Finite Group Databasebeta

← #41012403

Subgroups — #41012403

Order 27008.

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

normal subgroups only

This group has 254 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#27213460C212yesmaximal, complemented
3#27213076C211yesmaximal, 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
12C2×Dic211⋊C4—21no—
13#20269911—22no—
14#20269899—22no—
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—
32Dic211⋊C4—42no—
33(C2×D211)⋊C4—22no—
34(C2×D211)⋊C4—22no—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211C2311yescharacteristic
38C2×C8×C211—22no—
39C4⋊C4×C211—22no—
40C22×D211—22no—
41Dic422—42no—
42D844—42no—
43D844—42no—
44D844—81no—
45C22⋊D211—42no—
46C22⋊D211—42no—
47C2×Dic211—22no—
48C2×Dic211—22no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C2×Dic211—41no—
53C2×Dic211—41no—
54C4×D211—44no—
55C23×C211C2×D411yescharacteristic
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211D812yes—
62C2×C4×C211C2×D412yes—
63C8×C211—42no—
64C22×C211—21no—
65C22×C211—21no—
66C22×C211#12011yescharacteristic
67C22×C211C2×D811yescharacteristic
68C22×C211AGL(1,ℤ/8)11yescharacteristic
69C2×D211—42no—
70C2×D211—42no—
71C2×D211—42no—
72C4×C211—22no—
73C4×C211—22no—
74C4×C211—41no—
75Dic211—42no—
76Dic211—42no—
77Dic211—81no—
78C2×C211—21no—
79C2×C211—21no—
80C2×C211#44711yescharacteristic
81C2×C211#45611yescharacteristic
82C2×C211#45811yescharacteristic
83D211—82no—
84C211#161011yescharacteristic, complemented, Sylow
85#1610—2111nomaximal, Sylow
86#537—2111no—
87#480—2112no—
88#398—2111no—
89#327—2112no—
90#324D21111yescharacteristic, complemented
91C2×#41—4221no—
92C22⋊Q8—4222no—
93#121—4222no—
94#121—4221no—
95#121—2112no—
96C2×C4⋊C4—4221no—
97C2×C4⋊C4—2111no—
98C2×C4⋊C4C2×D21111yescharacteristic
99#107—4222no—
100#102—4222no—
101C22⋊C8C2×D21112yes—
102#95—4221no—
103#95—2111no—
104#41—8442no—
105#41—8442no—
106C2×Q8—4221no—
107C2×D4—4221no—
108C2×D4—4221no—
109C2×D4—4222no—
110C2×D4—2112no—
111C22×C4—4222no—
112C22×C4—4221no—
113C22×C4—4221no—
114C22×C4D84411yescharacteristic
115C22×C4D84411yescharacteristic
116C22×C4C22×D21111yescharacteristic
117C2×C8—22no—
118C4⋊C4—22no—
119C4⋊C4—8442no—
120C4⋊C4—4221no—
121C4⋊C4—4222no—
122C4⋊C4—2112no—
123C22⋊C4—4222no—
124C22⋊C4—4222no—
125C23—4222no—
126C23C2×D84411yescharacteristic
127Q8—8442no—
128D4—8442no—
129D4—8442no—
130D4—8442no—
131D4—8442no—
132D4—16881no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—8444no—
139C2×C4—8441no—
140C2×C4—8441no—
141C2×C4—8441no—
142C2×C4—8441no—
143C2×C4—8441no—
144C2×C4—4222no—
145C2×C4—4222no—
146C2×C4D168812yes—
147C2×C4D4×D21112yes—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—8442no—
152C22—8442no—
153C22—8442no—
154C22#2026994711yescharacteristic
155C22C2×D168811yescharacteristic
156C22#2027000511yescharacteristic
157C4—22no—
158C4—22no—
159C4—41no—
160C4—8442no—
161C4—8442no—
162C4—16881no—
163C2—21no—
164C2—21no—
165C2—16882no—
166C2#2721328211yescharacteristic
167C2#2721333111yescharacteristic
168C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radical#27213727C2
hypercenter#324D211
hypocenterC211#1610
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. #27213460C2semidirect
3. #27213076C2semidirect
4. #27212881C2semidirect
5. #324×C211C2semidirect
84. C211#1610semidirect
90. #324D211semidirect

Maximal subgroups

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

subgroupindex
#272137272
#272134602
#272130762
#272128812
#324×C2112
#1610211