Finite Group Databasebeta

← #41012429

Subgroups — #41012429

Order 27008.

208 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213479C211yesmaximal, characteristic, complemented
2#27213462C211yesmaximal, characteristic
3#27213085C211yesmaximal, characteristic, complemented
4#27212883C211yesmaximal, characteristic, complemented
5#27212880C211yesmaximal, characteristic
6#27212872C211yesmaximal, characteristic, complemented
7#539×C211C211yesmaximal, characteristic
8C2×C4⋊Dic211C2211yescharacteristic
9#20269910—21no—
10#20269908—21no—
11#20269901—21no—
12#20269898—21no—
13C22⋊(C211⋊C8)C2211yescharacteristic
14C22⋊(C211⋊C8)C2211yescharacteristic
15C211⋊#95—21no—
16C211⋊#95—21no—
17C211⋊#95C2211yescharacteristic
18#126×C211—21no—
19#125×C211—21no—
20#123×C211—21no—
21#123×C211—21no—
22C22⋊Q8×C211—21no—
23C22⋊Q8×C211C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C4×D4×C211—21no—
26C2×C4⋊C4×C211C2211yescharacteristic
27C22×Dic211—21no—
28C22×Dic211—21no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C2×Q8×C211D411yescharacteristic
34C2×D4×C211—21no—
35C2×D4×C211D411yescharacteristic
36C22×C4×C211—21no—
37C22×C4×C211D411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211C2311yescharacteristic
40C4⋊C4×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D411yescharacteristic
47C4⋊C4×C211D411yescharacteristic
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C22⋊C4×C211—21no—
52C22⋊C4×C211—21no—
53C42×C211—21no—
54C2×Dic211—41no—
55C2×Dic211—41no—
56C2×Dic211—42no—
57C2×Dic211—41no—
58C2×Dic211—41no—
59C211⋊C8—41no—
60C211⋊C8—41no—
61C23×C211—21no—
62C23×C211C2×D411yescharacteristic
63Q8×C211—41no—
64D4×C211—41no—
65D4×C211—41no—
66D4×C211—41no—
67C2×C4×C211—21no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—21no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—81no—
77C2×C4×C211C2×D411yescharacteristic
78C2×C4×C211C2×D411yescharacteristic
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—41no—
91C4×C211—41no—
92C4×C211—41no—
93C4×C211—41no—
94Dic211—81no—
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#539D21111yescharacteristic
105#484—2111no—
106#481—2111no—
107#400—2111no—
108#328—2111no—
109#327—2111no—
110#324—2111no—
111#126—21no—
112#125—21no—
113#123—21no—
114#123—21no—
115C22⋊Q8—21no—
116C22⋊Q8C2×D21111yescharacteristic
117#121C2×D21111yescharacteristic
118C4×D4—21no—
119C2×C4⋊C4—2111no—
120C2×C4⋊C4C2×D21111yescharacteristic
121#107—4221no—
122#106—4221no—
123#103—4221no—
124#102—4221no—
125C22⋊C8—2111no—
126C22⋊C8—2111no—
127#95—4221no—
128#95—4221no—
129#95—2111no—
130C2×Q8C22⋊D21111yescharacteristic
131C2×D4—21no—
132C2×D4C22⋊D21111yescharacteristic
133C22×C4—21no—
134C22×C4—4221no—
135C22×C4—4221no—
136C22×C4C22⋊D21111yescharacteristic
137C22×C4C22⋊D21111yescharacteristic
138C22×C4C22×D21111yescharacteristic
139C2×C8—4221no—
140C2×C8—4221no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—4221no—
148C4⋊C4—4221no—
149C4⋊C4C22⋊D21111yescharacteristic
150C4⋊C4C22⋊D21111yescharacteristic
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—21no—
156C42—21no—
157C23—21no—
158C23C2×C22⋊D21111yescharacteristic
159Q8—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—81no—
173C2×C4—8441no—
174C2×C4—8441no—
175C2×C4—8442no—
176C2×C4—8441no—
177C2×C4—8441no—
178C2×C4C2×C22⋊D21111yescharacteristic
179C2×C4C2×C22⋊D21111yescharacteristic
180C8—8441no—
181C8—8441no—
182C22—21no—
183C22—21no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22#2026994611yescharacteristic
188C22#2026999711yescharacteristic
189C22#2027000711yescharacteristic
190C4—21no—
191C4—21no—
192C4—41no—
193C4—41no—
194C4—41no—
195C4—41no—
196C4—41no—
197C4—16881no—
198C4—16881no—
199C2—21no—
200C2—21no—
201C2—81no—
202C2#2721330811yescharacteristic
203C2#2721331911yescharacteristic
204C2#2721333611yescharacteristic
subgroupG/H
centerC22#20269946
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#539×C211C2
socleC22×C211#120
prime radical#121×C211C22
hypercenter#539D211
hypocenterC211#1615
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #27213479C2semidirect
3. #27213085C2semidirect
4. #27212883C2semidirect
6. #27212872C2semidirect
102. C211#1615semidirect

Maximal subgroups

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

subgroupindex
#272134792
#272134622
#272130852
#272128832
#272128802
#272128722
#539×C2112
#1615211