Finite Group Databasebeta

← C3×#76298 #11803849

Subgroups — C3×#76298 #11803849

Order 1152.

1609 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#940C211yesmaximal, characteristic
2C2×C3×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#893C211yesmaximal, characteristic, complemented
4C32×#390C211yesmaximal, characteristic, complemented
5C3×#4977C211yesmaximal, characteristic, complemented
6C3×#4958C214yesmaximal, complemented
7C3×#4956C214yesmaximal, complemented
8C3×#4939C211yesmaximal, characteristic, complemented
9C3×#4932C211yesmaximal, characteristic, complemented
10C3×#2328—31nomaximal
11#76298C311yesmaximal, characteristic, direct product factor
12C22×C32×D4C2211yescharacteristic
13C22×C3×#67C2214yes—
14C23×C3×Dic3C2212yes—
15C22×C3×D12C2211yescharacteristic
16C22×C3×C4×D3C2212yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4C2214yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×#217C2212yes—
21C3×#940C418yescomplemented
22C3×D4×Dic3—24no—
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×C4×Dic3C2211yescharacteristic
27C2×C3×C4×Dic3C2211yescharacteristic
28C3×#893—24no—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2214yes—
32C2×C3×#127—31no—
33C2×C3×C4×D4—31no—
34C2×C3×C4×D4—31no—
35C2×#940C2×C311yescharacteristic
36C2×D4×Dic3C2×C311yescharacteristic, complemented
37C2×#893C2×C311yescharacteristic, complemented
38C3×#390—21no—
39C3×#390—31no—
40C3×#390D311yescharacteristic, complemented
41C3×#390C2×C311yescharacteristic, complemented
42C3×#386—34no—
43C3×#386—34no—
44C3×#382—31no—
45C3×#376—31no—
46#4977C2×C311yescharacteristic, complemented
47#4958C2×C314yescomplemented
48#4956C2×C314yescomplemented
49#4939C2×C311yescharacteristic, complemented
50#4932C2×C311yescharacteristic, complemented
51C24×C32C2312yes—
52C23×C3×D3C2312yes—
53C2×C32×D4—22no—
54C2×C32×D4—22no—
55C2×C32×D4C2×C414yescomplemented
56C22×C32×C4C2312yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2311yescharacteristic
59C2×C3×#67—44no—
60C2×C3×#67—44no—
61C2×C3×#67C2×C4116yescomplemented
62C22×C3×Dic3—24no—
63C22×C3×Dic3—44no—
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2311yescharacteristic
67C22×C3×Dic3C2311yescharacteristic
68C2×C3×D12—22no—
69C2×C3×D12—22no—
70C2×C3×D12C2×C414yescomplemented
71C2×C3×C4×D3—24no—
72C2×C3×C4×D3—44no—
73C32×C4⋊C4—22no—
74C32×C22⋊C4—44no—
75C32×C42—22no—
76C3×#217—24no—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—44no—
79C3×C4⋊Dic3—22no—
80C3×C4×Dic3C2×C414yescomplemented
81C3×C4×Dic3D414yescomplemented
82#2328—31noSylow
83C22×C3×D4—21no—
84C22×C3×D4—34no—
85C22×C3×D4—31no—
86C22×C3×D4C2×D311yescharacteristic
87C22×C3×D4C22×C311yescharacteristic
88C23×C3×C4—32no—
89C23×C3×C4—32no—
90C22×#67C22×C314yes—
91C23×Dic3C22×C312yes—
92C22×D12C22×C311yescharacteristic
93C22×C4×D3C22×C312yes—
94C3×#127—38no—
95C3×C4×D4—64no—
96C3×C4×D4—64no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4C2×D311yescharacteristic
100C2×C3×C4⋊C4C22×C311yescharacteristic
101C2×C3×C22⋊C4—24no—
102C2×C3×C22⋊C4—32no—
103C2×C3×C22⋊C4—32no—
104C2×C3×C22⋊C4—34no—
105C2×C3×C22⋊C4C2×D314yes—
106C2×C3×C22⋊C4C22×C314yes—
107C2×C3×C42—21no—
108C2×C3×C42—31no—
109C2×C3×C42—31no—
110C2×C3×C42C2×D311yescharacteristic
111C2×C3×C42C22×C311yescharacteristic
112C2×#217C22×C312yes—
113#940C3×C418yescomplemented
114D4×Dic3—24no—
115C2×(C2×D3)⋊C4C22×C312yes—
116C2×(C2×D3)⋊C4C22×C314yes—
117C2×C4⋊Dic3C22×C311yescharacteristic
118C2×C4×Dic3C22×C311yescharacteristic
119C2×C4×Dic3C22×C311yescharacteristic
120#893—24no—
121C3×#95—32no—
122C3×#95—32no—
123C3×#95—34no—
124C3⋊#95C22×C312yes—
125C3⋊#95C22×C312yes—
126C3⋊#95C22×C314yes—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—24no—
130C23×C32—24no—
131C23×C32C22×C414yes—
132C23×C32C2411yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—24no—
136C22×C3×D3—24no—
137C22×C3×D3C22×C414yes—
138C32×D4—28no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—24no—
144C2×C32×C4C22×C412yes—
145C2×C32×C4C2×D412yes—
146C2×C32×C4#4112yes—
147C3×#67—416no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—48no—
155C2×C3×Dic3C22×C414yes—
156C2×C3×Dic3C2×D412yes—
157C2×C3×Dic3C2×D412yes—
158C3×D12—28no—
159C3×C4×D3—48no—
160C2×#127—31no—
161C2×C4×D4—31no—
162C2×C4×D4—31no—
163#390—31no—
164#390C3×D311yescharacteristic, complemented
165#386—34no—
166#386—34no—
167#382—31no—
168#376—31no—
169C24×C3—22no—
170C24×C3—32no—
171C24×C3C22×D312yes—
172C24×C3C23×C312yes—
173C23×D3C23×C312yes—
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—316no—
180C2×C3×D4—34no—
181C2×C3×D4—42no—
182C2×C3×D4—42no—
183C2×C3×D4—62no—
184C2×C3×D4—62no—
185C2×C3×D4—124no—
186C2×C3×D4—124no—
187C2×C3×D4C4×D314yescomplemented
188C2×C3×D4C2×C3×C414yescomplemented
189C22×C3×C4—21no—
190C22×C3×C4—22no—
191C22×C3×C4—22no—
192C22×C3×C4—32no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—32no—
196C22×C3×C4—64no—
197C22×C3×C4—64no—
198C22×C3×C4—124no—
199C22×C3×C4—124no—
200C22×C3×C4C22×D312yes—
201C22×C3×C4C22×D312yes—
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C23×C312yes—
204C22×C3×C4C23×C312yes—
205C22×C3×C4C23×C311yescharacteristic
206C2×#67—44no—
207C2×#67—44no—
208C2×#67C2×C3×C4116yescomplemented
209C22×Dic3—24no—
210C22×Dic3—44no—
211C22×Dic3C23×C312yes—
212C22×Dic3C23×C312yes—
213C22×Dic3C23×C311yescharacteristic
214C22×Dic3C23×C311yescharacteristic
215C2×D12—22no—
216C2×D12—22no—
217C2×D12C2×C3×C414yescomplemented
218C2×C4×D3—24no—
219C2×C4×D3—44no—
220C3×C4⋊C4—22no—
221C3×C4⋊C4—22no—
222C3×C4⋊C4—42no—
223C3×C4⋊C4—62no—
224C3×C22⋊C4—44no—
225C3×C22⋊C4—44no—
226C3×C22⋊C4—48no—
227C3×C22⋊C4—64no—
228C3×C22⋊C4—64no—
229C3×C22⋊C4—124no—
230C3×C42—22no—
231C3×C42—22no—
232C3×C42—34no—
233C3×C42—34no—
234C3×C42—42no—
235#217—24no—
236(C2×D3)⋊C4—24no—
237(C2×D3)⋊C4—44no—
238C4⋊Dic3—22no—
239C4×Dic3C2×C3×C414yescomplemented
240C4×Dic3C3×D414yescomplemented
241C22×C32—24no—
242C22×C32—28no—
243C22×C32—42no—
244C22×C32—42no—
245C22×C32—44no—
246C22×C32C23×C411yescharacteristic
247C22×C32C22×D411yescharacteristic
248C22×C32C2×#4111yescharacteristic
249C22×C32H(5,𝔽2)12yes—
250C22×C32H(5,𝔽2)12yes—
251C2×C3×D3—24no—
252C2×C3×D3—28no—
253C2×C3×D3—42no—
254C2×C3×D3—42no—
255C2×C3×D3—44no—
256C32×C4—42no—
257C32×C4—42no—
258C32×C4C4×D414yes—
259C3×Dic3—24no—
260C3×Dic3—22no—
261C3×Dic3—22no—
262C3×Dic3—42no—
263C22×D4—34no—
264C22×D4—31no—
265C22×D4C2×C3×D311yescharacteristic
266C23×C4—32no—
267C23×C4—32no—
268#127—38no—
269C4×D4—64no—
270C4×D4—64no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4C2×C3×D311yescharacteristic
273C2×C22⋊C4—32no—
274C2×C22⋊C4—32no—
275C2×C22⋊C4—34no—
276C2×C22⋊C4C2×C3×D314yes—
277C2×C42—31no—
278C2×C42—31no—
279C2×C42C2×C3×D311yescharacteristic
280#95—32no—
281#95—32no—
282#95—34no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—21no—
292C23×C3—24no—
293C23×C3—34no—
294C23×C3—44no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—44no—
298C23×C3—62no—
299C23×C3—62no—
300C23×C3—64no—
301C23×C3—64no—
302C23×C3C2×C4×D314yes—
303C23×C3C22×C3×C414yes—
304C23×C3C23×D311yescharacteristic
305C23×C3C24×C311yescharacteristic
306C22×D3—22no—
307C22×D3—22no—
308C22×D3—24no—
309C22×D3—24no—
310C22×D3C22×C3×C414yes—
311C3×D4—28no—
312C3×D4—28no—
313C3×D4—48no—
314C3×D4—68no—
315C3×D4—1216no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—22no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—34no—
331C2×C3×C4—32no—
332C2×C3×C4—32no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4—42no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—62no—
341C2×C3×C4—62no—
342C2×C3×C4—64no—
343C2×C3×C4—128no—
344C2×C3×C4—128no—
345C2×C3×C4C2×C4×D312yes—
346C2×C3×C4D3×D412yes—
347C2×C3×C4#23712yes—
348C2×C3×C4C22×C3×C412yes—
349C2×C3×C4C2×C3×D412yes—
350C2×C3×C4C3×#4112yes—
351#67—416no—
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—22no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—48no—
359C2×Dic3C22×C3×C414yes—
360C2×Dic3C2×C3×D412yes—
361C2×Dic3C2×C3×D412yes—
362D12—28no—
363C4×D3—48no—
364C2×C32—44no—
365C2×C32C2×C4×D411yescharacteristic
366C2×C32#51812yes—
367C2×C32#53511yescharacteristic
368C2×C32#53511yescharacteristic
369C2×C32#54012yes—
370C3×D3—44no—
371C24—32no—
372C24C22×C3×D312yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—316no—
376C2×D4—34no—
377C2×D4—62no—
378C2×D4—62no—
379C2×D4—124no—
380C2×D4—124no—
381C2×D4C3×C4×D314yescomplemented
382C22×C4—32no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—32no—
386C22×C4—64no—
387C22×C4—64no—
388C22×C4—124no—
389C22×C4—124no—
390C22×C4C22×C3×D312yes—
391C22×C4C22×C3×D312yes—
392C22×C4C22×C3×D311yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—62no—
395C22⋊C4—44no—
396C22⋊C4—64no—
397C22⋊C4—64no—
398C22⋊C4—124no—
399C42—22no—
400C42—34no—
401C42—34no—
402C22×C3—24no—
403C22×C3—28no—
404C22×C3—24no—
405C22×C3—28no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—22no—
411C22×C3—44no—
412C22×C3—48no—
413C22×C3—42no—
414C22×C3—42no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—44no—
418C22×C3—44no—
419C22×C3—64no—
420C22×C3—68no—
421C22×C3—84no—
422C22×C3—82no—
423C22×C3—82no—
424C22×C3—122no—
425C22×C3—122no—
426C22×C3—124no—
427C22×C3C22×C4×D311yescharacteristic
428C22×C3C2×D3×D411yescharacteristic
429C22×C3C2×#23711yescharacteristic
430C22×C3#100412yes—
431C22×C3#100912yes—
432C22×C3C23×C3×C411yescharacteristic
433C22×C3C22×C3×D411yescharacteristic
434C22×C3C2×C3×#4111yescharacteristic
435C22×C3C3×H(5,𝔽2)12yes—
436C22×C3C3×H(5,𝔽2)12yes—
437C2×D3—24no—
438C2×D3—28no—
439C2×D3—42no—
440C2×D3—42no—
441C2×D3—44no—
442C3×C4—24no—
443C3×C4—44no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—64no—
451C3×C4—82no—
452C3×C4—122no—
453C3×C4#67⋊C414yes—
454C3×C4C3×C4×D414yes—
455Dic3—24no—
456Dic3—22no—
457Dic3—22no—
458Dic3—42no—
459C32#232811yescharacteristic, complemented, Sylow
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—34no—
465C23—62no—
466C23—62no—
467C23—64no—
468C23—64no—
469C23C2×C3×C4×D314yes—
470C23C23×C3×D311yescharacteristic
471D4—28no—
472D4—68no—
473D4—1216no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—34no—
480C2×C4—32no—
481C2×C4—32no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—128no—
489C2×C4—128no—
490C2×C4C2×C3×C4×D312yes—
491C2×C4C3×D3×D412yes—
492C2×C4C3×#23712yes—
493C2×C3—22no—
494C2×C3—22no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—44no—
499C2×C3—44no—
500C2×C3—84no—
501C2×C3—124no—
502C2×C3C2×#67⋊C411yescharacteristic
503C2×C3#554512yes—
504C2×C3#561912yes—
505C2×C3#567411yescharacteristic
506C2×C3#571511yescharacteristic
507C2×C3C2×C3×C4×D411yescharacteristic
508C2×C3C3×#51812yes—
509C2×C3C3×#53511yescharacteristic
510C2×C3C3×#53511yescharacteristic
511C2×C3C3×#54012yes—
512D3—44no—
513C22—28no—
514C22—24no—
515C22—44no—
516C22—42no—
517C22—42no—
518C22—68no—
519C22—64no—
520C22—124no—
521C22—122no—
522C22—122no—
523C22C22×C3×C4×D311yescharacteristic
524C22C2×C3×D3×D411yescharacteristic
525C22C2×C3×#23711yescharacteristic
526C22C3×#100412yes—
527C22C3×#100912yes—
528C4—42no—
529C4—42no—
530C4—64no—
531C4—62no—
532C4—62no—
533C4—122no—
534C4C3×#67⋊C414yes—
535C3—21no—
536C3#7629811yescharacteristic, direct product factor
537C3C3×#232811yescharacteristic, complemented
538C2—44no—
539C2—124no—
540C2C2×C3×#67⋊C411yescharacteristic
541C2C3×#554512yes—
542C2C3×#561912yes—
543C2C3×#567411yescharacteristic
544C2C3×#571511yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#390C2
socleC23×C32C24
prime radicalC2×C3×#940C2
cosocleC23×C3C24×C3
hypercenterC3×#390D3
hypocenterC3C3×#2328
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#390

Complements

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

NH ≅ G/Ntype
2. C2×C3×D4×Dic3C2semidirect
3. C2×C3×#893C2semidirect
4. C32×#390C2semidirect
5. C3×#4977C2semidirect
6. C3×#4958C2semidirect
7. C3×#4956C2semidirect
8. C3×#4939C2semidirect
9. C3×#4932C2semidirect
11. #76298C3direct (N × H)
21. C3×#940C4semidirect
36. C2×D4×Dic3C2×C3semidirect
37. C2×#893C2×C3semidirect
40. C3×#390D3semidirect
41. C3×#390C2×C3semidirect
46. #4977C2×C3semidirect
47. #4958C2×C3semidirect
48. #4956C2×C3semidirect
49. #4939C2×C3semidirect
50. #4932C2×C3semidirect
55. C2×C32×D4C2×C4semidirect
61. C2×C3×#67C2×C4semidirect
70. C2×C3×D12C2×C4semidirect
80. C3×C4×Dic3C2×C4semidirect
81. C3×C4×Dic3D4semidirect
113. #940C3×C4semidirect
164. #390C3×D3semidirect
187. C2×C3×D4C4×D3semidirect
188. C2×C3×D4C2×C3×C4semidirect
208. C2×#67C2×C3×C4semidirect
217. C2×D12C2×C3×C4semidirect
239. C4×Dic3C2×C3×C4semidirect
240. C4×Dic3C3×D4semidirect
381. C2×D4C3×C4×D3semidirect
459. C32#2328semidirect
536. C3#76298direct (N × H)
537. C3C3×#2328semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9402
C2×C3×D4×Dic32
C2×C3×#8932
C32×#3902
C3×#49772
C3×#49582
C3×#49562
C3×#49392
C3×#49322
C3×#23283
#762983