Finite Group Databasebeta

← C3×#10590910 #13157703

Subgroups — C3×#10590910 #13157703

Order 1728.

694 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3743C211yesmaximal, characteristic
2C3×#63874C211yesmaximal, characteristic
3C2×C3×#3710C211yesmaximal, characteristic
4C32×#815C211yesmaximal, characteristic, complemented
5C32×C8×Dic3C211yesmaximal, characteristic, complemented
6C3×#63554C211yesmaximal, characteristic, complemented
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×#4708—31nomaximal
9C3×#4689—31nomaximal
10#10590910C311yesmaximal, characteristic, direct product factor
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×#3743C414yescomplemented
15C32×C4×Dic3C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×#3724C412yes—
20C3×#3722C412yes—
21C3×#3710C414yescomplemented
22C2×#3743C2×C311yescharacteristic
23C2×C3×ΓL(1,𝔽25)—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C2×C3×ΓL(1,𝔽25)—61no—
26C2×C3×D3×C8—31no—
27C2×C3×D3×C8—31no—
28C3×#872—31no—
29C3×#872—31no—
30#63874C2×C311yescharacteristic
31C2×#3710C2×C311yescharacteristic
32C32×#97—31no—
33C32×C4×C8—31no—
34C3×#815—21no—
35C3×#815—31no—
36C3×#815D311yescharacteristic, complemented
37C3×#815C2×C311yescharacteristic, complemented
38C3×C8×Dic3—21no—
39C3×C8×Dic3—31no—
40C3×C8×Dic3D311yescharacteristic, complemented
41C3×C8×Dic3C2×C311yescharacteristic, complemented
42C3×#803—31no—
43C3×C4×C3⋊C8—31no—
44#63554C2×C311yescharacteristic, complemented
45Dic3×C3⋊C8C2×C311yescharacteristic, complemented
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C3×C4×C3⋊D3C2×C412yes—
50C3×C4×C3⋊D3C2×C412yes—
51C2×C32×Dic3C2×C412yes—
52C2×C32×Dic3C2×C412yes—
53C33×C8C2×C412yes—
54C3×C3⋊(C3⋊C8)C2×C412yes—
55C32×C3⋊C8C2×C412yes—
56C32×C3⋊C8C2×C412yes—
57C3×#405—91no—
58#4708—31no—
59#4689—31no—
60C2×C4×C3⋊D3C22×C311yescharacteristic
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—31no—
63C2×C3×C4×D3—61no—
64C2×C32×C8—21no—
65C2×C32×C8—21no—
66C2×C32×C8—21no—
67C2×C32×C8—31no—
68C2×C32×C8—31no—
69C2×C32×C8—41no—
70C2×C32×C8C2×D311yescharacteristic
71C2×C32×C8C2×D311yescharacteristic
72C2×C32×C8C22×C311yescharacteristic
73C32×C42—31no—
74C32×C42—31no—
75C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
76#3743C3×C414yescomplemented
77C3×(C2×D3)⋊C4—32no—
78C3×(C2×D3)⋊C4—32no—
79C3×Dic3⋊C4—32no—
80C3×Dic3⋊C4—32no—
81C3×C4×Dic3—21no—
82C3×C4×Dic3—21no—
83C3×C4×Dic3C2×D311yescharacteristic
84C3×C4×Dic3C2×D311yescharacteristic
85C3×C4×Dic3C22×C311yescharacteristic
86C3×C4×Dic3C22×C311yescharacteristic
87C2×C3×C3⋊C8—21no—
88C2×C3×C3⋊C8—21no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8C2×D311yescharacteristic
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C22×C311yescharacteristic
95C2×C3×C3⋊C8C22×C311yescharacteristic
96C3×ΓL(1,𝔽25)—34no—
97C3×ΓL(1,𝔽25)—34no—
98C3×ΓL(1,𝔽25)—64no—
99C3×D3×C8—34no—
100C3×D3×C8—34no—
101#3724C3×C412yes—
102#3722C3×C412yes—
103#3710C3×C414yescomplemented
104C22×C33C22×C411yescharacteristic
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3C4212yes—
107C33×C4C22×C411yescharacteristic
108C33×C4C22×C411yescharacteristic
109C3×C3⋊Dic3C4212yes—
110C32×Dic3—22no—
111C32×Dic3—22no—
112C2×C3×M16—91no—
113C22×C3×C8—91no—
114C3×#117—91no—
115C2×ΓL(1,𝔽25)—31no—
116C2×ΓL(1,𝔽25)—31no—
117C2×ΓL(1,𝔽25)—61no—
118C2×D3×C8—31no—
119C2×D3×C8—31no—
120#872—31no—
121#872—31no—
122C3×#97—31no—
123C3×#97—31no—
124C3×#97—61no—
125C3×#97—91no—
126C3×C4×C8—31no—
127C3×C4×C8—31no—
128C3×C4×C8—61no—
129C3×C4×C8—91no—
130#815—31no—
131#815C3×D311yescharacteristic, complemented
132C8×Dic3—31no—
133C8×Dic3C3×D311yescharacteristic, complemented
134#803—31no—
135C4×C3⋊C8—31no—
136C22×C3⋊D3C2×C3×C411yescharacteristic
137C22×C3×D3—31no—
138C22×C3×D3—31no—
139C22×C3×D3—61no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—32no—
144C2×C32×C4—32no—
145C2×C32×C4—41no—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C311yescharacteristic
149C2×C3⋊Dic3C2×C3×C411yescharacteristic
150C4×C3⋊D3C2×C3×C412yes—
151C4×C3⋊D3C2×C3×C412yes—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—31no—
155C2×C3×Dic3—31no—
156C2×C3×Dic3—61no—
157C2×C3×Dic3C4×D312yes—
158C2×C3×Dic3C4×D312yes—
159C2×C3×Dic3C2×C3×C412yes—
160C2×C3×Dic3C2×C3×C412yes—
161C3×C4×D3—32no—
162C3×C4×D3—32no—
163C3×C4×D3—32no—
164C3×C4×D3—32no—
165C3×C4×D3—62no—
166C3×C4×D3—62no—
167C32×C8—22no—
168C32×C8—22no—
169C32×C8—22no—
170C32×C8—32no—
171C32×C8—32no—
172C32×C8—42no—
173C32×C8C4×D312yes—
174C32×C8C4×D312yes—
175C32×C8C2×C3×C412yes—
176C3⋊(C3⋊C8)C2×C3×C412yes—
177C3×C3⋊C8—22no—
178C3×C3⋊C8—22no—
179C3×C3⋊C8—32no—
180C3×C3⋊C8—32no—
181C3×C3⋊C8—62no—
182C3×C3⋊C8C4×D312yes—
183C3×C3⋊C8C4×D312yes—
184C3×C3⋊C8C2×C3×C412yes—
185C3×C3⋊C8C2×C3×C412yes—
186#405—91noSylow
187C2×C33C2×C4211yescharacteristic
188C2×C33#13112yes—
189C3×C3⋊D3—22no—
190C22×C3×C4—91no—
191C2×C4×D3—31no—
192C2×C4×D3—31no—
193C2×C4×D3—61no—
194C3×M16—94no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—21no—
198C2×C3×C8—31no—
199C2×C3×C8—31no—
200C2×C3×C8—31no—
201C2×C3×C8—31no—
202C2×C3×C8—41no—
203C2×C3×C8—61no—
204C2×C3×C8—61no—
205C2×C3×C8—94no—
206C2×C3×C8—91no—
207C2×C3×C8D3211yescharacteristic
208C2×C3×C8C2×C3×D311yescharacteristic
209C2×C3×C8C2×C3×D311yescharacteristic
210C3×C4⋊C4—92no—
211C3×C22⋊C4—92no—
212C3×C42—31no—
213C3×C42—31no—
214C3×C42—31no—
215C3×C42—31no—
216C3×C42—61no—
217C3×C42—61no—
218(C2×D3)⋊C4—32no—
219(C2×D3)⋊C4—32no—
220Dic3⋊C4—32no—
221Dic3⋊C4—32no—
222C4×Dic3C2×C3×D311yescharacteristic
223C4×Dic3C2×C3×D311yescharacteristic
224C2×C3⋊C8—31no—
225C2×C3⋊C8—31no—
226C2×C3⋊C8—61no—
227C2×C3⋊C8C2×C3×D311yescharacteristic
228C2×C3⋊C8C2×C3×D311yescharacteristic
229ΓL(1,𝔽25)—34no—
230ΓL(1,𝔽25)—34no—
231ΓL(1,𝔽25)—64no—
232D3×C8—34no—
233D3×C8—34no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32C2×C4×D311yescharacteristic
239C22×C32C2×C4×D311yescharacteristic
240C22×C32C22×C3×C411yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3C3×C4212yes—
243C2×C3×D3—32no—
244C2×C3×D3—32no—
245C2×C3×D3—62no—
246C2×C3×D3—62no—
247C2×C3×D3—62no—
248C2×C3×D3—122no—
249C32×C4—21no—
250C32×C4—21no—
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—41no—
256C32×C4—41no—
257C32×C4—62no—
258C32×C4—62no—
259C32×C4C2×C4×D311yescharacteristic
260C32×C4C2×C4×D311yescharacteristic
261C32×C4C2×C4×D311yescharacteristic
262C32×C4C2×C4×D311yescharacteristic
263C32×C4C22×C3×C411yescharacteristic
264C32×C4C22×C3×C411yescharacteristic
265C3⋊Dic3C3×C4212yes—
266C3×Dic3—24no—
267C3×Dic3—22no—
268C3×Dic3—22no—
269C3×Dic3—22no—
270C3×Dic3—22no—
271C3×Dic3—32no—
272C3×Dic3—32no—
273C3×Dic3—42no—
274C3×Dic3—62no—
275C2×M16—91no—
276C22×C8—91no—
277#117—91no—
278#97—31no—
279#97—91no—
280C4×C8—31no—
281C4×C8—91no—
282C33#40511yescharacteristic, complemented, Sylow
283C23×C3—91no—
284C22×D3—31no—
285C22×D3—31no—
286C22×D3—61no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—32no—
291C2×C3×C4—32no—
292C2×C3×C4—32no—
293C2×C3×C4—32no—
294C2×C3×C4—41no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—91no—
298C2×C3×C4—92no—
299C2×C3×C4—92no—
300C2×C3×C4C2×D3211yescharacteristic
301C2×C3×C4C22×C3×D311yescharacteristic
302C2×C3×C4C22×C3×D311yescharacteristic
303C2×Dic3—31no—
304C2×Dic3—31no—
305C2×Dic3—61no—
306C2×Dic3C3×C4×D312yes—
307C2×Dic3C3×C4×D312yes—
308C4×D3—32no—
309C4×D3—32no—
310C4×D3—32no—
311C4×D3—32no—
312C4×D3—62no—
313C4×D3—62no—
314C3×C8—22no—
315C3×C8—22no—
316C3×C8—22no—
317C3×C8—32no—
318C3×C8—32no—
319C3×C8—32no—
320C3×C8—32no—
321C3×C8—42no—
322C3×C8—62no—
323C3×C8—62no—
324C3×C8—92no—
325C3×C8#62412yes—
326C3×C8C3×C4×D312yes—
327C3×C8C3×C4×D312yes—
328C3⋊C8—32no—
329C3⋊C8—32no—
330C3⋊C8—62no—
331C3⋊C8C3×C4×D312yes—
332C3⋊C8C3×C4×D312yes—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—41no—
340C2×C32—42no—
341C2×C32C42×D311yescharacteristic
342C2×C32C42×D311yescharacteristic
343C2×C32#90112yes—
344C2×C32#90712yes—
345C2×C32C2×C3×C4211yescharacteristic
346C2×C32C3×#13112yes—
347C3⋊D3—22no—
348C3×D3—62no—
349C3×D3—62no—
350C3×D3—122no—
351C22×C4—91no—
352M16—94no—
353C2×C8—31no—
354C2×C8—31no—
355C2×C8—94no—
356C2×C8—91no—
357C2×C8C3×D3211yescharacteristic
358C4⋊C4—92no—
359C22⋊C4—92no—
360C42—31no—
361C42—31no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—92no—
367C22×C3—182no—
368C22×C3C4×D3211yescharacteristic
369C22×C3C2×C3×C4×D311yescharacteristic
370C22×C3C2×C3×C4×D311yescharacteristic
371C2×D3—32no—
372C2×D3—32no—
373C2×D3—62no—
374C2×D3—62no—
375C2×D3—62no—
376C2×D3—122no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4—64no—
386C3×C4—62no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4—62no—
390C3×C4—92no—
391C3×C4—122no—
392C3×C4C4×D3211yescharacteristic
393C3×C4C2×#62411yescharacteristic
394C3×C4C2×C3×C4×D311yescharacteristic
395C3×C4C2×C3×C4×D311yescharacteristic
396C3×C4C2×C3×C4×D311yescharacteristic
397C3×C4C2×C3×C4×D311yescharacteristic
398Dic3—22no—
399Dic3—22no—
400Dic3—32no—
401Dic3—32no—
402Dic3—62no—
403C32—21no—
404C32—21no—
405C32—21no—
406C32—41no—
407C32#468911yescharacteristic, complemented
408C32#470811yescharacteristic, complemented
409C32C3×#40511yescharacteristic, complemented
410C23—91no—
411C2×C4—32no—
412C2×C4—32no—
413C2×C4—91no—
414C2×C4—92no—
415C2×C4—92no—
416C2×C4C2×C3×D3211yescharacteristic
417C8—32no—
418C8—32no—
419C8—92no—
420C8C3×#62412yes—
421C2×C3—22no—
422C2×C3—22no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—42no—
428C2×C3—41no—
429C2×C3—182no—
430C2×C3#6380412yes—
431C2×C3C4×#62411yescharacteristic
432C2×C3C3×C42×D311yescharacteristic
433C2×C3C3×C42×D311yescharacteristic
434C2×C3C3×#90112yes—
435C2×C3C3×#90712yes—
436D3—62no—
437D3—62no—
438D3—122no—
439C22—92no—
440C22—182no—
441C22C3×C4×D3211yescharacteristic
442C4—62no—
443C4—62no—
444C4—92no—
445C4C3×C4×D3211yescharacteristic
446C4C2×C3×#62411yescharacteristic
447C3—21no—
448C3—21no—
449C3—21no—
450C3—41no—
451C3#1059091011yescharacteristic, direct product factor
452C3C3×#468911yescharacteristic, complemented
453C3C3×#470811yescharacteristic, complemented
454C2—182no—
455C2C3×#6380412yes—
456C2C3×C4×#62411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C2×C3×C42
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#405
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. C32×#815C2semidirect
5. C32×C8×Dic3C2semidirect
6. C3×#63554C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10590910C3direct (N × H)
14. C3×#3743C4semidirect
21. C3×#3710C4semidirect
36. C3×#815D3semidirect
37. C3×#815C2×C3semidirect
40. C3×C8×Dic3D3semidirect
41. C3×C8×Dic3C2×C3semidirect
44. #63554C2×C3semidirect
45. Dic3×C3⋊C8C2×C3semidirect
76. #3743C3×C4semidirect
103. #3710C3×C4semidirect
131. #815C3×D3semidirect
133. C8×Dic3C3×D3semidirect
282. C33#405semidirect
407. C32#4689semidirect
408. C32#4708semidirect
409. C32C3×#405semidirect
451. C3#10590910direct (N × H)
452. C3C3×#4689semidirect
453. C3C3×#4708semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37432
C3×#638742
C2×C3×#37102
C32×#8152
C32×C8×Dic32
C3×#635542
C3×Dic3×C3⋊C82
C3×#47083
C3×#46893
#105909103