Finite Group Databasebeta

← #10594973

Subgroups — #10594973

Order 576.

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

normal subgroups only

This group has 473 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#64133C211yesmaximal, characteristic, complemented
2C3×#930C211yesmaximal, characteristic
3C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
4Dic3×#67C212yesmaximal, complemented
5#63961C212yesmaximal, complemented
6#63896C212yesmaximal, complemented
7#63884C212yesmaximal, complemented
8#63861C212yesmaximal, complemented
9#63847C212yesmaximal, complemented
10#5602—31nomaximal
11#5515—31nomaximal
12C22×C32×C4C2211yescharacteristic
13C2×#634C2211yescharacteristic
14C2×C3×#67C2212yes—
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C2×C3×D12C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19#3757C2212yes—
20C4⋊(C3⋊Dic3)C2211yescharacteristic
21#3750C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2211yescharacteristic
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26#3724C2212yes—
27#3724C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32Dic32C2212yes—
33C3×#121—31no—
34C2×C3×C4⋊C4—31no—
35#935—32no—
36#935—32no—
37D4×Dic3—32no—
38#930D311yescharacteristic
39#929—32no—
40C4×#67—32no—
41#924—31no—
42#924—31no—
43#924—61no—
44C2×Dic3⋊C4D311yescharacteristic, complemented
45#898—32no—
46#896—32no—
47#894—32no—
48#889—32no—
49#889—32no—
50#879—32no—
51#878—32no—
52C23×C32C2311yescharacteristic
53C22×C3×D3C2312yes—
54C2×C32×C4—41no—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C2×C3⋊Dic3C2312yes—
59#634—41no—
60C3×#67—24no—
61C2×C3×Dic3—22no—
62C2×C3×Dic3—41no—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66C3×D12—41no—
67D3×Dic3—42no—
68#539—91noSylow
69C2×C3×D4—32no—
70C2×C3×D4—31no—
71C22×C3×C4—21no—
72C22×C3×C4—31no—
73C22×C3×C4—31no—
74C22×C3×C4C2×D311yescharacteristic
75C22×C3×C4C2×D311yescharacteristic
76C2×#67C2×D312yes—
77C22×Dic3—32no—
78C22×Dic3C2×D311yescharacteristic
79C22×Dic3C2×D311yescharacteristic
80C2×D12C2×D311yescharacteristic
81C2×C4×D3—32no—
82C2×Dic6—31no—
83C2×Dic6—31no—
84C2×Dic6—61no—
85C3×C4⋊C4—31no—
86C3×C4⋊C4—32no—
87C3×C4⋊C4—32no—
88C3×C22⋊C4—32no—
89#217—32no—
90#217—32no—
91#217—32no—
92#217—32no—
93#217—32no—
94#217—62no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4C2×D312yes—
99C4⋊Dic3—31no—
100C4⋊Dic3—31no—
101C4⋊Dic3—32no—
102C4⋊Dic3—61no—
103C4⋊Dic3C2×D311yescharacteristic
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—62no—
112Dic3⋊C4C2×D312yes—
113Dic3⋊C4C2×D312yes—
114C4×Dic3—32no—
115C4×Dic3—32no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32#4112yes—
119C22×C32C2411yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C32×C4—21no—
124C32×C4—21no—
125C3⋊Dic3—22no—
126C3⋊Dic3—22no—
127C3×Dic3—22no—
128C3×Dic3—22no—
129C3×Dic3—22no—
130#126—92no—
131#125—92no—
132#123—92no—
133#123—92no—
134C22⋊Q8—92no—
135C22⋊Q8—91no—
136#121—31no—
137C4×D4—92no—
138C2×C4⋊C4—31no—
139C23×C3—21no—
140C23×C3—32no—
141C23×C3C22×D311yescharacteristic
142C23×C3C22×D311yescharacteristic
143C22×D3C22×D312yes—
144C3×D4—64no—
145C3×D4—121no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—32no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—62no—
155C2×C3×C4—121no—
156C2×C3×C4C22×D311yescharacteristic
157C2×C3×C4C22×D311yescharacteristic
158C2×C3×C4C22×D311yescharacteristic
159C2×C3×C4C22×D311yescharacteristic
160#67—24no—
161C2×Dic3—22no—
162C2×Dic3—32no—
163C2×Dic3—32no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—41no—
167C2×Dic3—62no—
168C2×Dic3—62no—
169C2×Dic3—122no—
170C2×Dic3C22×D312yes—
171C2×Dic3C22×D312yes—
172C2×Dic3C22×D312yes—
173D12—41no—
174C4×D3—122no—
175Dic6—121no—
176Dic6—121no—
177Dic6—122no—
178C2×C32—22no—
179C2×C32C2×#4111yescharacteristic
180C2×C32H(5,𝔽2)11yescharacteristic
181C2×C322−1+411yescharacteristic
182C3×D3—42no—
183C2×Q8—91no—
184C2×D4—32no—
185C2×D4—31no—
186C22×C4—31no—
187C22×C4—31no—
188C22×C4—92no—
189C22×C4D3211yescharacteristic
190C4⋊C4—32no—
191C4⋊C4—32no—
192C4⋊C4—31no—
193C4⋊C4—92no—
194C4⋊C4—92no—
195C4⋊C4—92no—
196C4⋊C4—92no—
197C4⋊C4—91no—
198C22⋊C4—32no—
199C22⋊C4—92no—
200C22⋊C4—92no—
201C22⋊C4—92no—
202C22⋊C4—92no—
203C42—92no—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—22no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—62no—
213C22×C3—62no—
214C22×C3—62no—
215C22×C3#23512yes—
216C22×C3#23712yes—
217C22×C3C23×D311yescharacteristic
218C22×C3C23×D311yescharacteristic
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C3×C4—22no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—41no—
228C3×C4—62no—
229C3×C4—62no—
230C3×C4—62no—
231Dic3—22no—
232Dic3—22no—
233Dic3—22no—
234Dic3—62no—
235Dic3—62no—
236Dic3—62no—
237Dic3—62no—
238Dic3—122no—
239Dic3—122no—
240C32#53911yescharacteristic, complemented, Sylow
241C23—32no—
242C23C2×D3211yescharacteristic
243Q8—361no—
244D4—64no—
245D4—121no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—32no—
249C2×C4—41no—
250C2×C4—62no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4—121no—
254C2×C4—362no—
255C2×C4C2×D3211yescharacteristic
256C2×C4C2×D3211yescharacteristic
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—21no—
261C2×C3—21no—
262C2×C3—42no—
263C2×C3—122no—
264C2×C3C2×#23511yescharacteristic
265C2×C3C2×#23711yescharacteristic
266C2×C3#100411yescharacteristic
267C2×C3#100411yescharacteristic
268C2×C3#100711yescharacteristic
269C2×C3#101011yescharacteristic
270D3—42no—
271C22—21no—
272C22—21no—
273C22—62no—
274C22—62no—
275C22—62no—
276C22#379512yes—
277C22C22×D3211yescharacteristic
278C4—21no—
279C4—21no—
280C4—62no—
281C4—62no—
282C4—62no—
283C4—182no—
284C4—182no—
285C3—21no—
286C3#551511yescharacteristic, complemented
287C3#560211yescharacteristic, complemented
288C2—22no—
289C2—122no—
290C2C2×#379511yescharacteristic
291C2#6429711yescharacteristic
292C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC3×#930C2
hypercenterC22×C4D32
hypocenterC32#539
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64133C2semidirect
3. C2×C3×Dic3⋊C4C2semidirect
4. Dic3×#67C2semidirect
5. #63961C2semidirect
6. #63896C2semidirect
7. #63884C2semidirect
8. #63861C2semidirect
9. #63847C2semidirect
44. C2×Dic3⋊C4D3semidirect
240. C32#539semidirect
286. C3#5515semidirect
287. C3#5602semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×#9302
C2×C3×Dic3⋊C42
Dic3×#672
#639612
#638962
#638842
#638612
#638472
#56023
#55153