Finite Group Databasebeta

← #10594345

Subgroups — #10594345

Order 576.

412 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 128 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
1#64099C211yesmaximal, characteristic, complemented
2C3×#897C211yesmaximal, characteristic
3C3×#887C211yesmaximal, characteristic, complemented
4#63873C211yesmaximal, characteristic, complemented
5#63869C212yesmaximal, complemented
6#63861C212yesmaximal
7#63859C212yesmaximal
8#63856C211yesmaximal, characteristic
9#63855C211yesmaximal, characteristic
10#63847C212yesmaximal, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5625—31nomaximal
13#5583—31nomaximal
14C2×C3×C4×D3C2211yescharacteristic
15C2×C3×Dic6C2211yescharacteristic
16C2×C3×Dic6C2211yescharacteristic
17C32×C4⋊C4C2211yescharacteristic
18C4⋊(C3⋊Dic3)C2211yescharacteristic
19C4⋊(C3⋊Dic3)C2212yes—
20#3750C2212yes—
21C4×C3⋊Dic3C2211yescharacteristic
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×C4×Dic3C2211yescharacteristic
26C3×C4×Dic3C2212yes—
27#3724C2212yes—
28#3724C2212yes—
29#3724C2212yes—
30Dic3⋊Dic3C2212yes—
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2211yescharacteristic
33(C2×D3)⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C3×C4×Q8—31no—
39#946—31no—
40Q8×Dic3—31no—
41#944—31no—
42#944—31no—
43#926—31no—
44#924—31no—
45#898—32no—
46#897D311yescharacteristic
47#890—31no—
48#890—31no—
49#890—32no—
50#890—61no—
51#889—32no—
52#889—32no—
53#887D311yescharacteristic, complemented
54#879—32no—
55#879—32no—
56#876—32no—
57#872—31no—
58#870—32no—
59C4×Dic6—31no—
60C22×C3×D3C2311yescharacteristic
61C2×C32×C4C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C3⋊Dic3C2312yes—
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3×Dic3C2311yescharacteristic
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C3×C4×D3—22no—
72C3×Dic6—22no—
73C3×Dic6—22no—
74#541—91noSylow
75C2×C3×Q8—31no—
76C2×C3×Q8—31no—
77C22×C3×C4—31no—
78C2×C4×D3C2×D311yescharacteristic
79C2×Dic6C2×D311yescharacteristic
80C2×Dic6C2×D311yescharacteristic
81C3×C4⋊C4—21no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4—32no—
84C3×C4⋊C4C2×D311yescharacteristic
85C3×C4⋊C4C2×D311yescharacteristic
86C3×C22⋊C4—32no—
87C3×C42—31no—
88C3×C42—32no—
89#217—32no—
90#217—31no—
91#217—31no—
92(C2×D3)⋊C4—31no—
93(C2×D3)⋊C4—31no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4C2×D312yes—
96C4⋊Dic3—31no—
97C4⋊Dic3—32no—
98C4⋊Dic3—32no—
99C4⋊Dic3—31no—
100C4⋊Dic3—32no—
101C4⋊Dic3—31no—
102C4⋊Dic3—31no—
103C4⋊Dic3—61no—
104C4⋊Dic3—62no—
105C4⋊Dic3C2×D312yes—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—31no—
110Dic3⋊C4—32no—
111Dic3⋊C4—32no—
112Dic3⋊C4—31no—
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—62no—
118Dic3⋊C4C2×D312yes—
119C4×Dic3—31no—
120C4×Dic3—31no—
121C4×Dic3—32no—
122C4×Dic3—32no—
123C4×Dic3—61no—
124C4×Dic3C2×D312yes—
125C4×Dic3C2×D311yescharacteristic
126C22×C32C2411yescharacteristic
127C2×C3×D3—21no—
128C2×C3×D3—21no—
129C2×C3×D3—21no—
130C32×C4—22no—
131C32×C4#4112yes—
132C3⋊Dic3—22no—
133C3⋊Dic3—21no—
134C3⋊Dic3—21no—
135C3×Dic3—22no—
136C3×Dic3—22no—
137C3×Dic3—21no—
138C3×Dic3—21no—
139C3×Dic3—21no—
140C4⋊Q8—91no—
141#126—92no—
142#126—92no—
143#125—92no—
144#125—92no—
145#125—91no—
146C22⋊Q8—31no—
147C22⋊Q8—91no—
148C4×Q8—31no—
149C4×Q8—91no—
150#117—91no—
151C23×C3—31no—
152C22×D3C22×D311yescharacteristic
153C3×Q8—62no—
154C3×Q8—62no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—31no—
158C2×C3×C4—32no—
159C2×C3×C4—31no—
160C2×C3×C4—32no—
161C2×C3×C4—31no—
162C2×C3×C4—62no—
163C2×C3×C4C22×D312yes—
164C2×C3×C4C22×D312yes—
165C2×C3×C4C22×D311yescharacteristic
166C2×C3×C4C22×D311yescharacteristic
167C2×Dic3—31no—
168C2×Dic3—31no—
169C2×Dic3—32no—
170C2×Dic3—31no—
171C2×Dic3—32no—
172C2×Dic3—31no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—62no—
176C2×Dic3C22×D311yescharacteristic
177C2×Dic3C22×D312yes—
178C2×Dic3C22×D311yescharacteristic
179C2×Dic3C22×D312yes—
180C2×Dic3C22×D311yescharacteristic
181C4×D3—22no—
182Dic6—22no—
183Dic6—22no—
184C2×C32C2×#4111yescharacteristic
185C2×C322−1+411yescharacteristic
186C2×C322−1+411yescharacteristic
187C3×D3—41no—
188C2×Q8—31no—
189C2×Q8—31no—
190C22×C4—31no—
191C4⋊C4—32no—
192C4⋊C4—32no—
193C4⋊C4—91no—
194C4⋊C4—91no—
195C4⋊C4—92no—
196C4⋊C4—92no—
197C4⋊C4—92no—
198C4⋊C4—92no—
199C4⋊C4—91no—
200C4⋊C4—92no—
201C4⋊C4—92no—
202C4⋊C4D3211yescharacteristic
203C22⋊C4—32no—
204C22⋊C4—91no—
205C22⋊C4—91no—
206C22⋊C4—92no—
207C42—31no—
208C42—32no—
209C42—92no—
210C42—91no—
211C22×C3—21no—
212C22×C3—61no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3C23×D311yescharacteristic
216C22×C3C23×D311yescharacteristic
217C2×D3—21no—
218C2×D3—21no—
219C2×D3—21no—
220C3×C4—22no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—42no—
224C3×C4—61no—
225C3×C4—62no—
226C3×C4—62no—
227C3×C4—61no—
228C3×C4—61no—
229C3×C4#23512yes—
230C3×C4#23712yes—
231Dic3—22no—
232Dic3—22no—
233Dic3—21no—
234Dic3—21no—
235Dic3—21no—
236Dic3—62no—
237Dic3—61no—
238Dic3—62no—
239Dic3—61no—
240Dic3—61no—
241Dic3—61no—
242Dic3—121no—
243Dic3—121no—
244Dic3—122no—
245C32#54111yescharacteristic, complemented, Sylow
246C23—31no—
247Q8—62no—
248Q8—62no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—31no—
254C2×C4—62no—
255C2×C4—91no—
256C2×C4—92no—
257C2×C4—91no—
258C2×C4C2×D3212yes—
259C2×C4C2×D3211yescharacteristic
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—121no—
264C2×C3C2×#23511yescharacteristic
265C2×C3C2×#23711yescharacteristic
266C2×C3#100711yescharacteristic
267C2×C3#100711yescharacteristic
268C2×C3#101011yescharacteristic
269C2×C3#101011yescharacteristic
270D3—41no—
271C22—61no—
272C22—61no—
273C22—61no—
274C22C22×D3211yescharacteristic
275C4—22no—
276C4—61no—
277C4—62no—
278C4—62no—
279C4—61no—
280C4—61no—
281C4—181no—
282C4—182no—
283C4—181no—
284C4#379512yes—
285C3—21no—
286C3#558311yescharacteristic, complemented
287C3#562511yescharacteristic, complemented
288C2—121no—
289C2C2×#379511yescharacteristic
290C2#6430911yescharacteristic
291C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×C3×D3C23
hypercenterC4⋊C4D32
hypocenterC32#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
3. C3×#887C2semidirect
4. #63873C2semidirect
5. #63869C2semidirect
10. #63847C2semidirect
11. Dic3×Dic6C2semidirect
53. #887D3semidirect
245. C32#541semidirect
286. C3#5583semidirect
287. C3#5625semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8972
C3×#8872
#638732
#638692
#638612
#638592
#638562
#638552
#638472
Dic3×Dic62
#56253
#55833