Finite Group Databasebeta

← #41012006

Subgroups — #41012006

Order 27008.

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

normal subgroups only

This group has 216 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
2C4×#16809946C211yesmaximal, characteristic, complemented
3#27212887C212yesmaximal, complemented
4#27212885C212yesmaximal, complemented
5#432×C211C211yesmaximal, characteristic, complemented
6C2×#16809946—21no—
7#20269971—21no—
8#20269969C2211yescharacteristic
9#20269965C2211yescharacteristic
10#20269960—21no—
11#20269956C411yescharacteristic
12#20269951C411yescharacteristic
13C4×Dic422C411yescharacteristic
14C4×D844C411yescharacteristic
15C4×C22⋊D211—21no—
16#20269934—21no—
17#20269888C2212yes—
18C4×C211⋊C8—21no—
19C2×C42×C211C2211yescharacteristic
20C4⋊C8×C211—21no—
21C22⋊C8×C211—21no—
22#97×C211C2212yes—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—21no—
26C2×C4×D211—21no—
27#16809946—24no—
28C2×C211⋊C8—22no—
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×C211C2×C411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C22×C4×C211D411yescharacteristic
44C2×C8×C211—22no—
45C42×C211C2×C411yescharacteristic
46C42×C211D411yescharacteristic
47C42×C211D411yescharacteristic
48C42×C211C2311yescharacteristic
49C22×D211—21no—
50Dic422—41no—
51D844—41no—
52C22⋊D211—42no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C4×D211—42no—
57C211⋊C8—24no—
58C23×C211C22⋊C411yescharacteristic
59C2×C4×C211—22no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22×C411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C8×C211—42no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211C2×C22⋊C411yescharacteristic
73C2×D211—41no—
74C2×D211—41no—
75C2×D211—41no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211#10412yes—
82Dic211—41no—
83Dic211—41no—
84Dic211—41no—
85C2×C211—41no—
86C2×C211#41311yescharacteristic
87C2×C211C2×#10411yescharacteristic
88C2×C211#42111yescharacteristic
89D211—81no—
90C211#152911yescharacteristic, complemented, Sylow
91#1529—2111nomaximal, Sylow
92#529—2111no—
93#432D21111yescharacteristic, complemented
94C4×M16—2111no—
95#330—2112no—
96#329—2112no—
97C2×M16—4221no—
98#126—4221no—
99#125—2111no—
100#124—2111no—
101#123—4221no—
102C22⋊Q8—2111no—
103#121—2111no—
104C4×Q8—2111no—
105C4×D4—4221no—
106C4×D4—2111no—
107#117—4221no—
108C2×C42C2×D21111yescharacteristic
109C4⋊C8—21no—
110C22⋊C8—21no—
111#97—2112no—
112#97C2×D21112yes—
113C4×C8—4221no—
114C2×Q8—2111no—
115C2×D4—4221no—
116C2×D4—2111no—
117C22×C4—4221no—
118C22×C4C4×D21111yescharacteristic
119C22×C4C22⋊D21111yescharacteristic
120C22×C4D84411yescharacteristic
121M16—4224no—
122C2×C8—22no—
123C2×C8—4222no—
124C4⋊C4—4221no—
125C4⋊C4—4221no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—2111no—
129C4⋊C4—2111no—
130C22⋊C4—4221no—
131C22⋊C4—4221no—
132C22⋊C4—4221no—
133C22⋊C4—4221no—
134C22⋊C4—4221no—
135C42—4221no—
136C42C4×D21111yescharacteristic
137C42C22⋊D21111yescharacteristic
138C42D84411yescharacteristic
139C42C22×D21111yescharacteristic
140C23—4221no—
141C23(C2×D211)⋊C411yescharacteristic
142Q8—8441no—
143D4—8441no—
144D4—8442no—
145C2×C4—22no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—8442no—
149C2×C4—4221no—
150C2×C4—4221no—
151C2×C4—4221no—
152C2×C4(C2×D211)⋊C411yescharacteristic
153C2×C4(C2×D211)⋊C411yescharacteristic
154C2×C4(C2×D211)⋊C411yescharacteristic
155C2×C4C2×C4×D21111yescharacteristic
156C2×C4C2×C22⋊D21111yescharacteristic
157C2×C4C2×D84411yescharacteristic
158C8—42no—
159C8—4224no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—8441no—
164C22—8441no—
165C22—8441no—
166C22C2×(C2×D211)⋊C411yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—8441no—
173C4—8441no—
174C4—8441no—
175C4#2026990312yes—
176C2—41no—
177C2—16881no—
178C2#2721312811yescharacteristic
179C2C2×#2026990311yescharacteristic
180C2#2721316411yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#432×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C4
cosocleC42×C211C23
hypercenter#432D211
hypocenterC211#1529
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. C4×#16809946C2semidirect
3. #27212887C2semidirect
4. #27212885C2semidirect
5. #432×C211C2semidirect
90. C211#1529semidirect
93. #432D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
C4×#168099462
#272128872
#272128852
#432×C2112
#1529211