Finite Group Databasebeta

← #41012320

Subgroups — #41012320

Order 27008.

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

normal subgroups only

This group has 316 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#27213772C211yesmaximal, characteristic, complemented
2#27213297C212yesmaximal, complemented
3#27212882C212yesmaximal, complemented
4#27212870C211yesmaximal, characteristic, complemented
5#392×C211C211yesmaximal, characteristic
6C2×#16809950—22no—
7#20269956—21no—
8#20269956C2212yes—
9#20269954—22no—
10#20269954—22no—
11C2×C22⋊Dic211—21no—
12#20269898—22no—
13C22⋊(C211⋊C8)C2212yes—
14C22×D4×C211—21no—
15C22×D4×C211C2211yescharacteristic
16C2×C22⋊C4×C211—21no—
17C2×C22⋊C4×C211C2211yescharacteristic
18#95×C211C2211yescharacteristic
19C2×Dic422—21no—
20C2×Dic422D412yescomplemented
21C22×Dic211—21no—
22C22×Dic211—21no—
23#16809950—44no—
24C2×C211⋊C8—22no—
25C4⋊Dic211—21no—
26C4⋊Dic211D412yescomplemented
27Dic211⋊C4—22no—
28Dic211⋊C4—22no—
29C22⋊Dic211—22no—
30C22⋊Dic211—22no—
31C24×C211—21no—
32C24×C211D411yescharacteristic
33C2×D4×C211—22no—
34C2×D4×C211—42no—
35C2×D4×C211—42no—
36C2×D4×C211—42no—
37C2×D4×C211—42no—
38C22×C4×C211—21no—
39C22×C4×C211D411yescharacteristic
40C22×C4×C211C2311yescharacteristic
41C22⋊C4×C211—22no—
42C22⋊C4×C211—42no—
43Dic422—42no—
44Dic422—81no—
45C2×Dic211—22no—
46C2×Dic211—22no—
47C2×Dic211—81no—
48C2×Dic211—81no—
49C211⋊C8—42no—
50C23×C211—21no—
51C23×C211—21no—
52C23×C211—22no—
53C23×C211—41no—
54C23×C211—41no—
55C23×C211—41no—
56C23×C211—41no—
57C23×C211—41no—
58C23×C211—41no—
59C23×C211—41no—
60C23×C211—41no—
61C23×C211SD1612yescomplemented
62C23×C211C2×D411yescharacteristic
63D4×C211—24no—
64D4×C211—42no—
65D4×C211—48no—
66C2×C4×C211—22no—
67C2×C4×C211—21no—
68C2×C4×C211—41no—
69C2×C4×C211—41no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211C2×D412yes—
73C22×C211—21no—
74C22×C211—21no—
75C22×C211—22no—
76C22×C211—42no—
77C22×C211—42no—
78C22×C211—42no—
79C22×C211—41no—
80C22×C211—41no—
81C22×C211—42no—
82C22×C211—81no—
83C22×C211—81no—
84C22×C211—81no—
85C22×C211—81no—
86C22×C211—81no—
87C22×C211#12011yescharacteristic
88C22×C211C2×SD1611yescharacteristic
89C22×C211#13711yescharacteristic
90C4×C211—22no—
91C4×C211—42no—
92C4×C211—41no—
93Dic211—42no—
94Dic211—42no—
95C2×C211—21no—
96C2×C211—21no—
97C2×C211—42no—
98C2×C211—82no—
99C2×C211#44811yescharacteristic
100C2×C211#45311yescharacteristic
101C2×C211B(4,𝔽2)11yescharacteristic
102C211#159611yescharacteristic, complemented, Sylow
103#1596—2111nomaximal, Sylow
104#543—2111no—
105#450—2112no—
106#392D21111yescharacteristic
107#328—2112no—
108#323—2111no—
109C22×D4—21no—
110C22×D4C2×D21111yescharacteristic
111C2×SD16—4222no—
112C22⋊Q8—4222no—
113C22⋊Q8—4222no—
114C22⋊Q8—4221no—
115C22⋊Q8—2112no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—4221no—
118C2×C22⋊C4C2×D21111yescharacteristic
119#102—4222no—
120C22⋊C8—2112no—
121#95C2×D21111yescharacteristic
122C24—21no—
123C24C22⋊D21111yescharacteristic
124C2×Q8—4221no—
125C2×Q8—2112no—
126C2×D4—22no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—42no—
130C2×D4—42no—
131C22×C4—21no—
132C22×C4—4221no—
133C22×C4—4221no—
134C22×C4C22⋊D21111yescharacteristic
135C22×C4C22×D21111yescharacteristic
136SD16—8444no—
137C2×C8—4222no—
138C4⋊C4—4222no—
139C4⋊C4—4221no—
140C4⋊C4—4222no—
141C4⋊C4—2112no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—4222no—
145C22⋊C4—4222no—
146C23—21no—
147C23—21no—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23#1680995012yescomplemented
158C23C2×C22⋊D21111yescharacteristic
159Q8—8442no—
160Q8—16881no—
161D4—24no—
162D4—42no—
163D4—48no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—16881no—
171C2×C4—16881no—
172C2×C4—4222no—
173C2×C4—4222no—
174C2×C4D4×D21112yes—
175C8—8442no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—41no—
183C22—41no—
184C22—42no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22—81no—
189C22—81no—
190C22#2026994511yescharacteristic
191C22C2×#1680995011yescharacteristic
192C22#2027000711yescharacteristic
193C4—22no—
194C4—42no—
195C4—41no—
196C4—8442no—
197C4—8442no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—82no—
202C2#2721328411yescharacteristic
203C2#2721331311yescharacteristic
204C2#2721333211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting, prime radical#392×C211C2
socleC22×C211#120
hypercenter#392D211
hypocenterC211#1596
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #27213772C2semidirect
2. #27213297C2semidirect
3. #27212882C2semidirect
4. #27212870C2semidirect
20. C2×Dic422D4semidirect
26. C4⋊Dic211D4semidirect
61. C23×C211SD16semidirect
102. C211#1596semidirect
157. C23#16809950semidirect

Maximal subgroups

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

subgroupindex
#272137722
#272132972
#272128822
#272128702
#392×C2112
#1596211