Finite Group Databasebeta

← C3×#10590327 #13157342

Subgroups — C3×#10590327 #13157342

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C3×#3710C212yesmaximal, complemented
3C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63557C212yesmaximal, complemented
6C3×#4834—31nomaximal
7C3×C8×D12—31nomaximal
8#10590327C311yesmaximal, characteristic, direct product factor
9C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
10C2×C3×C4×C3⋊D3C2212yes—
11C33×C42C2211yescharacteristic
12C3×#3752C412yes—
13C3×C4⋊(C3⋊Dic3)C411yescharacteristic
14C2×C32×C3⋊C8C2211yescharacteristic
15C2×C32×C3⋊C8C2212yes—
16C2×C32×C3⋊C8C2211yescharacteristic
17C3×#3710—24no—
18C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C2×C3×D3×C8—32no—
20C2×C3×D3×C8—32no—
21C3×C4×D12—31no—
22C3×C4×D12—31no—
23C3×C4×D12—61no—
24C2×#3710C2×C312yescomplemented
25C32×C4⋊C8—31no—
26C32×C4×C8—31no—
27C3×(C2×D3)⋊C8—32no—
28C3×(C2×D3)⋊C8—32no—
29C3×C4⋊(C3⋊C8)—21no—
30C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
31C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
32C3×C4×C3⋊C8—21no—
33C3×C4×C3⋊C8D311yescharacteristic, complemented
34C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
35#63557C2×C312yescomplemented
36C22×C3×C3⋊D3C2×C412yes—
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3C2×C412yes—
41C3×C4⋊(C3⋊D3)C814yescomplemented
42C3×C4×C3⋊D3—22no—
43C3×C4×C3⋊D3—22no—
44C32×C3⋊C8—22no—
45C32×C3⋊C8—21no—
46C32×C3⋊C8D412yescomplemented
47C3×C8×D4—91no—
48#4834—31no—
49C8×D12—31no—
50C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
51C2×C4×C3⋊D3C22×C312yes—
52C2×C3×D12—31no—
53C2×C3×D12—31no—
54C2×C3×D12—61no—
55C2×C3×C4×D3—32no—
56C2×C3×C4×D3—32no—
57C2×C3×C4×D3—62no—
58C2×C32×C8—31no—
59C2×C32×C8—32no—
60C2×C32×C8—31no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42—41no—
65C32×C42C2×D311yescharacteristic
66C32×C42C2×D311yescharacteristic
67C32×C42C22×C311yescharacteristic
68#3752C3×C412yes—
69C4⋊(C3⋊Dic3)C3×C411yescharacteristic
70C3×(C2×D3)⋊C4—32no—
71C3×(C2×D3)⋊C4—32no—
72C3×(C2×D3)⋊C4—62no—
73C3×C4⋊Dic3—31no—
74C3×C4⋊Dic3—31no—
75C3×C4⋊Dic3—61no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—22no—
79C2×C3×C3⋊C8C2×D312yes—
80C2×C3×C3⋊C8C2×D311yescharacteristic
81C2×C3×C3⋊C8C2×D311yescharacteristic
82C2×C3×C3⋊C8C22×C311yescharacteristic
83C2×C3×C3⋊C8C22×C312yes—
84C2×C3×C3⋊C8C22×C311yescharacteristic
85C3×D3×C8—64no—
86C3×D3×C8—64no—
87#3710—24no—
88C22×C33C22×C411yescharacteristic
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3C2×C814yescomplemented
92C33×C4—21no—
93C33×C4C2×C812yescomplemented
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—22no—
97C22×C3×C8—92no—
98C3×C4×D4—91no—
99C2×D3×C8—32no—
100C2×D3×C8—32no—
101C4×D12—31no—
102C4×D12—31no—
103C4×D12—61no—
104C3×C4⋊C8—31no—
105C3×C4⋊C8—31no—
106C3×C4⋊C8—61no—
107C3×C22⋊C8—92no—
108C3×C4×C8—31no—
109C3×C4×C8—31no—
110C3×C4×C8—61no—
111(C2×D3)⋊C8—32no—
112(C2×D3)⋊C8—32no—
113C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
114C4×C3⋊C8C3×D311yescharacteristic, complemented
115C22×C3⋊D3C2×C3×C412yes—
116C22×C3×D3—32no—
117C22×C3×D3—32no—
118C22×C3×D3—62no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—41no—
129C2×C32×C4—41no—
130C2×C32×C4—41no—
131C2×C32×C4C4×D311yescharacteristic
132C2×C32×C4C4×D311yescharacteristic
133C2×C32×C4C4×D311yescharacteristic
134C2×C32×C4C4×D311yescharacteristic
135C2×C32×C4C2×C3×C411yescharacteristic
136C2×C32×C4C2×C3×C411yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3C2×C3×C412yes—
141C4⋊(C3⋊D3)C3×C814yescomplemented
142C4×C3⋊D3—22no—
143C4×C3⋊D3—22no—
144C2×C3×Dic3—32no—
145C2×C3×Dic3—32no—
146C2×C3×Dic3—62no—
147C3×D12—34no—
148C3×D12—34no—
149C3×D12—64no—
150C3×C4×D3—62no—
151C3×C4×D3—62no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×C4×D3—122no—
155C3×C4×D3—122no—
156C32×C8—32no—
157C32×C8—61no—
158C32×C8—62no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—21no—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8—21no—
164C3×C3⋊C8—22no—
165C3×C3⋊C8—42no—
166C3×C3⋊C8D1212yescomplemented
167C3×C3⋊C8C3×D412yescomplemented
168C8×D4—91noSylow
169C2×C33C4×D411yescharacteristic
170C2×C33C22×C811yescharacteristic
171C2×C33#13111yescharacteristic
172C3×C3⋊D3—24no—
173C2×C3×D4—91no—
174C22×C3×C4—92no—
175C2×D12—31no—
176C2×D12—31no—
177C2×D12—61no—
178C2×C4×D3—32no—
179C2×C4×D3—32no—
180C2×C4×D3—62no—
181C2×C3×C8—31no—
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—32no—
185C2×C3×C8—31no—
186C2×C3×C8—32no—
187C2×C3×C8—61no—
188C2×C3×C8—61no—
189C2×C3×C8—62no—
190C2×C3×C8—184no—
191C3×C4⋊C4—91no—
192C3×C22⋊C4—92no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—41no—
197C3×C42D3211yescharacteristic
198C3×C42C2×C3×D311yescharacteristic
199C3×C42C2×C3×D311yescharacteristic
200(C2×D3)⋊C4—32no—
201(C2×D3)⋊C4—32no—
202(C2×D3)⋊C4—62no—
203C4⋊Dic3—31no—
204C4⋊Dic3—31no—
205C4⋊Dic3—61no—
206C2×C3⋊C8C2×C3×D311yescharacteristic
207C2×C3⋊C8C2×C3×D312yes—
208C2×C3⋊C8C2×C3×D311yescharacteristic
209D3×C8—64no—
210D3×C8—64no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—41no—
215C22×C32C2×C4×D311yescharacteristic
216C22×C32C2×C4×D311yescharacteristic
217C22×C32C22×C3×C411yescharacteristic
218C2×C3⋊D3—22no—
219C2×C3⋊D3—22no—
220C2×C3⋊D3C2×C3×C814yescomplemented
221C2×C3×D3—34no—
222C2×C3×D3—34no—
223C2×C3×D3—64no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—122no—
229C2×C3×D3—122no—
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—42no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—42no—
249C32×C4D3×C812yescomplemented
250C32×C4D3×C812yescomplemented
251C32×C4C2×C3×C812yescomplemented
252C32×C4C2×D1211yescharacteristic
253C32×C4#23511yescharacteristic
254C32×C4D3×D411yescharacteristic
255C32×C4#23911yescharacteristic
256C32×C4C2×C3×D411yescharacteristic
257C32×C4C3×#4111yescharacteristic
258C3⋊Dic3—22no—
259C3×Dic3—62no—
260C3×Dic3—62no—
261C3×Dic3—122no—
262C22×C8—92no—
263C4×D4—91no—
264C4⋊C8—31no—
265C22⋊C8—92no—
266C4×C8—31no—
267C33C8×D411yescharacteristic, complemented, Sylow
268C23×C3—92no—
269C22×D3—32no—
270C22×D3—32no—
271C22×D3—62no—
272C3×D4—94no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—92no—
286C2×C3×C4—182no—
287C2×C3×C4—182no—
288C2×C3×C4#62411yescharacteristic
289C2×C3×C4#62411yescharacteristic
290C2×C3×C4C3×C4×D311yescharacteristic
291C2×C3×C4C3×C4×D311yescharacteristic
292C2×C3×C4C3×C4×D311yescharacteristic
293C2×C3×C4C3×C4×D311yescharacteristic
294C2×C3×C4C2×D3211yescharacteristic
295C2×C3×C4C22×C3×D311yescharacteristic
296C2×C3×C4C22×C3×D311yescharacteristic
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—62no—
300D12—34no—
301D12—34no—
302D12—64no—
303C4×D3—62no—
304C4×D3—62no—
305C4×D3—62no—
306C4×D3—62no—
307C4×D3—122no—
308C4×D3—122no—
309C3×C8—32no—
310C3×C8—32no—
311C3×C8—62no—
312C3×C8—62no—
313C3×C8—61no—
314C3×C8—62no—
315C3×C8—62no—
316C3×C8—61no—
317C3×C8—122no—
318C3⋊C8—21no—
319C3⋊C8—22no—
320C3⋊C8C3×D1212yescomplemented
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32C4×D1211yescharacteristic
334C2×C32#89311yescharacteristic
335C2×C32C2×D3×C811yescharacteristic
336C2×C32C2×D3×C811yescharacteristic
337C2×C32#90111yescharacteristic
338C2×C32#90711yescharacteristic
339C2×C32C3×C4×D411yescharacteristic
340C2×C32C22×C3×C811yescharacteristic
341C2×C32C3×#13111yescharacteristic
342C3⋊D3—24no—
343C3×D3—64no—
344C3×D3—64no—
345C3×D3—124no—
346C2×D4—91no—
347C22×C4—92no—
348C2×C8—31no—
349C2×C8—31no—
350C2×C8—32no—
351C2×C8—184no—
352C4⋊C4—91no—
353C22⋊C4—92no—
354C42C3×D3211yescharacteristic
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—41no—
359C22×C3—94no—
360C22×C3—182no—
361C22×C3—182no—
362C22×C3C2×#62411yescharacteristic
363C22×C3C2×C3×C4×D311yescharacteristic
364C22×C3C2×C3×C4×D311yescharacteristic
365C2×D3—34no—
366C2×D3—34no—
367C2×D3—64no—
368C2×D3—62no—
369C2×D3—62no—
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—122no—
373C2×D3—122no—
374C3×C4—22no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—42no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—42no—
393C3×C4—182no—
394C3×C4#371012yescomplemented
395C3×C4C3×D3×C812yescomplemented
396C3×C4C3×D3×C812yescomplemented
397C3×C4#379911yescharacteristic
398C3×C4D3×D1211yescharacteristic
399C3×C4C2×C3×D1211yescharacteristic
400C3×C4C3×#23511yescharacteristic
401C3×C4C3×D3×D411yescharacteristic
402C3×C4C3×#23911yescharacteristic
403Dic3—62no—
404Dic3—62no—
405Dic3—122no—
406C32—21no—
407C32—21no—
408C32—21no—
409C32—41no—
410C32C8×D1211yescharacteristic, complemented
411C32#483411yescharacteristic, complemented
412C32C3×C8×D411yescharacteristic, complemented
413C23—92no—
414D4—94no—
415C2×C4—92no—
416C2×C4—182no—
417C2×C4—182no—
418C2×C4C3×#62411yescharacteristic
419C2×C4C3×#62411yescharacteristic
420C2×C4C2×C3×D3211yescharacteristic
421C8—32no—
422C8—61no—
423C8—62no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—184no—
437C2×C3C2×#371011yescharacteristic
438C2×C3#6381711yescharacteristic
439C2×C3#6389411yescharacteristic
440C2×C3C3×C4×D1211yescharacteristic
441C2×C3C3×#89311yescharacteristic
442C2×C3C2×C3×D3×C811yescharacteristic
443C2×C3C2×C3×D3×C811yescharacteristic
444C2×C3C3×#90111yescharacteristic
445C2×C3C3×#90711yescharacteristic
446D3—64no—
447D3—64no—
448D3—124no—
449C22—94no—
450C22—182no—
451C22—182no—
452C22C2×C3×#62411yescharacteristic
453C4—21no—
454C4—182no—
455C4C3×#371012yescomplemented
456C4C3×#379911yescharacteristic
457C4C3×D3×D1211yescharacteristic
458C3—21no—
459C3—21no—
460C3—21no—
461C3—41no—
462C3#1059032711yescharacteristic, direct product factor
463C3C3×C8×D1211yescharacteristic, complemented
464C3C3×#483411yescharacteristic, complemented
465C2—184no—
466C2C2×C3×#371011yescharacteristic
467C2C3×#6381711yescharacteristic
468C2C3×#6389411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×C8×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3710C2semidirect
3. C32×C4⋊(C3⋊C8)C2semidirect
4. C32×C4×C3⋊C8C2semidirect
5. C3×#63557C2semidirect
8. #10590327C3direct (N × H)
24. C2×#3710C2×C3semidirect
30. C3×C4⋊(C3⋊C8)D3semidirect
31. C3×C4⋊(C3⋊C8)C2×C3semidirect
33. C3×C4×C3⋊C8D3semidirect
34. C3×C4×C3⋊C8C2×C3semidirect
35. #63557C2×C3semidirect
41. C3×C4⋊(C3⋊D3)C8semidirect
46. C32×C3⋊C8D4semidirect
91. C2×C3×C3⋊D3C2×C8semidirect
93. C33×C4C2×C8semidirect
113. C4⋊(C3⋊C8)C3×D3semidirect
114. C4×C3⋊C8C3×D3semidirect
141. C4⋊(C3⋊D3)C3×C8semidirect
166. C3×C3⋊C8D12semidirect
167. C3×C3⋊C8C3×D4semidirect
220. C2×C3⋊D3C2×C3×C8semidirect
249. C32×C4D3×C8semidirect
250. C32×C4D3×C8semidirect
251. C32×C4C2×C3×C8semidirect
267. C33C8×D4semidirect
320. C3⋊C8C3×D12semidirect
394. C3×C4#3710semidirect
395. C3×C4C3×D3×C8semidirect
396. C3×C4C3×D3×C8semidirect
410. C32C8×D12semidirect
411. C32#4834semidirect
412. C32C3×C8×D4semidirect
455. C4C3×#3710semidirect
462. C3#10590327direct (N × H)
463. C3C3×C8×D12semidirect
464. C3C3×#4834semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C2×C3×#37102
C32×C4⋊(C3⋊C8)2
C32×C4×C3⋊C82
C3×#635572
C3×#48343
C3×C8×D123
#105903273