Finite Group Databasebeta

← #41011899

Subgroups — #41011899

Order 27008.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213656C211yesmaximal, characteristic
2#27213108C211yesmaximal, characteristic, complemented
3#95×D211C211yesmaximal, characteristic, complemented
4#27212884C211yesmaximal, characteristic, complemented
5#27212882C211yesmaximal, characteristic, complemented
6#27212880C211yesmaximal, characteristic, complemented
7#327×C211C211yesmaximal, characteristic, complemented
8C2×#16809961C411yescharacteristic, complemented
9C22×C4×D211—21no—
10C22×C4×D211C2211yescharacteristic
11C2×#16809945—21no—
12#20269959—21no—
13#20269957C411yescharacteristic, complemented
14#20269956C2211yescharacteristic
15#20269948C411yescharacteristic, complemented
16#20269945—21no—
17D4×Dic211—21no—
18(C22⋊D211)⋊C4—21no—
19#20269935C411yescharacteristic, complemented
20C22⋊(C211⋊C8)C2211yescharacteristic
21(C2×D211)⋊C8—21no—
22C211⋊#95—21no—
23C211⋊#95C2211yescharacteristic
24#121×C211C2211yescharacteristic
25C22⋊C8×C211C2211yescharacteristic
26#95×C211C2211yescharacteristic
27C23×D211D411yescharacteristic
28#16809961—42no—
29C2×Dic422C2×C411yescharacteristic, complemented
30C2×C22⋊D211—21no—
31C2×C22⋊D211—21no—
32C22×Dic211—21no—
33C22×Dic211—21no—
34C22×Dic211D411yescharacteristic
35C2×C4×D211—24no—
36C2×C4×D211—42no—
37C2×C4×D211—42no—
38C2×C4×D211—42no—
39C2×C4×D211C2×C411yescharacteristic
40C2×C4×D211C2×C411yescharacteristic
41C2×C4×D211D411yescharacteristic
42C2×C4×D211D411yescharacteristic
43#16809945—42no—
44C2×C211⋊C8—21no—
45C4⋊Dic211—21no—
46C4⋊Dic211C2×C411yescharacteristic, complemented
47Dic211⋊C4—21no—
48C22⋊Dic211—21no—
49C22⋊Dic211—21no—
50(C2×D211)⋊C4—21no—
51(C2×D211)⋊C4—21no—
52C4×Dic211—21no—
53C2×D4×C211—21no—
54C2×D4×C211C2×C411yescharacteristic, complemented
55C22×C4×C211—21no—
56C22×C4×C211C2311yescharacteristic
57C2×C8×C211—21no—
58C4⋊C4×C211C2×C411yescharacteristic, complemented
59C22⋊C4×C211—21no—
60C22×D211—21no—
61C22×D211—21no—
62C22×D211—22no—
63C22×D211—22no—
64C22×D211C22⋊C411yescharacteristic
65C22×D211C22⋊C411yescharacteristic
66Dic422—41no—
67C22⋊D211—42no—
68C22⋊D211—42no—
69C2×Dic211—22no—
70C2×Dic211—21no—
71C2×Dic211—21no—
72C2×Dic211—41no—
73C2×Dic211—42no—
74C2×Dic211—41no—
75C2×Dic211—41no—
76C2×Dic211C22⋊C411yescharacteristic
77C2×Dic211C22⋊C411yescharacteristic
78C4×D211—22no—
79C4×D211—22no—
80C4×D211—41no—
81C4×D211—41no—
82C4×D211—48no—
83C211⋊C8—41no—
84C23×C211—21no—
85C23×C211C2×D411yescharacteristic
86D4×C211—41no—
87D4×C211—41no—
88D4×C211—41no—
89C2×C4×C211—22no—
90C2×C4×C211—21no—
91C2×C4×C211—41no—
92C2×C4×C211—41no—
93C2×C4×C211—41no—
94C2×C4×C211C22×C411yescharacteristic
95C2×C4×C211C2×D411yescharacteristic
96C8×C211—41no—
97C22×C211—21no—
98C22×C211—21no—
99C22×C211—41no—
100C22×C211—41no—
101C22×C211—41no—
102C22×C211C2×C22⋊C411yescharacteristic
103C22×C211AGL(1,ℤ/8)11yescharacteristic
104C22×C211#13711yescharacteristic
105C2×D211—24no—
106C2×D211—24no—
107C2×D211—21no—
108C2×D211—21no—
109C2×D211—21no—
110C2×D211—21no—
111C2×D211#9912yes—
112C2×D211#10412yes—
113C4×C211—21no—
114C4×C211—21no—
115C4×C211—42no—
116C4×C211—41no—
117Dic211—21no—
118Dic211—21no—
119Dic211—42no—
120Dic211—41no—
121Dic211—41no—
122C2×C211—21no—
123C2×C211—21no—
124C2×C211—81no—
125C2×C211C2×#9911yescharacteristic
126C2×C211#41711yescharacteristic
127C2×C211C2×#10411yescharacteristic
128D211—22no—
129D211—22no—
130C211#150411yescharacteristic, complemented, Sylow
131#1504—2111nomaximal, Sylow
132#525—2111no—
133#407—2111no—
134C2×#95—2111no—
135#328—2111no—
136#328—2111no—
137#327—2111no—
138#327D21111yescharacteristic, complemented
139C2×#41—2111no—
140C23×C4—4221no—
141C23×C4—2111no—
142C2×M16—4221no—
143#123—4221no—
144C22⋊Q8—2111no—
145C22⋊Q8—2111no—
146#121—2111no—
147#121C2×D21111yescharacteristic
148#120—4221no—
149C4×D4—4221no—
150C4×D4—4221no—
151#117—2111no—
152C22⋊C8—4221no—
153C22⋊C8—2111no—
154C22⋊C8C2×D21111yescharacteristic
155#95—4221no—
156#95—2111no—
157#95C2×D21111yescharacteristic
158C24—2111no—
159#41—8442no—
160C2×Q8—2111no—
161C2×D4—21no—
162C2×D4—4221no—
163C2×D4—4221no—
164C2×D4C4×D21111yescharacteristic, complemented
165C22×C4—21no—
166C22×C4—8442no—
167C22×C4—8442no—
168C22×C4—8442no—
169C22×C4—4224no—
170C22×C4—4221no—
171C22×C4—4221no—
172C22×C4—2111no—
173C22×C4—2111no—
174C22×C4—2111no—
175C22×C4—2111no—
176C22×C4—2111no—
177C22×C4C22×D21111yescharacteristic
178M16—8442no—
179C2×C8—21no—
180C2×C8—4221no—
181C4⋊C4—4221no—
182C4⋊C4—4221no—
183C4⋊C4—2111no—
184C4⋊C4C4×D21111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—4221no—
187C22⋊C4—4221no—
188C22⋊C4—4221no—
189C22⋊C4—4221no—
190C42—4221no—
191C23—21no—
192C23—4222no—
193C23—4222no—
194C23—4221no—
195C23—4221no—
196C23—2111no—
197C23—2111no—
198C23D4×D21111yescharacteristic
199Q8—8441no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—8442no—
204D4—8442no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—8441no—
211C2×C4—8441no—
212C2×C4—8448no—
213C2×C4—8441no—
214C2×C4—8441no—
215C2×C4—8442no—
216C2×C4—8441no—
217C2×C4—4222no—
218C2×C4—4222no—
219C2×C4—4222no—
220C2×C4—4221no—
221C2×C4—4221no—
222C2×C4—2111no—
223C2×C4—2111no—
224C2×C4C2×C4×D21111yescharacteristic
225C2×C4D4×D21111yescharacteristic
226C8—41no—
227C8—8441no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—4224no—
234C22—4224no—
235C22—4221no—
236C22—4221no—
237C22—4221no—
238C22—4221no—
239C22—2112no—
240C22—2112no—
241C22C22⋊C4×D21111yescharacteristic
242C22#2027000111yescharacteristic
243C22#2027000611yescharacteristic
244C4—21no—
245C4—21no—
246C4—42no—
247C4—41no—
248C4—8442no—
249C4—8441no—
250C4—8441no—
251C4—4221no—
252C4—4221no—
253C2—21no—
254C2—21no—
255C2—81no—
256C2—4222no—
257C2—4222no—
258C2#99×D21111yescharacteristic
259C2#2721314411yescharacteristic
260C2#104×D21111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#327×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213656C2
cosocleC22×C4×C211C23
hypercenter#327D211
hypocenterC211#1504
LCSG ▷ C2×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. #27213108C2semidirect
3. #95×D211C2semidirect
4. #27212884C2semidirect
5. #27212882C2semidirect
6. #27212880C2semidirect
7. #327×C211C2semidirect
8. C2×#16809961C4semidirect
13. #20269957C4semidirect
15. #20269948C4semidirect
19. #20269935C4semidirect
29. C2×Dic422C2×C4semidirect
46. C4⋊Dic211C2×C4semidirect
54. C2×D4×C211C2×C4semidirect
58. C4⋊C4×C211C2×C4semidirect
130. C211#1504semidirect
138. #327D211semidirect
164. C2×D4C4×D211semidirect
184. C4⋊C4C4×D211semidirect

Maximal subgroups

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

subgroupindex
#272136562
#272131082
#95×D2112
#272128842
#272128822
#272128802
#327×C2112
#1504211