Finite Group Databasebeta

← #10593969

Subgroups — #10593969

Order 576.

658 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 152 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×#3799C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64084C211yesmaximal, characteristic, complemented
4C3×#875C211yesmaximal, characteristic, complemented
5C3×C42×D3C211yesmaximal, characteristic, complemented
6#63911C211yesmaximal, characteristic, complemented
7#63906C214yesmaximal, complemented
8#63880C214yesmaximal, complemented
9#63866C211yesmaximal, characteristic, complemented
10#5668—31nomaximal
11#5477—31nomaximal
12C2×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#634C2211yescharacteristic
14C2×C3×D12C2211yescharacteristic
15C2×C3×C4×D3C2212yes—
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×#626C2212yes—
19C2×#626C2214yes—
20C2×#625C2212yes—
21C2×#624C2212yes—
22C2×D3×Dic3C2212yes—
23#3799—24no—
24#3795—24no—
25C32×C42C2211yescharacteristic
26#3752C2214yes—
27C3×(C2×D3)⋊C4C2214yes—
28C3×C4×Dic3C2211yescharacteristic
29C3×C4×Dic3C2212yes—
30Dic3⋊Dic3C2214yes—
31C2×#239—31no—
32C2×#237—31no—
33C2×#235—31no—
34C2×#235—31no—
35C3×#124—31no—
36C2×C3×C42—31no—
37#944—31no—
38#940—31no—
39#930—34no—
40#884—34no—
41#67⋊C4—34no—
42#875—31no—
43#875—31no—
44#875—61no—
45#875D311yescharacteristic, complemented
46#874—31no—
47C4×D12—34no—
48C42×D3D311yescharacteristic, complemented
49#869—31no—
50C22×C3⋊D3C2312yes—
51C22×C3×D3C2312yes—
52C22×C3×D3C2311yescharacteristic
53C2×C32×C4C2312yes—
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3C2312yes—
56C4⋊(C3⋊D3)—22no—
57#634—22no—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2311yescharacteristic
60C2×C3×Dic3C2312yes—
61C3×D12—22no—
62C3×C4×D3—24no—
63C3×C4×D3D414yescomplemented
64C3×Dic6—22no—
65#626—24no—
66#626—28no—
67#625—24no—
68#624—24no—
69D3×Dic3—24no—
70#532—91noSylow
71C2×C3×Q8—31no—
72C2×C3×D4—31no—
73C22×C3×C4—31no—
74C22×C3×C4—32no—
75C2×#67—32no—
76C2×#67—32no—
77C2×#67—34no—
78C2×#67—32no—
79C22×Dic3—32no—
80#239—64no—
81#237—64no—
82#235—64no—
83#235—64no—
84C2×D12—34no—
85C2×D12—32no—
86C2×D12—31no—
87C2×D12—31no—
88C2×D12—61no—
89C2×D12C2×D311yescharacteristic
90C2×C4×D3—32no—
91C2×C4×D3—32no—
92C2×C4×D3—32no—
93C2×C4×D3C2×D312yes—
94C2×C4×D3C2×D311yescharacteristic
95C2×Dic6—31no—
96C2×Dic6—31no—
97C2×Dic6—61no—
98C2×Dic6C2×D311yescharacteristic
99C3×C22⋊C4—34no—
100C3×C42—21no—
101C3×C42—31no—
102C3×C42—32no—
103C3×C42C2×D311yescharacteristic
104C3×C42C2×D311yescharacteristic
105(C2×D3)⋊C4—34no—
106(C2×D3)⋊C4—34no—
107(C2×D3)⋊C4—64no—
108(C2×D3)⋊C4C2×D314yes—
109C4⋊Dic3—34no—
110Dic3⋊C4—34no—
111C4×Dic3C2×D312yes—
112C4×Dic3C2×D311yescharacteristic
113C22×C32C2411yescharacteristic
114C2×C3⋊D3—22no—
115C2×C3⋊D3—24no—
116C2×C3×D3—22no—
117C2×C3×D3—22no—
118C2×C3×D3—24no—
119C2×C3×D3C2×D412yes—
120C32×C4—22no—
121C32×C4C2×D412yes—
122C3⋊Dic3—22no—
123C3×Dic3—22no—
124C3×Dic3C2×D412yes—
125C3×Dic3#4114yescomplemented
126C2×#41—91no—
127C2×#41—91no—
128C4⋊Q8—91no—
129#127—91no—
130#124—31no—
131#124—91no—
132#121—94no—
133C4×D4—94no—
134C2×C42—31no—
135C23×C3—31no—
136C23×C3—32no—
137C22×D3—32no—
138C22×D3—32no—
139C22×D3—62no—
140C22×D3C22×D312yes—
141C22×D3C22×D311yescharacteristic
142C3×Q8—62no—
143C3×D4—62no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4—32no—
147C2×C3×C4—31no—
148C2×C3×C4—34no—
149C2×C3×C4—32no—
150C2×C3×C4—64no—
151C2×C3×C4C22×D312yes—
152C2×C3×C4C22×D312yes—
153C2×C3×C4C22×D311yescharacteristic
154C2×C3×C4C22×D311yescharacteristic
155#67—64no—
156#67—64no—
157#67—64no—
158#67—68no—
159C2×Dic3—32no—
160C2×Dic3—32no—
161C2×Dic3—62no—
162C2×Dic3—64no—
163C2×Dic3C22×D312yes—
164C2×Dic3C22×D311yescharacteristic
165C2×Dic3C22×D312yes—
166D12—22no—
167D12—68no—
168D12—64no—
169D12—62no—
170D12—62no—
171D12—122no—
172C4×D3—24no—
173C4×D3—64no—
174C4×D3—64no—
175C4×D3—64no—
176C4×D3D1214yescomplemented
177Dic6—22no—
178Dic6—62no—
179Dic6—62no—
180Dic6—122no—
181C2×C32C22×D411yescharacteristic
182C2×C32C2×#4112yes—
183C3⋊D3—42no—
184C3×D3—22no—
185C3×D3—42no—
186#41—184no—
187#41—184no—
188C2×Q8—31no—
189C2×Q8—91no—
190C2×D4—31no—
191C2×D4—94no—
192C2×D4—92no—
193C2×D4—92no—
194C2×D4—91no—
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—92no—
198C22×C4—92no—
199C4⋊C4—94no—
200C22⋊C4—34no—
201C22⋊C4—94no—
202C42—32no—
203C42—31no—
204C42D3211yescharacteristic
205C22×C3—21no—
206C22×C3—32no—
207C22×C3—62no—
208C22×C3—62no—
209C22×C3—64no—
210C22×C3C23×D311yescharacteristic
211C22×C3C23×D311yescharacteristic
212C2×D3—22no—
213C2×D3—22no—
214C2×D3—24no—
215C2×D3—62no—
216C2×D3—64no—
217C2×D3—62no—
218C2×D3—64no—
219C2×D3—122no—
220C2×D3—124no—
221C2×D3C2×D1212yes—
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—32no—
226C3×C4—34no—
227C3×C4—42no—
228C3×C4—62no—
229C3×C4C2×D1212yes—
230C3×C4D3×D412yes—
231Dic3—22no—
232Dic3—62no—
233Dic3—62no—
234Dic3—122no—
235Dic3C2×D1212yes—
236Dic3#23514yescomplemented
237C32#53211yescharacteristic, complemented, Sylow
238C23—31no—
239C23—32no—
240C23—92no—
241Q8—62no—
242Q8—182no—
243D4—62no—
244D4—188no—
245D4—184no—
246D4—184no—
247D4—182no—
248C2×C4—32no—
249C2×C4—31no—
250C2×C4—34no—
251C2×C4—32no—
252C2×C4—64no—
253C2×C4—92no—
254C2×C4—184no—
255C2×C4—184no—
256C2×C4C2×D3212yes—
257C2×C4C2×D3211yescharacteristic
258C2×C3—21no—
259C2×C3—22no—
260C2×C3—62no—
261C2×C3—122no—
262C2×C3C22×D1211yescharacteristic
263C2×C3C2×#23512yes—
264C2×C3C2×D3×D411yescharacteristic
265C2×C3D3×#4112yes—
266D3—22no—
267D3—42no—
268D3—122no—
269D3—122no—
270D3—242no—
271C22—32no—
272C22—62no—
273C22—62no—
274C22—64no—
275C22—182no—
276C22—184no—
277C22C22×D3211yescharacteristic
278C4—22no—
279C4—32no—
280C4—34no—
281C4—62no—
282C4—182no—
283C4D3×D1212yes—
284C3—21no—
285C3#547711yescharacteristic, complemented
286C3#566811yescharacteristic, complemented
287C2—62no—
288C2—122no—
289C2—362no—
290C2C2×D3×D1211yescharacteristic
291C2D3×#23512yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
hypercenterC42D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#3799C2semidirect
2. C2×#3795C2semidirect
3. #64084C2semidirect
4. C3×#875C2semidirect
5. C3×C42×D3C2semidirect
6. #63911C2semidirect
7. #63906C2semidirect
8. #63880C2semidirect
9. #63866C2semidirect
45. #875D3semidirect
48. C42×D3D3semidirect
63. C3×C4×D3D4semidirect
125. C3×Dic3#41semidirect
176. C4×D3D12semidirect
236. Dic3#235semidirect
237. C32#532semidirect
285. C3#5477semidirect
286. C3#5668semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
C2×#37952
#640842
C3×#8752
C3×C42×D32
#639112
#639062
#638802
#638662
#56683
#54773