Finite Group Databasebeta

← #41011804

Subgroups — #41011804

Order 27008.

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

normal subgroups only

This group has 212 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#27213199C211yesmaximal, characteristic, complemented
3#27212876C212yesmaximal, complemented
4#27212873C212yesmaximal, complemented
5#431×C211C211yesmaximal, characteristic, complemented
6#20269971—21no—
7#20269969C411yescharacteristic
8#20269965C411yescharacteristic
9#20269960—21no—
10#20269956C411yescharacteristic
11#20269951C411yescharacteristic
12C4×Dic422C2211yescharacteristic
13C4×D844C2211yescharacteristic
14C4×C22⋊D211—21no—
15#20269934—21no—
16C4⋊(C211⋊C8)—21no—
17C22⋊(C211⋊C8)—21no—
18C4×C211⋊C8C2212yes—
19C2×C42×C211C2211yescharacteristic
20C4⋊C8×C211C2212yes—
21C22⋊C8×C211—21no—
22C4×C8×C211—21no—
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×C211C2×C411yescharacteristic
41C22×C4×C211D411yescharacteristic
42C22×C4×C211D411yescharacteristic
43C2×C8×C211—22no—
44C42×C211C2×C411yescharacteristic
45C42×C211D411yescharacteristic
46C42×C211D411yescharacteristic
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49Dic422—22no—
50D844—22no—
51C22⋊D211—42no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C4×D211—42no—
56C211⋊C8—42no—
57C23×C211C22⋊C411yescharacteristic
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—41no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—42no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211C2×C22⋊C411yescharacteristic
72C2×D211—41no—
73C2×D211—41no—
74C2×D211—41no—
75C4×C211—21no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211#13112yes—
80C4×C211#13512yes—
81Dic211—41no—
82Dic211—41no—
83Dic211—41no—
84C2×C211—41no—
85C2×C211#40811yescharacteristic
86C2×C211#41611yescharacteristic
87C2×C211#42111yescharacteristic
88D211—81no—
89C211#148211yescharacteristic, complemented, Sylow
90#1482—2111nomaximal, Sylow
91#529—2111no—
92#431—2111no—
93#431D21111yescharacteristic, complemented
94Q8⋊C8—2112no—
95D4⋊C8—2112no—
96#126—4221no—
97#125—2111no—
98#124—2111no—
99#123—4221no—
100C22⋊Q8—2111no—
101#121—2111no—
102C4×Q8—2111no—
103C4×D4—4221no—
104C4×D4—2111no—
105#117—4221no—
106C2×C42C2×D21111yescharacteristic
107C4⋊C8—4221no—
108C4⋊C8C2×D21112yes—
109C22⋊C8—21no—
110C22⋊C8—4221no—
111C4×C8—21no—
112C4×C8—2112no—
113C2×Q8—2111no—
114C2×D4—4221no—
115C2×D4—2111no—
116C22×C4—4221no—
117C22×C4C4×D21111yescharacteristic
118C22×C4C22⋊D21111yescharacteristic
119C22×C4D84411yescharacteristic
120C2×C8—22no—
121C2×C8—4222no—
122C4⋊C4—4221no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4—4221no—
126C4⋊C4—2111no—
127C4⋊C4—2111no—
128C22⋊C4—4221no—
129C22⋊C4—4221no—
130C22⋊C4—4221no—
131C22⋊C4—4221no—
132C22⋊C4—4221no—
133C42—4221no—
134C42C4×D21111yescharacteristic
135C42C22⋊D21111yescharacteristic
136C42D84411yescharacteristic
137C42C22×D21111yescharacteristic
138C23—4221no—
139C23(C2×D211)⋊C411yescharacteristic
140Q8—4222no—
141D4—8442no—
142D4—4222no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—41no—
146C2×C4—8442no—
147C2×C4—4221no—
148C2×C4—4221no—
149C2×C4—4221no—
150C2×C4(C2×D211)⋊C411yescharacteristic
151C2×C4(C2×D211)⋊C411yescharacteristic
152C2×C4(C2×D211)⋊C411yescharacteristic
153C2×C4C2×C4×D21111yescharacteristic
154C2×C4C2×C22⋊D21111yescharacteristic
155C2×C4C2×D84411yescharacteristic
156C8—42no—
157C8—8442no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—8441no—
162C22—8441no—
163C22—8441no—
164C22C2×(C2×D211)⋊C411yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—8441no—
170C4—8441no—
171C4—8441no—
172C4#2026998312yes—
173C4#2026999712yes—
174C2—41no—
175C2—16881no—
176C2#2721311211yescharacteristic
177C2#2721314011yescharacteristic
178C2#2721316211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#431×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C4
cosocleC42×C211C23
hypercenter#431D211
hypocenterC211#1482
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
2. #27213199C2semidirect
3. #27212876C2semidirect
4. #27212873C2semidirect
5. #431×C211C2semidirect
89. C211#1482semidirect
93. #431D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
#272131992
#272128762
#272128732
#431×C2112
#1482211