Finite Group Databasebeta

← #41014607

Subgroups — #41014607

Order 27008.

326 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 69 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#27213710C211yesmaximal, characteristic
2#27213188C211yesmaximal, characteristic, complemented
3#27213147C211yesmaximal, characteristic, complemented
4C2×#20269893C211yesmaximal, characteristic, complemented
5#27213048C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7C2×#104×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10#20269973—21no—
11#20269966—21no—
12#20269951—21no—
13#20269951—21no—
14#20269945—22no—
15#20269936C2211yescharacteristic
16C2×C4⋊Dic211C2211yescharacteristic
17C2×(C2×D211)⋊C4—22no—
18#20269911—21no—
19#20269909—21no—
20#20269901—21no—
21#20269898—21no—
22#20269893—22no—
23C2×#41×C211C2211yescharacteristic
24C2×M16×C211C2211yescharacteristic
25C2×C42×C211C2211yescharacteristic
26#104×C211—22no—
27C23×D211D412yes—
28C2×D844—21no—
29C2×D844—21no—
30C2×D844—21no—
31C2×D844—21no—
32C2×D844—21no—
33C2×D844—21no—
34C2×C22⋊D211—22no—
35C2×C22⋊D211—22no—
36C22×Dic211—21no—
37#16809946—21no—
38#16809946—21no—
39C2×C211⋊C8—21no—
40C4⋊Dic211—21no—
41C4⋊Dic211—21no—
42C4⋊Dic211D411yescharacteristic
43C4⋊Dic211D411yescharacteristic
44C22⋊Dic211—21no—
45(C2×D211)⋊C4—22no—
46(C2×D211)⋊C4—22no—
47(C2×D211)⋊C4—44no—
48C4×Dic211—21no—
49#41×C211—21no—
50#41×C211—21no—
51#41×C211—21no—
52#41×C211—21no—
53C2×Q8×C211D411yescharacteristic
54C2×D4×C211—21no—
55C2×D4×C211D411yescharacteristic
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59M16×C211—21no—
60M16×C211D412yescomplemented
61C2×C8×C211—21no—
62C42×C211—21no—
63C42×C211—21no—
64C22×D211—22no—
65C22×D211—22no—
66C22×D211—22no—
67C22×D211—42no—
68C22×D211—42no—
69D844—41no—
70D844—41no—
71D844—41no—
72D844—41no—
73C22⋊D211—82no—
74C22⋊D211—82no—
75C2×Dic211—21no—
76C2×Dic211—21no—
77C2×Dic211—42no—
78C2×Dic211—41no—
79C211⋊C8—41no—
80C23×C211—21no—
81C23×C211C2×D411yescharacteristic
82Q8×C211—21no—
83Q8×C211—21no—
84D4×C211—21no—
85D4×C211—21no—
86D4×C211—41no—
87D4×C211—41no—
88C2×C4×C211—22no—
89C2×C4×C211—21no—
90C2×C4×C211—41no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211C2×D411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211#4111yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C8×C211—22no—
101C22×C211—21no—
102C22×C211—21no—
103C22×C211—41no—
104C22×C211—41no—
105C22×C211—41no—
106C22×C211#12011yescharacteristic
107C22×C211#12111yescharacteristic
108C22×C211#12311yescharacteristic
109C2×D211—44no—
110C2×D211—42no—
111C2×D211—42no—
112C2×D211—42no—
113C2×D211—82no—
114C4×C211—42no—
115C4×C211—41no—
116C4×C211#12111yescharacteristic
117C4×C211#12111yescharacteristic
118C4×C211#12311yescharacteristic
119C4×C211#12411yescharacteristic
120Dic211—41no—
121Dic211—41no—
122Dic211—81no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—81no—
126C2×C211#39411yescharacteristic
127C2×C211#45311yescharacteristic
128C2×C211#45511yescharacteristic
129D211—82no—
130C211#203711yescharacteristic, complemented, Sylow
131#2037—2111nomaximal, Sylow
132#535—2111no—
133#428—2111no—
134C2×#104D21111yescharacteristic, complemented
135#417—2111no—
136C2×#100—2111no—
137#390—2111no—
138#339—2111no—
139C2×#41C2×D21111yescharacteristic
140C22×D4—2111no—
141C2×M16—2111no—
142C2×M16C2×D21111yescharacteristic
143#127—4221no—
144#124—4221no—
145#121—4221no—
146#121—4221no—
147#120—4222no—
148#117—2111no—
149C2×C4⋊C4—2111no—
150C2×C22⋊C4—4222no—
151C2×C42C2×D21111yescharacteristic
152#107—4221no—
153#106—4221no—
154#104—22no—
155#103—4221no—
156#102—4221no—
157#100—4222no—
158C24—2112no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163C2×Q8D84411yescharacteristic
164C2×D4—21no—
165C2×D4—4222no—
166C2×D4—4222no—
167C2×D4—4221no—
168C2×D4—4221no—
169C2×D4—4221no—
170C2×D4—4221no—
171C2×D4—4221no—
172C2×D4—4221no—
173C2×D4D84411yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178M16—21no—
179M16—4221no—
180M16—4221no—
181M16C22⋊D21112yescomplemented
182C2×C8—21no—
183C2×C8—4221no—
184C4⋊C4—4221no—
185C4⋊C4—4221no—
186C4⋊C4—2111no—
187C4⋊C4—2111no—
188C22⋊C4—8444no—
189C22⋊C4—4222no—
190C22⋊C4—4222no—
191C22⋊C4—4221no—
192C42—21no—
193C42—21no—
194C42—4221no—
195C23—21no—
196C23—8442no—
197C23—8442no—
198C23—4222no—
199C23—4222no—
200C23—4222no—
201C23D4×D21111yescharacteristic
202Q8—21no—
203Q8—21no—
204D4—21no—
205D4—21no—
206D4—41no—
207D4—41no—
208D4—8441no—
209D4—8441no—
210D4—8441no—
211D4—8441no—
212D4—16882no—
213D4—16882no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—8442no—
221C2×C4—8441no—
222C2×C4—4221no—
223C2×C4—4221no—
224C2×C4D4×D21111yescharacteristic
225C2×C4C2×C22⋊D21111yescharacteristic
226C2×C4C2×D84411yescharacteristic
227C2×C4#1680996111yescharacteristic
228C2×C4#1680996111yescharacteristic
229C2×C4#1680996211yescharacteristic
230C8—22no—
231C8—8441no—
232C22—21no—
233C22—21no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—8444no—
238C22—8442no—
239C22—8442no—
240C22—8442no—
241C22—16882no—
242C22#2026994711yescharacteristic
243C22#2026994811yescharacteristic
244C22#2026995911yescharacteristic
245C4—42no—
246C4—41no—
247C4—8441no—
248C4—8441no—
249C4—16881no—
250C4#2026994911yescharacteristic
251C4#2026995111yescharacteristic
252C4#2026996111yescharacteristic
253C4#2026996311yescharacteristic
254C2—21no—
255C2—21no—
256C2—81no—
257C2—16882no—
258C2#2721306211yescharacteristic
259C2#2721331511yescharacteristic
260C2#2721332511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#104×C211C2
socleC22×C211#120
prime radical#27213710C2
hypercenterC2×#104D211
hypocenterC211#2037
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #27213188C2semidirect
3. #27213147C2semidirect
4. C2×#20269893C2semidirect
5. #27213048C2semidirect
6. #27212906C2semidirect
7. C2×#104×C211C2semidirect
60. M16×C211D4semidirect
130. C211#2037semidirect
134. C2×#104D211semidirect
181. M16C22⋊D211semidirect

Maximal subgroups

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

subgroupindex
#272137102
#272131882
#272131472
C2×#202698932
#272130482
#272129062
C2×#104×C2112
#2037211