Finite Group Databasebeta

← C3×#74226 #11803033

Subgroups — C3×#74226 #11803033

Order 1152.

1213 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×#382C211yesmaximal, characteristic, complemented
3C3×#4990C212yesmaximal, complemented
4C3×#4989C212yesmaximal, complemented
5C3×#4937C211yesmaximal, characteristic, complemented
6C3×#4667C211yesmaximal, characteristic, complemented
7C3×#4666C211yesmaximal, characteristic, complemented
8C3×#4665C212yesmaximal
9C3×#4664C211yesmaximal, characteristic, complemented
10C3×D3×#95C211yesmaximal, characteristic, complemented
11C3×#4653C212yesmaximal, complemented
12C3×#2318—31nomaximal
13#74226C311yesmaximal, characteristic, direct product factor
14C22×C3×D12C2211yescharacteristic
15C22×C3×C4×D3C2212yes—
16C22×C3×C4×D3C2211yescharacteristic
17C22×C3×C4×D3C2211yescharacteristic
18C2×C32×C4⋊C4C2212yes—
19C2×C32×C42C2211yescharacteristic
20C2×C3×(C2×D3)⋊C4C2212yes—
21C2×C3×(C2×D3)⋊C4C2212yes—
22C2×C3×(C2×D3)⋊C4C2212yes—
23C2×C3×(C2×D3)⋊C4C2211yescharacteristic
24C2×C3×(C2×D3)⋊C4C2211yescharacteristic
25C2×C3×C4⋊Dic3C2211yescharacteristic
26C2×C3×Dic3⋊C4C2212yes—
27C2×C3×C4×Dic3C2212yes—
28C3×C4×D12C418yescomplemented
29C32×#95C2212yes—
30C32×#95C2211yescharacteristic
31C32×#95C2211yescharacteristic
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36C3×C3⋊#95C2211yescharacteristic
37C3×C3⋊#95C2212yes—
38C2×C3×C4×D4—31no—
39C2×C4×D12C2×C311yescharacteristic
40C3×#388—32no—
41C3×#387—32no—
42C3×#386—32no—
43C3×#385—32no—
44C3×#383—31no—
45C3×#382—21no—
46C3×#382D311yescharacteristic, complemented
47C3×#382C2×C311yescharacteristic, complemented
48C3×#381—31no—
49C3×#380—31no—
50C3×C4×C22⋊C4—31no—
51C2×C3×#95—31no—
52#4990C2×C312yescomplemented
53#4989C2×C312yescomplemented
54#4937C2×C311yescharacteristic, complemented
55#4667C2×C311yescharacteristic, complemented
56#4666C2×C311yescharacteristic, complemented
57#4665C2×C312yes—
58#4664C2×C311yescharacteristic, complemented
59D3×#95C2×C311yescharacteristic, complemented
60#4653C2×C312yescomplemented
61C23×C3×D3C2311yescharacteristic
62C23×C3×D3C2311yescharacteristic
63C22×C32×C4C2312yes—
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2311yescharacteristic
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×D12—24no—
73C2×C3×D12C2×C414yes—
74C2×C3×C4×D3—24no—
75C2×C3×C4×D3—42no—
76C2×C3×C4×D3—42no—
77C2×C3×C4×D3—42no—
78C2×C3×C4×D3—42no—
79C2×C3×C4×D3C2×C414yes—
80C2×C3×C4×D3C2×C414yescomplemented
81C32×C4⋊C4—42no—
82C32×C42C2×C414yescomplemented
83C3×(C2×D3)⋊C4—24no—
84C3×(C2×D3)⋊C4—42no—
85C3×(C2×D3)⋊C4—42no—
86C3×(C2×D3)⋊C4C2×C414yes—
87C3×(C2×D3)⋊C4C2×C414yescomplemented
88C3×C4⋊Dic3C2×C414yescomplemented
89C3×Dic3⋊C4—42no—
90C3×C4×Dic3—42no—
91#2318—31noSylow
92C22×C3×D4—31no—
93C23×C3×C4—31no—
94C23×C3×C4—31no—
95C23×C3×C4—32no—
96C22×D12C22×C311yescharacteristic
97C22×C4×D3C22×C312yes—
98C22×C4×D3C22×C311yescharacteristic
99C22×C4×D3C22×C311yescharacteristic
100C3×C4×D4—38no—
101C2×C3×C4⋊C4—22no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—32no—
104C2×C3×C4⋊C4C2×D312yes—
105C2×C3×C4⋊C4C22×C312yes—
106C2×C3×C22⋊C4—31no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—32no—
109C2×C3×C22⋊C4—32no—
110C2×C3×C22⋊C4—32no—
111C2×C3×C42—21no—
112C2×C3×C42—32no—
113C2×C3×C42C2×D311yescharacteristic
114C2×C3×C42C22×C311yescharacteristic
115C2×(C2×D3)⋊C4C22×C312yes—
116C2×(C2×D3)⋊C4C22×C312yes—
117C2×(C2×D3)⋊C4C22×C312yes—
118C2×(C2×D3)⋊C4C22×C311yescharacteristic
119C2×(C2×D3)⋊C4C22×C311yescharacteristic
120C2×C4⋊Dic3C22×C311yescharacteristic
121C2×Dic3⋊C4C22×C312yes—
122C2×C4×Dic3C22×C312yes—
123C4×D12C3×C418yescomplemented
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—22no—
127C3×#95—31no—
128C3×#95—32no—
129C3×#95—31no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95—32no—
133C3×#95C2×D311yescharacteristic
134C3×#95C2×D311yescharacteristic
135C3×#95C2×D312yes—
136C3×#95C22×C312yes—
137C3×#95C22×C311yescharacteristic
138C3×#95C22×C311yescharacteristic
139C3⋊#95C22×C312yes—
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C311yescharacteristic
143C3⋊#95C22×C311yescharacteristic
144C3⋊#95C22×C312yes—
145C23×C32C2411yescharacteristic
146C22×C3×D3—21no—
147C22×C3×D3—22no—
148C22×C3×D3—21no—
149C22×C3×D3—22no—
150C22×C3×D3—21no—
151C22×C3×D3—22no—
152C22×C3×D3—21no—
153C22×C3×D3C22×C412yes—
154C22×C3×D3C22×C412yes—
155C22×C3×D3#4114yes—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—22no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4—22no—
165C2×C32×C4—22no—
166C2×C32×C4C22×C412yes—
167C2×C32×C4C22×C412yes—
168C2×C32×C4C22×C412yes—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3C22×C412yes—
178C2×C3×Dic3C22×C412yes—
179C3×D12—28no—
180C3×C4×D3—44no—
181C3×C4×D3—44no—
182C3×C4×D3—48no—
183C2×C4×D4—31no—
184#388—32no—
185#387—32no—
186#386—32no—
187#385—32no—
188#383—31no—
189#382C3×D311yescharacteristic, complemented
190#381—31no—
191#380—31no—
192C4×C22⋊C4—31no—
193C2×#95—31no—
194C24×C3—31no—
195C24×C3—31no—
196C23×D3C23×C311yescharacteristic
197C23×D3C23×C311yescharacteristic
198C2×C3×D4—34no—
199C2×C3×D4—64no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—22no—
205C22×C3×C4—31no—
206C22×C3×C4—31no—
207C22×C3×C4—34no—
208C22×C3×C4—34no—
209C22×C3×C4—32no—
210C22×C3×C4—32no—
211C22×C3×C4—64no—
212C22×C3×C4—122no—
213C22×C3×C4—122no—
214C22×C3×C4—122no—
215C22×C3×C4—122no—
216C22×C3×C4C22×D312yes—
217C22×C3×C4C22×D312yes—
218C22×C3×C4C22×D311yescharacteristic
219C22×C3×C4C22×D311yescharacteristic
220C22×C3×C4C22×D311yescharacteristic
221C22×C3×C4C23×C312yes—
222C22×C3×C4C23×C312yes—
223C22×C3×C4C23×C311yescharacteristic
224C22×C3×C4C23×C311yescharacteristic
225C22×C3×C4C23×C311yescharacteristic
226C22×Dic3C23×C312yes—
227C22×Dic3C23×C312yes—
228C22×Dic3C23×C311yescharacteristic
229C22×Dic3C23×C311yescharacteristic
230C2×D12—24no—
231C2×D12C2×C3×C414yes—
232C2×C4×D3—24no—
233C2×C4×D3—42no—
234C2×C4×D3—42no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3C2×C3×C414yes—
238C2×C4×D3C2×C3×C414yescomplemented
239C3×C4⋊C4—34no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4—42no—
242C3×C4⋊C4—44no—
243C3×C4⋊C4—122no—
244C3×C22⋊C4—34no—
245C3×C22⋊C4—34no—
246C3×C22⋊C4—64no—
247C3×C22⋊C4—122no—
248C3×C22⋊C4—122no—
249C3×C42—24no—
250C3×C42—122no—
251C3×C42C4×D314yescomplemented
252C3×C42C2×C3×C414yescomplemented
253(C2×D3)⋊C4—24no—
254(C2×D3)⋊C4—42no—
255(C2×D3)⋊C4—42no—
256(C2×D3)⋊C4C2×C3×C414yes—
257(C2×D3)⋊C4C2×C3×C414yescomplemented
258C4⋊Dic3C2×C3×C414yescomplemented
259Dic3⋊C4—42no—
260C4×Dic3—42no—
261C22×C32C23×C411yescharacteristic
262C22×C32C2×#4112yes—
263C22×C32H(5,𝔽2)11yescharacteristic
264C22×C32H(5,𝔽2)12yes—
265C22×C322−1+411yescharacteristic
266C2×C3×D3—22no—
267C2×C3×D3—22no—
268C2×C3×D3—24no—
269C2×C3×D3—24no—
270C2×C3×D3—22no—
271C2×C3×D3—22no—
272C2×C3×D3—22no—
273C2×C3×D3—42no—
274C2×C3×D3—42no—
275C2×C3×D3#11714yes—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—42no—
280C32×C4—42no—
281C3×Dic3—22no—
282C3×Dic3—22no—
283C3×Dic3—42no—
284C3×Dic3—42no—
285C22×D4—31no—
286C23×C4—31no—
287C23×C4—31no—
288C23×C4—32no—
289C4×D4—38no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4C2×C3×D312yes—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—32no—
296C2×C22⋊C4—32no—
297C2×C22⋊C4—32no—
298C2×C42—32no—
299C2×C42C2×C3×D311yescharacteristic
300#95—31no—
301#95—32no—
302#95—31no—
303#95—32no—
304#95—32no—
305#95—32no—
306#95C2×C3×D311yescharacteristic
307#95C2×C3×D311yescharacteristic
308#95C2×C3×D312yes—
309C23×C3—21no—
310C23×C3—32no—
311C23×C3—34no—
312C23×C3—32no—
313C23×C3—61no—
314C23×C3—61no—
315C23×C3—62no—
316C23×C3—61no—
317C23×C3—61no—
318C23×C3—62no—
319C23×C3—62no—
320C23×C3C23×D311yescharacteristic
321C23×C3C24×C311yescharacteristic
322C22×D3—21no—
323C22×D3—22no—
324C22×D3—21no—
325C22×D3—22no—
326C22×D3—21no—
327C22×D3—22no—
328C22×D3—21no—
329C22×D3C22×C3×C412yes—
330C22×D3C22×C3×C412yes—
331C22×D3C3×#4114yes—
332C3×D4—68no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—22no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—62no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—124no—
377C2×C3×C4—124no—
378C2×C3×C4—128no—
379C2×C3×C4C2×C4×D312yes—
380C2×C3×C4C2×C4×D312yes—
381C2×C3×C4C2×C4×D312yes—
382C2×C3×C4C22×C3×C412yes—
383C2×C3×C4C22×C3×C412yes—
384C2×C3×C4C22×C3×C412yes—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—22no—
393C2×Dic3C22×C3×C412yes—
394C2×Dic3C22×C3×C412yes—
395D12—28no—
396C4×D3—44no—
397C4×D3—44no—
398C4×D3—48no—
399C2×C32C2×#11711yescharacteristic
400C2×C32#51811yescharacteristic
401C2×C32#52011yescharacteristic
402C2×C32#53812yes—
403C2×C32#53912yes—
404C3×D3—24no—
405C3×D3—42no—
406C24—31no—
407C24—31no—
408C2×D4—34no—
409C2×D4—64no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—64no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4C22×C3×D312yes—
422C22×C4C22×C3×D312yes—
423C22×C4C22×C3×D311yescharacteristic
424C22×C4C22×C3×D311yescharacteristic
425C22×C4C22×C3×D311yescharacteristic
426C4⋊C4—34no—
427C4⋊C4—42no—
428C4⋊C4—122no—
429C22⋊C4—34no—
430C22⋊C4—34no—
431C22⋊C4—64no—
432C22⋊C4—122no—
433C22⋊C4—122no—
434C42—122no—
435C42C3×C4×D314yescomplemented
436C22×C3—21no—
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—22no—
440C22×C3—22no—
441C22×C3—34no—
442C22×C3—64no—
443C22×C3—64no—
444C22×C3—62no—
445C22×C3—62no—
446C22×C3—62no—
447C22×C3—62no—
448C22×C3—62no—
449C22×C3—122no—
450C22×C3—122no—
451C22×C3C22×C4×D311yescharacteristic
452C22×C3#100411yescharacteristic
453C22×C3#100711yescharacteristic
454C22×C3D3×#4112yes—
455C22×C3#100912yes—
456C22×C3C23×C3×C411yescharacteristic
457C22×C3C2×C3×#4112yes—
458C22×C3C3×H(5,𝔽2)11yescharacteristic
459C22×C3C3×H(5,𝔽2)12yes—
460C22×C3C3×2−1+411yescharacteristic
461C2×D3—22no—
462C2×D3—22no—
463C2×D3—24no—
464C2×D3—24no—
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—42no—
469C2×D3—42no—
470C2×D3C3×#11714yes—
471C3×C4—24no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—62no—
485C3×C4—62no—
486C3×C4—82no—
487C3×C4—82no—
488C3×C4—122no—
489C3×C4—122no—
490Dic3—22no—
491Dic3—22no—
492Dic3—42no—
493Dic3—42no—
494C32#231811yescharacteristic, complemented, Sylow
495C23—32no—
496C23—34no—
497C23—32no—
498C23—61no—
499C23—61no—
500C23—62no—
501C23—61no—
502C23—61no—
503C23—62no—
504C23—62no—
505C23C23×C3×D311yescharacteristic
506D4—68no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—32no—
518C2×C4—32no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—62no—
524C2×C4—62no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—124no—
528C2×C4—124no—
529C2×C4—128no—
530C2×C4C2×C3×C4×D312yes—
531C2×C4C2×C3×C4×D312yes—
532C2×C4C2×C3×C4×D312yes—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—22no—
537C2×C3—22no—
538C2×C3—64no—
539C2×C3—122no—
540C2×C3#550411yescharacteristic
541C2×C3D3×#11711yescharacteristic
542C2×C3#552111yescharacteristic
543C2×C3#567812yes—
544C2×C3#569212yes—
545C2×C3C2×C3×#11711yescharacteristic
546C2×C3C3×#51811yescharacteristic
547C2×C3C3×#52011yescharacteristic
548C2×C3C3×#53812yes—
549C2×C3C3×#53912yes—
550D3—24no—
551D3—42no—
552C22—34no—
553C22—62no—
554C22—62no—
555C22—64no—
556C22—64no—
557C22—62no—
558C22—62no—
559C22—62no—
560C22—122no—
561C22—122no—
562C22C22×C3×C4×D311yescharacteristic
563C22C3×#100411yescharacteristic
564C22C3×#100711yescharacteristic
565C22C3×D3×#4112yes—
566C22C3×#100912yes—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—42no—
571C4—42no—
572C4—62no—
573C4—62no—
574C4—122no—
575C4—122no—
576C3—21no—
577C3#7422611yescharacteristic, direct product factor
578C3C3×#231811yescharacteristic, complemented
579C2—64no—
580C2—122no—
581C2C3×#550411yescharacteristic
582C2C3×D3×#11711yescharacteristic
583C2C3×#552111yescharacteristic
584C2C3×#567812yes—
585C2C3×#569212yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#382C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#382D3
hypocenterC3C3×#2318
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#382

Complements

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

NH ≅ G/Ntype
2. C32×#382C2semidirect
3. C3×#4990C2semidirect
4. C3×#4989C2semidirect
5. C3×#4937C2semidirect
6. C3×#4667C2semidirect
7. C3×#4666C2semidirect
9. C3×#4664C2semidirect
10. C3×D3×#95C2semidirect
11. C3×#4653C2semidirect
13. #74226C3direct (N × H)
28. C3×C4×D12C4semidirect
46. C3×#382D3semidirect
47. C3×#382C2×C3semidirect
52. #4990C2×C3semidirect
53. #4989C2×C3semidirect
54. #4937C2×C3semidirect
55. #4667C2×C3semidirect
56. #4666C2×C3semidirect
58. #4664C2×C3semidirect
59. D3×#95C2×C3semidirect
60. #4653C2×C3semidirect
80. C2×C3×C4×D3C2×C4semidirect
82. C32×C42C2×C4semidirect
87. C3×(C2×D3)⋊C4C2×C4semidirect
88. C3×C4⋊Dic3C2×C4semidirect
123. C4×D12C3×C4semidirect
189. #382C3×D3semidirect
238. C2×C4×D3C2×C3×C4semidirect
251. C3×C42C4×D3semidirect
252. C3×C42C2×C3×C4semidirect
257. (C2×D3)⋊C4C2×C3×C4semidirect
258. C4⋊Dic3C2×C3×C4semidirect
435. C42C3×C4×D3semidirect
494. C32#2318semidirect
577. C3#74226direct (N × H)
578. C3C3×#2318semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C32×#3822
C3×#49902
C3×#49892
C3×#49372
C3×#46672
C3×#46662
C3×#46652
C3×#46642
C3×D3×#952
C3×#46532
C3×#23183
#742263