Finite Group Databasebeta

← C3×#69027 #11788464

Subgroups — C3×#69027 #11788464

Order 1152.

737 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#908C211yesmaximal, characteristic
2C3×#5521C211yesmaximal, characteristic, complemented
3C3×#5139C211yesmaximal, characteristic, complemented
4C3×#5131C211yesmaximal, characteristic, complemented
5C3×#5006C211yesmaximal, characteristic, complemented
6C32×#344C211yesmaximal, characteristic, complemented
7C3×#4557C211yesmaximal, characteristic, complemented
8C3×#1904—31nomaximal
9#69027C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C22×C3×D12C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C32×#117C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C3×#926C2211yescharacteristic
17C3×#908—22no—
18C3×#908—22no—
19C2×C3×D24C412yescomplemented
20C2×C3×#204C412yescomplemented
21C3×#893—22no—
22C3×D3×C22⋊C4—22no—
23C3×C4×D12—22no—
24C3×#825—22no—
25C3×#821—22no—
26C3×#806—22no—
27C2×C3×AGL(1,ℤ/8)—31no—
28C3×#518—31no—
29C2×#908C2×C311yescharacteristic
30#5521C2×C311yescharacteristic, complemented
31C3×#421—31no—
32C3×#418—31no—
33C3×#410—31no—
34#5139C2×C311yescharacteristic, complemented
35#5131C2×C311yescharacteristic, complemented
36#5006C2×C311yescharacteristic, complemented
37C3×#344—21no—
38C3×#344D311yescharacteristic, complemented
39C3×#344C2×C311yescharacteristic, complemented
40C3×H(3,ℤ/4)—31no—
41#4557C2×C311yescharacteristic, complemented
42C23×C3×D3—21no—
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—21no—
45C3×#235—21no—
46C3×#235—21no—
47C3×#235—21no—
48C3×#235—21no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12C2×C412yes—
53C2×C3×D12C2×C411yescharacteristic, complemented
54C2×C3×D12D412yes—
55C2×C3×D12D412yes—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—24no—
58C2×C3×Dic6C2×C411yescharacteristic, complemented
59C32×M16—21no—
60C32×M16—21no—
61C32×M16—21no—
62C32×M16—21no—
63C2×C32×C8—21no—
64C2×C32×C8C2×C412yescomplemented
65C32×C4⋊C4—21no—
66C32×C22⋊C4D412yes—
67C32×C42—21no—
68C3×#217—21no—
69C3×#217—21no—
70C3×(C2×D3)⋊C4—24no—
71C3×C4⋊Dic3—21no—
72C3×C4⋊Dic3—21no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3D412yes—
75C3×D24—24no—
76C3×#204—24no—
77#1904—31noSylow
78C2×C3×#41—31no—
79C22×C3×D4—31no—
80C2×#235C22×C311yescharacteristic
81C22×D12C22×C311yescharacteristic
82C3×AGL(1,ℤ/8)—62no—
83C3×AGL(1,ℤ/8)—62no—
84C2×C3×SD16—32no—
85C2×C3×D8—32no—
86C2×C3×M16—21no—
87C2×C3×M16—21no—
88C2×C3×M16C2×D311yescharacteristic
89C2×C3×M16C2×D311yescharacteristic
90C2×C3×M16C22×C311yescharacteristic
91C2×C3×M16C22×C311yescharacteristic
92C3×C4×D4—62no—
93C3×C4×D4—62no—
94C3×#117—21no—
95C3×#117—31no—
96C3×#117—31no—
97C3×#117C2×D311yescharacteristic
98C3×#117C22×C311yescharacteristic
99C2×C3×C22⋊C4—62no—
100#926C22×C311yescharacteristic
101#926C22×C311yescharacteristic
102#908—22no—
103#908—22no—
104C2×D24C3×C412yescomplemented
105C2×#204C3×C412yescomplemented
106#893—22no—
107D3×C22⋊C4—22no—
108C4×D12—22no—
109C3×#104—62no—
110C3×#102—62no—
111C3×#99—62no—
112#825—22no—
113#821—22no—
114#806—22no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3—22no—
117C22×C3×D3—22no—
118C22×C3×D3—22no—
119C22×C3×D3—21no—
120C22×C3×D3—82no—
121C2×C32×C4—22no—
122C2×C32×C4C22×C411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C2×C32×C4#4111yescharacteristic
128C3×#67—41no—
129C3×#67—41no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—22no—
132C2×C3×Dic3—41no—
133C3×D12—21no—
134C3×D12—22no—
135C3×D12—21no—
136C3×D12—22no—
137C3×D12—22no—
138C3×D12C22⋊C414yes—
139C3×C4×D3—41no—
140C3×C4×D3—41no—
141C3×C4×D3—84no—
142C3×Dic6—21no—
143C3×Dic6—21no—
144C32×C8—22no—
145C32×C8—41no—
146C2×AGL(1,ℤ/8)—31no—
147#518—31no—
148#421—31no—
149#418—31no—
150#410—31no—
151#344C3×D311yescharacteristic, complemented
152H(3,ℤ/4)—31no—
153C24×C3—61no—
154C23×D3—21no—
155C3×#41—61no—
156C3×#41—61no—
157C3×#41—61no—
158C3×#41—61no—
159C2×C3×Q8—31no—
160C2×C3×D4—32no—
161C2×C3×D4—31no—
162C2×C3×D4—32no—
163C2×C3×D4—32no—
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C22×C3×C4—21no—
169C22×C3×C4—64no—
170C22×C3×C4—61no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C23×C311yescharacteristic
173C2×#67—21no—
174#235—21no—
175#235—21no—
176#235—21no—
177#235—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12C2×C3×C412yes—
182C2×D12C2×C3×C411yescharacteristic, complemented
183C2×D12C3×D412yes—
184C2×D12C3×D412yes—
185C2×C4×D3—21no—
186C2×C4×D3—24no—
187C2×Dic6C2×C3×C411yescharacteristic, complemented
188C3×SD16—64no—
189C3×D8—64no—
190C3×M16—21no—
191C3×M16—21no—
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—22no—
195C3×M16—21no—
196C3×M16—21no—
197C3×M16—21no—
198C3×M16—21no—
199C3×M16—41no—
200C3×M16—41no—
201C3×M16—41no—
202C2×C3×C8—21no—
203C2×C3×C8—21no—
204C2×C3×C8—22no—
205C2×C3×C8—41no—
206C2×C3×C8C4×D312yescomplemented
207C2×C3×C8C2×C3×C412yescomplemented
208C3×C4⋊C4—21no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—41no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—64no—
215C3×C22⋊C4—61no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4D1212yes—
218C3×C22⋊C4C3×D412yes—
219C3×C42—21no—
220C3×C42—21no—
221C3×C42—32no—
222C3×C42—41no—
223C3×C42—61no—
224#217—21no—
225#217—21no—
226(C2×D3)⋊C4—24no—
227C4⋊Dic3—21no—
228C4⋊Dic3—21no—
229C4×Dic3—21no—
230C4×Dic3C3×D412yes—
231D24—24no—
232#204—24no—
233C22×C32—41no—
234C22×C32C4×D411yescharacteristic
235C22×C32#12011yescharacteristic
236C22×C32#12111yescharacteristic
237C2×C3×D3—24no—
238C2×C3×D3—44no—
239C2×C3×D3—41no—
240C2×C3×D3—82no—
241C2×C3×D3—82no—
242C2×C3×D3—81no—
243C32×C4—42no—
244C32×C4C2×C22⋊C411yescharacteristic
245C32×C4C4×D411yescharacteristic
246C32×C4#12111yescharacteristic
247C32×C4#12311yescharacteristic
248C3×Dic3—42no—
249C3×Dic3—41no—
250C3×Dic3—81no—
251C2×#41—31no—
252C22×D4—31no—
253AGL(1,ℤ/8)—62no—
254AGL(1,ℤ/8)—62no—
255C2×SD16—32no—
256C2×D8—32no—
257C2×M16C2×C3×D311yescharacteristic
258C2×M16C2×C3×D311yescharacteristic
259C4×D4—62no—
260C4×D4—62no—
261#117—31no—
262#117—31no—
263#117C2×C3×D311yescharacteristic
264C2×C22⋊C4—62no—
265#104—62no—
266#102—62no—
267#99—62no—
268C23×C3—21no—
269C23×C3—62no—
270C23×C3—62no—
271C23×C3—62no—
272C23×C3—61no—
273C23×C3—242no—
274C23×C3C2×D1211yescharacteristic
275C23×C3C2×C3×D411yescharacteristic
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—22no—
279C22×D3—21no—
280C22×D3—82no—
281C3×Q8—61no—
282C3×Q8—61no—
283C3×D4—34no—
284C3×D4—61no—
285C3×D4—62no—
286C3×D4—62no—
287C3×D4—61no—
288C3×D4—62no—
289C3×D4—121no—
290C3×D4—121no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—42no—
300C2×C3×C4—61no—
301C2×C3×C4—62no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—244no—
306C2×C3×C4C2×C4×D311yescharacteristic
307C2×C3×C4#23511yescharacteristic
308C2×C3×C4D3×D411yescharacteristic
309C2×C3×C4D3×D411yescharacteristic
310C2×C3×C4D3×D411yescharacteristic
311C2×C3×C4#23911yescharacteristic
312C2×C3×C4C22×C3×C411yescharacteristic
313C2×C3×C4C2×C3×D411yescharacteristic
314C2×C3×C4C2×C3×D411yescharacteristic
315C2×C3×C4C2×C3×D411yescharacteristic
316C2×C3×C4C3×#4111yescharacteristic
317C2×C3×C4C3×#4111yescharacteristic
318#67—41no—
319#67—41no—
320C2×Dic3—21no—
321C2×Dic3—22no—
322C2×Dic3—41no—
323D12—21no—
324D12—22no—
325D12—21no—
326D12—22no—
327D12—22no—
328D12C3×C22⋊C414yes—
329C4×D3—41no—
330C4×D3—41no—
331C4×D3—84no—
332Dic6—21no—
333Dic6—21no—
334C3×C8—22no—
335C3×C8—22no—
336C3×C8—42no—
337C3×C8—42no—
338C3×C8—41no—
339C3×C8—41no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32#38611yescharacteristic
344C3×D3—44no—
345C3×D3—81no—
346C24—61no—
347#41—61no—
348#41—61no—
349#41—61no—
350#41—61no—
351C2×Q8—31no—
352C2×D4—32no—
353C2×D4—31no—
354C2×D4—32no—
355C2×D4—32no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C22×C4—64no—
361C22×C4—61no—
362C22×C4C22×C3×D311yescharacteristic
363SD16—64no—
364D8—64no—
365M16—21no—
366M16—21no—
367M16—21no—
368M16—21no—
369C2×C8—21no—
370C2×C8C3×C4×D312yescomplemented
371C4⋊C4—21no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C22⋊C4—64no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4C3×D1212yes—
378C42—21no—
379C42—32no—
380C42—61no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—64no—
387C22×C3—81no—
388C22×C3—124no—
389C22×C3—121no—
390C22×C3—242no—
391C22×C3—242no—
392C22×C3—241no—
393C22×C3C4×D1211yescharacteristic
394C22×C3#88211yescharacteristic
395C22×C3#89511yescharacteristic
396C22×C3C3×C4×D411yescharacteristic
397C22×C3C3×#12011yescharacteristic
398C22×C3C3×#12111yescharacteristic
399C2×D3—24no—
400C2×D3—44no—
401C2×D3—41no—
402C2×D3—82no—
403C2×D3—82no—
404C2×D3—81no—
405C3×C4—21no—
406C3×C4—21no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—82no—
412C3×C4—122no—
413C3×C4—121no—
414C3×C4—241no—
415C3×C4D3×C22⋊C411yescharacteristic
416C3×C4#88411yescharacteristic
417C3×C4#89311yescharacteristic
418C3×C4#89411yescharacteristic
419C3×C4C2×C3×C22⋊C411yescharacteristic
420C3×C4C3×C4×D411yescharacteristic
421C3×C4C3×#12111yescharacteristic
422C3×C4C3×#12311yescharacteristic
423Dic3—42no—
424Dic3—41no—
425Dic3—81no—
426C32#190411yescharacteristic, complemented, Sylow
427C23—62no—
428C23—62no—
429C23—62no—
430C23—61no—
431C23—242no—
432C23C2×C3×D1211yescharacteristic
433Q8—61no—
434Q8—61no—
435D4—34no—
436D4—61no—
437D4—62no—
438D4—62no—
439D4—61no—
440D4—62no—
441D4—121no—
442D4—121no—
443C2×C4—22no—
444C2×C4—62no—
445C2×C4—61no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—244no—
450C2×C4C2×C3×C4×D311yescharacteristic
451C2×C4C3×#23511yescharacteristic
452C2×C4C3×D3×D411yescharacteristic
453C2×C4C3×D3×D411yescharacteristic
454C2×C4C3×D3×D411yescharacteristic
455C2×C4C3×#23911yescharacteristic
456C8—22no—
457C8—41no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—41no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—124no—
469C2×C3—241no—
470C2×C3#466511yescharacteristic
471C2×C3C3×#38611yescharacteristic
472D3—44no—
473D3—81no—
474C22—41no—
475C22—64no—
476C22—124no—
477C22—121no—
478C22—242no—
479C22—242no—
480C22—241no—
481C22C3×C4×D1211yescharacteristic
482C22C3×#88211yescharacteristic
483C22C3×#89511yescharacteristic
484C4—42no—
485C4—122no—
486C4—121no—
487C4—241no—
488C4C3×D3×C22⋊C411yescharacteristic
489C4C3×#88411yescharacteristic
490C4C3×#89311yescharacteristic
491C4C3×#89411yescharacteristic
492C3—21no—
493C3#6902711yescharacteristic, direct product factor
494C3C3×#190411yescharacteristic, complemented
495C2—21no—
496C2—21no—
497C2—21no—
498C2—124no—
499C2—241no—
500C2C3×#466511yescharacteristic
subgroupG/H
centerC2×C3#4665
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#344C2
socleC2×C32#386
prime radicalC2×C3×#908C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#344D3
hypocenterC3C3×#1904
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#344

Complements

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

NH ≅ G/Ntype
2. C3×#5521C2semidirect
3. C3×#5139C2semidirect
4. C3×#5131C2semidirect
5. C3×#5006C2semidirect
6. C32×#344C2semidirect
7. C3×#4557C2semidirect
9. #69027C3direct (N × H)
19. C2×C3×D24C4semidirect
20. C2×C3×#204C4semidirect
30. #5521C2×C3semidirect
34. #5139C2×C3semidirect
35. #5131C2×C3semidirect
36. #5006C2×C3semidirect
38. C3×#344D3semidirect
39. C3×#344C2×C3semidirect
41. #4557C2×C3semidirect
53. C2×C3×D12C2×C4semidirect
58. C2×C3×Dic6C2×C4semidirect
64. C2×C32×C8C2×C4semidirect
104. C2×D24C3×C4semidirect
105. C2×#204C3×C4semidirect
151. #344C3×D3semidirect
182. C2×D12C2×C3×C4semidirect
187. C2×Dic6C2×C3×C4semidirect
206. C2×C3×C8C4×D3semidirect
207. C2×C3×C8C2×C3×C4semidirect
370. C2×C8C3×C4×D3semidirect
426. C32#1904semidirect
493. C3#69027direct (N × H)
494. C3C3×#1904semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9082
C3×#55212
C3×#51392
C3×#51312
C3×#50062
C32×#3442
C3×#45572
C3×#19043
#690273