Finite Group Databasebeta

← #41013471

Subgroups — #41013471

Order 27008.

504 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 81 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×C22⋊C4×D211C211yesmaximal, characteristic
2#27213001C212yesmaximal
3#27212913C211yesmaximal, characteristic, complemented
4#27212912C212yesmaximal, complemented
5#342×C211C211yesmaximal, characteristic, complemented
6C24×D211C2211yescharacteristic
7C22×C4×D211C412yescomplemented
8C2×C22⋊Dic211C2212yes—
9C2×C22⋊Dic211C2211yescharacteristic
10C2×(C2×D211)⋊C4—24no—
11C2×(C2×D211)⋊C4C412yescomplemented
12C22⋊C4×D211—24no—
13C2×C22⋊C4×C211C2212yes—
14C2×C22⋊C4×C211C2211yescharacteristic
15C23×D211—21no—
16C23×D211—22no—
17C23×D211—21no—
18C23×D211—44no—
19C23×D211C2×C411yescharacteristic
20C23×D211C2×C411yescharacteristic
21C23×D211D414yes—
22C22×Dic211—22no—
23C22×Dic211C2×C412yes—
24C2×C4×D211—22no—
25C2×C4×D211—22no—
26C2×C4×D211—44no—
27C22⋊Dic211—22no—
28C22⋊Dic211—22no—
29C22⋊Dic211—22no—
30(C2×D211)⋊C4—24no—
31(C2×D211)⋊C4—48no—
32C24×C211C2311yescharacteristic
33C22×C4×C211—22no—
34C22×C4×C211C2×C412yes—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C22×D211—21no—
39C22×D211—21no—
40C22×D211—24no—
41C22×D211—21no—
42C22×D211—21no—
43C22×D211—24no—
44C22×D211—24no—
45C22×D211—28no—
46C22×D211—42no—
47C22×D211—44no—
48C22×D211—44no—
49C22×D211—48no—
50C22×D211—44no—
51C22×D211C22⋊C412yes—
52C22×D211C22⋊C412yes—
53C2×Dic211—22no—
54C2×Dic211—42no—
55C2×Dic211—44no—
56C2×Dic211—42no—
57C4×D211—44no—
58C4×D211—44no—
59C23×C211—42no—
60C23×C211C22×C411yescharacteristic
61C23×C211C2×D412yes—
62C23×C211#4111yescharacteristic
63C23×C211#4112yes—
64C23×C211#4111yescharacteristic
65C2×C4×C211—22no—
66C2×C4×C211—42no—
67C2×C4×C211—44no—
68C2×C4×C211—42no—
69C22×C211—22no—
70C22×C211—24no—
71C22×C211—41no—
72C22×C211—42no—
73C22×C211—41no—
74C22×C211C2×C22⋊C411yescharacteristic
75C22×C211#11711yescharacteristic
76C22×C211#11711yescharacteristic
77C22×C211#12312yes—
78C22×C211#12312yes—
79C2×D211—24no—
80C2×D211—24no—
81C2×D211—22no—
82C2×D211—22no—
83C2×D211—22no—
84C2×D211—24no—
85C2×D211—48no—
86C2×D211—48no—
87C2×D211—42no—
88C2×D211—42no—
89C2×D211#9914yes—
90C4×C211—42no—
91C4×C211—82no—
92Dic211—42no—
93Dic211—82no—
94C2×C211—22no—
95C2×C211—22no—
96C2×C211—21no—
97C2×C211—21no—
98C2×C211#38111yescharacteristic
99C2×C211C2×#9912yes—
100D211—24no—
101D211—42no—
102C211#181911yescharacteristic, complemented, Sylow
103#1819—2111nomaximal, Sylow
104C22×C22⋊C4—2111no—
105#379—2112no—
106#342—2112no—
107#342—2111no—
108#342D21111yescharacteristic, complemented
109C25—2111no—
110C23×C4—2112no—
111C2×C22⋊C4—4224no—
112C2×C22⋊C4—4224no—
113C2×C22⋊C4—2112no—
114C2×C22⋊C4—2111no—
115C2×C22⋊C4—2112no—
116C2×C22⋊C4C2×D21112yes—
117C2×C22⋊C4C2×D21111yescharacteristic
118C24—8444no—
119C24—4221no—
120C24—4222no—
121C24—4221no—
122C24—2111no—
123C24—2114no—
124C24—2111no—
125C24C22×D21111yescharacteristic
126C22×C4—22no—
127C22×C4—8444no—
128C22×C4—4222no—
129C22×C4—4222no—
130C22×C4—4222no—
131C22×C4—2112no—
132C22×C4C4×D21112yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—8448no—
137C22⋊C4—4224no—
138C22⋊C4—4222no—
139C22⋊C4—4222no—
140C22⋊C4—4222no—
141C23—42no—
142C23—8442no—
143C23—8444no—
144C23—8444no—
145C23—8448no—
146C23—8444no—
147C23—4228no—
148C23—4224no—
149C23—4221no—
150C23—4221no—
151C23—4224no—
152C23—4224no—
153C23—4221no—
154C23—4221no—
155C23—2112no—
156C23—2112no—
157C23C2×C4×D21111yescharacteristic
158C23D4×D21112yes—
159C23#1680996111yescharacteristic
160C23#1680996212yes—
161C23#1680996311yescharacteristic
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—42no—
166C2×C4—8444no—
167C2×C4—8444no—
168C2×C4—8442no—
169C2×C4—8444no—
170C2×C4—8442no—
171C2×C4—4222no—
172C22—22no—
173C22—24no—
174C22—41no—
175C22—42no—
176C22—41no—
177C22—8442no—
178C22—8442no—
179C22—8448no—
180C22—8448no—
181C22—4224no—
182C22—4224no—
183C22—4224no—
184C22—4222no—
185C22—4222no—
186C22—4222no—
187C22—2114no—
188C22C22⋊C4×D21111yescharacteristic
189C22#2026993411yescharacteristic
190C22#2026993511yescharacteristic
191C22#2026995912yes—
192C22#2026996212yes—
193C4—42no—
194C4—82no—
195C4—8442no—
196C4—16882no—
197C2—22no—
198C2—22no—
199C2—21no—
200C2—21no—
201C2—8442no—
202C2—4224no—
203C2#2721300811yescharacteristic
204C2#99×D21112yes—
subgroupG/H
centerC22C22⋊C4×D211
commutatorC23×C211C22×C4
FrattiniC24C22×D211
Fitting#342×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC24×D211C22
cosocleC24×C211C23
hypercenter#342D211
hypocenterC211#1819
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #27212913C2semidirect
4. #27212912C2semidirect
5. #342×C211C2semidirect
7. C22×C4×D211C4semidirect
11. C2×(C2×D211)⋊C4C4semidirect
102. C211#1819semidirect
108. #342D211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊C4×D2112
#272130012
#272129132
#272129122
#342×C2112
#1819211