Finite Group Databasebeta

← C3×#80767 #11878402

Subgroups — C3×#80767 #11878402

Order 1152.

813 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5628C212yesmaximal
2C32×#467C211yesmaximal, characteristic, complemented
3C3×#5186C212yesmaximal, complemented
4C3×#5116C211yesmaximal, characteristic, complemented
5C3×#5110C211yesmaximal, characteristic
6C3×#5052C212yesmaximal, complemented
7C3×#4888C212yesmaximal, complemented
8C3×#4814C212yesmaximal, complemented
9C3×#4809C212yesmaximal, complemented
10C3×#3202—31nomaximal
11#80767C311yesmaximal, characteristic, direct product factor
12C2×C3×D3×Q8C2212yes—
13C2×C3×#235C2211yescharacteristic
14C2×C32×Q16C2211yescharacteristic
15C22×C32×C8C2211yescharacteristic
16C32×C22⋊Q8C2212yes—
17C3×#946C2212yes—
18C2×C3×#216C2212yes—
19C3×#924C2212yes—
20C2×C3×#208C2211yescharacteristic
21C2×C3×Dic12C2211yescharacteristic
22C3×#897C2212yes—
23C3×#894—22no—
24C3×#892C2212yes—
25C3×#888—22no—
26C3×#885—22no—
27C3×#883—22no—
28C32×#107C2211yescharacteristic
29C32×#103C2212yes—
30C3×#835C2212yes—
31C3×(C2×D3)⋊C8C2212yes—
32C3×(C2×D3)⋊C8C2212yes—
33C3×#818C2211yescharacteristic
34C3×#816C2212yes—
35C3×#810C2212yes—
36C3×#808C2212yes—
37C3×#536—32no—
38#5628C2×C312yes—
39C3×#485—32no—
40C3×#485—32no—
41C3×#470—32no—
42C3×#467—21no—
43C3×#467—31no—
44C3×#467D311yescharacteristic, complemented
45C3×#467C2×C311yescharacteristic, complemented
46C3×#451—32no—
47C3×#451—32no—
48C3×#408—31no—
49#5186C2×C312yescomplemented
50#5116C2×C311yescharacteristic, complemented
51#5110C2×C311yescharacteristic
52#5052C2×C312yescomplemented
53#4888C2×C312yescomplemented
54#4814C2×C312yescomplemented
55#4809C2×C312yescomplemented
56C2×C32×Q8C2312yes—
57C22×C32×C4C2311yescharacteristic
58C2×C3×#67D412yes—
59C3×D3×Q8—42no—
60C3×D3×Q8—42no—
61C3×#235—41no—
62C3×#235—41no—
63C2×C3×D12D411yescharacteristic
64C2×C3×C4×D3—22no—
65C2×C3×C4×D3C2312yes—
66C2×C3×Dic6—22no—
67C2×C3×Dic6D411yescharacteristic
68C2×C3×Dic6C2312yes—
69C32×Q16—41no—
70C2×C32×C8—41no—
71C2×C32×C8C2311yescharacteristic
72C2×C32×C8C2311yescharacteristic
73C32×C4⋊C4—22no—
74C32×C4⋊C4C2312yes—
75C32×C22⋊C4—22no—
76C3×#217—22no—
77C3×#216—42no—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—22no—
80C3×(C2×D3)⋊C4—22no—
81C3×C4⋊Dic3—22no—
82C3×C4⋊Dic3C2312yes—
83C3×Dic3⋊C4—22no—
84C3×Dic3⋊C4—22no—
85C3×C4×Dic3—22no—
86C3×#208—41no—
87C2×C3×C3⋊C8C2312yes—
88C3×Dic12—41no—
89#3202—31noSylow
90C2×C3×#41—31no—
91C22×C3×Q8—32no—
92C2×D3×Q8C22×C312yes—
93C2×#235C22×C311yescharacteristic
94C2×C3×Q16—21no—
95C2×C3×Q16—32no—
96C2×C3×Q16—31no—
97C2×C3×Q16C2×D311yescharacteristic
98C2×C3×Q16C22×C311yescharacteristic
99C2×C3×M16—31no—
100C22×C3×C8—21no—
101C22×C3×C8C2×D311yescharacteristic
102C22×C3×C8C22×C311yescharacteristic
103C3×C4⋊Q8—62no—
104C3×#124—62no—
105C3×#123—62no—
106C3×#123—62no—
107C3×C22⋊Q8—22no—
108C3×C22⋊Q8—32no—
109C3×C22⋊Q8—32no—
110C3×C22⋊Q8—32no—
111C3×C22⋊Q8C2×D312yes—
112C3×C22⋊Q8C22×C312yes—
113C3×#117—32no—
114#946C22×C312yes—
115C2×#216C22×C312yes—
116#924C22×C312yes—
117C2×#208C22×C311yescharacteristic
118C2×Dic12C22×C311yescharacteristic
119#897C22×C312yes—
120#894—22no—
121#892C22×C312yes—
122#888—22no—
123#885—22no—
124#883—22no—
125C3×#107—21no—
126C3×#107—31no—
127C3×#107C2×D311yescharacteristic
128C3×#107C22×C311yescharacteristic
129C3×#106—32no—
130C3×#103—22no—
131C3×#103—32no—
132C3×#103—32no—
133C3×#103—32no—
134C3×#103C2×D312yes—
135C3×#103C22×C312yes—
136C3×C22⋊C8—32no—
137C3×C22⋊C8—32no—
138#835C22×C312yes—
139(C2×D3)⋊C8C22×C312yes—
140(C2×D3)⋊C8C22×C312yes—
141#818C22×C311yescharacteristic
142#816C22×C312yes—
143#810C22×C312yes—
144#808C22×C312yes—
145C23×C32C2×D411yescharacteristic
146C22×C3×D3C2×D412yes—
147C32×Q8—22no—
148C32×Q8—22no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4C2×D411yescharacteristic
154C2×C32×C4C2411yescharacteristic
155C3×#67—42no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3C2×D412yes—
159C3×D12—41no—
160C3×C4×D3—22no—
161C3×C4×D3—22no—
162C3×C4×D3—44no—
163C3×Dic6—22no—
164C3×Dic6—22no—
165C3×Dic6—41no—
166C3×Dic6—44no—
167C32×C8—21no—
168C32×C8—21no—
169C3×C3⋊C8—22no—
170#536—32no—
171#485—32no—
172#485—32no—
173#470—32no—
174#467—31no—
175#467C3×D311yescharacteristic, complemented
176#451—32no—
177#451—32no—
178#408—31no—
179C3×#41—121no—
180C3×#41—121no—
181C2×C3×Q8—22no—
182C2×C3×Q8—31no—
183C2×C3×Q8—32no—
184C2×C3×Q8—62no—
185C2×C3×Q8—122no—
186C2×C3×Q8—122no—
187C2×C3×Q8C22×D312yes—
188C2×C3×Q8C23×C312yes—
189C2×C3×D4—32no—
190C2×C3×D4—31no—
191C22×C3×C4—21no—
192C22×C3×C4—32no—
193C22×C3×C4—62no—
194C22×C3×C4C22×D311yescharacteristic
195C22×C3×C4C23×C311yescharacteristic
196C2×#67C3×D412yes—
197D3×Q8—42no—
198D3×Q8—42no—
199#235—41no—
200#235—41no—
201C2×D12C3×D411yescharacteristic
202C2×C4×D3—22no—
203C2×C4×D3C23×C312yes—
204C2×Dic6—22no—
205C2×Dic6C3×D411yescharacteristic
206C2×Dic6C23×C312yes—
207C3×Q16—41no—
208C3×Q16—41no—
209C3×Q16—42no—
210C3×Q16—121no—
211C3×Q16—122no—
212C3×M16—121no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—32no—
216C2×C3×C8—41no—
217C2×C3×C8—41no—
218C2×C3×C8—42no—
219C2×C3×C8C22×D311yescharacteristic
220C2×C3×C8C22×D311yescharacteristic
221C2×C3×C8C23×C311yescharacteristic
222C2×C3×C8C23×C311yescharacteristic
223C3×C4⋊C4—22no—
224C3×C4⋊C4—22no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—32no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—62no—
229C3×C4⋊C4—62no—
230C3×C4⋊C4—62no—
231C3×C4⋊C4C22×D312yes—
232C3×C4⋊C4C23×C312yes—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—42no—
236C3×C22⋊C4—62no—
237C3×C22⋊C4—62no—
238C3×C22⋊C4—62no—
239C3×C22⋊C4—62no—
240C3×C42—62no—
241#217—22no—
242#216—42no—
243(C2×D3)⋊C4—22no—
244(C2×D3)⋊C4—22no—
245(C2×D3)⋊C4—22no—
246C4⋊Dic3—22no—
247C4⋊Dic3C23×C312yes—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250C4×Dic3—22no—
251#208—41no—
252C2×C3⋊C8C23×C312yes—
253Dic12—41no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32C22×D411yescharacteristic
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C32×C4—21no—
262C32×C4—42no—
263C32×C4—42no—
264C32×C4H(5,𝔽2)11yescharacteristic
265C32×C4H(5,𝔽2)11yescharacteristic
266C3×Dic3—22no—
267C3×Dic3—42no—
268C3×Dic3—42no—
269C2×#41—31no—
270C22×Q8—32no—
271C2×Q16—32no—
272C2×Q16—31no—
273C2×Q16C2×C3×D311yescharacteristic
274C2×M16—31no—
275C22×C8C2×C3×D311yescharacteristic
276C4⋊Q8—62no—
277#124—62no—
278#123—62no—
279#123—62no—
280C22⋊Q8—32no—
281C22⋊Q8—32no—
282C22⋊Q8—32no—
283C22⋊Q8C2×C3×D312yes—
284#117—32no—
285#107—31no—
286#107C2×C3×D311yescharacteristic
287#106—32no—
288#103—32no—
289#103—32no—
290#103—32no—
291#103C2×C3×D312yes—
292C22⋊C8—32no—
293C22⋊C8—32no—
294C23×C3—21no—
295C23×C3—32no—
296C23×C3D3×D411yescharacteristic
297C23×C3C2×C3×D411yescharacteristic
298C22×D3C2×C3×D412yes—
299C3×Q8—22no—
300C3×Q8—22no—
301C3×Q8—22no—
302C3×Q8—22no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—62no—
306C3×Q8—62no—
307C3×Q8—121no—
308C3×Q8—124no—
309C3×D4—121no—
310C3×D4—122no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—32no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—124no—
331C2×C3×C4D3×D411yescharacteristic
332C2×C3×C4C2×C3×D411yescharacteristic
333C2×C3×C4C23×D311yescharacteristic
334C2×C3×C4C24×C311yescharacteristic
335#67—42no—
336C2×Dic3—22no—
337C2×Dic3—22no—
338C2×Dic3C2×C3×D412yes—
339D12—41no—
340C4×D3—22no—
341C4×D3—22no—
342C4×D3—44no—
343Dic6—22no—
344Dic6—22no—
345Dic6—41no—
346Dic6—44no—
347C3×C8—22no—
348C3×C8—21no—
349C3×C8—21no—
350C3×C8—21no—
351C3×C8—21no—
352C3×C8—41no—
353C3×C8—62no—
354C3⋊C8—22no—
355C2×C32—41no—
356C2×C32#53411yescharacteristic
357C2×C32#57811yescharacteristic
358C2×C32#57811yescharacteristic
359C3×D3—42no—
360#41—121no—
361#41—121no—
362C2×Q8—31no—
363C2×Q8—32no—
364C2×Q8—62no—
365C2×Q8—122no—
366C2×Q8—122no—
367C2×Q8C22×C3×D312yes—
368C2×D4—32no—
369C2×D4—31no—
370C22×C4—32no—
371C22×C4—62no—
372C22×C4C22×C3×D311yescharacteristic
373Q16—41no—
374Q16—121no—
375Q16—122no—
376M16—121no—
377C2×C8—32no—
378C2×C8—41no—
379C2×C8C22×C3×D311yescharacteristic
380C2×C8C22×C3×D311yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—32no—
383C4⋊C4—62no—
384C4⋊C4—62no—
385C4⋊C4—62no—
386C4⋊C4C22×C3×D312yes—
387C22⋊C4—22no—
388C22⋊C4—62no—
389C22⋊C4—62no—
390C22⋊C4—62no—
391C22⋊C4—62no—
392C42—62no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3C2×D3×D411yescharacteristic
407C22×C3C22×C3×D411yescharacteristic
408C2×D3—22no—
409C2×D3—22no—
410C2×D3—22no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—41no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—62no—
421C3×C4—82no—
422C3×C4—82no—
423C3×C4—122no—
424C3×C4—122no—
425C3×C4#100411yescharacteristic
426C3×C4#100911yescharacteristic
427C3×C4C3×H(5,𝔽2)11yescharacteristic
428C3×C4C3×H(5,𝔽2)11yescharacteristic
429Dic3—22no—
430Dic3—42no—
431Dic3—42no—
432C32#320211yescharacteristic, complemented, Sylow
433C23—32no—
434C23C3×D3×D411yescharacteristic
435Q8—22no—
436Q8—22no—
437Q8—62no—
438Q8—62no—
439Q8—121no—
440Q8—124no—
441D4—121no—
442D4—122no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—32no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—124no—
453C2×C4C3×D3×D411yescharacteristic
454C2×C4C23×C3×D311yescharacteristic
455C8—21no—
456C8—21no—
457C8—62no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—42no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—122no—
465C2×C3#560511yescharacteristic
466C2×C3#576611yescharacteristic
467C2×C3#577111yescharacteristic
468C2×C3C3×#53411yescharacteristic
469C2×C3C3×#57811yescharacteristic
470C2×C3C3×#57811yescharacteristic
471D3—42no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22C2×C3×D3×D411yescharacteristic
479C4—21no—
480C4—42no—
481C4—42no—
482C4—62no—
483C4—122no—
484C4—122no—
485C4C3×#100411yescharacteristic
486C4C3×#100911yescharacteristic
487C3—21no—
488C3#8076711yescharacteristic, direct product factor
489C3C3×#320211yescharacteristic, complemented
490C2—41no—
491C2—122no—
492C2C3×#560511yescharacteristic
493C2C3×#576611yescharacteristic
494C2C3×#577111yescharacteristic
subgroupG/H
centerC22×C3C2×D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC32×#467C2
socleC22×C32C22×D4
prime radicalC2×C3×#235C22
hypercenterC3×#467D3
hypocenterC3C3×#3202
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#467

Complements

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

NH ≅ G/Ntype
2. C32×#467C2semidirect
3. C3×#5186C2semidirect
4. C3×#5116C2semidirect
6. C3×#5052C2semidirect
7. C3×#4888C2semidirect
8. C3×#4814C2semidirect
9. C3×#4809C2semidirect
11. #80767C3direct (N × H)
44. C3×#467D3semidirect
45. C3×#467C2×C3semidirect
49. #5186C2×C3semidirect
50. #5116C2×C3semidirect
52. #5052C2×C3semidirect
53. #4888C2×C3semidirect
54. #4814C2×C3semidirect
55. #4809C2×C3semidirect
175. #467C3×D3semidirect
432. C32#3202semidirect
488. C3#80767direct (N × H)
489. C3C3×#3202semidirect

Maximal subgroups

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

subgroupindex
C3×#56282
C32×#4672
C3×#51862
C3×#51162
C3×#51102
C3×#50522
C3×#48882
C3×#48142
C3×#48092
C3×#32023
#807673