Finite Group Databasebeta

← #41011885

Subgroups — #41011885

Order 27008.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#27212992C211yesmaximal, characteristic, complemented
3#27212884C212yesmaximal, complemented
4#27212881C212yesmaximal
5#431×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—
22C4×C8×C211—21no—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—21no—
26C22×Dic211—22no—
27C2×C4×D211—21no—
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—24no—
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—42no—
53C2×Dic211—24no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—42no—
58C2×Dic211—42no—
59C4×D211—42no—
60C23×C211C2×D411yescharacteristic
61C2×C4×C211—22no—
62C2×C4×C211—22no—
63C2×C4×C211—41no—
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22⋊C411yescharacteristic
67C2×C4×C211C22⋊C411yescharacteristic
68C2×C4×C211C22×C411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—42no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211C2×C22⋊C411yescharacteristic
74C22×C211#13512yes—
75C2×D211—41no—
76C2×D211—41no—
77C2×D211—41no—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—21no—
83C4×C211#10412yes—
84Dic211—44no—
85Dic211—41no—
86Dic211—41no—
87Dic211—41no—
88C2×C211—22no—
89C2×C211#41011yescharacteristic
90C2×C211#41611yescharacteristic
91C2×C211C2×#10411yescharacteristic
92D211—81no—
93C211#150011yescharacteristic, complemented, Sylow
94#1500—2111nomaximal, Sylow
95#529—2111no—
96#431D21111yescharacteristic, complemented
97#376—2111no—
98#328—2112no—
99#327—2112no—
100#126—4221no—
101#125—2111no—
102#124—2111no—
103#123—4221no—
104C22⋊Q8—2111no—
105#121—2111no—
106C4×Q8—2111no—
107C4×D4—4221no—
108C4×D4—2111no—
109#117—4221no—
110C2×C42—4221no—
111C2×C42C2×D21111yescharacteristic
112C4⋊C8—21no—
113C22⋊C8C2×D21112yes—
114C4×C8—21no—
115#95—4221no—
116#95—2112no—
117C2×Q8—2111no—
118C2×D4—4221no—
119C2×D4—2111no—
120C22×C4—4222no—
121C22×C4—4221no—
122C22×C4D84411yescharacteristic
123C22×C4D84411yescharacteristic
124C22×C4C22×D21111yescharacteristic
125C2×C8—22no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—4221no—
129C4⋊C4—4221no—
130C4⋊C4—2111no—
131C4⋊C4—2111no—
132C22⋊C4—4221no—
133C22⋊C4—4221no—
134C22⋊C4—4221no—
135C22⋊C4—4221no—
136C22⋊C4—4221no—
137C42—4224no—
138C42—4221no—
139C42C4×D21111yescharacteristic
140C42C4×D21111yescharacteristic
141C42C22⋊D21111yescharacteristic
142C42C22⋊D21111yescharacteristic
143C23—4221no—
144C23C2×D84411yescharacteristic
145Q8—8441no—
146D4—8441no—
147D4—8442no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—8442no—
152C2×C4—8442no—
153C2×C4—8442no—
154C2×C4—4224no—
155C2×C4—4221no—
156C2×C4—4221no—
157C2×C4—4221no—
158C2×C4(C2×D211)⋊C411yescharacteristic
159C2×C4(C2×D211)⋊C411yescharacteristic
160C2×C4(C2×D211)⋊C411yescharacteristic
161C2×C4(C2×D211)⋊C411yescharacteristic
162C2×C4C2×C4×D21111yescharacteristic
163C2×C4C2×C22⋊D21111yescharacteristic
164C8—42no—
165C22—21no—
166C22—21no—
167C22—8441no—
168C22—8441no—
169C22—8441no—
170C22C2×(C2×D211)⋊C411yescharacteristic
171C22#2026999912yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—8444no—
178C4—8441no—
179C4—8441no—
180C4—8441no—
181C4#2026990312yes—
182C2—22no—
183C2—16881no—
184C2#2721311911yescharacteristic
185C2#2721314311yescharacteristic
186C2C2×#2026990311yescharacteristic
subgroupG/H
centerC2×C4(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#431×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C22
cosocleC22×C4×C211C23
hypercenter#431D211
hypocenterC211#1500
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #27212992C2semidirect
3. #27212884C2semidirect
5. #431×C211C2semidirect
7. #20269969C4semidirect
8. #20269965C4semidirect
12. C4×Dic422C4semidirect
13. C4×D844C4semidirect
93. C211#1500semidirect
96. #431D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
#272129922
#272128842
#272128812
#431×C2112
#1500211