Finite Group Databasebeta

← #10594721

Subgroups — #10594721

Order 576.

526 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64298C211yesmaximal, characteristic, complemented
2#64297C211yesmaximal, characteristic, complemented
3#64118C211yesmaximal, characteristic, complemented
4C3×#909C211yesmaximal, characteristic, complemented
5C3×#907C211yesmaximal, characteristic, complemented
6C2×#3719C211yesmaximal, characteristic, complemented
7#63829C211yesmaximal, characteristic, complemented
8#63806C212yesmaximal, complemented
9#63801C212yesmaximal, complemented
10#63800C212yesmaximal, complemented
11D3×Dic12C212yesmaximal, complemented
12#5779—31nomaximal
13#5753—31nomaximal
14#3828C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C3×#235C2211yescharacteristic
17C3×#235C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19#3805—22no—
20#3804—22no—
21#3799C2212yes—
22#3797—21no—
23#3796—21no—
24#3795—21no—
25#3795C2212yes—
26D3×Dic6—21no—
27D3×Dic6C2212yes—
28D3×Dic6C2212yes—
29C32×M16C2211yescharacteristic
30#3746C2212yes—
31#3744C2212yes—
32C2×C3×C3⋊C8C2211yescharacteristic
33C3×Dic12C2212yes—
34C3×#204C2212yes—
35C3×ΓL(1,𝔽25)C2212yes—
36C3×D3×C8C2212yes—
37#3719C2211yescharacteristic
38#3719C2212yes—
39#3719C2212yes—
40#3717C2211yescharacteristic
41#3716C2211yescharacteristic
42#3714C2211yescharacteristic
43#1010—31no—
44#1010—31no—
45#1010—31no—
46#1007—31no—
47C3×#137—31no—
48C3×#131—31no—
49#950—31no—
50C2×#216—31no—
51#918—32no—
52D3×Q16—32no—
53#916—32no—
54#915—32no—
55#909—31no—
56#909—31no—
57#909—32no—
58#909—32no—
59#909—61no—
60#909D311yescharacteristic, complemented
61#907D311yescharacteristic, complemented
62C2×Dic12—31no—
63C2×Dic12—32no—
64#904—31no—
65#904—32no—
66C2×C32×C4C2311yescharacteristic
67#638—21no—
68C2×C3⋊Dic3—21no—
69C4⋊(C3⋊D3)C2311yescharacteristic
70C4×C3⋊D3—21no—
71#634—21no—
72#634C2311yescharacteristic
73#634C2312yes—
74C3×#67—21no—
75C3×#67D412yes—
76C2×C3×Dic3—21no—
77C3×D12D411yescharacteristic
78C3×D12C2311yescharacteristic
79C3×C4×D3—21no—
80C3×C4×D3C2312yes—
81C3×Dic6—21no—
82C3×Dic6D411yescharacteristic
83C3×Dic6C2311yescharacteristic
84C3×Dic6C2312yes—
85#627—22no—
86#627—22no—
87#626—22no—
88#625—22no—
89#624—22no—
90D3×Dic3—22no—
91D3×Dic3—22no—
92D3×Dic3—22no—
93C32×C8C2312yes—
94C3×C3⋊C8C2311yescharacteristic
95C3×C3⋊C8C2311yescharacteristic
96#578—91noSylow
97C3×#41—31no—
98C3×#41—31no—
99C2×C3×Q8—31no—
100#239—32no—
101#239—61no—
102D3×Q8—32no—
103D3×Q8—32no—
104D3×Q8—61no—
105D3×Q8—61no—
106D3×Q8—61no—
107#237—32no—
108#237—61no—
109#237—61no—
110#237—62no—
111#237—62no—
112#237—62no—
113#235—31no—
114#235—31no—
115#235—32no—
116#235—32no—
117#235—61no—
118#235—62no—
119#235—61no—
120#235C2×D311yescharacteristic
121#235C2×D311yescharacteristic
122C2×Dic6—31no—
123C2×Dic6—31no—
124C2×Dic6—32no—
125C2×Dic6—32no—
126C2×Dic6—61no—
127C2×Dic6—61no—
128C2×Dic6C2×D311yescharacteristic
129C3×Q16—32no—
130C3×SD16—32no—
131C3×M16—21no—
132C3×M16—32no—
133C3×M16C2×D311yescharacteristic
134C3×M16C2×D311yescharacteristic
135C2×C3×C8—31no—
136C2×C3×C8—32no—
137#216—32no—
138#216—32no—
139#216—31no—
140#215—31no—
141#214—31no—
142#213—31no—
143C2×C3⋊C8C2×D311yescharacteristic
144Dic12—31no—
145Dic12—32no—
146Dic12—32no—
147Dic12—32no—
148Dic12—32no—
149Dic12—62no—
150Dic12C2×D312yes—
151D24—31no—
152#204—31no—
153#204—31no—
154#204—32no—
155#204—32no—
156#204—62no—
157#204C2×D312yes—
158ΓL(1,𝔽25)C2×D312yes—
159D3×C8C2×D312yes—
160C22×C32C2×D411yescharacteristic
161C2×C3⋊D3—21no—
162C2×C3×D3—21no—
163C2×C3×D3C2×D412yes—
164C32×C4C2×D411yescharacteristic
165C32×C4C2411yescharacteristic
166C3⋊Dic3—22no—
167C3⋊Dic3—21no—
168C3×Dic3—21no—
169C3×Dic3—22no—
170C3×Dic3C2×D412yes—
1712−1+4—91no—
1722−1+4—91no—
173#137—31no—
174#137—92no—
175#137—92no—
176#137—91no—
177#135—92no—
178#135—91no—
179C2×Q16—92no—
180C2×Q16—91no—
181#131—31no—
182C3×Q8—31no—
183C3×Q8—31no—
184C3×Q8—32no—
185C3×Q8—61no—
186C3×D4—32no—
187C3×D4—31no—
188C3×D4—31no—
189C3×D4—61no—
190C2×C3×C4—21no—
191C2×C3×C4—32no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4C22×D311yescharacteristic
195C2×C3×C4C22×D311yescharacteristic
196#67—21no—
197#67—62no—
198#67—62no—
199#67—61no—
200#67—61no—
201#67—62no—
202#67—121no—
203#67D1212yes—
204C2×Dic3—21no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—61no—
208C2×Dic3—61no—
209C2×Dic3—62no—
210C2×Dic3—121no—
211D12—31no—
212D12—31no—
213D12—61no—
214D12—62no—
215D12D1211yescharacteristic
216D12C22×D311yescharacteristic
217C4×D3—21no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—61no—
223C4×D3—61no—
224C4×D3—62no—
225C4×D3—121no—
226C4×D3C22×D312yes—
227Dic6—21no—
228Dic6—31no—
229Dic6—32no—
230Dic6—32no—
231Dic6—31no—
232Dic6—62no—
233Dic6—61no—
234Dic6—62no—
235Dic6—62no—
236Dic6—61no—
237Dic6—61no—
238Dic6—62no—
239Dic6—62no—
240Dic6—121no—
241Dic6D1211yescharacteristic
242Dic6C22×D311yescharacteristic
243Dic6C22×D312yes—
244C3×C8—22no—
245C3×C8—31no—
246C3×C8—31no—
247C3×C8C22×D312yes—
248C3×C8C22×D312yes—
249C3⋊C8C22×D311yescharacteristic
250C3⋊C8C22×D311yescharacteristic
251C2×C32—21no—
252C2×C32C22×D411yescharacteristic
253C3⋊D3—41no—
254C3×D3—22no—
255C3×D3—41no—
256#41—31no—
257#41—31no—
258#41—92no—
259#41—92no—
260#41—91no—
261#41—182no—
262#41—181no—
263#41—182no—
264#41—181no—
265C2×Q8—31no—
266C2×Q8—92no—
267C2×Q8—92no—
268C2×Q8—91no—
269C2×Q8—181no—
270C2×Q8—181no—
271Q16—32no—
272Q16—92no—
273Q16—91no—
274Q16—92no—
275Q16—92no—
276SD16—32no—
277SD16—91no—
278SD16—91no—
279SD16—92no—
280D8—91no—
281M16—32no—
282M16D3211yescharacteristic
283C2×C8—31no—
284C2×C8—32no—
285C22×C3—21no—
286C22×C3—32no—
287C22×C3—61no—
288C22×C3C2×D1211yescharacteristic
289C22×C3D3×D411yescharacteristic
290C2×D3—21no—
291C2×D3—61no—
292C2×D3—61no—
293C2×D3—121no—
294C2×D3C2×D1212yes—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—32no—
298C3×C4—61no—
299C3×C4—62no—
300C3×C4C2×D1211yescharacteristic
301C3×C4D3×D411yescharacteristic
302C3×C4C23×D311yescharacteristic
303C3×C4C23×D311yescharacteristic
304Dic3—22no—
305Dic3—21no—
306Dic3—61no—
307Dic3—62no—
308Dic3—61no—
309Dic3—62no—
310Dic3—122no—
311Dic3—121no—
312Dic3C2×D1212yes—
313C32#57811yescharacteristic, complemented, Sylow
314Q8—31no—
315Q8—32no—
316Q8—31no—
317Q8—61no—
318Q8—91no—
319Q8—92no—
320Q8—182no—
321Q8—182no—
322Q8—181no—
323D4—32no—
324D4—31no—
325D4—31no—
326D4—61no—
327D4—91no—
328D4—182no—
329D4—182no—
330D4—181no—
331C2×C4—32no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—182no—
335C2×C4—182no—
336C2×C4—182no—
337C2×C4—182no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4C2×D3211yescharacteristic
341C8—31no—
342C8—31no—
343C8C2×D3212yes—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—21no—
347C2×C3—41no—
348C2×C3—62no—
349C2×C3—121no—
350C2×C3C22×D1211yescharacteristic
351C2×C3C2×D3×D411yescharacteristic
352D3—22no—
353D3—41no—
354D3—121no—
355D3—121no—
356D3—241no—
357C22—32no—
358C22—61no—
359C22—181no—
360C22D3×D1211yescharacteristic
361C4—32no—
362C4—61no—
363C4—62no—
364C4—182no—
365C4—181no—
366C4D3×D1211yescharacteristic
367C4C22×D3211yescharacteristic
368C3—21no—
369C3#575311yescharacteristic, complemented
370C3#577911yescharacteristic, complemented
371C2—21no—
372C2—62no—
373C2—121no—
374C2—361no—
375C2C2×D3×D1211yescharacteristic
subgroupG/H
centerC2C2×D3×D12
commutator, cosocleC32×C4C24
FrattiniC4C22×D32
FittingC32×M16C22
socleC2×C32C22×D4
hypercenterM16D32
hypocenterC32#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #64298C2semidirect
2. #64297C2semidirect
3. #64118C2semidirect
4. C3×#909C2semidirect
5. C3×#907C2semidirect
6. C2×#3719C2semidirect
7. #63829C2semidirect
8. #63806C2semidirect
9. #63801C2semidirect
10. #63800C2semidirect
11. D3×Dic12C2semidirect
60. #909D3semidirect
61. #907D3semidirect
313. C32#578semidirect
369. C3#5753semidirect
370. C3#5779semidirect

Maximal subgroups

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

subgroupindex
#642982
#642972
#641182
C3×#9092
C3×#9072
C2×#37192
#638292
#638062
#638012
#638002
D3×Dic122
#57793
#57533