Finite Group Databasebeta

← #72220323

Subgroups — #72220323

Order 2592.

2112 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 2112 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C22×C3⋊D32C211yesmaximal, characteristic
2C2×#10620198C212yesmaximal, complemented
3C22⋊#66472C212yesmaximal, complemented
4#13049410C212yesmaximal, complemented
5C22×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic
6C22×C32×C3⋊D3C412yescomplemented
7C2×C3⋊D32—24no—
8C2×C3⋊D32C412yescomplemented
9C2×C3⋊D32C2211yescharacteristic
10C2×C3⋊D32C2211yescharacteristic
11C2×#66472C2214yescomplemented
12#10620198—24no—
13C22×D3×C3⋊D3—64no—
14C22×C34C2×C411yescharacteristic, complemented
15C2×C3⋊(C3⋊(C3⋊D3))—22no—
16C2×C3⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic, complemented
17C2×C3⋊(C3⋊(C3⋊D3))C2311yescharacteristic
18C2×C32×C3⋊D3—24no—
19C2×C32×C3⋊D3C2×C412yes—
20C2×C32×C3⋊D3C2×C412yescomplemented
21C3⋊D32—22no—
22C3⋊D32—24no—
23C3⋊D32D414yes—
24#66472—24no—
25C2×C22⋊#157—92nomaximal
26C22×C3⋊(C3⋊D3)—64no—
27C22×C3⋊(C3⋊D3)—128no—
28C22×C3×C3⋊D3—24no—
29C22×C32×D3—64no—
30C2×D3×C3⋊D3—64no—
31C2×D3×C3⋊D3—64no—
32C2×D3×C3⋊D3—64no—
33C2×D3×C3⋊D3—64no—
34C2×D3×C3⋊D3—632no—
35C2×C34—21no—
36C2×C34C22×C411yescharacteristic
37C3⋊(C3⋊(C3⋊D3))—21no—
38C3⋊(C3⋊(C3⋊D3))C2×D412yes—
39C32×C3⋊D3—22no—
40C32×C3⋊D3C22⋊C414yescomplemented
41C23×C3⋊D3—92no—
42C22×D32—188no—
43C22×#157—94no—
44C22⋊#157—94no—
45C22⋊#157—94no—
46C22⋊#157—188no—
47C22×C33—24no—
48C22×C33—48no—
49C2×C3⋊(C3⋊D3)—64no—
50C2×C3⋊(C3⋊D3)—64no—
51C2×C3⋊(C3⋊D3)—616no—
52C2×C3⋊(C3⋊D3)—128no—
53C2×C3⋊(C3⋊D3)—128no—
54C2×C3⋊(C3⋊D3)—1232no—
55C2×C3×C3⋊D3—216no—
56C2×C3×C3⋊D3—24no—
57C2×C3×C3⋊D3—24no—
58C2×C32×D3—64no—
59C2×C32×D3—64no—
60C2×C32×D3—616no—
61D3×C3⋊D3—616no—
62D3×C3⋊D3—616no—
63D3×C3⋊D3—616no—
64D3×C3⋊D3—616no—
65C34C2×C22⋊C411yescharacteristic, complemented, Sylow
66C23×C32—92no—
67C22×C3⋊D3—92no—
68C22×C3⋊D3—92no—
69C22×C3⋊D3—92no—
70C22×C3⋊D3—92no—
71C22×C3⋊D3—92no—
72C22×C3⋊D3—188no—
73C22×C3⋊D3—188no—
74C22×C3⋊D3—188no—
75C22×C3⋊D3—3616no—
76C22×C3⋊D3—368no—
77C22×C3⋊D3#15712yescomplemented
78C22×C3×D3—616no—
79C2×D32—1816no—
80C2×D32—188no—
81C2×D32—188no—
82C2×D32—1864no—
83C2×#157—98no—
84C2×#157—1816no—
85C2×#157—188no—
86C2×C33—28no—
87C2×C33—24no—
88C2×C33—48no—
89C2×C33—416no—
90C3⋊(C3⋊D3)—68no—
91C3⋊(C3⋊D3)—68no—
92C3⋊(C3⋊D3)—1216no—
93C3⋊(C3⋊D3)—1216no—
94C3×C3⋊D3—28no—
95C3×C3⋊D3—28no—
96C32×D3—68no—
97C32×D3—68no—
98C23×D3—544no—
99C22×C32—28no—
100C22×C32—28no—
101C22×C32—416no—
102C22×C32—48no—
103C22×C32—92no—
104C22×C32—92no—
105C22×C32—184no—
106C22×C32C2×#15712yescomplemented
107C2×C3⋊D3—24no—
108C2×C3⋊D3—92no—
109C2×C3⋊D3—98no—
110C2×C3⋊D3—92no—
111C2×C3⋊D3—184no—
112C2×C3⋊D3—184no—
113C2×C3⋊D3—184no—
114C2×C3⋊D3—184no—
115C2×C3⋊D3—188no—
116C2×C3⋊D3—188no—
117C2×C3⋊D3—1832no—
118C2×C3⋊D3—188no—
119C2×C3⋊D3—188no—
120C2×C3⋊D3—3616no—
121C2×C3⋊D3—3616no—
122C2×C3⋊D3—3664no—
123C2×C3⋊D3—3616no—
124C2×C3⋊D3—368no—
125C2×C3⋊D3—368no—
126C2×C3⋊D3—3632no—
127C2×C3⋊D3C2×#15712yes—
128C2×C3⋊D3C2×#15712yescomplemented
129C2×C3×D3—616no—
130C2×C3×D3—616no—
131C2×C3×D3—664no—
132D32—1832no—
133D32—1864no—
134D32—1832no—
135#157—188no—
136#157—3616no—
137C2×C22⋊C4—811noSylow
138C33—24no—
139C33—48no—
140C23×C3—184no—
141C22×D3—64no—
142C22×D3—544no—
143C22×D3—544no—
144C22×D3—544no—
145C22×D3—544no—
146C22×D3—544no—
147C22×D3—5432no—
148C22×D3—1088no—
149C2×C32—22no—
150C2×C32—216no—
151C2×C32—28no—
152C2×C32—28no—
153C2×C32—432no—
154C2×C32—416no—
155C2×C32—48no—
156C2×C32—416no—
157C2×C32—48no—
158C2×C32—94no—
159C2×C32—182no—
160C2×C32C22×#15712yes—
161C3⋊D3—22no—
162C3⋊D3—94no—
163C3⋊D3—182no—
164C3⋊D3—1816no—
165C3⋊D3—1816no—
166C3⋊D3—1816no—
167C3⋊D3—3632no—
168C3⋊D3—3632no—
169C3⋊D3—368no—
170C3⋊D3—3616no—
171C3⋊D3—3616no—
172C3⋊D3C22⋊#15714yescomplemented
173C3×D3—632no—
174C3×D3—632no—
175C24—811no—
176C22×C4—812no—
177C22⋊C4—812no—
178C22⋊C4—812no—
179C22×C3—24no—
180C22×C3—48no—
181C22×C3—184no—
182C22×C3—184no—
183C22×C3—1816no—
184C2×D3—64no—
185C2×D3—64no—
186C2×D3—616no—
187C2×D3—5416no—
188C2×D3—544no—
189C2×D3—544no—
190C2×D3—5416no—
191C2×D3—5416no—
192C2×D3—5416no—
193C2×D3—5416no—
194C2×D3—1088no—
195C2×D3—1088no—
196C2×D3—10832no—
197C32—28no—
198C32—28no—
199C32—416no—
200C32—48no—
201C32C2×C22⋊#15712yescomplemented
202C23—92no—
203C23—811no—
204C23—811no—
205C23—812no—
206C23—811no—
207C23—1624no—
208C2×C4—814no—
209C2×C4—1624no—
210C2×C3—28no—
211C2×C3—24no—
212C2×C3—416no—
213C2×C3—48no—
214C2×C3—188no—
215C2×C3—188no—
216D3—68no—
217D3—68no—
218D3—548no—
219D3—548no—
220D3—10816no—
221D3—10816no—
222C22—92no—
223C22—92no—
224C22—184no—
225C22—814no—
226C22—811no—
227C22—811no—
228C22—1622no—
229C22—1624no—
230C22—1622no—
231C22#1062019811yescharacteristic, complemented
232C4—1624no—
233C3—24no—
234C3—48no—
235C2—21no—
236C2—94no—
237C2—182no—
238C2—812no—
239C2—1621no—
240C2C2×#1062019811yescharacteristic
subgroupG/H
center, FrattiniC2C2×#10620198
commutator, socleC2×C34C22×C4
FittingC22×C34C2×C4
prime radicalC22×C3⋊D32C2
cosocleC2×C3⋊(C3⋊(C3⋊D3))C23
hypercenterC22#10620198
hypocenterC34C2×C22⋊C4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×#10620198C2semidirect
3. C22⋊#66472C2semidirect
4. #13049410C2semidirect
6. C22×C32×C3⋊D3C4semidirect
8. C2×C3⋊D32C4semidirect
11. C2×#66472C22semidirect
14. C22×C34C2×C4semidirect
16. C2×C3⋊(C3⋊(C3⋊D3))C2×C4semidirect
20. C2×C32×C3⋊D3C2×C4semidirect
40. C32×C3⋊D3C22⋊C4semidirect
65. C34C2×C22⋊C4semidirect
77. C22×C3⋊D3#157semidirect
106. C22×C32C2×#157semidirect
128. C2×C3⋊D3C2×#157semidirect
172. C3⋊D3C22⋊#157semidirect
201. C32C2×C22⋊#157semidirect
231. C22#10620198semidirect

Maximal subgroups

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

subgroupindex
C22×C3⋊D322
C2×#106201982
C22⋊#664722
#130494102
C2×C22⋊#1579