Finite Group Databasebeta

← #41012379

Subgroups — #41012379

Order 27008.

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

normal subgroups only

This group has 260 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#27213304C211yesmaximal, characteristic, complemented
3#27213284C211yesmaximal, characteristic, complemented
4#27213059C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212869C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C2×Dic844—21no—
10C2×#16809950—21no—
11#20269972—21no—
12#20269963—21no—
13#20269961—21no—
14#20269956C2211yescharacteristic
15#20269949—21no—
16#20269945—21no—
17D4×Dic211—21no—
18C2×(C2×D211)⋊C4—21no—
19C2×(C2×D211)⋊C4C2211yescharacteristic
20#20269902—21no—
21#20269900—21no—
22C22⋊(C211⋊C8)C2211yescharacteristic
23C211⋊#95—21no—
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28C2×Dic422—21no—
29C2×Dic422—21no—
30C2×Dic422—42no—
31C2×Dic422D411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C22×Dic211—21no—
35C22×Dic211—21no—
36C22×Dic211D411yescharacteristic
37Dic844—42no—
38#16809950—42no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211D411yescharacteristic
42Dic211⋊C4—21no—
43C22⋊Dic211—21no—
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—21no—
46(C2×D211)⋊C4—21no—
47(C2×D211)⋊C4—21no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—42no—
50C4×Dic211—21no—
51C2×D4×C211—21no—
52C2×D4×C211D411yescharacteristic
53C22×C4×C211—21no—
54C22×C4×C211C2311yescharacteristic
55C2×C8×C211—21no—
56C4⋊C4×C211D411yescharacteristic
57C22⋊C4×C211—21no—
58C22×D211—21no—
59C22×D211—21no—
60C22×D211—21no—
61C22×D211—41no—
62C22×D211—41no—
63Dic422—22no—
64Dic422—22no—
65Dic422—41no—
66Dic422—41no—
67Dic422—41no—
68C22⋊D211—81no—
69C22⋊D211—81no—
70C2×Dic211—22no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—21no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—81no—
78C211⋊C8—41no—
79C23×C211—21no—
80C23×C211C2×D411yescharacteristic
81D4×C211—41no—
82D4×C211—41no—
83D4×C211—41no—
84C2×C4×C211—21no—
85C2×C4×C211—41no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211—41no—
89C2×C4×C211C2×D411yescharacteristic
90C2×C4×C211C2×D411yescharacteristic
91C8×C211—41no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211—41no—
95C22×C211—41no—
96C22×C211—41no—
97C22×C211#12011yescharacteristic
98C22×C211#13511yescharacteristic
99C22×C211#13711yescharacteristic
100C2×D211—42no—
101C2×D211—41no—
102C2×D211—41no—
103C2×D211—41no—
104C2×D211—81no—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—41no—
108C4×C211—81no—
109Dic211—41no—
110Dic211—41no—
111Dic211—42no—
112Dic211—41no—
113C2×C211—21no—
114C2×C211—21no—
115C2×C211—81no—
116C2×C211#45211yescharacteristic
117C2×C211#45511yescharacteristic
118C2×C211B(4,𝔽2)11yescharacteristic
119D211—81no—
120C211#160611yescharacteristic, complemented, Sylow
121#1606—2111nomaximal, Sylow
122#538—2111no—
123#451—2111no—
124#448—2111no—
125#393—2111no—
126#328—2111no—
127#327D21111yescharacteristic, complemented
128#323—2111no—
129C22×Q8—2111no—
130C2×Q16—4221no—
131C2×SD16—4221no—
132#126—4221no—
133#124—4221no—
134#123—4221no—
135C22⋊Q8—2111no—
136#121—4221no—
137#121C2×D21111yescharacteristic
138#120—4221no—
139C4×D4—4221no—
140C2×C22⋊C4—4221no—
141C2×C22⋊C4—2111no—
142#103—4221no—
143#103—4221no—
144C22⋊C8—2111no—
145C22⋊C8C2×D21111yescharacteristic
146#95—4221no—
147#95C2×D21111yescharacteristic
148C24—2111no—
149C2×Q8—8442no—
150C2×Q8—4221no—
151C2×Q8—4221no—
152C2×Q8—2111no—
153C2×D4—21no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4D84411yescharacteristic
157C22×C4—21no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4—2111no—
161C22×C4C22×D21111yescharacteristic
162Q16—8442no—
163SD16—8442no—
164C2×C8—21no—
165C2×C8—4221no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C4⋊C4—2111no—
169C4⋊C4D84411yescharacteristic
170C22⋊C4—21no—
171C22⋊C4—8442no—
172C22⋊C4—4221no—
173C22⋊C4—4221no—
174C22⋊C4—4221no—
175C22⋊C4—4221no—
176C22⋊C4—4221no—
177C22⋊C4—4221no—
178C42—4221no—
179C23—21no—
180C23—8441no—
181C23—8441no—
182C23—4221no—
183C23—4221no—
184C23—4221no—
185C23D4×D21111yescharacteristic
186Q8—8441no—
187Q8—8441no—
188Q8—8441no—
189Q8—4222no—
190Q8—4222no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—16881no—
195D4—16881no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—8441no—
202C2×C4—8441no—
203C2×C4—8441no—
204C2×C4—16881no—
205C2×C4—4222no—
206C2×C4—4221no—
207C2×C4—4221no—
208C2×C4—4221no—
209C2×C4D4×D21111yescharacteristic
210C2×C4C2×D84411yescharacteristic
211C8—41no—
212C8—8441no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—8442no—
219C22—8441no—
220C22—8441no—
221C22—8441no—
222C22—16881no—
223C22#2026994711yescharacteristic
224C22#2026999611yescharacteristic
225C22#2027000611yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—81no—
230C4—8441no—
231C4—8441no—
232C4—8442no—
233C4—8441no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—16881no—
238C2#2721330911yescharacteristic
239C2#2721332511yescharacteristic
240C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211#120
prime radical#27213733C2
hypercenter#327D211
hypocenterC211#1606
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. #27213304C2semidirect
3. #27213284C2semidirect
4. #27213059C2semidirect
5. #27212882C2semidirect
6. #27212869C2semidirect
7. #327×C211C2semidirect
120. C211#1606semidirect
127. #327D211semidirect

Maximal subgroups

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

subgroupindex
#272137332
#272133042
#272132842
#272130592
#272128822
#272128692
#327×C2112
#1606211