Finite Group Databasebeta

← #41011973

Subgroups — #41011973

Order 27008.

218 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213680C211yesmaximal, characteristic
2#27213199C211yesmaximal, characteristic, complemented
3#27212882C212yesmaximal, complemented
4#27212879C212yesmaximal
5#383×C211C211yesmaximal, characteristic, complemented
6#20269971—21no—
7#20269969C411yescharacteristic, complemented
8#20269965C411yescharacteristic, complemented
9#20269960—21no—
10#20269956C2211yescharacteristic
11#20269951C2211yescharacteristic
12C4×Dic422C411yescharacteristic, complemented
13C4×D844C411yescharacteristic, complemented
14C4×C22⋊D211—21no—
15#20269934—21no—
16C4⋊(C211⋊C8)—21no—
17C22⋊(C211⋊C8)C2212yes—
18C4×C211⋊C8—21no—
19C2×C42×C211C2211yescharacteristic
20#95×C211—21no—
21#95×C211—21no—
22#95×C211C2212yes—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—21no—
26C2×C4×D211—21no—
27C2×C211⋊C8—22no—
28C4⋊Dic211C2×C411yescharacteristic
29C4⋊Dic211C2×C411yescharacteristic
30Dic211⋊C4—21no—
31Dic211⋊C4—21no—
32Dic211⋊C4—21no—
33Dic211⋊C4—21no—
34C22⋊Dic211—21no—
35(C2×D211)⋊C4—21no—
36(C2×D211)⋊C4—21no—
37(C2×D211)⋊C4—21no—
38(C2×D211)⋊C4—21no—
39C4×Dic211—21no—
40C22×C4×C211—22no—
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C22×C4×C211C2311yescharacteristic
44C42×C211C2×C411yescharacteristic
45C42×C211C2×C411yescharacteristic
46C42×C211D411yescharacteristic
47C42×C211D411yescharacteristic
48C22×D211—21no—
49Dic422—41no—
50D844—41no—
51C22⋊D211—42no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C4×D211—42no—
56C211⋊C8—42no—
57C23×C211C2×D411yescharacteristic
58C2×C4×C211—22no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—42no—
62C2×C4×C211—42no—
63C2×C4×C211—42no—
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22×C411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C22×C211—21no—
71C22×C211—21no—
72C22×C211C2×C22⋊C411yescharacteristic
73C22×C211#13512yes—
74C2×D211—41no—
75C2×D211—41no—
76C2×D211—41no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—21no—
83C4×C211—82no—
84Dic211—41no—
85Dic211—41no—
86Dic211—41no—
87C2×C211—22no—
88C2×C211#41011yescharacteristic
89C2×C211#41611yescharacteristic
90C2×C211#42111yescharacteristic
91D211—81no—
92C211#152111yescharacteristic, complemented, Sylow
93#1521—2111nomaximal, Sylow
94#529—2111no—
95#431—2111no—
96#383D21111yescharacteristic, complemented
97#328—2112no—
98#327—2112no—
99#126—4221no—
100#125—2111no—
101#124—2111no—
102#123—4221no—
103C22⋊Q8—2111no—
104#121—2111no—
105C4×Q8—2111no—
106C4×D4—4221no—
107C4×D4—2111no—
108#117—4221no—
109C2×C42C2×D21111yescharacteristic
110C4⋊C8—4221no—
111C22⋊C8—2112no—
112C4×C8—4221no—
113#95—21no—
114#95—21no—
115#95C2×D21112yes—
116C2×Q8—2111no—
117C2×D4—4221no—
118C2×D4—2111no—
119C22×C4—22no—
120C22×C4—4221no—
121C22×C4C22⋊D21111yescharacteristic
122C22×C4C22⋊D21111yescharacteristic
123C22×C4C22×D21111yescharacteristic
124C2×C8—4222no—
125C4⋊C4—4221no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—4221no—
129C4⋊C4—2111no—
130C4⋊C4—2111no—
131C22⋊C4—4221no—
132C22⋊C4—4221no—
133C22⋊C4—4221no—
134C22⋊C4—4221no—
135C22⋊C4—4221no—
136C42—4221no—
137C42C4×D21111yescharacteristic
138C42C4×D21111yescharacteristic
139C42D84411yescharacteristic
140C42D84411yescharacteristic
141C23—4221no—
142C23C2×C22⋊D21111yescharacteristic
143Q8—8441no—
144D4—8441no—
145D4—8442no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—8442no—
153C2×C4—4221no—
154C2×C4—4221no—
155C2×C4—4221no—
156C2×C4(C2×D211)⋊C411yescharacteristic
157C2×C4(C2×D211)⋊C411yescharacteristic
158C2×C4(C2×D211)⋊C411yescharacteristic
159C2×C4(C2×D211)⋊C411yescharacteristic
160C2×C4C2×C4×D21111yescharacteristic
161C2×C4C2×D84411yescharacteristic
162C8—8442no—
163C22—21no—
164C22—21no—
165C22—8441no—
166C22—8441no—
167C22—8441no—
168C22C2×(C2×D211)⋊C411yescharacteristic
169C22#2026999712yes—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—82no—
177C4—8441no—
178C4—8441no—
179C4—8441no—
180C2—22no—
181C2—16881no—
182C2#2721311911yescharacteristic
183C2#2721314011yescharacteristic
184C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#383×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C22
cosocleC22×C4×C211C23
hypercenter#383D211
hypocenterC211#1521
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #383

Complements

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

NH ≅ G/Ntype
2. #27213199C2semidirect
3. #27212882C2semidirect
5. #383×C211C2semidirect
7. #20269969C4semidirect
8. #20269965C4semidirect
12. C4×Dic422C4semidirect
13. C4×D844C4semidirect
92. C211#1521semidirect
96. #383D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
#272131992
#272128822
#272128792
#383×C2112
#1521211