Finite Group Databasebeta

← #41012443

Subgroups — #41012443

Order 27008.

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

normal subgroups only

This group has 240 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#27213733C211yesmaximal, characteristic
2#27213481C211yesmaximal, characteristic, complemented
3#27213081C211yesmaximal, characteristic, complemented
4#27212884C211yesmaximal, characteristic, complemented
5#27212880C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#481×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269963—21no—
10#20269961—21no—
11#20269956C2211yescharacteristic
12#20269949—21no—
13#20269945—21no—
14D4×Dic211—21no—
15C2×Dic211⋊C4—21no—
16C2×(C2×D211)⋊C4—21no—
17C2×(C2×D211)⋊C4C2211yescharacteristic
18#20269910—21no—
19#20269900—21no—
20C22⋊(C211⋊C8)C2211yescharacteristic
21C211⋊#95C2211yescharacteristic
22#121×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24#106×C211—21no—
25#102×C211—21no—
26C22⋊C8×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×Dic422D411yescharacteristic
29C2×C22⋊D211—21no—
30C2×C22⋊D211—21no—
31C22×Dic211—21no—
32C22×Dic211—21no—
33C22×Dic211D411yescharacteristic
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211D411yescharacteristic
37Dic211⋊C4—21no—
38Dic211⋊C4—42no—
39C22⋊Dic211—21no—
40C22⋊Dic211—21no—
41(C2×D211)⋊C4—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—42no—
46C4×Dic211—21no—
47C2×D4×C211—21no—
48C2×D4×C211D411yescharacteristic
49C22×C4×C211—21no—
50C22×C4×C211C2311yescharacteristic
51C2×C8×C211—21no—
52C4⋊C4×C211—21no—
53C4⋊C4×C211—21no—
54C4⋊C4×C211D411yescharacteristic
55C22⋊C4×C211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—21no—
59C22×D211—41no—
60C22×D211—41no—
61Dic422—41no—
62C22⋊D211—81no—
63C22⋊D211—81no—
64C2×Dic211—22no—
65C2×Dic211—21no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211—81no—
72C211⋊C8—41no—
73C23×C211—21no—
74C23×C211C2×D411yescharacteristic
75D4×C211—41no—
76D4×C211—41no—
77D4×C211—41no—
78C2×C4×C211—21no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211—41no—
83C2×C4×C211C2×D411yescharacteristic
84C2×C4×C211C2×D411yescharacteristic
85C8×C211—41no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211—41no—
91C22×C211#12011yescharacteristic
92C22×C211#13511yescharacteristic
93C22×C211#13711yescharacteristic
94C2×D211—42no—
95C2×D211—41no—
96C2×D211—41no—
97C2×D211—41no—
98C2×D211—81no—
99C4×C211—21no—
100C4×C211—21no—
101C4×C211—41no—
102C4×C211—81no—
103Dic211—42no—
104Dic211—41no—
105Dic211—41no—
106Dic211—41no—
107C2×C211—21no—
108C2×C211—21no—
109C2×C211—81no—
110C2×C211#45211yescharacteristic
111C2×C211#45311yescharacteristic
112C2×C211#45811yescharacteristic
113D211—81no—
114C211#161811yescharacteristic, complemented, Sylow
115#1618—2111nomaximal, Sylow
116#538—2111no—
117#484—2111no—
118#481D21111yescharacteristic, complemented
119#399—2111no—
120#328—2111no—
121#327—2111no—
122#323—2111no—
123#126—4221no—
124#124—4221no—
125#123—4221no—
126C22⋊Q8—2111no—
127#121—4221no—
128#121C2×D21111yescharacteristic
129#120—4221no—
130C4×D4—4221no—
131C2×C4⋊C4—4221no—
132C2×C4⋊C4C2×D21111yescharacteristic
133C2×C22⋊C4—4221no—
134C2×C22⋊C4—2111no—
135#107—4221no—
136#106—21no—
137#103—4221no—
138#102—21no—
139C22⋊C8—2111no—
140C22⋊C8C2×D21111yescharacteristic
141#95—2111no—
142C24—2111no—
143C2×Q8—2111no—
144C2×D4—21no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4C22⋊D21111yescharacteristic
148C22×C4—21no—
149C22×C4—4221no—
150C22×C4—4221no—
151C22×C4—2111no—
152C22×C4C22×D21111yescharacteristic
153C2×C8—21no—
154C2×C8—4221no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—8442no—
158C4⋊C4—4221no—
159C4⋊C4—4221no—
160C4⋊C4—2111no—
161C4⋊C4C22⋊D21111yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—8442no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C22⋊C4—4221no—
167C22⋊C4—4221no—
168C22⋊C4—4221no—
169C22⋊C4—4221no—
170C42—4221no—
171C23—21no—
172C23—8441no—
173C23—8441no—
174C23—4221no—
175C23—4221no—
176C23—4221no—
177C23D4×D21111yescharacteristic
178Q8—8441no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—16881no—
183D4—16881no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—8441no—
192C2×C4—16881no—
193C2×C4—4222no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4D4×D21111yescharacteristic
198C2×C4C2×C22⋊D21111yescharacteristic
199C8—41no—
200C8—8441no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—8442no—
207C22—8441no—
208C22—8441no—
209C22—8441no—
210C22—16881no—
211C22#2026994511yescharacteristic
212C22#2026999811yescharacteristic
213C22#2027000611yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—81no—
218C4—8442no—
219C4—8441no—
220C4—8441no—
221C4—8441no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—16881no—
226C2#2721330511yescharacteristic
227C2#2721331211yescharacteristic
228C2#2721333511yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#481×C211C2
socleC22×C211#120
prime radical#27213733C2
hypercenter#481D211
hypocenterC211#1618
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #27213481C2semidirect
3. #27213081C2semidirect
4. #27212884C2semidirect
5. #27212880C2semidirect
6. #27212869C2semidirect
7. #481×C211C2semidirect
114. C211#1618semidirect
118. #481D211semidirect

Maximal subgroups

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

subgroupindex
#272137332
#272134812
#272130812
#272128842
#272128802
#272128692
#481×C2112
#1618211