Finite Group Databasebeta

← C3×#70334 #11788509

Subgroups — C3×#70334 #11788509

Order 1152.

695 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5812C211yesmaximal, characteristic, complemented
2C2×C3×#942C211yesmaximal, characteristic
3C32×#419C211yesmaximal, characteristic, complemented
4C3×#5139C211yesmaximal, characteristic
5C3×#5006C211yesmaximal, characteristic
6C3×#4559C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#1905—31nomaximal
9#70334C311yesmaximal, characteristic, direct product factor
10C22×C32×Q8C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×Q8×Dic3—22no—
15C3×Q8×Dic3—22no—
16C2×C3×#216C412yescomplemented
17C3×#942—22no—
18C3×#942—22no—
19C2×C3×#215C412yescomplemented
20C3×#926—22no—
21C3×#926C2211yescharacteristic
22C3×#926C2211yescharacteristic
23C2×C3×#208C2211yescharacteristic
24C32×#103—22no—
25C3×#825—22no—
26C3×#806—22no—
27C2×C3×#137—31no—
28C3×#519—31no—
29#5812C2×C311yescharacteristic, complemented
30C2×#942C2×C311yescharacteristic
31C3×#421—31no—
32C3×#419—21no—
33C3×#419D311yescharacteristic, complemented
34C3×#419C2×C311yescharacteristic, complemented
35C3×#410—31no—
36#5139C2×C311yescharacteristic
37#5006C2×C311yescharacteristic
38C3×#344—31no—
39C3×H(3,ℤ/4)—31no—
40#4559C2×C311yescharacteristic, complemented
41#4531C2×C311yescharacteristic, complemented
42C2×C32×Q8—21no—
43C2×C32×Q8—21no—
44C2×C32×Q8—21no—
45C2×C32×Q8C2×C412yes—
46C2×C32×Q8D412yes—
47C2×C32×Q8D412yes—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C3×#235—21no—
55C2×C3×D12C2×C411yescharacteristic
56C2×C3×C4×D3—21no—
57C2×C3×Dic6C2×C411yescharacteristic
58C32×M16—21no—
59C32×M16—21no—
60C2×C32×C8—21no—
61C32×C4⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C42—21no—
64C3×#217—21no—
65C3×#217D412yes—
66C3×#216—24no—
67C3×#215—24no—
68C3×C4⋊Dic3—24no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4×Dic3—24no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3D412yes—
74C3×#208—21no—
75C3×#208—21no—
76C2×C3×C3⋊C8C2×C412yes—
77#1905—31noSylow
78C2×C3×#41—31no—
79C22×C3×Q8—21no—
80C22×C3×Q8C2×D311yescharacteristic
81C22×C3×Q8C22×C311yescharacteristic
82C2×#235C22×C311yescharacteristic
83C3×#137—62no—
84C3×#137—62no—
85C2×C3×Q16—32no—
86C2×C3×SD16—32no—
87C2×C3×M16—21no—
88C2×C3×M16—31no—
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C22×C311yescharacteristic
91C3×C4×Q8—62no—
92C3×C4×Q8—62no—
93C3×#117—21no—
94C3×#117—31no—
95C3×#117—31no—
96C3×#117—62no—
97C3×#117C2×D311yescharacteristic
98C3×#117C22×C311yescharacteristic
99Q8×Dic3—22no—
100Q8×Dic3—22no—
101C2×#216C3×C412yescomplemented
102#942—22no—
103#942—22no—
104C2×#215C3×C412yescomplemented
105#926—22no—
106#926C22×C311yescharacteristic
107#926C22×C311yescharacteristic
108C2×#208C22×C311yescharacteristic
109C3×#104—62no—
110C3×#103—24no—
111C3×#103—22no—
112C3×#103—22no—
113C3×#99—62no—
114#825—22no—
115#806—22no—
116C23×C32C2×D411yescharacteristic
117C22×C3×D3—21no—
118C32×Q8—22no—
119C32×Q8—22no—
120C32×Q8—22no—
121C32×Q8C22⋊C414yes—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—41no—
126C2×C32×C4C22×C411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4#4111yescharacteristic
131C2×C32×C4#4111yescharacteristic
132C3×#67—41no—
133C3×#67—41no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—22no—
137C3×D12—21no—
138C3×D12—21no—
139C3×C4×D3—41no—
140C3×C4×D3—41no—
141C3×Dic6—21no—
142C3×Dic6—21no—
143C32×C8—41no—
144C3×C3⋊C8—22no—
145C2×#137—31no—
146#519—31no—
147#421—31no—
148#419C3×D311yescharacteristic, complemented
149#410—31no—
150#344—31no—
151H(3,ℤ/4)—31no—
152C3×#41—61no—
153C3×#41—61no—
154C3×#41—61no—
155C3×#41—61no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8—22no—
164C2×C3×Q8—22no—
165C2×C3×Q8—31no—
166C2×C3×Q8—41no—
167C2×C3×Q8—41no—
168C2×C3×Q8—41no—
169C2×C3×Q8C4×D312yes—
170C2×C3×Q8D1212yes—
171C2×C3×Q8#6712yes—
172C2×C3×Q8C2×C3×C412yes—
173C2×C3×Q8C3×D412yes—
174C2×C3×Q8C3×D412yes—
175C2×C3×D4—31no—
176C2×C3×D4—61no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—41no—
181C22×C3×C4—61no—
182C22×C3×C4C22×D311yescharacteristic
183C22×C3×C4C23×C311yescharacteristic
184C2×#67—21no—
185#235—21no—
186#235—21no—
187#235—21no—
188#235—21no—
189C2×D12C2×C3×C411yescharacteristic
190C2×C4×D3—21no—
191C2×Dic6C2×C3×C411yescharacteristic
192C3×Q16—64no—
193C3×SD16—64no—
194C3×M16—21no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—41no—
199C3×M16—41no—
200C3×M16—61no—
201C3×M16—61no—
202C2×C3×C8—22no—
203C2×C3×C8—21no—
204C2×C3×C8—21no—
205C2×C3×C8—32no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—21no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—64no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C22⋊C4—21no—
213C3×C22⋊C4—21no—
214C3×C22⋊C4—32no—
215C3×C22⋊C4—41no—
216C3×C22⋊C4—61no—
217C3×C42—21no—
218C3×C42—21no—
219C3×C42—32no—
220C3×C42—41no—
221C3×C42—64no—
222C3×C42—61no—
223#217—21no—
224#217C3×D412yes—
225#216—24no—
226#215—24no—
227C4⋊Dic3—24no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230C4×Dic3—24no—
231C4×Dic3—21no—
232C4×Dic3C3×D412yes—
233#208—21no—
234#208—21no—
235C2×C3⋊C8C2×C3×C412yes—
236C22×C32—41no—
237C22×C32C4×D411yescharacteristic
238C22×C32#12011yescharacteristic
239C22×C32#12111yescharacteristic
240C2×C3×D3—41no—
241C2×C3×D3—81no—
242C32×C4—24no—
243C32×C4—81no—
244C32×C4C2×C22⋊C411yescharacteristic
245C32×C4C4×D411yescharacteristic
246C32×C4#12111yescharacteristic
247C32×C4#12311yescharacteristic
248C3×Dic3—42no—
249C3×Dic3—42no—
250C3×Dic3—41no—
251C2×#41—31no—
252C22×Q8C2×C3×D311yescharacteristic
253#137—62no—
254#137—62no—
255C2×Q16—32no—
256C2×SD16—32no—
257C2×M16—31no—
258C2×M16C2×C3×D311yescharacteristic
259C4×Q8—62no—
260C4×Q8—62no—
261#117—31no—
262#117—31no—
263#117—62no—
264#117C2×C3×D311yescharacteristic
265#104—62no—
266#103—22no—
267#99—62no—
268C23×C3—21no—
269C23×C3—61no—
270C23×C3D3×D411yescharacteristic
271C23×C3C2×C3×D411yescharacteristic
272C22×D3—21no—
273C3×Q8—22no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—22no—
277C3×Q8—22no—
278C3×Q8—22no—
279C3×Q8—24no—
280C3×Q8—42no—
281C3×Q8—42no—
282C3×Q8—42no—
283C3×Q8—61no—
284C3×Q8—61no—
285C3×Q8(C2×D3)⋊C414yes—
286C3×Q8C3×C22⋊C414yes—
287C3×D4—61no—
288C3×D4—61no—
289C3×D4—121no—
290C3×D4—121no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—61no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4C2×C4×D311yescharacteristic
315C2×C3×C4C2×D1211yescharacteristic
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4#23711yescharacteristic
318C2×C3×C4#23711yescharacteristic
319C2×C3×C4C2×#6711yescharacteristic
320C2×C3×C4C22×C3×C411yescharacteristic
321C2×C3×C4C2×C3×D411yescharacteristic
322C2×C3×C4C2×C3×D411yescharacteristic
323C2×C3×C4C2×C3×D411yescharacteristic
324C2×C3×C4C3×#4111yescharacteristic
325C2×C3×C4C3×#4111yescharacteristic
326#67—41no—
327#67—41no—
328C2×Dic3—21no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331D12—21no—
332D12—21no—
333C4×D3—41no—
334C4×D3—41no—
335Dic6—21no—
336Dic6—21no—
337C3×C8—42no—
338C3×C8—41no—
339C3×C8—41no—
340C3×C8—62no—
341C3⋊C8—22no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32#38611yescharacteristic
346C3×D3—81no—
347#41—61no—
348#41—61no—
349#41—61no—
350#41—61no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—31no—
355C2×Q8C3×C4×D312yes—
356C2×Q8C3×D1212yes—
357C2×Q8C3×#6712yes—
358C2×D4—31no—
359C2×D4—61no—
360C22×C4—21no—
361C22×C4—61no—
362C22×C4C22×C3×D311yescharacteristic
363Q16—64no—
364SD16—64no—
365M16—21no—
366M16—21no—
367M16—61no—
368M16—61no—
369C2×C8—21no—
370C2×C8—32no—
371C4⋊C4—21no—
372C4⋊C4—64no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C22⋊C4—21no—
376C22⋊C4—32no—
377C22⋊C4—61no—
378C42—21no—
379C42—32no—
380C42—64no—
381C42—61no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—41no—
386C22×C3—41no—
387C22×C3—42no—
388C22×C3—121no—
389C22×C3—241no—
390C22×C3#67⋊C411yescharacteristic
391C22×C3#88211yescharacteristic
392C22×C3#93911yescharacteristic
393C22×C3C3×C4×D411yescharacteristic
394C22×C3C3×#12011yescharacteristic
395C22×C3C3×#12111yescharacteristic
396C2×D3—41no—
397C2×D3—81no—
398C3×C4—24no—
399C3×C4—24no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—44no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4—81no—
408C3×C4—122no—
409C3×C4—122no—
410C3×C4—121no—
411C3×C4#67⋊C411yescharacteristic
412C3×C4#88611yescharacteristic
413C3×C4C2×(C2×D3)⋊C411yescharacteristic
414C3×C4#93911yescharacteristic
415C3×C4C2×C3×C22⋊C411yescharacteristic
416C3×C4C3×C4×D411yescharacteristic
417C3×C4C3×#12111yescharacteristic
418C3×C4C3×#12311yescharacteristic
419Dic3—42no—
420Dic3—42no—
421Dic3—41no—
422C32#190511yescharacteristic, complemented, Sylow
423C23—61no—
424C23C3×D3×D411yescharacteristic
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—61no—
429Q8—61no—
430Q8C3×(C2×D3)⋊C414yes—
431D4—61no—
432D4—61no—
433D4—121no—
434D4—121no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—61no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—121no—
443C2×C4—121no—
444C2×C4C2×C3×C4×D311yescharacteristic
445C2×C4C2×C3×D1211yescharacteristic
446C2×C4C3×D3×D411yescharacteristic
447C2×C4C3×#23711yescharacteristic
448C2×C4C3×#23711yescharacteristic
449C2×C4C2×C3×#6711yescharacteristic
450C8—41no—
451C8—62no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—41no—
461C2×C3—41no—
462C2×C3—241no—
463C2×C3#495811yescharacteristic
464C2×C3C3×#38611yescharacteristic
465D3—81no—
466C22—41no—
467C22—121no—
468C22—241no—
469C22C3×#67⋊C411yescharacteristic
470C22C3×#88211yescharacteristic
471C22C3×#93911yescharacteristic
472C4—24no—
473C4—81no—
474C4—122no—
475C4—122no—
476C4—121no—
477C4C3×#67⋊C411yescharacteristic
478C4C3×#88611yescharacteristic
479C4C2×C3×(C2×D3)⋊C411yescharacteristic
480C4C3×#93911yescharacteristic
481C3—21no—
482C3#7033411yescharacteristic, direct product factor
483C3C3×#190511yescharacteristic, complemented
484C2—21no—
485C2—21no—
486C2—21no—
487C2—241no—
488C2C3×#495811yescharacteristic
subgroupG/H
centerC2×C3#4958
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#419C2
socleC2×C32#386
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#419D3
hypocenterC3C3×#1905
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#419

Complements

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

NH ≅ G/Ntype
1. C3×#5812C2semidirect
3. C32×#419C2semidirect
6. C3×#4559C2semidirect
7. C3×#4531C2semidirect
9. #70334C3direct (N × H)
16. C2×C3×#216C4semidirect
19. C2×C3×#215C4semidirect
29. #5812C2×C3semidirect
33. C3×#419D3semidirect
34. C3×#419C2×C3semidirect
40. #4559C2×C3semidirect
41. #4531C2×C3semidirect
101. C2×#216C3×C4semidirect
104. C2×#215C3×C4semidirect
148. #419C3×D3semidirect
422. C32#1905semidirect
482. C3#70334direct (N × H)
483. C3C3×#1905semidirect

Maximal subgroups

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

subgroupindex
C3×#58122
C2×C3×#9422
C32×#4192
C3×#51392
C3×#50062
C3×#45592
C3×#45312
C3×#19053
#703343