Finite Group Databasebeta

← #41012430

Subgroups — #41012430

Order 27008.

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

normal subgroups only

This group has 208 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#27213741C211yesmaximal, characteristic
2#27213482C211yesmaximal, characteristic, complemented
3#27213462C211yesmaximal, characteristic
4#27213086C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212871C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic
8#20269970—21no—
9#20269967—21no—
10#20269958—21no—
11#20269958—21no—
12#20269956C2211yescharacteristic
13#20269954—21no—
14D4×Dic211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×Dic211⋊C4C2211yescharacteristic
17#20269911—21no—
18#20269908—21no—
19#20269902—21no—
20#20269898—21no—
21C22⋊(C211⋊C8)C2211yescharacteristic
22C211⋊#95—21no—
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C2×Dic422D411yescharacteristic
28C22×Dic211—21no—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—21no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42C22⋊Dic211—21no—
43C22⋊Dic211—21no—
44C22⋊Dic211—21no—
45C22⋊Dic211—21no—
46C4×Dic211—21no—
47C2×D4×C211—21no—
48C2×D4×C211D411yescharacteristic
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C4⋊C4×C211D411yescharacteristic
53C22⋊C4×C211—21no—
54Dic422—41no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—41no—
62C2×Dic211—42no—
63C2×Dic211—41no—
64C2×Dic211—81no—
65C211⋊C8—41no—
66C23×C211—21no—
67C23×C211C2×D411yescharacteristic
68D4×C211—41no—
69D4×C211—41no—
70D4×C211—41no—
71C2×C4×C211—21no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C8×C211—41no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211—41no—
82C22×C211—41no—
83C22×C211—41no—
84C22×C211#12011yescharacteristic
85C22×C211#13511yescharacteristic
86C22×C211#13711yescharacteristic
87C4×C211—21no—
88C4×C211—21no—
89C4×C211—41no—
90C4×C211—81no—
91Dic211—41no—
92Dic211—41no—
93Dic211—41no—
94Dic211—41no—
95Dic211—81no—
96C2×C211—21no—
97C2×C211—21no—
98C2×C211—81no—
99C2×C211#45211yescharacteristic
100C2×C211#45411yescharacteristic
101C2×C211#45811yescharacteristic
102C211#161511yescharacteristic, complemented, Sylow
103#1615—2111nomaximal, Sylow
104#539—2111no—
105#484—2111no—
106#481—2111no—
107#400—2111no—
108#328—2111no—
109#327D21111yescharacteristic
110#324—2111no—
111#126—4221no—
112#125—4221no—
113#123—4221no—
114#123—4221no—
115C22⋊Q8—4221no—
116C22⋊Q8—2111no—
117#121C2×D21111yescharacteristic
118C4×D4—4221no—
119C2×C4⋊C4—2111no—
120C2×C4⋊C4—2111no—
121#107—4221no—
122#106—4221no—
123#103—4221no—
124#102—4221no—
125C22⋊C8—2111no—
126C22⋊C8C2×D21111yescharacteristic
127#95—4221no—
128#95—4221no—
129#95C2×D21111yescharacteristic
130C2×Q8—2111no—
131C2×D4—21no—
132C2×D4D84411yescharacteristic
133C22×C4—21no—
134C22×C4—4221no—
135C22×C4—4221no—
136C22×C4—2111no—
137C22×C4—2111no—
138C22×C4C22×D21111yescharacteristic
139C2×C8—21no—
140C2×C8—4221no—
141C4⋊C4—4221no—
142C4⋊C4—4221no—
143C4⋊C4—4221no—
144C4⋊C4—4221no—
145C4⋊C4—4221no—
146C4⋊C4—4221no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4—2111no—
150C4⋊C4D84411yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—4221no—
153C22⋊C4—4221no—
154C22⋊C4—4221no—
155C22⋊C4—4221no—
156C42—4221no—
157C23—21no—
158C23D4×D21111yescharacteristic
159Q8—8441no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—8441no—
169C2×C4—8442no—
170C2×C4—8441no—
171C2×C4—16881no—
172C2×C4—4221no—
173C2×C4—4221no—
174C2×C4—4221no—
175C2×C4—4221no—
176C2×C4—4221no—
177C2×C4—4221no—
178C2×C4D4×D21111yescharacteristic
179C2×C4C2×D84411yescharacteristic
180C8—41no—
181C8—8441no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#2026994711yescharacteristic
188C22#2026999611yescharacteristic
189C22#2027000611yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—81no—
194C4—8441no—
195C4—8441no—
196C4—8441no—
197C4—8441no—
198C4—16881no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2721330911yescharacteristic
203C2#2721332011yescharacteristic
204C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#121×C211C22
hypercenter#327D211
hypocenterC211#1615
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27213482C2semidirect
4. #27213086C2semidirect
5. #27212882C2semidirect
6. #27212871C2semidirect
102. C211#1615semidirect

Maximal subgroups

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

subgroupindex
#272137412
#272134822
#272134622
#272130862
#272128822
#272128712
#327×C2112
#1615211