Finite Group Databasebeta

← #41011937

Subgroups — #41011937

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#27213012C211yesmaximal, characteristic, complemented
3#27212884C212yesmaximal, complemented
4#27212881C212yesmaximal
5#432×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—
16C2×C4⋊Dic211—21no—
17C211⋊#95—21no—
18C211⋊#95C2212yes—
19C2×C42×C211C2211yescharacteristic
20C4⋊C8×C211—21no—
21C22⋊C8×C211C2212yes—
22#97×C211—21no—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—21no—
26C22×Dic211—22no—
27C2×C4×D211—21no—
28C4⋊Dic211—42no—
29C4⋊Dic211C2×C411yescharacteristic
30C4⋊Dic211C2×C411yescharacteristic
31Dic211⋊C4—21no—
32Dic211⋊C4—21no—
33Dic211⋊C4—21no—
34Dic211⋊C4—21no—
35C22⋊Dic211—21no—
36(C2×D211)⋊C4—21no—
37(C2×D211)⋊C4—21no—
38(C2×D211)⋊C4—21no—
39(C2×D211)⋊C4—21no—
40C4×Dic211—21no—
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C22×C4×C211C2311yescharacteristic
44C2×C8×C211—22no—
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—
57C2×Dic211—42no—
58C2×Dic211—42no—
59C2×Dic211—42no—
60C4×D211—42no—
61C23×C211C2×D411yescharacteristic
62C2×C4×C211—21no—
63C2×C4×C211—21no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22⋊C411yescharacteristic
69C2×C4×C211C22⋊C411yescharacteristic
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C8×C211—42no—
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—
87Dic211—41no—
88Dic211—41no—
89Dic211—41no—
90Dic211—82no—
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#432D21111yescharacteristic, complemented
101#382—2111no—
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—4221no—
115C2×C42C2×D21111yescharacteristic
116C4⋊C8—21no—
117C22⋊C8C2×D21112yes—
118#97—21no—
119#95—4221no—
120#95—2112no—
121C2×Q8—2111no—
122C2×D4—4221no—
123C2×D4—2111no—
124C22×C4—4222no—
125C22×C4—4221no—
126C22×C4D84411yescharacteristic
127C22×C4D84411yescharacteristic
128C22×C4C22×D21111yescharacteristic
129C2×C8—22no—
130C4⋊C4—8442no—
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
145C42C22⋊D21111yescharacteristic
146C42C22⋊D21111yescharacteristic
147C23—4221no—
148C23C2×D84411yescharacteristic
149Q8—8441no—
150D4—8441no—
151D4—8441no—
152D4—8441no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—8442no—
158C2×C4—8442no—
159C2×C4—8442no—
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×C22⋊D21111yescharacteristic
170C8—42no—
171C22—21no—
172C22—21no—
173C22—8441no—
174C22—8441no—
175C22—8441no—
176C22C2×(C2×D211)⋊C411yescharacteristic
177C22#2027000511yescharacteristic
178C22#2027001011yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—8441no—
186C4—8441no—
187C4—8441no—
188C4—16882no—
189C2—21no—
190C2—21no—
191C2—16881no—
192C2#2721311911yescharacteristic
193C2#2721314811yescharacteristic
194C2#2721316211yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C22
cosocleC22×C4×C211C23
hypercenter#432D211
hypocenterC211#1512
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #27213012C2semidirect
3. #27212884C2semidirect
5. #432×C211C2semidirect
7. #20269969C4semidirect
8. #20269965C4semidirect
12. C4×Dic422C4semidirect
13. C4×D844C4semidirect
97. C211#1512semidirect
100. #432D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
#272130122
#272128842
#272128812
#432×C2112
#1512211