Finite Group Databasebeta

← #41012729

Subgroups — #41012729

Order 27008.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#27213900C211yesmaximal, characteristic
2#27213552C211yesmaximal, characteristic, complemented
3#27213508C211yesmaximal, characteristic
4#27212888C211yesmaximal, characteristic, complemented
5#27212887C211yesmaximal, characteristic, complemented
6#27212885C211yesmaximal, characteristic
7#491×C211C211yesmaximal, characteristic, complemented
8#20269976—21no—
9#20269972—21no—
10#20269972—21no—
11#20269969C2211yescharacteristic
12#20269965C2211yescharacteristic
13#20269962—21no—
14#20269957—21no—
15#20269955—21no—
16#20269910—21no—
17#20269908—21no—
18#20269900—21no—
19#20269897—21no—
20#20269888C2211yescharacteristic
21#20269888C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#125×C211C2211yescharacteristic
24#103×C211—21no—
25#103×C211—21no—
26#97×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422D411yescharacteristic
29C2×D844D411yescharacteristic
30C2×C4×D211—21no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—21no—
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44C4×Dic211—21no—
45C2×Q8×C211—21no—
46C2×C8×C211—21no—
47C4⋊C4×C211—21no—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211D411yescharacteristic
52C4⋊C4×C211D411yescharacteristic
53C42×C211C2311yescharacteristic
54C22×D211—21no—
55Dic422—41no—
56Dic422—81no—
57D844—41no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C4×D211—81no—
62C211⋊C8—22no—
63C211⋊C8—41no—
64Q8×C211—42no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211—41no—
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C2×C4×C211C2×D411yescharacteristic
71C8×C211—41no—
72C22×C211#12011yescharacteristic
73C2×D211—41no—
74C2×D211—41no—
75C2×D211—41no—
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—41no—
80C4×C211—41no—
81C4×C211—81no—
82Dic211—41no—
83Dic211—41no—
84Dic211—41no—
85C2×C211#45411yescharacteristic
86C2×C211#45511yescharacteristic
87C2×C211#45611yescharacteristic
88D211—81no—
89C211#166411yescharacteristic, complemented, Sylow
90#1664—2111nomaximal, Sylow
91#563—2111no—
92#501—2111no—
93#491D21111yescharacteristic, complemented
94#489—2111no—
95#330—2111no—
96#330—2111no—
97#329—2111no—
98C4⋊Q8—4221no—
99C4⋊Q8C2×D21111yescharacteristic
100#126—4221no—
101#126—4221no—
102#125—2111no—
103#125C2×D21111yescharacteristic
104#124—2111no—
105#123—4221no—
106C22⋊Q8—4221no—
107C22⋊Q8—4221no—
108#107—4221no—
109#106—4221no—
110#103—21no—
111#103—21no—
112#103—4221no—
113#102—4221no—
114#97—2111no—
115#97—2111no—
116#97C2×D21111yescharacteristic
117C2×Q8—21no—
118C2×Q8—4221no—
119C2×Q8—2111no—
120C2×D4—2111no—
121C22×C4—4221no—
122C2×C8—21no—
123C2×C8—4221no—
124C2×C8—4221no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—4221no—
130C4⋊C4—4221no—
131C4⋊C4—4221no—
132C4⋊C4—4221no—
133C4⋊C4—2111no—
134C4⋊C4—2111no—
135C4⋊C4C22⋊D21111yescharacteristic
136C4⋊C4C22⋊D21111yescharacteristic
137C22⋊C4—4221no—
138C22⋊C4—4221no—
139C22⋊C4—4221no—
140C22⋊C4—4221no—
141C22⋊C4—4221no—
142C42—4221no—
143C42C22×D21111yescharacteristic
144C23—4221no—
145Q8—42no—
146Q8—8441no—
147Q8—16881no—
148D4—8441no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—16881no—
153C2×C4—4221no—
154C2×C4—4221no—
155C2×C4—4221no—
156C2×C4D4×D21111yescharacteristic
157C2×C4D4×D21111yescharacteristic
158C2×C4C2×C22⋊D21111yescharacteristic
159C8—41no—
160C8—8441no—
161C8—4222no—
162C22—8441no—
163C22—8441no—
164C22—8441no—
165C22#2026994511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—81no—
172C4—8441no—
173C4—8441no—
174C4—8441no—
175C2—16881no—
176C2#2721331811yescharacteristic
177C2#2721332211yescharacteristic
178C2#2721332711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#491×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#491D211
hypocenterC211#1664
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #27213552C2semidirect
4. #27212888C2semidirect
5. #27212887C2semidirect
7. #491×C211C2semidirect
89. C211#1664semidirect
93. #491D211semidirect

Maximal subgroups

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

subgroupindex
#272139002
#272135522
#272135082
#272128882
#272128872
#272128852
#491×C2112
#1664211