Finite Group Databasebeta

← C3×#10590853 #13157821

Subgroups — C3×#10590853 #13157821

Order 1728.

626 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3746C211yesmaximal, characteristic, complemented
2C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic
3C3×#63857C212yesmaximal
4C32×#817C211yesmaximal, characteristic, complemented
5C3×#63567C212yesmaximal, complemented
6C3×#5137—31nomaximal
7C3×#4867—31nomaximal
8#10590853C311yesmaximal, characteristic, direct product factor
9C2×C3×#634C2212yes—
10C2×C32×C4×D3C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C3×#3746—22no—
13C32×C4⋊Dic3C2212yes—
14C2×C32×C3⋊C8C2211yescharacteristic
15C32×ΓL(1,𝔽25)—22no—
16C3×Dic3⋊Dic3—22no—
17C3×(C2×D3)⋊Dic3—22no—
18C2×C32×M16—31no—
19C2×#3746C2×C311yescharacteristic, complemented
20C3×#924—32no—
21C2×C3×Dic12—31no—
22C2×C3×Dic12—31no—
23C2×C3×Dic12—61no—
24C2×C3×ΓL(1,𝔽25)—21no—
25C2×C3×ΓL(1,𝔽25)D311yescharacteristic
26C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
27C3×#897—32no—
28#63857C2×C312yes—
29C32×#106—31no—
30C3×#817—21no—
31C3×#817D311yescharacteristic, complemented
32C3×#817C2×C311yescharacteristic, complemented
33C3×#816—32no—
34C3×#810—32no—
35#63567C2×C312yescomplemented
36C22×C32×D3D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3—22no—
39C3×#634—22no—
40C3×#634—22no—
41C2×C32×Dic3—22no—
42C2×C32×Dic3D411yescharacteristic
43C32×C4×D3—21no—
44C32×C4×D3—21no—
45C33×C8D412yes—
46C32×C3⋊C8—21no—
47C3×#470—91no—
48#5137—31no—
49#4867—31no—
50C22×C32×C4—31no—
51C2×#634C22×C312yes—
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3C2×D311yescharacteristic
54C2×C3×C4×D3C22×C311yescharacteristic
55C2×C3×Dic6—32no—
56C2×C3×Dic6—32no—
57C2×C3×Dic6—62no—
58C32×M16—62no—
59C2×C32×C8—21no—
60C2×C32×C8—21no—
61C2×C32×C8—21no—
62C2×C32×C8—31no—
63C2×C32×C8—41no—
64C2×C32×C8C2×D311yescharacteristic
65C2×C32×C8C2×D311yescharacteristic
66C2×C32×C8C22×C311yescharacteristic
67C32×C4⋊C4—32no—
68#3746—22no—
69C3×#217—62no—
70C3×(C2×D3)⋊C4—62no—
71C3×C4⋊Dic3—22no—
72C3×C4⋊Dic3—62no—
73C3×C4⋊Dic3C2×D312yes—
74C3×C4⋊Dic3C22×C312yes—
75C3×Dic3⋊C4—62no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8C2×D311yescharacteristic
78C2×C3×C3⋊C8C22×C311yescharacteristic
79C3×Dic12—64no—
80C3×Dic12—62no—
81C3×Dic12—62no—
82C3×ΓL(1,𝔽25)—24no—
83C3×ΓL(1,𝔽25)—22no—
84C3×ΓL(1,𝔽25)—22no—
85Dic3⋊Dic3—22no—
86(C2×D3)⋊Dic3—22no—
87C22×C33C2×D411yescharacteristic
88C2×C32×D3—21no—
89C2×C32×D3—22no—
90C33×C4C2×D411yescharacteristic
91C33×C4#4111yescharacteristic
92C3×C3⋊Dic3—42no—
93C32×Dic3—21no—
94C32×Dic3—42no—
95C2×C3×Q16—91no—
96C2×C3×M16—31no—
97C2×C3×M16—31no—
98C2×C3×M16—61no—
99C3×C22⋊Q8—92no—
100#924—32no—
101C2×Dic12—31no—
102C2×Dic12—31no—
103C2×Dic12—61no—
104C2×ΓL(1,𝔽25)C3×D311yescharacteristic
105#897—32no—
106C3×#106—31no—
107C3×#106—31no—
108C3×#106—61no—
109C3×#103—92no—
110#817C3×D311yescharacteristic, complemented
111#816—32no—
112#810—32no—
113C23×C32—31no—
114C22×C3×D3—21no—
115C22×C3×D3D1211yescharacteristic
116C22×C3×D3C3×D411yescharacteristic
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—31no—
121C2×C32×C4—41no—
122C2×C32×C4—61no—
123C2×C32×C4—61no—
124C2×C32×C4—62no—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3—22no—
129#634—22no—
130#634—22no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—21no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—62no—
137C2×C3×Dic3—122no—
138C2×C3×Dic3D1211yescharacteristic
139C2×C3×Dic3C3×D411yescharacteristic
140C3×C4×D3—22no—
141C3×C4×D3—22no—
142C3×C4×D3—21no—
143C3×C4×D3—21no—
144C3×C4×D3—21no—
145C3×C4×D3—21no—
146C3×Dic6—64no—
147C3×Dic6—62no—
148C3×Dic6—62no—
149C3×Dic6—62no—
150C3×Dic6—62no—
151C3×Dic6—122no—
152C32×C8—22no—
153C32×C8—22no—
154C32×C8—22no—
155C32×C8—42no—
156C32×C8—61no—
157C32×C8D1212yes—
158C32×C8#6712yes—
159C32×C8C3×D412yes—
160C3×C3⋊C8—22no—
161C3×C3⋊C8—21no—
162C3×C3⋊C8—21no—
163#470—91noSylow
164C2×C33#12111yescharacteristic
165C2×C33#13712yes—
166C32×D3—41no—
167C2×C3×Q8—92no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—61no—
171C2×C4×D3C2×C3×D311yescharacteristic
172C2×Dic6—32no—
173C2×Dic6—32no—
174C2×Dic6—62no—
175C3×Q16—182no—
176C3×M16—64no—
177C3×M16—62no—
178C3×M16—62no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—41no—
185C2×C3×C8—61no—
186C2×C3×C8D3211yescharacteristic
187C2×C3×C8C2×C3×D311yescharacteristic
188C2×C3×C8C2×C3×D311yescharacteristic
189C3×C4⋊C4—32no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—62no—
192C3×C4⋊C4—182no—
193C3×C22⋊C4—182no—
194#217—62no—
195(C2×D3)⋊C4—62no—
196C4⋊Dic3—62no—
197C4⋊Dic3C2×C3×D312yes—
198Dic3⋊C4—62no—
199C2×C3⋊C8C2×C3×D311yescharacteristic
200Dic12—64no—
201Dic12—62no—
202Dic12—62no—
203ΓL(1,𝔽25)—22no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—21no—
207C22×C32—41no—
208C22×C32—61no—
209C22×C32—62no—
210C22×C32C2×D1211yescharacteristic
211C22×C32D3×D411yescharacteristic
212C22×C32C2×C3×D411yescharacteristic
213C2×C3×D3—22no—
214C2×C3×D3—22no—
215C2×C3×D3—21no—
216C2×C3×D3—21no—
217C2×C3×D3—22no—
218C2×C3×D3—42no—
219C32×C4—21no—
220C32×C4—21no—
221C32×C4—21no—
222C32×C4—21no—
223C32×C4—21no—
224C32×C4—21no—
225C32×C4—41no—
226C32×C4—41no—
227C32×C4—61no—
228C32×C4—122no—
229C32×C4C2×D1211yescharacteristic
230C32×C4#23511yescharacteristic
231C32×C4#23911yescharacteristic
232C32×C4C2×#6711yescharacteristic
233C32×C4C2×C3×D411yescharacteristic
234C32×C4C3×#4111yescharacteristic
235C3⋊Dic3—42no—
236C3×Dic3—22no—
237C3×Dic3—21no—
238C3×Dic3—21no—
239C3×Dic3—42no—
240C3×Dic3—42no—
241C3×Dic3—82no—
242C3×Dic3—124no—
243C3×Dic3—122no—
244C3×Dic3—122no—
245C2×Q16—91no—
246C2×M16—31no—
247C22⋊Q8—92no—
248#106—31no—
249#103—92no—
250C33#47011yescharacteristic, complemented, Sylow
251C23×C3—31no—
252C23×C3—31no—
253C23×C3—61no—
254C22×D3C3×D1211yescharacteristic
255C3×Q8—182no—
256C3×Q8—182no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—41no—
263C2×C3×C4—61no—
264C2×C3×C4—62no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—62no—
271C2×C3×C4—62no—
272C2×C3×C4—122no—
273C2×C3×C4—182no—
274C2×C3×C4C2×D3211yescharacteristic
275C2×C3×C4C22×C3×D311yescharacteristic
276C2×C3×C4C22×C3×D311yescharacteristic
277C2×Dic3—22no—
278C2×Dic3—62no—
279C2×Dic3—62no—
280C2×Dic3—122no—
281C2×Dic3C3×D1211yescharacteristic
282C4×D3—21no—
283C4×D3—21no—
284Dic6—64no—
285Dic6—62no—
286Dic6—62no—
287Dic6—62no—
288Dic6—62no—
289Dic6—122no—
290C3×C8—22no—
291C3×C8—22no—
292C3×C8—22no—
293C3×C8—42no—
294C3×C8—62no—
295C3×C8—61no—
296C3×C8—61no—
297C3×C8#62612yes—
298C3×C8C3×D1212yes—
299C3×C8C3×#6712yes—
300C3⋊C8—21no—
301C2×C32—21no—
302C2×C32—22no—
303C2×C32—22no—
304C2×C32—21no—
305C2×C32—22no—
306C2×C32—21no—
307C2×C32—41no—
308C2×C32—42no—
309C2×C32—121no—
310C2×C32#89511yescharacteristic
311C2×C32#90912yes—
312C2×C32#91512yes—
313C2×C32#93011yescharacteristic
314C2×C32C3×#12111yescharacteristic
315C2×C32C3×#13712yes—
316C3×D3—42no—
317C3×D3—41no—
318C3×D3—41no—
319C2×Q8—92no—
320C22×C4—31no—
321Q16—182no—
322M16—62no—
323C2×C8—31no—
324C2×C8C3×D3211yescharacteristic
325C4⋊C4—32no—
326C4⋊C4—182no—
327C22⋊C4—182no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—41no—
332C22×C3—62no—
333C22×C3—61no—
334C22×C3—62no—
335C22×C3—62no—
336C22×C3—61no—
337C22×C3—122no—
338C22×C3D3×D1211yescharacteristic
339C22×C3C2×C3×D1211yescharacteristic
340C22×C3C3×D3×D411yescharacteristic
341C2×D3—22no—
342C2×D3—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—62no—
352C3×C4—61no—
353C3×C4—61no—
354C3×C4—122no—
355C3×C4—122no—
356C3×C4—242no—
357C3×C4—362no—
358C3×C4#379911yescharacteristic
359C3×C4C2×#62611yescharacteristic
360C3×C4C2×C3×D1211yescharacteristic
361C3×C4C3×#23511yescharacteristic
362C3×C4C3×#23911yescharacteristic
363C3×C4C2×C3×#6711yescharacteristic
364Dic3—21no—
365Dic3—42no—
366Dic3—124no—
367Dic3—122no—
368Dic3—122no—
369C32—21no—
370C32—21no—
371C32—21no—
372C32—41no—
373C32#486711yescharacteristic, complemented
374C32#513711yescharacteristic, complemented
375C32C3×#47011yescharacteristic, complemented
376C23—31no—
377Q8—182no—
378Q8—182no—
379C2×C4—31no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—62no—
383C2×C4—182no—
384C2×C4C2×C3×D3211yescharacteristic
385C8—61no—
386C8C3×#62612yes—
387C2×C3—22no—
388C2×C3—22no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—42no—
394C2×C3—41no—
395C2×C3—122no—
396C2×C3—121no—
397C2×C3—121no—
398C2×C3#6380012yes—
399C2×C3#6390911yescharacteristic
400C2×C3C3×#89511yescharacteristic
401C2×C3C3×#90912yes—
402C2×C3C3×#91512yes—
403C2×C3C3×#93011yescharacteristic
404D3—41no—
405C22—62no—
406C22—61no—
407C22C3×D3×D1211yescharacteristic
408C4—61no—
409C4—122no—
410C4—362no—
411C4C3×#379911yescharacteristic
412C4C2×C3×#62611yescharacteristic
413C3—21no—
414C3—21no—
415C3—21no—
416C3—41no—
417C3#1059085311yescharacteristic, direct product factor
418C3C3×#486711yescharacteristic, complemented
419C3C3×#513711yescharacteristic, complemented
420C2—121no—
421C2C3×#6380012yes—
422C2C3×#6390911yescharacteristic
subgroupG/H
centerC22×C3D3×D12
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C3×C8D32
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3746C2semidirect
4. C32×#817C2semidirect
5. C3×#63567C2semidirect
8. #10590853C3direct (N × H)
19. C2×#3746C2×C3semidirect
31. C3×#817D3semidirect
32. C3×#817C2×C3semidirect
35. #63567C2×C3semidirect
110. #817C3×D3semidirect
250. C33#470semidirect
373. C32#4867semidirect
374. C32#5137semidirect
375. C32C3×#470semidirect
417. C3#10590853direct (N × H)
418. C3C3×#4867semidirect
419. C3C3×#5137semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37462
C2×C32×ΓL(1,𝔽25)2
C3×#638572
C32×#8172
C3×#635672
C3×#51373
C3×#48673
#105908533