Finite Group Databasebeta

← #41012435

Subgroups — #41012435

Order 27008.

246 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213462C212yesmaximal, complemented
2#27213085C211yesmaximal, characteristic, complemented
3#27212880C212yesmaximal, complemented
4#27212872C211yesmaximal, characteristic, complemented
5#537×C211C211yesmaximal, characteristic
6C2×C4⋊Dic211C2211yescharacteristic
7#20269908—22no—
8#20269898—22no—
9C22⋊(C211⋊C8)C2212yes—
10C211⋊#95—21no—
11C211⋊#95—21no—
12C211⋊#95C2211yescharacteristic
13C2×#41×C211—21no—
14C22⋊Q8×C211—22no—
15#121×C211—22no—
16#121×C211—21no—
17#121×C211C2212yes—
18C2×C4⋊C4×C211C2211yescharacteristic
19C22×Dic211—21no—
20C22×Dic211—21no—
21C2×C211⋊C8—22no—
22C4⋊Dic211—22no—
23#41×C211—42no—
24#41×C211—42no—
25C2×Q8×C211—21no—
26C2×D4×C211—21no—
27C2×D4×C211—22no—
28C2×D4×C211—21no—
29C2×D4×C211D412yes—
30C22×C4×C211—22no—
31C22×C4×C211D411yescharacteristic
32C22×C4×C211D411yescharacteristic
33C22×C4×C211C2311yescharacteristic
34C4⋊C4×C211—21no—
35C4⋊C4×C211—22no—
36C4⋊C4×C211D412yes—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—22no—
39C2×Dic211—41no—
40C2×Dic211—41no—
41C2×Dic211—42no—
42C2×Dic211—41no—
43C2×Dic211—41no—
44C211⋊C8—42no—
45C23×C211—22no—
46C23×C211C2×D411yescharacteristic
47Q8×C211—42no—
48D4×C211—42no—
49D4×C211—42no—
50D4×C211—42no—
51D4×C211—42no—
52D4×C211—81no—
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—22no—
57C2×C4×C211—41no—
58C2×C4×C211—44no—
59C2×C4×C211—41no—
60C2×C4×C211SD1612yes—
61C2×C4×C211C2×D412yes—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211—42no—
65C22×C211—42no—
66C22×C211—42no—
67C22×C211#12011yescharacteristic
68C22×C211C2×SD1611yescharacteristic
69C22×C211AGL(1,ℤ/8)11yescharacteristic
70C4×C211—22no—
71C4×C211—22no—
72C4×C211—42no—
73C4×C211—41no—
74Dic211—81no—
75Dic211—81no—
76C2×C211—21no—
77C2×C211—21no—
78C2×C211—82no—
79C2×C211#45011yescharacteristic
80C2×C211#45411yescharacteristic
81C2×C211#45811yescharacteristic
82C211#161611yescharacteristic, complemented, Sylow
83#1616—2111nomaximal, Sylow
84#537D21111yescharacteristic
85#481—2112no—
86#400—2111no—
87#327—2112no—
88#324—2111no—
89C2×#41—21no—
90C22⋊Q8—22no—
91#121—22no—
92#121—21no—
93#121C2×D21112yes—
94C2×C4⋊C4—2111no—
95C2×C4⋊C4C2×D21111yescharacteristic
96#106—4222no—
97#102—4222no—
98C22⋊C8—2112no—
99#95—4221no—
100#95—4221no—
101#95—2111no—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—21no—
108C2×D4C22⋊D21112yes—
109C22×C4—22no—
110C22×C4—4221no—
111C22×C4—4221no—
112C22×C4C22⋊D21111yescharacteristic
113C22×C4C22⋊D21111yescharacteristic
114C22×C4C22×D21111yescharacteristic
115C2×C8—4222no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—4222no—
119C4⋊C4C22⋊D21112yes—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C23—22no—
123C23C2×C22⋊D21111yescharacteristic
124Q8—42no—
125D4—42no—
126D4—42no—
127D4—42no—
128D4—42no—
129D4—81no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—41no—
135C2×C4—44no—
136C2×C4—41no—
137C2×C4—8441no—
138C2×C4—8441no—
139C2×C4—8442no—
140C2×C4—8441no—
141C2×C4—8441no—
142C2×C4#1680995012yes—
143C2×C4C2×C22⋊D21112yes—
144C8—8442no—
145C22—21no—
146C22—21no—
147C22—42no—
148C22—42no—
149C22—42no—
150C22#2026994611yescharacteristic
151C22C2×#1680995011yescharacteristic
152C22#2027000311yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—42no—
157C4—16881no—
158C4—16881no—
159C2—21no—
160C2—21no—
161C2—82no—
162C2#2721329711yescharacteristic
163C2#2721331911yescharacteristic
164C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#537×C211C2
socleC22×C211#120
hypercenter#537D211
hypocenterC211#1616
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #27213462C2semidirect
2. #27213085C2semidirect
3. #27212880C2semidirect
4. #27212872C2semidirect
82. C211#1616semidirect

Maximal subgroups

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

subgroupindex
#272134622
#272130852
#272128802
#272128722
#537×C2112
#1616211