Finite Group Databasebeta

← #41011987

Subgroups — #41011987

Order 27008.

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

normal subgroups only

This group has 234 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
2C2×#20269888C211yesmaximal, characteristic, complemented
3#27212887C212yesmaximal, complemented
4#27212885C212yesmaximal, complemented
5C4×M16×C211C211yesmaximal, characteristic, complemented
6C22×C211⋊C8—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#20269888—21no—
18#20269888C2212yes—
19C2×M16×C211—21no—
20C2×C42×C211C2211yescharacteristic
21#97×C211C2212yes—
22C4×C8×C211—21no—
23C2×Dic422C2×C411yescharacteristic
24C2×D844C2×C411yescharacteristic
25C2×C22⋊D211—21no—
26C2×C4×D211—21no—
27C2×C211⋊C8—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
44M16×C211—24no—
45C2×C8×C211—22no—
46C42×C211C2×C411yescharacteristic
47C42×C211D411yescharacteristic
48C42×C211D411yescharacteristic
49C42×C211C2311yescharacteristic
50C22×D211—21no—
51Dic422—41no—
52D844—41no—
53C22⋊D211—42no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C4×D211—42no—
58C211⋊C8—24no—
59C23×C211C22⋊C411yescharacteristic
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—41no—
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22×C411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C8×C211—24no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211#10412yes—
73C22×C211C2×C22⋊C411yescharacteristic
74C2×D211—41no—
75C2×D211—41no—
76C2×D211—41no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211#10412yes—
83Dic211—41no—
84Dic211—41no—
85Dic211—41no—
86C2×C211—22no—
87C2×C211#41311yescharacteristic
88C2×C211C2×#10411yescharacteristic
89C2×C211C2×#10411yescharacteristic
90D211—81no—
91C211#152411yescharacteristic, complemented, Sylow
92#1524—2111nomaximal, Sylow
93#529—2111no—
94C4×M16D21111yescharacteristic, complemented
95C2×#97—2111no—
96#330—2112no—
97#329—2112no—
98C2×M16—21no—
99C22×C8—4221no—
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×C42C2×D21111yescharacteristic
111#97—4221no—
112#97—2112no—
113#97C2×D21112yes—
114C4×C8—21no—
115C2×Q8—2111no—
116C2×D4—4221no—
117C2×D4—2111no—
118C22×C4—4221no—
119C22×C4C4×D21111yescharacteristic
120C22×C4C22⋊D21111yescharacteristic
121C22×C4D84411yescharacteristic
122M16—24no—
123C2×C8—22no—
124C2×C8—4224no—
125C2×C8—4222no—
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—4221no—
138C42C4×D21111yescharacteristic
139C42C22⋊D21111yescharacteristic
140C42D84411yescharacteristic
141C42C22×D21111yescharacteristic
142C23—4221no—
143C23(C2×D211)⋊C411yescharacteristic
144Q8—8441no—
145D4—8441no—
146D4—8442no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—8442no—
151C2×C4—4221no—
152C2×C4—4221no—
153C2×C4—4221no—
154C2×C4(C2×D211)⋊C411yescharacteristic
155C2×C4(C2×D211)⋊C411yescharacteristic
156C2×C4(C2×D211)⋊C411yescharacteristic
157C2×C4C2×C4×D21111yescharacteristic
158C2×C4C2×C22⋊D21111yescharacteristic
159C2×C4C2×D84411yescharacteristic
160C8—24no—
161C8—4224no—
162C22—21no—
163C22—21no—
164C22—8441no—
165C22—8441no—
166C22—8441no—
167C22#2026990512yes—
168C22C2×(C2×D211)⋊C411yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—8441no—
175C4—8441no—
176C4—8441no—
177C4#2026990312yes—
178C2—22no—
179C2—16881no—
180C2#2721312811yescharacteristic
181C2C2×#2026990311yescharacteristic
182C2C2×#2026990511yescharacteristic
subgroupG/H
centerC2×C4(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
FittingC4×M16×C211C2
socleC22×C211C2×C22⋊C4
prime radical#20269951C4
cosocleC42×C211C23
hypercenterC4×M16D211
hypocenterC211#1524
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#20269888C2semidirect
3. #27212887C2semidirect
4. #27212885C2semidirect
5. C4×M16×C211C2semidirect
91. C211#1524semidirect
94. C4×M16D211semidirect

Maximal subgroups

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

subgroupindex
#272136802
C2×#202698882
#272128872
#272128852
C4×M16×C2112
#1524211