Finite Group Databasebeta

← #10595152

Subgroups — #10595152

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×C3×#237C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64145C211yesmaximal, characteristic, complemented
4C3×#940C211yesmaximal, characteristic, complemented
5Dic3×#67C214yesmaximal, complemented
6#63970C214yesmaximal, complemented
7C4×D3×Dic3C211yesmaximal, characteristic, complemented
8#63866C211yesmaximal, characteristic, complemented
9#63863C211yesmaximal, characteristic, complemented
10#5832—31nomaximal
11#5716—31nomaximal
12C2×C32×D4C2211yescharacteristic
13C2×#634C2211yescharacteristic
14C2×C3×#67C2214yes—
15C2×C3×#67C2212yes—
16C22×C3×Dic3C2212yes—
17C3×#237—24no—
18C2×C3×D12C2211yescharacteristic
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2211yescharacteristic
21C2×#625C2212yes—
22C2×D3×Dic3C2212yes—
23C2×D3×Dic3C2212yes—
24#3795—24no—
25#3757C2214yes—
26C4×C3⋊Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28Dic3⋊Dic3C2214yes—
29(C2×D3)⋊Dic3C2214yes—
30Dic32C2212yes—
31C2×C3×#41—31no—
32C2×#237—31no—
33C2×#237D311yescharacteristic, complemented
34C2×#235—31no—
35C3×#127—31no—
36#947—31no—
37#944—31no—
38#940D311yescharacteristic, complemented
39#939—34no—
40#938—34no—
41#936—31no—
42#936—31no—
43#936—31no—
44#936—61no—
45D4×Dic3—34no—
46C4×#67—34no—
47C2×C4×Dic3—31no—
48C42×D3—31no—
49#869—31no—
50C23×C32C2312yes—
51C22×C3×D3C2312yes—
52C22×C3×D3C2311yescharacteristic
53C32×D4—22no—
54C2×C32×C4C2311yescharacteristic
55C2×C3⋊Dic3C2312yes—
56C2×C3⋊Dic3C2312yes—
57#634—22no—
58C3×#67—24no—
59C3×#67—28no—
60C2×C3×Dic3—24no—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2311yescharacteristic
63C2×C3×Dic3C2312yes—
64C3×D12—22no—
65C3×C4×D3D414yescomplemented
66C3×Dic6—22no—
67#625—24no—
68D3×Dic3—24no—
69D3×Dic3—24no—
70#532—91noSylow
71C3×#41—64no—
72C2×C3×Q8—31no—
73C2×C3×D4—21no—
74C2×C3×D4—34no—
75C2×C3×D4—32no—
76C2×C3×D4—31no—
77C2×C3×D4C2×D311yescharacteristic
78C2×C3×D4C2×D311yescharacteristic
79C22×C3×C4—31no—
80C22×C3×C4—32no—
81C2×#67—32no—
82C2×#67—32no—
83C2×#67C2×D314yes—
84C2×#67C2×D312yes—
85C22×Dic3—32no—
86C22×Dic3—32no—
87C22×Dic3C2×D312yes—
88#237—24no—
89#237—64no—
90#235—64no—
91C2×D12C2×D311yescharacteristic
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3C2×D311yescharacteristic
95C2×Dic6—31no—
96C2×Dic6—31no—
97C2×Dic6—61no—
98C2×Dic6C2×D311yescharacteristic
99C3×C42—31no—
100#217—34no—
101#217—34no—
102#217—34no—
103#217—64no—
104(C2×D3)⋊C4—34no—
105C4⋊Dic3—34no—
106Dic3⋊C4—34no—
107C4×Dic3—31no—
108C4×Dic3—31no—
109C4×Dic3—32no—
110C4×Dic3—32no—
111C4×Dic3—61no—
112C4×Dic3C2×D311yescharacteristic
113C22×C32—22no—
114C22×C32—24no—
115C22×C32C2411yescharacteristic
116C2×C3×D3—22no—
117C2×C3×D3—22no—
118C2×C3×D3—24no—
119C2×C3×D3C2×D412yes—
120C32×C4C2×D412yes—
121C3⋊Dic3—22no—
122C3⋊Dic3—22no—
123C3×Dic3—22no—
124C3×Dic3C2×D412yes—
125C3×Dic3#4114yescomplemented
126C2×#41—31no—
127C2×#41—91no—
128C4⋊Q8—91no—
129#127—31no—
130#124—91no—
131#124—91no—
132#121—94no—
133C4×D4—94no—
134C2×C42—91no—
135C23×C3—22no—
136C23×C3—31no—
137C23×C3—32no—
138C23×C3C22×D312yes—
139C23×C3C22×D312yes—
140C22×D3C22×D312yes—
141C22×D3C22×D311yescharacteristic
142C3×Q8—62no—
143C3×D4—22no—
144C3×D4—22no—
145C3×D4—42no—
146C3×D4—68no—
147C3×D4—64no—
148C3×D4—62no—
149C2×C3×C4—21no—
150C2×C3×C4—31no—
151C2×C3×C4—32no—
152C2×C3×C4—34no—
153C2×C3×C4—32no—
154C2×C3×C4—64no—
155C2×C3×C4C22×D311yescharacteristic
156C2×C3×C4C22×D311yescharacteristic
157#67—24no—
158#67—28no—
159#67—64no—
160#67—64no—
161C2×Dic3—24no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—62no—
167C2×Dic3—62no—
168C2×Dic3—64no—
169C2×Dic3—64no—
170C2×Dic3C22×D312yes—
171C2×Dic3C22×D312yes—
172C2×Dic3C22×D311yescharacteristic
173D12—22no—
174C4×D3—64no—
175C4×D3—64no—
176C4×D3#6714yescomplemented
177Dic6—22no—
178Dic6—62no—
179Dic6—62no—
180Dic6—122no—
181C2×C32—42no—
182C2×C32C22×D411yescharacteristic
183C2×C32C2×#4112yes—
184C3×D3—22no—
185C3×D3—42no—
186#41—64no—
187#41—184no—
188C2×Q8—31no—
189C2×Q8—91no—
190C2×D4—34no—
191C2×D4—32no—
192C2×D4—31no—
193C2×D4—92no—
194C2×D4D3211yescharacteristic
195C22×C4—31no—
196C22×C4—32no—
197C22×C4—92no—
198C22×C4—92no—
199C4⋊C4—94no—
200C22⋊C4—94no—
201C22⋊C4—94no—
202C42—31no—
203C42—92no—
204C42—91no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—22no—
209C22×C3—21no—
210C22×C3—32no—
211C22×C3—42no—
212C22×C3—44no—
213C22×C3—62no—
214C22×C3—64no—
215C22×C3—62no—
216C22×C3C23×D311yescharacteristic
217C22×C3C23×D311yescharacteristic
218C2×D3—22no—
219C2×D3—22no—
220C2×D3—24no—
221C2×D3C2×#6712yes—
222C3×C4—22no—
223C3×C4—32no—
224C3×C4—34no—
225C3×C4—62no—
226C3×C4D3×D412yes—
227C3×C4C2×#6712yes—
228Dic3—22no—
229Dic3—62no—
230Dic3—62no—
231Dic3—62no—
232Dic3—62no—
233Dic3—122no—
234Dic3—122no—
235Dic3#23714yescomplemented
236Dic3C2×#6712yes—
237C32#53211yescharacteristic, complemented, Sylow
238C23—31no—
239C23—32no—
240C23C2×D3212yes—
241Q8—62no—
242Q8—182no—
243D4—22no—
244D4—68no—
245D4—64no—
246D4—62no—
247D4—184no—
248C2×C4—32no—
249C2×C4—31no—
250C2×C4—34no—
251C2×C4—32no—
252C2×C4—64no—
253C2×C4—92no—
254C2×C4—92no—
255C2×C4—184no—
256C2×C4—184no—
257C2×C4C2×D3211yescharacteristic
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—62no—
263C2×C3—82no—
264C2×C3—122no—
265C2×C3C2×D3×D411yescharacteristic
266C2×C3C2×#23712yes—
267C2×C3D3×#4112yes—
268C2×C3C22×#6711yescharacteristic
269D3—22no—
270D3—42no—
271C22—24no—
272C22—22no—
273C22—32no—
274C22—62no—
275C22—64no—
276C22—62no—
277C22C22×D3211yescharacteristic
278C4—34no—
279C4—32no—
280C4—62no—
281C4—182no—
282C4—182no—
283C4D3×#6712yes—
284C3—21no—
285C3#571611yescharacteristic, complemented
286C3#583211yescharacteristic, complemented
287C2—42no—
288C2—62no—
289C2—122no—
290C2D3×#23712yes—
291C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#237C2semidirect
2. C2×#3795C2semidirect
3. #64145C2semidirect
4. C3×#940C2semidirect
5. Dic3×#67C2semidirect
6. #63970C2semidirect
7. C4×D3×Dic3C2semidirect
8. #63866C2semidirect
9. #63863C2semidirect
33. C2×#237D3semidirect
38. #940D3semidirect
65. C3×C4×D3D4semidirect
125. C3×Dic3#41semidirect
176. C4×D3#67semidirect
235. Dic3#237semidirect
237. C32#532semidirect
285. C3#5716semidirect
286. C3#5832semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2372
C2×#37952
#641452
C3×#9402
Dic3×#672
#639702
C4×D3×Dic32
#638662
#638632
#58323
#57163