Finite Group Databasebeta

← #41012437

Subgroups — #41012437

Order 27008.

246 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213727C211yesmaximal, characteristic
2#27213465C212yesmaximal, complemented
3#27213085C211yesmaximal, characteristic, complemented
4#27212881C212yesmaximal, complemented
5#324×C211C211yesmaximal, characteristic, complemented
6C2×#16809962—21no—
7#20269957—22no—
8#20269952—22no—
9#20269951—21no—
10#20269951C2212yes—
11C2×C4⋊Dic211C2211yescharacteristic
12#20269909—22no—
13#20269899—22no—
14C211⋊#95—21no—
15C211⋊#95—21no—
16C211⋊#95C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19#16809962—42no—
20#16809962—42no—
21C2×Dic422—21no—
22C2×D844—21no—
23C2×D844—21no—
24C2×D844D412yes—
25C2×C22⋊D211—22no—
26C22×Dic211—21no—
27C22×Dic211—21no—
28C2×C4×D211—22no—
29C4⋊Dic211—21no—
30C4⋊Dic211—22no—
31C4⋊Dic211D412yes—
32(C2×D211)⋊C4—22no—
33(C2×D211)⋊C4—22no—
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37C2×C8×C211—22no—
38C4⋊C4×C211—22no—
39C22×D211—22no—
40Dic422—42no—
41D844—42no—
42D844—42no—
43D844—81no—
44C22⋊D211—42no—
45C22⋊D211—42no—
46C2×Dic211—22no—
47C2×Dic211—41no—
48C2×Dic211—41no—
49C2×Dic211—42no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C4×D211—44no—
53C23×C211C2×D411yescharacteristic
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211SD1612yes—
60C2×C4×C211C2×D412yes—
61C8×C211—42no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211#12011yescharacteristic
65C22×C211C2×SD1611yescharacteristic
66C22×C211AGL(1,ℤ/8)11yescharacteristic
67C2×D211—42no—
68C2×D211—42no—
69C2×D211—42no—
70C4×C211—22no—
71C4×C211—22no—
72C4×C211—41no—
73Dic211—42no—
74Dic211—81no—
75Dic211—81no—
76C2×C211—21no—
77C2×C211—21no—
78C2×C211#45011yescharacteristic
79C2×C211#45411yescharacteristic
80C2×C211#45811yescharacteristic
81D211—82no—
82C211#161611yescharacteristic, complemented, Sylow
83#1616—2111nomaximal, Sylow
84#537—2111no—
85#481—2112no—
86#400—2111no—
87#327—2112no—
88#324D21111yescharacteristic, complemented
89C2×#41—4221no—
90C22⋊Q8—4222no—
91#121—4222no—
92#121—4221no—
93#121—2112no—
94C2×C4⋊C4—2111no—
95C2×C4⋊C4C2×D21111yescharacteristic
96#106—4222no—
97#102—4222no—
98C22⋊C8C2×D21112yes—
99#95—4221no—
100#95—4221no—
101#95—2111no—
102#41—8442no—
103#41—8442no—
104C2×Q8—4221no—
105C2×D4—4221no—
106C2×D4—4221no—
107C2×D4—4222no—
108C2×D4—2112no—
109C22×C4—4222no—
110C22×C4—4221no—
111C22×C4—4221no—
112C22×C4D84411yescharacteristic
113C22×C4D84411yescharacteristic
114C22×C4C22×D21111yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—4221no—
118C4⋊C4—4222no—
119C4⋊C4—2112no—
120C22⋊C4—4222no—
121C22⋊C4—4222no—
122C23—4222no—
123C23C2×D84411yescharacteristic
124Q8—8442no—
125D4—8442no—
126D4—8442no—
127D4—8442no—
128D4—8442no—
129D4—16881no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—8444no—
136C2×C4—8441no—
137C2×C4—8441no—
138C2×C4—8442no—
139C2×C4—8441no—
140C2×C4—8441no—
141C2×C4—4222no—
142C2×C4#1680995112yes—
143C2×C4D4×D21112yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—8442no—
148C22—8442no—
149C22—8442no—
150C22#2026994711yescharacteristic
151C22C2×#1680995111yescharacteristic
152C22#2027000511yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—8442no—
157C4—16881no—
158C4—16881no—
159C2—21no—
160C2—21no—
161C2—16882no—
162C2#2721329911yescharacteristic
163C2#2721332111yescharacteristic
164C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radical#27213727C2
hypercenter#324D211
hypocenterC211#1616
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27213465C2semidirect
3. #27213085C2semidirect
4. #27212881C2semidirect
5. #324×C211C2semidirect
82. C211#1616semidirect
88. #324D211semidirect

Maximal subgroups

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

subgroupindex
#272137272
#272134652
#272130852
#272128812
#324×C2112
#1616211