Finite Group Databasebeta

← #41011934

Subgroups — #41011934

Order 27008.

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

normal subgroups only

This group has 222 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#27213203C211yesmaximal, characteristic, complemented
3#27212882C212yesmaximal, complemented
4#27212879C212yesmaximal
5#382×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—
18#20269888—21no—
19C2×C4⋊C4×C211—21no—
20C2×C42×C211C2211yescharacteristic
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
44C4⋊C4×C211—42no—
45C42×C211C2×C411yescharacteristic
46C42×C211C2×C411yescharacteristic
47C42×C211D411yescharacteristic
48C42×C211D411yescharacteristic
49C22×D211—21no—
50Dic422—41no—
51D844—41no—
52C22⋊D211—41no—
53C22⋊D211—41no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C4×D211—42no—
58C211⋊C8—42no—
59C23×C211C2×D411yescharacteristic
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—42no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22⋊C411yescharacteristic
69C2×C4×C211C22⋊C411yescharacteristic
70C2×C4×C211C22⋊C411yescharacteristic
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C22×C211—21no—
74C22×C211—21no—
75C22×C211C2×C22⋊C411yescharacteristic
76C22×C211AGL(1,ℤ/8)11yescharacteristic
77C22×C211#13711yescharacteristic
78C2×D211—41no—
79C2×D211—41no—
80C2×D211—41no—
81C4×C211—21no—
82C4×C211—21no—
83C4×C211—21no—
84C4×C211—21no—
85C4×C211—21no—
86C4×C211—21no—
87C4×C211—82no—
88Dic211—41no—
89Dic211—41no—
90Dic211—41no—
91C2×C211—21no—
92C2×C211—21no—
93C2×C211#41011yescharacteristic
94C2×C211#41711yescharacteristic
95C2×C211#42111yescharacteristic
96D211—81no—
97C211#151211yescharacteristic, complemented, Sylow
98#1512—2111nomaximal, Sylow
99#529—2111no—
100#432—2111no—
101#382D21111yescharacteristic, complemented
102#328—2112no—
103#327—2112no—
104#126—4221no—
105#125—2111no—
106#124—2111no—
107#123—4221no—
108C22⋊Q8—2111no—
109#121—2111no—
110C4×Q8—2111no—
111C4×D4—4221no—
112C4×D4—2111no—
113#117—4221no—
114C2×C4⋊C4—21no—
115C2×C42C2×D21111yescharacteristic
116C4⋊C8—4221no—
117C22⋊C8—2112no—
118#97—4221no—
119#95—21no—
120#95C2×D21112yes—
121C2×Q8—2111no—
122C2×D4—4221no—
123C2×D4—2111no—
124C22×C4—22no—
125C22×C4—4221no—
126C22×C4C22⋊D21111yescharacteristic
127C22×C4C22⋊D21111yescharacteristic
128C22×C4C22×D21111yescharacteristic
129C2×C8—4222no—
130C4⋊C4—42no—
131C4⋊C4—4221no—
132C4⋊C4—4221no—
133C4⋊C4—4221no—
134C4⋊C4—4221no—
135C4⋊C4—2111no—
136C4⋊C4—2111no—
137C22⋊C4—4221no—
138C22⋊C4—4221no—
139C22⋊C4—4221no—
140C22⋊C4—4221no—
141C22⋊C4—4221no—
142C42—4221no—
143C42C4×D21111yescharacteristic
144C42C4×D21111yescharacteristic
145C42D84411yescharacteristic
146C42D84411yescharacteristic
147C23—4221no—
148C23C2×C22⋊D21111yescharacteristic
149Q8—8441no—
150D4—8441no—
151D4—8441no—
152D4—8441no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—8442no—
161C2×C4—4221no—
162C2×C4—4221no—
163C2×C4—4221no—
164C2×C4(C2×D211)⋊C411yescharacteristic
165C2×C4(C2×D211)⋊C411yescharacteristic
166C2×C4(C2×D211)⋊C411yescharacteristic
167C2×C4(C2×D211)⋊C411yescharacteristic
168C2×C4C2×C4×D21111yescharacteristic
169C2×C4C2×D84411yescharacteristic
170C8—8442no—
171C22—21no—
172C22—21no—
173C22—8441no—
174C22—8441no—
175C22—8441no—
176C22C2×(C2×D211)⋊C411yescharacteristic
177C22#2027000311yescharacteristic
178C22#2027000811yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—82no—
186C4—8441no—
187C4—8441no—
188C4—8441no—
189C2—21no—
190C2—21no—
191C2—16881no—
192C2#2721311911yescharacteristic
193C2#2721314511yescharacteristic
194C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#382×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C22
cosocleC22×C4×C211C23
hypercenter#382D211
hypocenterC211#1512
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #27213203C2semidirect
3. #27212882C2semidirect
5. #382×C211C2semidirect
7. #20269969C4semidirect
8. #20269965C4semidirect
12. C4×Dic422C4semidirect
13. C4×D844C4semidirect
97. C211#1512semidirect
101. #382D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
#272132032
#272128822
#272128792
#382×C2112
#1512211