Finite Group Databasebeta

← #41011900

Subgroups — #41011900

Order 27008.

334 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 77 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#27213657C211yesmaximal, characteristic
2#27213110C211yesmaximal, characteristic, complemented
3#27212882C212yesmaximal, complemented
4#27212879C212yesmaximal, complemented
5C2×#95×C211C211yesmaximal, characteristic, complemented
6C2×#16809962C411yescharacteristic, complemented
7C2×#16809946—21no—
8#20269960—21no—
9#20269956C411yescharacteristic, complemented
10#20269956C2211yescharacteristic
11#20269951C411yescharacteristic, complemented
12#20269951C2211yescharacteristic
13#20269946—21no—
14C4×C22⋊D211—21no—
15C4×C22⋊D211—21no—
16#20269936C411yescharacteristic, complemented
17C22⋊(C211⋊C8)—21no—
18C22⋊(C211⋊C8)C2212yes—
19C23×C4×C211—21no—
20C23×C4×C211C2211yescharacteristic
21#95×C211—21no—
22#95×C211C2212yes—
23#16809962—42no—
24C2×Dic422C2×C411yescharacteristic, complemented
25C2×D844C2×C411yescharacteristic, complemented
26C2×C22⋊D211—21no—
27C2×C22⋊D211—21no—
28C2×C22⋊D211—21no—
29C2×C4×D211—21no—
30#16809946—42no—
31C2×C211⋊C8—22no—
32C4⋊Dic211C2×C411yescharacteristic, complemented
33C4⋊Dic211C2×C411yescharacteristic, complemented
34Dic211⋊C4—21no—
35Dic211⋊C4—21no—
36C22⋊Dic211—21no—
37C22⋊Dic211—21no—
38C22⋊Dic211—21no—
39(C2×D211)⋊C4—21no—
40(C2×D211)⋊C4—21no—
41C4×Dic211—21no—
42C24×C211D411yescharacteristic
43C22×C4×C211—24no—
44C22×C4×C211—22no—
45C22×C4×C211—42no—
46C22×C4×C211—42no—
47C22×C4×C211—42no—
48C22×C4×C211C2×C411yescharacteristic
49C22×C4×C211C2×C411yescharacteristic
50C22×C4×C211D411yescharacteristic
51C22×C4×C211D411yescharacteristic
52C22×C4×C211D411yescharacteristic
53C22×C4×C211C2311yescharacteristic
54C22×D211—21no—
55Dic422—41no—
56D844—41no—
57C22⋊D211—42no—
58C22⋊D211—42no—
59C22⋊D211—41no—
60C22⋊D211—41no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C2×Dic211—21no—
64C4×D211—42no—
65C211⋊C8—42no—
66C23×C211—22no—
67C23×C211—22no—
68C23×C211—21no—
69C23×C211—21no—
70C23×C211C22⋊C411yescharacteristic
71C23×C211C22⋊C411yescharacteristic
72C23×C211C2×D411yescharacteristic
73C2×C4×C211—22no—
74C2×C4×C211—22no—
75C2×C4×C211—24no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211—48no—
81C2×C4×C211—42no—
82C2×C4×C211—42no—
83C2×C4×C211C22⋊C411yescharacteristic
84C2×C4×C211C22⋊C411yescharacteristic
85C2×C4×C211C22×C411yescharacteristic
86C2×C4×C211C2×D411yescharacteristic
87C22×C211—24no—
88C22×C211—24no—
89C22×C211—21no—
90C22×C211—21no—
91C22×C211—21no—
92C22×C211—21no—
93C22×C211—21no—
94C22×C211—21no—
95C22×C211#9912yes—
96C22×C211#10412yes—
97C22×C211C2×C22⋊C411yescharacteristic
98C22×C211AGL(1,ℤ/8)11yescharacteristic
99C22×C211#13711yescharacteristic
100C2×D211—41no—
101C2×D211—41no—
102C2×D211—41no—
103C4×C211—21no—
104C4×C211—21no—
105C4×C211—21no—
106C4×C211—21no—
107C4×C211—44no—
108Dic211—41no—
109Dic211—41no—
110Dic211—41no—
111C2×C211—21no—
112C2×C211—22no—
113C2×C211—22no—
114C2×C211—21no—
115C2×C211C2×#9911yescharacteristic
116C2×C211#41711yescharacteristic
117C2×C211C2×#10411yescharacteristic
118D211—81no—
119C211#150411yescharacteristic, complemented, Sylow
120#1504—2111nomaximal, Sylow
121#525—2111no—
122#407—2111no—
123C2×#95D21111yescharacteristic, complemented
124#328—2112no—
125#327—2112no—
126C2×#41—2111no—
127C23×C4—21no—
128C23×C4C2×D21111yescharacteristic
129C2×M16—4221no—
130#123—4221no—
131C22⋊Q8—2111no—
132C22⋊Q8—2111no—
133#121—2111no—
134#121—2111no—
135#120—4221no—
136C4×D4—4221no—
137C4×D4—4221no—
138#117—2111no—
139C22⋊C8—4221no—
140C22⋊C8—2112no—
141#95—21no—
142#95C2×D21112yes—
143C24C22⋊D21111yescharacteristic
144#41—8442no—
145C2×Q8—2111no—
146C2×D4—4221no—
147C2×D4—4221no—
148C2×D4—4221no—
149C2×D4—2111no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—4221no—
156C22×C4C4×D21111yescharacteristic
157C22×C4C4×D21111yescharacteristic
158C22×C4C22⋊D21111yescharacteristic
159C22×C4D84411yescharacteristic
160C22×C4D84411yescharacteristic
161C22×C4C22×D21111yescharacteristic
162M16—8442no—
163C2×C8—4222no—
164C4⋊C4—4221no—
165C4⋊C4—4221no—
166C4⋊C4—2111no—
167C4⋊C4—2111no—
168C22⋊C4—4221no—
169C22⋊C4—4221no—
170C22⋊C4—4221no—
171C22⋊C4—4221no—
172C22⋊C4—4221no—
173C42—4221no—
174C23—22no—
175C23—22no—
176C23—21no—
177C23—21no—
178C23—4221no—
179C23(C2×D211)⋊C411yescharacteristic
180C23(C2×D211)⋊C411yescharacteristic
181C23C2×C22⋊D21111yescharacteristic
182Q8—8441no—
183D4—8441no—
184D4—8442no—
185D4—8442no—
186D4—8441no—
187D4—8441no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—48no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—8442no—
199C2×C4—4221no—
200C2×C4—4221no—
201C2×C4—4221no—
202C2×C4(C2×D211)⋊C411yescharacteristic
203C2×C4(C2×D211)⋊C411yescharacteristic
204C2×C4C2×C4×D21111yescharacteristic
205C2×C4C2×D84411yescharacteristic
206C8—8442no—
207C22—24no—
208C22—24no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—8441no—
216C22—8441no—
217C22—8441no—
218C22#2026989112yes—
219C22#2026990512yes—
220C22C2×(C2×D211)⋊C411yescharacteristic
221C22#2027000311yescharacteristic
222C22#2027000811yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—8441no—
229C4—8441no—
230C4—8441no—
231C2—21no—
232C2—22no—
233C2—22no—
234C2—21no—
235C2—16881no—
236C2C2×#2026989111yescharacteristic
237C2#2721314511yescharacteristic
238C2C2×#2026990511yescharacteristic
subgroupG/H
centerC22C2×(C2×D211)⋊C4
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#95×C211C2
socleC22×C211C2×C22⋊C4
prime radical#27213657C2
cosocleC22×C4×C211C23
hypercenterC2×#95D211
hypocenterC211#1504
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
2. #27213110C2semidirect
3. #27212882C2semidirect
4. #27212879C2semidirect
5. C2×#95×C211C2semidirect
6. C2×#16809962C4semidirect
9. #20269956C4semidirect
11. #20269951C4semidirect
16. #20269936C4semidirect
24. C2×Dic422C2×C4semidirect
25. C2×D844C2×C4semidirect
32. C4⋊Dic211C2×C4semidirect
33. C4⋊Dic211C2×C4semidirect
119. C211#1504semidirect
123. C2×#95D211semidirect

Maximal subgroups

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

subgroupindex
#272136572
#272131102
#272128822
#272128792
C2×#95×C2112
#1504211