximal, characteristic, complemented
2#27213423C211yesmaximal, characteristic, complemented
3#27213342C211yesmaximal, characteristic
4#27213223C211yesmaximal, characteristic, complemented
5#27212898C211yesmaximal, characteristic, complemented
6#27212889C211yesmaximal, characteristic, complemented
7#332×C211C211yesmaximal, characteristic, complemented
8C2×#16809949—21no—
9C2×#16809949—21no—
10C2×D1688—21no—
11C2×#16809945—21no—
12#20269974C2211yescharacteristic
13C4×D844C2211yescharacteristic
14#20269910—21no—
15C4⋊(C211⋊C8)C2211yescharacteristic
16#20269899—21no—
17#20269897—21no—
18(C2×D211)⋊C8—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4⋊C8×C211C2211yescharacteristic
22C4⋊C8×C211C2211yescharacteristic
23C2×D844—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844D411yescharacteristic
27C2×C4×D211—21no—
28#16809949—42no—
29#16809949—42no—
30D1688—42no—
31#16809945—42no—
32C2×C211⋊C8—21no—
33C2×C211⋊C8D411yescharacteristic
34C2×C211⋊C8D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36(C2×D211)⋊C4—21no—
37C2×Q8×C211—21no—
38C2×C8×C211—21no—
39C2×C8×C211—21no—
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C42×C211C2311yescharacteristic
43C22×D211—21no—
44C22×D211—41no—
45D844—21no—
46D844—21no—
47D844—41no—
48D844—41no—
49D844—41no—
50D844—41no—
51D844—81no—
52C2×Dic211—21no—
53C4×D211—41no—
54C4×D211—41no—
55C211⋊C8—21no—
56C211⋊C8—41no—
57C211⋊C8D812yescomplemented
58Q8×C211—41no—
59Q8×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211#4111yescharacteristic
64C8×C211—41no—
65C8×C211—41no—
66C22×C211#12111yescharacteristic
67C2×D211—41no—
68C2×D211—41no—
69C2×D211—41no—
70C2×D211—81no—
71C2×D211—81no—
72C2×D211—81no—
73C4×C211—21no—
74C4×C211—81no—
75C4×C211C2×D811yescharacteristic
76C4×C211AGL(1,ℤ/8)11yescharacteristic
77C4×C211AGL(1,ℤ/8)11yescharacteristic
78C4×C211#13711yescharacteristic
79Dic211—41no—
80C2×C211#45911yescharacteristic
81C2×C211#46811yescharacteristic
82C2×C211#47111yescharacteristic
83D211—81no—
84D211—161no—
85C211#168511yescharacteristic, complemented, Sylow
86#1685—2111nomaximal, Sylow
87#492—2111no—
88#474—2111no—
89#459—2111no—
90#436—2111no—
91#335—2111no—
92#332D21111yescharacteristic, complemented
93#331—2111no—
94C2×SD16—4221no—
95C2×SD16—4221no—
96C2×D8—4221no—
97C2×M16—4221no—
98C4⋊Q8C2×D21111yescharacteristic
99#127—2111no—
100C4×D4—2111no—
101#107—4221no—
102C4⋊C8—2111no—
103C4⋊C8C2×D21111yescharacteristic
104C4⋊C8C2×D21111yescharacteristic
105#102—4221no—
106#102—4221no—
107C22⋊C8—4221no—
108C4×C8—2111no—
109C2×Q8—21no—
110C2×D4—4221no—
111C2×D4—4221no—
112C2×D4—4221no—
113C2×D4—2111no—
114C22×C4—4221no—
115SD16—8442no—
116SD16—8442no—
117D8—8442no—
118M16—8442no—
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—4221no—
122C2×C8—2111no—
123C2×C8—2111no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—2111no—
127C22⋊C4—4221no—
128C42C22×D21111yescharacteristic
129C23—8441no—
130C23—4221no—
131Q8—41no—
132Q8—41no—
133D4—8441no—
134D4—8441no—
135D4—8441no—
136D4—8441no—
137D4—16881no—
138D4—4221no—
139D4—4221no—
140C2×C4—41no—
141C2×C4—8441no—
142C2×C4—8441no—
143C2×C4—4221no—
144C2×C4D4×D21111yescharacteristic
145C2×C4D4×D21111yescharacteristic
146C2×C4#1680996211yescharacteristic
147C8—41no—
148C8—41no—
149C8—8441no—
150C8—4221no—
151C8—2112no—
152C22—8441no—
153C22—8441no—
154C22—8441no—
155C22—16881no—
156C22—16881no—
157C22—16881no—
158C22#2026995011yescharacteristic
159C4—21no—
160C4—81no—
161C4—8441no—
162C4D8×D21111yescharacteristic
163C4#2027000411yescharacteristic
164C4#2027000411yescharacteristic
165C4#2027000911yescharacteristic
166C2—33761no—
167C2—16881no—
168C2#2721334011yescharacteristic
169C2#2721339511yescharacteristic
170C2#2721341211yescharacteristic
subgroupG/H
centerC22#20269950
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#121
prime radical#27213342C2
hypercenter#332D211
hypocenterC211#1685
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27213520C2semidirect
2. #27213423C2semidirect
4. #27213223C2semidirect
5. #27212898C2semidirect
6. #27212889C2semidirect
7. #332×C211C2semidirect
57. C211⋊C8D8semidirect
85. C211#1685semidirect
92. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272135202
#272134232
#272133422
#272132232
#272128982
#272128892
#332×C2112
#1685211