Finite Group Databasebeta

← #41012401

Subgroups — #41012401

Order 27008.

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

normal subgroups only

This group has 254 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#27213457C212yesmaximal, complemented
2#27213076C211yesmaximal, characteristic, complemented
3#27212880C212yesmaximal, complemented
4#27212872C211yesmaximal, characteristic, complemented
5#537×C211C211yesmaximal, characteristic
6C2×C4⋊Dic211C2211yescharacteristic
7C2×Dic211⋊C4—21no—
8#20269910—22no—
9#20269898—22no—
10C22⋊(C211⋊C8)C2212yes—
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—
23Dic211⋊C4—42no—
24#41×C211—42no—
25#41×C211—42no—
26C2×Q8×C211—21no—
27C2×D4×C211—21no—
28C2×D4×C211—22no—
29C2×D4×C211—21no—
30C2×D4×C211D412yes—
31C22×C4×C211—22no—
32C22×C4×C211D411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211C2311yescharacteristic
35C4⋊C4×C211—21no—
36C4⋊C4×C211—22no—
37C4⋊C4×C211D412yes—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C2×Dic211—22no—
41C2×Dic211—41no—
42C2×Dic211—41no—
43C2×Dic211—41no—
44C2×Dic211—41no—
45C2×Dic211—41no—
46C211⋊C8—42no—
47C23×C211—22no—
48C23×C211C2×D411yescharacteristic
49Q8×C211—42no—
50D4×C211—42no—
51D4×C211—42no—
52D4×C211—42no—
53D4×C211—42no—
54D4×C211—81no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—22no—
59C2×C4×C211—41no—
60C2×C4×C211—44no—
61C2×C4×C211—41no—
62C2×C4×C211D812yes—
63C2×C4×C211C2×D412yes—
64C22×C211—21no—
65C22×C211—21no—
66C22×C211—42no—
67C22×C211—42no—
68C22×C211—42no—
69C22×C211#12011yescharacteristic
70C22×C211C2×D811yescharacteristic
71C22×C211AGL(1,ℤ/8)11yescharacteristic
72C4×C211—22no—
73C4×C211—22no—
74C4×C211—42no—
75C4×C211—41no—
76Dic211—42no—
77Dic211—81no—
78C2×C211—21no—
79C2×C211—21no—
80C2×C211—82no—
81C2×C211#44711yescharacteristic
82C2×C211#45611yescharacteristic
83C2×C211#45811yescharacteristic
84C211#161011yescharacteristic, complemented, Sylow
85#1610—2111nomaximal, Sylow
86#537D21111yescharacteristic
87#480—2112no—
88#398—2111no—
89#327—2112no—
90#324—2111no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D21112yes—
96C2×C4⋊C4—4221no—
97C2×C4⋊C4—2111no—
98C2×C4⋊C4C2×D21111yescharacteristic
99#107—4222no—
100#102—4222no—
101C22⋊C8—2112no—
102#95—4221no—
103#95—2111no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D21112yes—
111C22×C4—22no—
112C22×C4—4221no—
113C22×C4—4221no—
114C22×C4C22⋊D21111yescharacteristic
115C22×C4C22⋊D21111yescharacteristic
116C22×C4C22×D21111yescharacteristic
117C2×C8—4222no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—8442no—
121C4⋊C4—4222no—
122C4⋊C4C22⋊D21112yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D21111yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—8441no—
141C2×C4—8441no—
142C2×C4—8441no—
143C2×C4—8441no—
144C2×C4—8441no—
145C2×C4—4222no—
146C2×C4#1680994712yes—
147C2×C4C2×C22⋊D21112yes—
148C8—8442no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#2026994611yescharacteristic
155C22C2×#1680994711yescharacteristic
156C22#2027000311yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—42no—
161C4—8442no—
162C4—16881no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2721328011yescharacteristic
167C2#2721332911yescharacteristic
168C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#537×C211C2
socleC22×C211#120
hypercenter#537D211
hypocenterC211#1610
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. #27213457C2semidirect
2. #27213076C2semidirect
3. #27212880C2semidirect
4. #27212872C2semidirect
84. C211#1610semidirect

Maximal subgroups

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

subgroupindex
#272134572
#272130762
#272128802
#272128722
#537×C2112
#1610211