Finite Group Databasebeta

← #41014024

Subgroups — #41014024

Order 27008.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C2×#20269899C211yesmaximal, characteristic
2C2×#20269897C211yesmaximal, characteristic
3#27213047C211yesmaximal, characteristic
4C4⋊C4×Dic211C211yesmaximal, characteristic, complemented
5#27212908C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×#106×C211C211yesmaximal, characteristic, complemented
8C22×D844C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10C2×C4⋊Dic211C2211yescharacteristic
11C2×(C2×D211)⋊C4—21no—
12C2×(C2×D211)⋊C4—21no—
13C2×C4×Dic211—21no—
14C2×C4×Dic211C2211yescharacteristic
15#20269899C414yescomplemented
16#20269897—22no—
17C211⋊#95—21no—
18C22×C8×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21#106×C211—22no—
22C23×D211—21no—
23C2×D844—21no—
24C2×D844—21no—
25C2×D844—21no—
26C2×D844—21no—
27C2×D844—21no—
28C2×D844C2×C412yes—
29C22×Dic211—21no—
30C22×Dic211D411yescharacteristic
31C22×Dic211D411yescharacteristic
32C2×C211⋊C8—21no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211C2×C412yes—
37(C2×D211)⋊C4—42no—
38(C2×D211)⋊C4—42no—
39C4×Dic211—21no—
40C4×Dic211—21no—
41C4×Dic211—42no—
42C22×C4×C211—21no—
43C22×C4×C211—21no—
44C22×C4×C211C2311yescharacteristic
45C2×C8×C211—21no—
46C2×C8×C211—21no—
47C2×C8×C211C2×C412yes—
48C4⋊C4×C211—21no—
49C4⋊C4×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211D412yes—
52C22×D211—22no—
53C22×D211—41no—
54C22×D211—41no—
55C22×D211—41no—
56C22×D211—41no—
57C22×D211—41no—
58C22×D211—41no—
59D844—22no—
60D844—22no—
61D844—41no—
62D844—41no—
63C2×Dic211—21no—
64C2×Dic211—21no—
65C2×Dic211—22no—
66C2×Dic211—21no—
67C2×Dic211—21no—
68C2×Dic211—21no—
69C2×Dic211—41no—
70C2×Dic211—41no—
71C2×Dic211SD1612yes—
72C211⋊C8—41no—
73C23×C211C2×D411yescharacteristic
74C2×C4×C211—22no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211—41no—
79C2×C4×C211—41no—
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C2×C4×C211#4111yescharacteristic
83C2×C4×C211#4111yescharacteristic
84C2×C4×C211#4111yescharacteristic
85C2×C4×C211#4111yescharacteristic
86C8×C211—22no—
87C22×C211C4×D411yescharacteristic
88C22×C211#12111yescharacteristic
89C22×C211#12311yescharacteristic
90C22×C211C2×SD1611yescharacteristic
91C22×C211#13511yescharacteristic
92C22×C211AGL(1,ℤ/8)11yescharacteristic
93C22×C211AGL(1,ℤ/8)11yescharacteristic
94C2×D211—42no—
95C2×D211—42no—
96C2×D211—42no—
97C2×D211—81no—
98C2×D211—81no—
99C2×D211—81no—
100C2×D211—81no—
101C4×C211—42no—
102C4×C211—81no—
103C4×C211#11711yescharacteristic
104C4×C211C4×D411yescharacteristic
105C4×C211#12411yescharacteristic
106C4×C211#12611yescharacteristic
107Dic211—22no—
108Dic211—42no—
109Dic211—41no—
110C2×C211#38811yescharacteristic
111C2×C211C4×SD1611yescharacteristic
112C2×C211#44211yescharacteristic
113C2×C211#46011yescharacteristic
114C2×C211#46311yescharacteristic
115C2×C211#48111yescharacteristic
116C2×C211#48211yescharacteristic
117D211—82no—
118C211#192611yescharacteristic, complemented, Sylow
119#1926—2111nomaximal, Sylow
120C2×#106D21111yescharacteristic, complemented
121C2×#102—2111no—
122C2×#102—2111no—
123#390—2111no—
124C4×C4⋊C4—2111no—
125#340—2111no—
126#336—2111no—
127C22×D4—2111no—
128C22×C8—2111no—
129C22×C8C2×D21111yescharacteristic
130C2×C4⋊C4—2111no—
131C2×C4⋊C4C2×D21111yescharacteristic
132C2×C4⋊C4C2×D21111yescharacteristic
133C2×C22⋊C4—4221no—
134C2×C22⋊C4—4221no—
135C2×C42—4221no—
136C2×C42—2111no—
137#106—22no—
138#102—4222no—
139#102—2114no—
140#95—4221no—
141C24—4221no—
142C2×D4—4221no—
143C2×D4—4221no—
144C2×D4—4221no—
145C2×D4—4221no—
146C2×D4—4221no—
147C2×D4—2112no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—4221no—
151C22×C4—2111no—
152C22×C4—2111no—
153C22×C4C22×D21111yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—4221no—
157C2×C8—4221no—
158C2×C8—4221no—
159C2×C8C4×D21112yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—4221no—
164C4⋊C4—2112no—
165C4⋊C4C22⋊D21112yes—
166C22⋊C4—8442no—
167C22⋊C4—8442no—
168C42—8442no—
169C42—4221no—
170C42—4221no—
171C23—8441no—
172C23—8441no—
173C23—8441no—
174C23—8441no—
175C23—8441no—
176C23—8441no—
177C23—4222no—
178C23D4×D21111yescharacteristic
179D4—8441no—
180D4—8441no—
181D4—4222no—
182D4—4222no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—8441no—
190C2×C4—8441no—
191C2×C4—4222no—
192C2×C4—4221no—
193C2×C4—4221no—
194C2×C4—4221no—
195C2×C4—4221no—
196C2×C4—4221no—
197C2×C4—2112no—
198C2×C4C2×C4×D21111yescharacteristic
199C2×C4C2×C22⋊D21111yescharacteristic
200C2×C4#1680996111yescharacteristic
201C2×C4#1680996211yescharacteristic
202C2×C4#1680996311yescharacteristic
203C2×C4#1680996311yescharacteristic
204C8—22no—
205C8—8441no—
206C22—8442no—
207C22—8442no—
208C22—8442no—
209C22—16881no—
210C22—16881no—
211C22—16881no—
212C22—16881no—
213C22#2026993911yescharacteristic
214C22#2026994911yescharacteristic
215C22#2026996211yescharacteristic
216C22SD16×D21111yescharacteristic
217C22#2026999811yescharacteristic
218C22#2027000411yescharacteristic
219C22#2027000411yescharacteristic
220C4—42no—
221C4—81no—
222C4—8441no—
223C4—8442no—
224C4—4222no—
225C4#2026993511yescharacteristic
226C4C4×C22⋊D21111yescharacteristic
227C4#2026996611yescharacteristic
228C4#2026997211yescharacteristic
229C2—16882no—
230C2#2721303611yescharacteristic
231C2#2721323311yescharacteristic
232C2#2721325711yescharacteristic
233C2#2721334411yescharacteristic
234C2#2721336311yescharacteristic
235C2#2721346611yescharacteristic
236C2#2721347211yescharacteristic
subgroupG/H
centerC23D4×D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC2×#106×C211C2
socleC23×C211C2×D4
prime radicalC22×D844C22
cosocleC22×C4×C211C23
hypercenterC2×#106D211
hypocenterC211#1926
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
4. C4⋊C4×Dic211C2semidirect
5. #27212908C2semidirect
6. #27212900C2semidirect
7. C2×#106×C211C2semidirect
15. #20269899C4semidirect
118. C211#1926semidirect
120. C2×#106D211semidirect

Maximal subgroups

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

subgroupindex
C2×#202698992
C2×#202698972
#272130472
C4⋊C4×Dic2112
#272129082
#272129002
C2×#106×C2112
#1926211