Finite Group Databasebeta

← #41012394

Subgroups — #41012394

Order 27008.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#27213745C211yesmaximal, characteristic
2#27213459C211yesmaximal, characteristic
3#27213077C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212881C211yesmaximal, characteristic
6#27212871C211yesmaximal, characteristic, complemented
7#483×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269968—21no—
10#20269962—21no—
11#20269961—21no—
12#20269953—21no—
13#20269951C2211yescharacteristic
14(C22⋊D211)⋊C4—21no—
15C2×Dic211⋊C4—21no—
16C2×Dic211⋊C4C2211yescharacteristic
17#20269910—21no—
18#20269897—21no—
19C22⋊(C211⋊C8)C2211yescharacteristic
20C211⋊#95—21no—
21C211⋊#95C2211yescharacteristic
22C22⋊Q8×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#106×C211—21no—
25#103×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C2×D844D411yescharacteristic
28C2×C22⋊D211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C4×D211—21no—
33C2×C211⋊C8—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—42no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45C4×Dic211—21no—
46C2×Q8×C211D411yescharacteristic
47C22×C4×C211—21no—
48C22×C4×C211C2311yescharacteristic
49C2×C8×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55C22×D211—21no—
56D844—41no—
57C22⋊D211—41no—
58C22⋊D211—41no—
59C2×Dic211—22no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—41no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C4×D211—81no—
69C211⋊C8—41no—
70C23×C211C2×D411yescharacteristic
71Q8×C211—41no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C2×D411yescharacteristic
79C2×C4×C211C2×D411yescharacteristic
80C8×C211—41no—
81C22×C211—21no—
82C22×C211—21no—
83C22×C211#12011yescharacteristic
84C22×C211#13511yescharacteristic
85C22×C211AGL(1,ℤ/8)11yescharacteristic
86C2×D211—41no—
87C2×D211—41no—
88C2×D211—41no—
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—41no—
92C4×C211—41no—
93C4×C211—81no—
94Dic211—42no—
95Dic211—41no—
96Dic211—41no—
97Dic211—41no—
98C2×C211—21no—
99C2×C211—21no—
100C2×C211#44911yescharacteristic
101C2×C211#45611yescharacteristic
102C2×C211#45811yescharacteristic
103D211—81no—
104C211#160911yescharacteristic, complemented, Sylow
105#1609—2111nomaximal, Sylow
106#539—2111no—
107#483D21111yescharacteristic, complemented
108#480—2111no—
109#398—2111no—
110#328—2111no—
111#327—2111no—
112#324—2111no—
113#126—4221no—
114#125—4221no—
115#123—4221no—
116#123—4221no—
117C22⋊Q8—4221no—
118C22⋊Q8C2×D21111yescharacteristic
119#121—2111no—
120C4×D4—4221no—
121C2×C4⋊C4—4221no—
122C2×C4⋊C4—2111no—
123C2×C4⋊C4C2×D21111yescharacteristic
124#107—4221no—
125#106—21no—
126#103—21no—
127#102—4221no—
128C22⋊C8—2111no—
129C22⋊C8C2×D21111yescharacteristic
130#95—4221no—
131#95—2111no—
132C2×Q8C22⋊D21111yescharacteristic
133C2×D4—4221no—
134C2×D4—2111no—
135C22×C4—21no—
136C22×C4—4221no—
137C22×C4—4221no—
138C22×C4—2111no—
139C22×C4—2111no—
140C22×C4C22×D21111yescharacteristic
141C2×C8—21no—
142C2×C8—4221no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—8442no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4—4221no—
150C4⋊C4—4221no—
151C4⋊C4—4221no—
152C4⋊C4—2111no—
153C4⋊C4C22⋊D21111yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—4221no—
156C22⋊C4—4221no—
157C22⋊C4—4221no—
158C22⋊C4—4221no—
159C42—4221no—
160C23—4221no—
161C23D4×D21111yescharacteristic
162Q8—41no—
163D4—8441no—
164D4—8441no—
165D4—8441no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—8441no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—16881no—
176C2×C4—4222no—
177C2×C4—4221no—
178C2×C4—4221no—
179C2×C4—4221no—
180C2×C4—4221no—
181C2×C4—4221no—
182C2×C4D4×D21111yescharacteristic
183C2×C4C2×C22⋊D21111yescharacteristic
184C8—41no—
185C8—8441no—
186C22—21no—
187C22—21no—
188C22—8441no—
189C22—8441no—
190C22—8441no—
191C22#2026994511yescharacteristic
192C22#2026999811yescharacteristic
193C22#2027000411yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4—8442no—
200C4—8441no—
201C4—8441no—
202C4—8441no—
203C2—21no—
204C2—21no—
205C2—16881no—
206C2#2721329011yescharacteristic
207C2#2721332711yescharacteristic
208C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#483×C211C2
socleC22×C211#120
prime radical#20269951C22
hypercenter#483D211
hypocenterC211#1609
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
3. #27213077C2semidirect
4. #27212883C2semidirect
6. #27212871C2semidirect
7. #483×C211C2semidirect
104. C211#1609semidirect
107. #483D211semidirect

Maximal subgroups

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

subgroupindex
#272137452
#272134592
#272130772
#272128832
#272128812
#272128712
#483×C2112
#1609211