Finite Group Databasebeta

← #10596121

Subgroups — #10596121

Order 576.

642 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#123C211yesmaximal, characteristic
2#64144C211yesmaximal, characteristic
3D4×C3⋊Dic3C211yesmaximal, characteristic
4#64135C211yesmaximal, characteristic, complemented
5C2×#3750C211yesmaximal, characteristic, complemented
6#64099C211yesmaximal, characteristic, complemented
7#64098C211yesmaximal, characteristic, complemented
8#64098C211yesmaximal, characteristic, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64088C211yesmaximal, characteristic, complemented
11#64088C211yesmaximal, characteristic, complemented
12#64087C211yesmaximal, characteristic, complemented
13#64087C211yesmaximal, characteristic, complemented
14#64086C211yesmaximal, characteristic, complemented
15#64086C211yesmaximal, characteristic, complemented
16#5650—34nomaximal
17C2×C32×D4C2211yescharacteristic
18C22×C32×C4C2211yescharacteristic
19C22×C3⋊Dic3C2211yescharacteristic
20C22×C3⋊Dic3C2211yescharacteristic
21C22×C3⋊Dic3C2211yescharacteristic
22C2×#634C2211yescharacteristic
23C32×C4⋊C4C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C32×C22⋊C4C2211yescharacteristic
26C32×C22⋊C4C2211yescharacteristic
27C32×C22⋊C4C2211yescharacteristic
28#3757C2211yescharacteristic
29#3757C2211yescharacteristic
30#3757C2211yescharacteristic
31#3757C2211yescharacteristic
32#3757C2211yescharacteristic
33C4⋊(C3⋊Dic3)C2211yescharacteristic
34C4⋊(C3⋊Dic3)C2211yescharacteristic
35C4⋊(C3⋊Dic3)C2211yescharacteristic
36C4⋊(C3⋊Dic3)C2211yescharacteristic
37#3750C2211yescharacteristic
38#3750C2211yescharacteristic
39#3750C2211yescharacteristic
40#3750C2211yescharacteristic
41#3750C2211yescharacteristic
42#3750C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47C4×C3⋊Dic3C2211yescharacteristic
48C4×C3⋊Dic3C2211yescharacteristic
49C4×C3⋊Dic3C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#123D314yes—
53#935—34no—
54D4×Dic3—34no—
55#926—34no—
56C2×Dic3⋊C4—34no—
57#890—34no—
58#889—34no—
59#889—34no—
60#887—34no—
61#879—34no—
62#879—34no—
63#878—34no—
64#878—34no—
65#877—34no—
66#877—34no—
67C23×C32C2311yescharacteristic
68C23×C32C2311yescharacteristic
69C32×D4—22no—
70C2×C32×C4—22no—
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C2×C3⋊Dic3C2311yescharacteristic
87#634—22no—
88#552—91noSylow
89C2×C3×D4C2×D314yes—
90C22×C3×C4C2×D314yes—
91C22×Dic3—34no—
92C22×Dic3—34no—
93C22×Dic3—34no—
94C2×Dic6—34no—
95C3×C4⋊C4C2×D314yes—
96C3×C4⋊C4C2×D314yes—
97C3×C22⋊C4C2×D314yes—
98C3×C22⋊C4C2×D314yes—
99C3×C22⋊C4C2×D314yes—
100#217—34no—
101#217—34no—
102#217—34no—
103#217—34no—
104#217—34no—
105C4⋊Dic3—34no—
106C4⋊Dic3—34no—
107C4⋊Dic3—34no—
108C4⋊Dic3—34no—
109Dic3⋊C4—34no—
110Dic3⋊C4—34no—
111Dic3⋊C4—34no—
112Dic3⋊C4—34no—
113Dic3⋊C4—34no—
114Dic3⋊C4—34no—
115Dic3⋊C4—34no—
116Dic3⋊C4—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119C4×Dic3—34no—
120C4×Dic3—34no—
121C4×Dic3—34no—
122C4×Dic3—34no—
123C4×Dic3—34no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32#4112yes—
130C22×C32C2411yescharacteristic
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3⋊Dic3#4112yes—
144#126—91no—
145#126—91no—
146#125—91no—
147#125—91no—
148#125—91no—
149#123—91no—
150#123C3⋊D311yescharacteristic
151C22⋊Q8—91no—
152C22⋊Q8—91no—
153C4×Q8—91no—
154C4×D4—91no—
155#117—91no—
156#117—91no—
157#117—91no—
158C2×C4⋊C4—91no—
159C23×C3C22×D314yes—
160C23×C3C22×D314yes—
161C3×D4—28no—
162C2×C3×C4—28no—
163C2×C3×C4C22×D314yes—
164C2×C3×C4C22×D314yes—
165C2×C3×C4C22×D314yes—
166C2×C3×C4C22×D314yes—
167C2×C3×C4C22×D314yes—
168C2×Dic3—34no—
169C2×Dic3—34no—
170C2×Dic3—34no—
171C2×Dic3—34no—
172C2×Dic3—34no—
173C2×Dic3—34no—
174C2×Dic3—34no—
175C2×Dic3—34no—
176C2×Dic3—68no—
177C2×Dic3—68no—
178C2×Dic3—68no—
179Dic6—68no—
180C2×C32—22no—
181C2×C32—41no—
182C2×C32C2×#4111yescharacteristic
183C2×C32C2×#4111yescharacteristic
184C2×C322−1+411yescharacteristic
185C2×Q8—91no—
186C2×D4C2×C3⋊D311yescharacteristic
187C22×C4—91no—
188C22×C4—91no—
189C22×C4—91no—
190C22×C4C2×C3⋊D311yescharacteristic
191C4⋊C4—91no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C4⋊C4—91no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—91no—
198C4⋊C4—91no—
199C4⋊C4—91no—
200C4⋊C4—91no—
201C4⋊C4—91no—
202C4⋊C4—91no—
203C4⋊C4—91no—
204C4⋊C4—91no—
205C4⋊C4C2×C3⋊D311yescharacteristic
206C4⋊C4C2×C3⋊D311yescharacteristic
207C22⋊C4—91no—
208C22⋊C4—91no—
209C22⋊C4—91no—
210C22⋊C4—91no—
211C22⋊C4—91no—
212C22⋊C4C2×C3⋊D311yescharacteristic
213C22⋊C4C2×C3⋊D311yescharacteristic
214C22⋊C4C2×C3⋊D311yescharacteristic
215C42—91no—
216C42—91no—
217C42—91no—
218C42—91no—
219C42—91no—
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3—24no—
224C22×C3—24no—
225C22×C3#23718yes—
226C22×C3C23×D314yes—
227C3×C4—24no—
228C3×C4—24no—
229C3×C4—24no—
230C3×C4—24no—
231C3×C4—24no—
232Dic3—38no—
233Dic3—64no—
234Dic3—64no—
235Dic3—64no—
236Dic3—64no—
237Dic3—64no—
238Dic3—64no—
239Dic3—64no—
240C32#55211yescharacteristic, complemented, Sylow
241C23C22×C3⋊D311yescharacteristic
242C23C22×C3⋊D311yescharacteristic
243Q8—182no—
244D4—22no—
245C2×C4—22no—
246C2×C4—91no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—91no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—182no—
255C2×C4—182no—
256C2×C4—182no—
257C2×C4C22×C3⋊D311yescharacteristic
258C2×C4C22×C3⋊D311yescharacteristic
259C2×C4C22×C3⋊D311yescharacteristic
260C2×C4C22×C3⋊D311yescharacteristic
261C2×C4C22×C3⋊D311yescharacteristic
262C2×C3—28no—
263C2×C3—44no—
264C2×C3C2×#23714yes—
265C2×C3D3×#4114yes—
266C2×C3#101014yes—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22#383012yes—
273C22C23×C3⋊D311yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—21no—
279C4—92no—
280C4—181no—
281C4—181no—
282C4—181no—
283C4—181no—
284C4—181no—
285C4—181no—
286C4—181no—
287C3#565014yescomplemented
288C2—22no—
289C2—41no—
290C2C2×#383011yescharacteristic
291C2#41×C3⋊D311yescharacteristic
292C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
prime radicalC2×C32×D4C22
hypercenter#123C3⋊D3
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #64135C2semidirect
5. C2×#3750C2semidirect
6. #64099C2semidirect
7. #64098C2semidirect
8. #64098C2semidirect
9. #64096C2semidirect
10. #64088C2semidirect
11. #64088C2semidirect
12. #64087C2semidirect
13. #64087C2semidirect
14. #64086C2semidirect
15. #64086C2semidirect
240. C32#552semidirect
287. C3#5650semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
#641442
D4×C3⋊Dic32
#641352
C2×#37502
#640992
#640982
#640962
#640882
#640872
#640862
#56503