Finite Group Databasebeta

← #41012380

Subgroups — #41012380

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#27213734C211yesmaximal, characteristic
2#27213304C211yesmaximal, characteristic, complemented
3#27213287C211yesmaximal, characteristic, complemented
4#27213058C211yesmaximal, characteristic, complemented
5#27212884C211yesmaximal, characteristic, complemented
6#27212881C211yesmaximal, characteristic, complemented
7#323×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C2×Dic844—21no—
10C2×#16809951—21no—
11#20269970—21no—
12#20269963—21no—
13#20269959—21no—
14#20269956C2211yescharacteristic
15#20269951C2211yescharacteristic
16#20269950—21no—
17#20269946—21no—
18C4×C22⋊D211—21no—
19C2×C22⋊Dic211—21no—
20#20269902—21no—
21#20269902—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24C2×C22⋊C4×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26C22⋊C8×C211C2211yescharacteristic
27C2×Dic422—21no—
28C2×Dic422—21no—
29C2×Dic422—42no—
30C2×Dic422D411yescharacteristic
31C2×D844D411yescharacteristic
32C2×C22⋊D211—21no—
33C2×C22⋊D211—21no—
34C2×C22⋊D211—21no—
35C22×Dic211—21no—
36C22×Dic211—21no—
37C2×C4×D211—21no—
38Dic844—42no—
39#16809951—42no—
40C4⋊Dic211D411yescharacteristic
41C4⋊Dic211D411yescharacteristic
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44C22⋊Dic211—21no—
45C22⋊Dic211—21no—
46C22⋊Dic211—21no—
47C22⋊Dic211—42no—
48(C2×D211)⋊C4—21no—
49(C2×D211)⋊C4—21no—
50C4×Dic211—21no—
51C24×C211D411yescharacteristic
52C22×C4×C211D411yescharacteristic
53C22×C4×C211C2311yescharacteristic
54C2×C8×C211—21no—
55C2×C8×C211—21no—
56C22⋊C4×C211—21no—
57C22⋊C4×C211—21no—
58C22×D211—21no—
59Dic422—22no—
60Dic422—22no—
61Dic422—41no—
62Dic422—41no—
63Dic422—41no—
64D844—41no—
65C22⋊D211—41no—
66C22⋊D211—41no—
67C22⋊D211—81no—
68C22⋊D211—81no—
69C2×Dic211—22no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—21no—
73C2×Dic211—41no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211—41no—
77C2×Dic211—41no—
78C4×D211—81no—
79C23×C211—21no—
80C23×C211—21no—
81C23×C211—21no—
82C23×C211—41no—
83C23×C211—41no—
84C23×C211C2×D411yescharacteristic
85C2×C4×C211—21no—
86C2×C4×C211—41no—
87C2×C4×C211—41no—
88C2×C4×C211C2×D411yescharacteristic
89C2×C4×C211C2×D411yescharacteristic
90C8×C211—41no—
91C8×C211—41no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211—42no—
95C22×C211—41no—
96C22×C211—41no—
97C22×C211—41no—
98C22×C211—81no—
99C22×C211#12011yescharacteristic
100C22×C211#13511yescharacteristic
101C22×C211#13711yescharacteristic
102C2×D211—41no—
103C2×D211—41no—
104C2×D211—41no—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—41no—
108Dic211—41no—
109Dic211—41no—
110Dic211—42no—
111Dic211—41no—
112Dic211—81no—
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#327—2111no—
128#323D21111yescharacteristic, complemented
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#121—2111no—
138#120—4221no—
139C4×D4—4221no—
140C2×C22⋊C4—4221no—
141C2×C22⋊C4C2×D21111yescharacteristic
142#103—4221no—
143#103—4221no—
144C22⋊C8C2×D21111yescharacteristic
145C22⋊C8C2×D21111yescharacteristic
146#95—4221no—
147#95—2111no—
148C24D84411yescharacteristic
149C2×Q8—8442no—
150C2×Q8—4221no—
151C2×Q8—4221no—
152C2×Q8—2111no—
153C2×D4—4221no—
154C2×D4—4221no—
155C2×D4—4221no—
156C2×D4—2111no—
157C22×C4—4221no—
158C22×C4—4221no—
159C22×C4—4221no—
160C22×C4D84411yescharacteristic
161C22×C4C22×D21111yescharacteristic
162Q16—8442no—
163SD16—8442no—
164C2×C8—21no—
165C2×C8—21no—
166C4⋊C4—4221no—
167C4⋊C4—4221no—
168C4⋊C4—2111no—
169C4⋊C4—2111no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—8442no—
173C22⋊C4—4221no—
174C22⋊C4—4221no—
175C22⋊C4—4221no—
176C22⋊C4—4221no—
177C22⋊C4—4221no—
178C42—4221no—
179C23—21no—
180C23—21no—
181C23—21no—
182C23—41no—
183C23—41no—
184C23—4221no—
185C23C2×D84411yescharacteristic
186Q8—8441no—
187Q8—8441no—
188Q8—8441no—
189Q8—4222no—
190Q8—4222no—
191D4—8441no—
192D4—8441no—
193D4—8441no—
194D4—16881no—
195D4—16881no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—8441no—
200C2×C4—8441no—
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×C4D4×D21111yescharacteristic
211C8—41no—
212C8—41no—
213C22—21no—
214C22—21no—
215C22—42no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22—8441no—
221C22—8441no—
222C22—8441no—
223C22#2026994711yescharacteristic
224C22#2026999911yescharacteristic
225C22#2027001011yescharacteristic
226C4—21no—
227C4—21no—
228C4—41no—
229C4—8441no—
230C4—8442no—
231C4—8441no—
232C4—8441no—
233C4—16881no—
234C2—21no—
235C2—21no—
236C2—81no—
237C2—16881no—
238C2#2721331011yescharacteristic
239C2#2721332611yescharacteristic
240C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#323×C211C2
socleC22×C211#120
prime radical#27213734C2
hypercenter#323D211
hypocenterC211#1606
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27213304C2semidirect
3. #27213287C2semidirect
4. #27213058C2semidirect
5. #27212884C2semidirect
6. #27212881C2semidirect
7. #323×C211C2semidirect
120. C211#1606semidirect
128. #323D211semidirect

Maximal subgroups

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

subgroupindex
#272137342
#272133042
#272132872
#272130582
#272128842
#272128812
#323×C2112
#1606211