Finite Group Databasebeta

← #41014608

Subgroups — #41014608

Order 27008.

326 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 67 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#27213713C211yesmaximal, characteristic
2#27213186C211yesmaximal, characteristic, complemented
3C2×#20269905C211yesmaximal, characteristic, complemented
4#27213145C211yesmaximal, characteristic, complemented
5C2×#20269894C211yesmaximal, characteristic, complemented
6#27212906C211yesmaximal, characteristic, complemented
7#390×C211C211yesmaximal, characteristic, complemented
8C2×#16809962C2211yescharacteristic
9C2×#16809946C2211yescharacteristic
10C2×#16809946C2211yescharacteristic
11#20269973—21no—
12#20269964—21no—
13#20269948—21no—
14#20269948—21no—
15#20269946—22no—
16#20269936C2211yescharacteristic
17#20269910—21no—
18#20269908—21no—
19#20269905—22no—
20#20269900—21no—
21#20269897—21no—
22#20269894—22no—
23C22×D4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C22⋊C4×C211—22no—
26C2×C42×C211C2211yescharacteristic
27#16809962—21no—
28#16809962—21no—
29#16809962—21no—
30#16809962—21no—
31C2×Dic422D411yescharacteristic
32C2×D844D411yescharacteristic
33C2×C22⋊D211—22no—
34C2×C22⋊D211—22no—
35C2×C22⋊D211—21no—
36C2×C4×D211—21no—
37#16809946—21no—
38#16809946—21no—
39#16809946—21no—
40#16809946D412yescomplemented
41C2×C211⋊C8—21no—
42C2×C211⋊C8—21no—
43C4⋊Dic211D411yescharacteristic
44C4⋊Dic211D411yescharacteristic
45C22⋊Dic211—22no—
46C22⋊Dic211—22no—
47C22⋊Dic211—21no—
48C4×Dic211—21no—
49C24×C211D412yes—
50C2×D4×C211—21no—
51C2×D4×C211—21no—
52C2×D4×C211—21no—
53C2×D4×C211—21no—
54C2×D4×C211—21no—
55C2×D4×C211—21no—
56C22×C4×C211—21no—
57C22×C4×C211—21no—
58C22×C4×C211C2311yescharacteristic
59C4⋊C4×C211—21no—
60C4⋊C4×C211—21no—
61C22⋊C4×C211—44no—
62C42×C211—21no—
63C42×C211—21no—
64C22×D211—21no—
65Dic422—21no—
66Dic422—21no—
67D844—21no—
68D844—21no—
69C22⋊D211—41no—
70C22⋊D211—41no—
71C22⋊D211—82no—
72C22⋊D211—82no—
73C2×Dic211—21no—
74C2×Dic211—21no—
75C2×Dic211—21no—
76C4×D211—41no—
77C4×D211—41no—
78C211⋊C8—22no—
79C211⋊C8—41no—
80C23×C211—22no—
81C23×C211—22no—
82C23×C211—22no—
83C23×C211—42no—
84C23×C211—42no—
85C23×C211C2×D411yescharacteristic
86D4×C211—41no—
87D4×C211—41no—
88D4×C211—41no—
89D4×C211—41no—
90C2×C4×C211—22no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—42no—
94C2×C4×C211—41no—
95C2×C4×C211C2×D411yescharacteristic
96C2×C4×C211C2×D411yescharacteristic
97C2×C4×C211C2×D411yescharacteristic
98C2×C4×C211#4111yescharacteristic
99C2×C4×C211#4111yescharacteristic
100C2×C4×C211#4111yescharacteristic
101C22×C211—21no—
102C22×C211—21no—
103C22×C211—44no—
104C22×C211—42no—
105C22×C211—42no—
106C22×C211—42no—
107C22×C211—82no—
108C22×C211#12011yescharacteristic
109C22×C211#12111yescharacteristic
110C22×C211#12311yescharacteristic
111C2×D211—41no—
112C2×D211—41no—
113C2×D211—41no—
114C4×C211—42no—
115C4×C211—81no—
116C4×C211#12111yescharacteristic
117C4×C211#12111yescharacteristic
118C4×C211#12311yescharacteristic
119C4×C211#12411yescharacteristic
120Dic211—41no—
121Dic211—41no—
122Dic211—41no—
123C2×C211—21no—
124C2×C211—21no—
125C2×C211—82no—
126C2×C211#39411yescharacteristic
127C2×C211#45311yescharacteristic
128C2×C211#45511yescharacteristic
129D211—81no—
130C211#203711yescharacteristic, complemented, Sylow
131#2037—2111nomaximal, Sylow
132#535—2111no—
133#428—2111no—
134C2×#104—2111no—
135#417—2111no—
136C2×#100—2111no—
137#390D21111yescharacteristic, complemented
138#339—2111no—
139C2×#41—2111no—
140C22×D4C2×D21111yescharacteristic
141C2×M16—2111no—
142C2×M16—2111no—
143#127—4221no—
144#124—4221no—
145#121—4221no—
146#121—4221no—
147#120—4222no—
148#117—2111no—
149C2×C4⋊C4C2×D21111yescharacteristic
150C2×C22⋊C4—22no—
151C2×C42C2×D21111yescharacteristic
152#107—4221no—
153#106—4221no—
154#104—4222no—
155#103—4221no—
156#102—4221no—
157#100—4222no—
158C24C22⋊D21112yes—
159#41—4221no—
160#41—4221no—
161#41—4221no—
162#41—4221no—
163C2×Q8—2111no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—4221no—
171C2×D4—4222no—
172C2×D4—4222no—
173C2×D4—2111no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—4221no—
177C22×C4C22×D21111yescharacteristic
178M16—4221no—
179M16—4221no—
180M16—4221no—
181M16—2112no—
182C2×C8—4221no—
183C2×C8—4221no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—2111no—
187C4⋊C4—2111no—
188C22⋊C4—44no—
189C22⋊C4—4222no—
190C22⋊C4—4222no—
191C22⋊C4—4221no—
192C42—21no—
193C42—21no—
194C42—4221no—
195C23—22no—
196C23—22no—
197C23—22no—
198C23—42no—
199C23—42no—
200C23—4221no—
201C23C2×C22⋊D21111yescharacteristic
202Q8—4221no—
203Q8—4221no—
204D4—41no—
205D4—41no—
206D4—41no—
207D4—41no—
208D4—8441no—
209D4—8441no—
210D4—16882no—
211D4—16882no—
212D4—4221no—
213D4—4221no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—8441no—
220C2×C4—8441no—
221C2×C4—4221no—
222C2×C4—4221no—
223C2×C4—4221no—
224C2×C4D4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C2×C4D4×D21111yescharacteristic
227C2×C4#1680996111yescharacteristic
228C2×C4#1680996211yescharacteristic
229C2×C4#1680996211yescharacteristic
230C8—8441no—
231C8—4222no—
232C22—21no—
233C22—21no—
234C22—44no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—82no—
239C22—8441no—
240C22—8441no—
241C22—8441no—
242C22#2026994511yescharacteristic
243C22#2026994911yescharacteristic
244C22#2026996011yescharacteristic
245C4—42no—
246C4—81no—
247C4—8441no—
248C4—8441no—
249C4—8441no—
250C4#2026995011yescharacteristic
251C4#2026995011yescharacteristic
252C4#2026995911yescharacteristic
253C4#2026996311yescharacteristic
254C2—21no—
255C2—21no—
256C2—82no—
257C2—16881no—
258C2#2721306111yescharacteristic
259C2#2721331311yescharacteristic
260C2#2721332311yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#390×C211C2
socleC22×C211#120
prime radical#27213713C2
hypercenter#390D211
hypocenterC211#2037
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #27213186C2semidirect
3. C2×#20269905C2semidirect
4. #27213145C2semidirect
5. C2×#20269894C2semidirect
6. #27212906C2semidirect
7. #390×C211C2semidirect
40. #16809946D4semidirect
130. C211#2037semidirect
137. #390D211semidirect

Maximal subgroups

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

subgroupindex
#272137132
#272131862
C2×#202699052
#272131452
C2×#202698942
#272129062
#390×C2112
#2037211