Finite Group Databasebeta

← #10594342

Subgroups — #10594342

Order 576.

440 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64099C211yesmaximal, characteristic, complemented
2C3×D3×C4⋊C4C211yesmaximal, characteristic
3C3×#888C211yesmaximal, characteristic, complemented
4#63919C211yesmaximal, characteristic, complemented
5#63869C211yesmaximal, characteristic, complemented
6#63856C212yesmaximal
7#63855C211yesmaximal, characteristic
8#63851C212yesmaximal
9#63845C212yesmaximal, complemented
10#63841C211yesmaximal, characteristic
11#63839C212yesmaximal, complemented
12#5729—31nomaximal
13#5500—31nomaximal
14C2×C3×C4×D3C2212yes—
15C2×C3×C4×D3C2211yescharacteristic
16C2×C3×Dic6C2212yes—
17C2×#627C2212yes—
18C32×C4⋊C4C2211yescharacteristic
19C4⋊(C3⋊Dic3)C2211yescharacteristic
20C4⋊(C3⋊Dic3)C2212yes—
21#3750C2212yes—
22C4×C3⋊Dic3C2211yescharacteristic
23C3×C4⋊Dic3C2211yescharacteristic
24C3×C4⋊Dic3C2211yescharacteristic
25C3×Dic3⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2211yescharacteristic
28#3724C2212yes—
29#3724C2212yes—
30Dic3⋊Dic3C2211yescharacteristic
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2211yescharacteristic
36Dic32C2212yes—
37C3×C4⋊Q8—31no—
38C2×C3×C4⋊C4—31no—
39#946—32no—
40#944—32no—
41#926—31no—
42#926—31no—
43#924—32no—
44#924—32no—
45#896—32no—
46#892—31no—
47D3×C4⋊C4D311yescharacteristic
48#890—31no—
49#890—31no—
50#890—32no—
51#890—31no—
52#890—61no—
53#889—32no—
54#888—32no—
55#888—31no—
56#888—31no—
57#888D311yescharacteristic, complemented
58#872—31no—
59#870—31no—
60C22×C3×D3C2311yescharacteristic
61C2×C32×C4C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C3⋊Dic3C2312yes—
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3×Dic3C2312yes—
67C2×C3×Dic3C2311yescharacteristic
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C3×C4×D3—42no—
72C3×C4×D3Q814yescomplemented
73C3×Dic6—42no—
74#627—42no—
75#544—91noSylow
76C2×C3×Q8—32no—
77C22×C3×C4—31no—
78C22×C3×C4—32no—
79C2×C4×D3C2×D312yes—
80C2×C4×D3C2×D311yescharacteristic
81C2×Dic6—32no—
82C2×Dic6—32no—
83C2×Dic6C2×D312yes—
84C3×C4⋊C4—21no—
85C3×C4⋊C4—32no—
86C3×C4⋊C4—31no—
87C3×C4⋊C4—31no—
88C3×C4⋊C4—32no—
89C3×C4⋊C4C2×D311yescharacteristic
90C3×C4⋊C4C2×D311yescharacteristic
91C3×C42—31no—
92#217—32no—
93#217—31no—
94#217—31no—
95(C2×D3)⋊C4—31no—
96(C2×D3)⋊C4—31no—
97(C2×D3)⋊C4—32no—
98C4⋊Dic3—31no—
99C4⋊Dic3—32no—
100C4⋊Dic3—32no—
101C4⋊Dic3—31no—
102C4⋊Dic3—31no—
103C4⋊Dic3—32no—
104C4⋊Dic3—31no—
105C4⋊Dic3—61no—
106C4⋊Dic3—62no—
107C4⋊Dic3C2×D311yescharacteristic
108C4⋊Dic3C2×D311yescharacteristic
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—31no—
114Dic3⋊C4—31no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—62no—
119Dic3⋊C4C2×D312yes—
120Dic3⋊C4C2×D312yes—
121C4×Dic3—31no—
122C4×Dic3—31no—
123C4×Dic3—32no—
124C4×Dic3—32no—
125C4×Dic3—61no—
126C4×Dic3C2×D311yescharacteristic
127C22×C32C2411yescharacteristic
128C2×C3×D3—21no—
129C2×C3×D3—21no—
130C2×C3×D3C2×Q812yes—
131C32×C4—22no—
132C32×C4C2×Q812yes—
133C3⋊Dic3—21no—
134C3⋊Dic3—22no—
135C3⋊Dic3—21no—
136C3×Dic3—22no—
137C3×Dic3—22no—
138C3×Dic3—21no—
139C3×Dic3—21no—
140C3×Dic3C2×Q812yes—
141C4⋊Q8—31no—
142C4⋊Q8—91no—
143C4⋊Q8—92no—
144#125—91no—
145#125—92no—
146#125—91no—
147C22⋊Q8—92no—
148C22⋊Q8—92no—
149#117—91no—
150#117—91no—
151C2×C4⋊C4—31no—
152C23×C3—31no—
153C22×D3C22×D311yescharacteristic
154C3×Q8—122no—
155C2×C3×C4—21no—
156C2×C3×C4—22no—
157C2×C3×C4—31no—
158C2×C3×C4—34no—
159C2×C3×C4—32no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—32no—
163C2×C3×C4—122no—
164C2×C3×C4C22×D312yes—
165C2×C3×C4C22×D312yes—
166C2×C3×C4C22×D311yescharacteristic
167C2×C3×C4C22×D311yescharacteristic
168C2×Dic3—32no—
169C2×Dic3—31no—
170C2×Dic3—31no—
171C2×Dic3—32no—
172C2×Dic3—31no—
173C2×Dic3—31no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—62no—
177C2×Dic3C22×D312yes—
178C2×Dic3C22×D311yescharacteristic
179C2×Dic3C22×D311yescharacteristic
180C2×Dic3C22×D312yes—
181C2×Dic3C22×D311yescharacteristic
182C4×D3—42no—
183C4×D3Dic614yescomplemented
184Dic6—42no—
185Dic6—122no—
186Dic6—122no—
187C2×C32C22×Q811yescharacteristic
188C2×C32H(5,𝔽2)11yescharacteristic
189C2×C322−1+411yescharacteristic
190C3×D3—22no—
191C2×Q8—32no—
192C2×Q8—92no—
193C22×C4—31no—
194C22×C4—32no—
195C4⋊C4—32no—
196C4⋊C4—31no—
197C4⋊C4—31no—
198C4⋊C4—32no—
199C4⋊C4—91no—
200C4⋊C4—91no—
201C4⋊C4—92no—
202C4⋊C4—92no—
203C4⋊C4—92no—
204C4⋊C4—91no—
205C4⋊C4—92no—
206C4⋊C4—92no—
207C4⋊C4D3211yescharacteristic
208C22⋊C4—91no—
209C22⋊C4—91no—
210C22⋊C4—92no—
211C42—31no—
212C42—92no—
213C42—91no—
214C22×C3—21no—
215C22×C3—32no—
216C22×C3—61no—
217C22×C3—61no—
218C22×C3C23×D311yescharacteristic
219C22×C3C23×D311yescharacteristic
220C2×D3—21no—
221C2×D3—21no—
222C2×D3C2×Dic612yes—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—32no—
227C3×C4—42no—
228C3×C4—62no—
229C3×C4—62no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4C2×Dic612yes—
233C3×C4D3×Q812yes—
234Dic3—21no—
235Dic3—22no—
236Dic3—22no—
237Dic3—21no—
238Dic3—61no—
239Dic3—61no—
240Dic3—62no—
241Dic3—61no—
242Dic3—62no—
243Dic3—61no—
244Dic3—121no—
245Dic3—121no—
246Dic3—122no—
247Dic3C2×Dic612yes—
248C32#54411yescharacteristic, complemented, Sylow
249C23—31no—
250Q8—122no—
251Q8—362no—
252C2×C4—31no—
253C2×C4—34no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—32no—
257C2×C4—32no—
258C2×C4—92no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—122no—
262C2×C4C2×D3212yes—
263C2×C4C2×D3211yescharacteristic
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—62no—
268C2×C3C22×Dic611yescharacteristic
269C2×C3#100411yescharacteristic
270C2×C3#100411yescharacteristic
271C2×C3C2×D3×Q811yescharacteristic
272C2×C3#100711yescharacteristic
273C2×C3#100711yescharacteristic
274D3—22no—
275C22—32no—
276C22—61no—
277C22—61no—
278C22C22×D3211yescharacteristic
279C4—22no—
280C4—32no—
281C4—62no—
282C4—61no—
283C4—62no—
284C4—61no—
285C4—181no—
286C4—182no—
287C4—181no—
288C4D3×Dic612yes—
289C3—21no—
290C3#550011yescharacteristic, complemented
291C3#572911yescharacteristic, complemented
292C2—62no—
293C2C2×D3×Dic611yescharacteristic
294C2#6431311yescharacteristic
295C2#6431611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC22×C3×D3C23
hypercenterC4⋊C4D32
hypocenterC32#544
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×#888C2semidirect
4. #63919C2semidirect
5. #63869C2semidirect
9. #63845C2semidirect
11. #63839C2semidirect
57. #888D3semidirect
72. C3×C4×D3Q8semidirect
183. C4×D3Dic6semidirect
248. C32#544semidirect
290. C3#5500semidirect
291. C3#5729semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×D3×C4⋊C42
C3×#8882
#639192
#638692
#638562
#638552
#638512
#638452
#638412
#638392
#57293
#55003