Finite Group Databasebeta

← #41012413

Subgroups — #41012413

Order 27008.

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

normal subgroups only

This group has 232 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#27213477C211yesmaximal, characteristic, complemented
3#27213457C211yesmaximal, characteristic, complemented
4#27213073C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8#20269972—21no—
9#20269963—21no—
10#20269961—21no—
11#20269956C2211yescharacteristic
12#20269949—21no—
13#20269945—21no—
14D4×Dic211—21no—
15C2×C4⋊Dic211C2211yescharacteristic
16C2×(C2×D211)⋊C4—21no—
17C2×(C2×D211)⋊C4C2211yescharacteristic
18#20269910—21no—
19#20269909—21no—
20#20269902—21no—
21#20269898—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×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⋊Dic211—21no—
37C4⋊Dic211—21no—
38C4⋊Dic211D411yescharacteristic
39Dic211⋊C4—21no—
40C22⋊Dic211—21no—
41C22⋊Dic211—21no—
42(C2×D211)⋊C4—21no—
43(C2×D211)⋊C4—21no—
44(C2×D211)⋊C4—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—42no—
47C4×Dic211—21no—
48C2×D4×C211—21no—
49C2×D4×C211D411yescharacteristic
50C22×C4×C211—21no—
51C22×C4×C211C2311yescharacteristic
52C2×C8×C211—21no—
53C4⋊C4×C211D411yescharacteristic
54C22⋊C4×C211—21no—
55C22×D211—21no—
56C22×D211—21no—
57C22×D211—21no—
58C22×D211—41no—
59C22×D211—41no—
60Dic422—41no—
61C22⋊D211—81no—
62C22⋊D211—81no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—21no—
66C2×Dic211—41no—
67C2×Dic211—41no—
68C2×Dic211—41no—
69C2×Dic211—41no—
70C2×Dic211—81no—
71C211⋊C8—41no—
72C23×C211—21no—
73C23×C211C2×D411yescharacteristic
74D4×C211—41no—
75D4×C211—41no—
76D4×C211—41no—
77C2×C4×C211—21no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—41no—
81C2×C4×C211—41no—
82C2×C4×C211C2×D411yescharacteristic
83C2×C4×C211C2×D411yescharacteristic
84C8×C211—41no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—41no—
88C22×C211—41no—
89C22×C211—41no—
90C22×C211#12011yescharacteristic
91C22×C211#13511yescharacteristic
92C22×C211AGL(1,ℤ/8)11yescharacteristic
93C2×D211—42no—
94C2×D211—41no—
95C2×D211—41no—
96C2×D211—41no—
97C2×D211—81no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—41no—
101C4×C211—81no—
102Dic211—41no—
103Dic211—41no—
104Dic211—41no—
105Dic211—81no—
106C2×C211—21no—
107C2×C211—21no—
108C2×C211—81no—
109C2×C211#44911yescharacteristic
110C2×C211#45511yescharacteristic
111C2×C211#45811yescharacteristic
112D211—81no—
113C211#161211yescharacteristic, complemented, Sylow
114#1612—2111nomaximal, Sylow
115#538—2111no—
116#483—2111no—
117#480—2111no—
118#397—2111no—
119#328—2111no—
120#327D21111yescharacteristic, complemented
121#323—2111no—
122#126—4221no—
123#124—4221no—
124#123—4221no—
125C22⋊Q8—2111no—
126#121—4221no—
127#121C2×D21111yescharacteristic
128#120—4221no—
129C4×D4—4221no—
130C2×C4⋊C4—2111no—
131C2×C22⋊C4—4221no—
132C2×C22⋊C4—2111no—
133#107—4221no—
134#106—4221no—
135#103—4221no—
136#102—4221no—
137C22⋊C8—2111no—
138C22⋊C8C2×D21111yescharacteristic
139#95—4221no—
140#95C2×D21111yescharacteristic
141C24—2111no—
142C2×Q8—2111no—
143C2×D4—21no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4D84411yescharacteristic
147C22×C4—21no—
148C22×C4—4221no—
149C22×C4—4221no—
150C22×C4—2111no—
151C22×C4C22×D21111yescharacteristic
152C2×C8—21no—
153C2×C8—4221no—
154C4⋊C4—4221no—
155C4⋊C4—4221no—
156C4⋊C4—4221no—
157C4⋊C4—4221no—
158C4⋊C4—2111no—
159C4⋊C4D84411yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—8442no—
162C22⋊C4—4221no—
163C22⋊C4—4221no—
164C22⋊C4—4221no—
165C22⋊C4—4221no—
166C22⋊C4—4221no—
167C22⋊C4—4221no—
168C42—4221no—
169C23—21no—
170C23—8441no—
171C23—8441no—
172C23—4221no—
173C23—4221no—
174C23—4221no—
175C23D4×D21111yescharacteristic
176Q8—8441no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—16881no—
181D4—16881no—
182C2×C4—21no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—8441no—
188C2×C4—8441no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—16881no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4D4×D21111yescharacteristic
196C2×C4C2×D84411yescharacteristic
197C8—41no—
198C8—8441no—
199C22—21no—
200C22—21no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—8442no—
205C22—8441no—
206C22—8441no—
207C22—8441no—
208C22—16881no—
209C22#2026994711yescharacteristic
210C22#2026999811yescharacteristic
211C22#2027000111yescharacteristic
212C4—21no—
213C4—21no—
214C4—41no—
215C4—81no—
216C4—8441no—
217C4—8441no—
218C4—8441no—
219C4—16881no—
220C2—21no—
221C2—21no—
222C2—81no—
223C2—16881no—
224C2#2721329211yescharacteristic
225C2#2721332511yescharacteristic
226C2#2721333711yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213733C2
hypercenter#327D211
hypocenterC211#1612
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. #27213477C2semidirect
3. #27213457C2semidirect
4. #27213073C2semidirect
5. #27212882C2semidirect
6. #27212869C2semidirect
7. #327×C211C2semidirect
113. C211#1612semidirect
120. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137332
#272134772
#272134572
#272130732
#272128822
#272128692
#327×C2112
#1612211