Finite Group Databasebeta

← C3×#73635 #11805359

Subgroups — C3×#73635 #11805359

Order 1152.

1105 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic
2C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#4990C212yesmaximal
4C3×#4976C212yesmaximal, complemented
5C3×#4940C212yesmaximal
6C3×#4937C211yesmaximal, characteristic, complemented
7C3×C4×Dic3⋊C4C212yesmaximal, complemented
8C3×#4667C211yesmaximal, characteristic
9C3×#4666C211yesmaximal, characteristic
10C3×#4651C211yesmaximal, characteristic, complemented
11C3×#4649C211yesmaximal, characteristic, complemented
12C3×#2347—31nomaximal
13#73635C311yesmaximal, characteristic, direct product factor
14C22×C3×C4×D3C2211yescharacteristic
15C2×C32×C4⋊C4C2212yes—
16C2×C32×C42C2212yes—
17C2×C32×C42C2211yescharacteristic
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×(C2×D3)⋊C4C2212yes—
20C2×C3×(C2×D3)⋊C4C2211yescharacteristic
21C2×C3×(C2×D3)⋊C4C2211yescharacteristic
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2211yescharacteristic
25C2×C3×Dic3⋊C4C2211yescharacteristic
26C2×C3×C4×Dic3C2212yes—
27C2×C3×C4×Dic3C2212yes—
28C2×C3×C4×Dic3C2211yescharacteristic
29C3×#872C418yescomplemented
30C32×#95C2211yescharacteristic
31C32×#95C2211yescharacteristic
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2211yescharacteristic
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C2×C3×#117—31no—
39C2×#872C2×C311yescharacteristic
40C3×#389—32no—
41C3×#388—32no—
42C3×#388—32no—
43C3×#383—31no—
44C3×#383—31no—
45C3×#380—31no—
46C3×C4×C4⋊C4—21no—
47C3×C4×C4⋊C4—32no—
48C3×C4×C4⋊C4—31no—
49C3×C4×C4⋊C4D311yescharacteristic, complemented
50C3×C4×C4⋊C4C2×C311yescharacteristic, complemented
51C3×C4×C22⋊C4—31no—
52#4990C2×C312yes—
53#4976C2×C312yescomplemented
54#4940C2×C312yes—
55#4937C2×C311yescharacteristic, complemented
56C4×Dic3⋊C4C2×C312yescomplemented
57#4667C2×C311yescharacteristic
58#4666C2×C311yescharacteristic
59#4651C2×C311yescharacteristic, complemented
60#4649C2×C311yescharacteristic, complemented
61C23×C3×D3C2311yescharacteristic
62C22×C32×C4C2312yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2312yes—
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2311yescharacteristic
71C22×C3×Dic3C2311yescharacteristic
72C2×C3×C4×D3—24no—
73C2×C3×C4×D3C2×C414yes—
74C32×C4⋊C4—24no—
75C32×C42—42no—
76C32×C42C2×C414yes—
77C3×(C2×D3)⋊C4—24no—
78C3×(C2×D3)⋊C4—42no—
79C3×(C2×D3)⋊C4C2×C414yes—
80C3×(C2×D3)⋊C4C2×C414yes—
81C3×Dic3⋊C4—24no—
82C3×Dic3⋊C4—24no—
83C3×Dic3⋊C4C2×C414yes—
84C3×Dic3⋊C4C2×C414yes—
85C3×C4×Dic3—24no—
86C3×C4×Dic3—42no—
87C3×C4×Dic3C2×C414yes—
88#2347—31noSylow
89C23×C3×C4—31no—
90C22×C4×D3C22×C311yescharacteristic
91C3×#117—38no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—31no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C4⋊C4—32no—
96C2×C3×C4⋊C4—32no—
97C2×C3×C4⋊C4C2×D312yes—
98C2×C3×C4⋊C4C22×C312yes—
99C2×C3×C22⋊C4—31no—
100C2×C3×C22⋊C4—31no—
101C2×C3×C22⋊C4—32no—
102C2×C3×C22⋊C4—32no—
103C2×C3×C42—21no—
104C2×C3×C42—22no—
105C2×C3×C42—31no—
106C2×C3×C42—32no—
107C2×C3×C42—32no—
108C2×C3×C42C2×D312yes—
109C2×C3×C42C2×D311yescharacteristic
110C2×C3×C42C22×C312yes—
111C2×C3×C42C22×C311yescharacteristic
112C2×(C2×D3)⋊C4C22×C312yes—
113C2×(C2×D3)⋊C4C22×C312yes—
114C2×(C2×D3)⋊C4C22×C311yescharacteristic
115C2×(C2×D3)⋊C4C22×C311yescharacteristic
116C2×Dic3⋊C4C22×C312yes—
117C2×Dic3⋊C4C22×C312yes—
118C2×Dic3⋊C4C22×C311yescharacteristic
119C2×Dic3⋊C4C22×C311yescharacteristic
120C2×C4×Dic3C22×C312yes—
121C2×C4×Dic3C22×C312yes—
122C2×C4×Dic3C22×C311yescharacteristic
123#872C3×C418yescomplemented
124C3×#95—21no—
125C3×#95—21no—
126C3×#95—31no—
127C3×#95—31no—
128C3×#95—32no—
129C3×#95—32no—
130C3×#95—32no—
131C3×#95—32no—
132C3×#95C2×D311yescharacteristic
133C3×#95C2×D311yescharacteristic
134C3×#95C22×C311yescharacteristic
135C3×#95C22×C311yescharacteristic
136C3⋊#95C22×C311yescharacteristic
137C3⋊#95C22×C311yescharacteristic
138C3⋊#95C22×C312yes—
139C3⋊#95C22×C312yes—
140C3⋊#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C23×C32C2411yescharacteristic
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C22×C3×D3C22×C412yes—
148C2×C32×C4—22no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4C22×C412yes—
158C2×C32×C4C22×C412yes—
159C2×C32×C4C22×C412yes—
160C2×C32×C4#4114yes—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—21no—
163C2×C3×Dic3—21no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—22no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3C22×C412yes—
171C2×C3×Dic3C22×C412yes—
172C2×C3×Dic3C22×C412yes—
173C2×C3×Dic3#4114yes—
174C3×C4×D3—28no—
175C2×#117—31no—
176#389—32no—
177#388—32no—
178#388—32no—
179#383—31no—
180#383—31no—
181#380—31no—
182C4×C4⋊C4—32no—
183C4×C4⋊C4—31no—
184C4×C4⋊C4C3×D311yescharacteristic, complemented
185C4×C22⋊C4—31no—
186C24×C3—31no—
187C23×D3C23×C311yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—22no—
192C22×C3×C4—22no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—34no—
197C22×C3×C4—32no—
198C22×C3×C4—32no—
199C22×C3×C4—64no—
200C22×C3×C4C22×D312yes—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C22×D312yes—
205C22×C3×C4C23×C312yes—
206C22×C3×C4C23×C311yescharacteristic
207C22×C3×C4C23×C311yescharacteristic
208C22×C3×C4C23×C311yescharacteristic
209C22×C3×C4C23×C312yes—
210C22×Dic3C23×C312yes—
211C22×Dic3C23×C312yes—
212C22×Dic3C23×C311yescharacteristic
213C22×Dic3C23×C311yescharacteristic
214C22×Dic3C23×C311yescharacteristic
215C2×C4×D3—24no—
216C2×C4×D3C2×C3×C414yes—
217C3×C4⋊C4—28no—
218C3×C4⋊C4—24no—
219C3×C4⋊C4—24no—
220C3×C4⋊C4—34no—
221C3×C4⋊C4—34no—
222C3×C4⋊C4—64no—
223C3×C4⋊C4—64no—
224C3×C22⋊C4—34no—
225C3×C22⋊C4—34no—
226C3×C22⋊C4—64no—
227C3×C22⋊C4—122no—
228C3×C42—24no—
229C3×C42—34no—
230C3×C42—42no—
231C3×C42—42no—
232C3×C42—44no—
233C3×C42—64no—
234C3×C42—122no—
235C3×C42C4×D314yes—
236C3×C42C2×C3×C414yes—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4C2×C3×C414yes—
240(C2×D3)⋊C4C2×C3×C414yes—
241Dic3⋊C4—24no—
242Dic3⋊C4—24no—
243Dic3⋊C4C2×C3×C414yes—
244Dic3⋊C4C2×C3×C414yes—
245C4×Dic3—24no—
246C4×Dic3—42no—
247C4×Dic3C2×C3×C414yes—
248C22×C32C23×C411yescharacteristic
249C22×C32C2×#4112yes—
250C22×C32C2×#4112yes—
251C22×C32H(5,𝔽2)11yescharacteristic
252C22×C322−1+411yescharacteristic
253C2×C3×D3—22no—
254C2×C3×D3—22no—
255C2×C3×D3—22no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C32×C4—22no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—24no—
262C32×C4—42no—
263C3×Dic3—22no—
264C3×Dic3—22no—
265C3×Dic3—42no—
266C3×Dic3—42no—
267C3×Dic3#11714yes—
268C23×C4—31no—
269#117—38no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—32no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4C2×C3×D312yes—
275C2×C22⋊C4—31no—
276C2×C22⋊C4—31no—
277C2×C22⋊C4—32no—
278C2×C22⋊C4—32no—
279C2×C42—31no—
280C2×C42—32no—
281C2×C42—32no—
282C2×C42C2×C3×D312yes—
283C2×C42C2×C3×D311yescharacteristic
284#95—31no—
285#95—31no—
286#95—32no—
287#95—32no—
288#95—32no—
289#95—32no—
290#95C2×C3×D311yescharacteristic
291#95C2×C3×D311yescharacteristic
292C23×C3—21no—
293C23×C3—32no—
294C23×C3—62no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—62no—
298C23×C3C23×D311yescharacteristic
299C23×C3C24×C311yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3—21no—
303C22×D3—21no—
304C22×D3C22×C3×C412yes—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—22no—
311C2×C3×C4—21no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
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—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—32no—
332C2×C3×C4—34no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—68no—
341C2×C3×C4—62no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4—62no—
347C2×C3×C4—62no—
348C2×C3×C4—62no—
349C2×C3×C4—62no—
350C2×C3×C4C2×C4×D312yes—
351C2×C3×C4C2×C4×D312yes—
352C2×C3×C4C2×C4×D312yes—
353C2×C3×C4#23514yes—
354C2×C3×C4C22×C3×C412yes—
355C2×C3×C4C22×C3×C412yes—
356C2×C3×C4C22×C3×C412yes—
357C2×C3×C4C3×#4114yes—
358C2×Dic3—22no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—22no—
365C2×Dic3—22no—
366C2×Dic3—22no—
367C2×Dic3C22×C3×C412yes—
368C2×Dic3C22×C3×C412yes—
369C2×Dic3C22×C3×C412yes—
370C2×Dic3C3×#4114yes—
371C4×D3—28no—
372C2×C32C2×#11711yescharacteristic
373C2×C32C4×#4111yescharacteristic
374C2×C32#52011yescharacteristic
375C2×C32#55312yes—
376C2×C32#55612yes—
377C3×D3—42no—
378C24—31no—
379C22×C4—31no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—34no—
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—64no—
386C22×C4C22×C3×D312yes—
387C22×C4C22×C3×D311yescharacteristic
388C22×C4C22×C3×D311yescharacteristic
389C22×C4C22×C3×D311yescharacteristic
390C22×C4C22×C3×D312yes—
391C4⋊C4—24no—
392C4⋊C4—34no—
393C4⋊C4—34no—
394C4⋊C4—64no—
395C4⋊C4—64no—
396C22⋊C4—34no—
397C22⋊C4—34no—
398C22⋊C4—64no—
399C22⋊C4—122no—
400C42—34no—
401C42—42no—
402C42—64no—
403C42—122no—
404C42C3×C4×D314yes—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—22no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—62no—
413C22×C3—122no—
414C22×C3—122no—
415C22×C3C22×C4×D311yescharacteristic
416C22×C3C2×#23512yes—
417C22×C3#100411yescharacteristic
418C22×C3#100711yescharacteristic
419C22×C3D3×#4112yes—
420C22×C3C23×C3×C411yescharacteristic
421C22×C3C2×C3×#4112yes—
422C22×C3C2×C3×#4112yes—
423C22×C3C3×H(5,𝔽2)11yescharacteristic
424C22×C3C3×2−1+411yescharacteristic
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—22no—
428C2×D3—42no—
429C2×D3—42no—
430C3×C4—24no—
431C3×C4—24no—
432C3×C4—22no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—24no—
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—24no—
440C3×C4—34no—
441C3×C4—42no—
442C3×C4—44no—
443C3×C4—44no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—122no—
449C3×C4—122no—
450Dic3—22no—
451Dic3—22no—
452Dic3—42no—
453Dic3—42no—
454Dic3C3×#11714yes—
455C32#234711yescharacteristic, complemented, Sylow
456C23—32no—
457C23—62no—
458C23—61no—
459C23—61no—
460C23—62no—
461C23C23×C3×D311yescharacteristic
462C2×C4—22no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—32no—
472C2×C4—34no—
473C2×C4—32no—
474C2×C4—32no—
475C2×C4—68no—
476C2×C4—62no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4C2×C3×C4×D312yes—
486C2×C4C2×C3×C4×D312yes—
487C2×C4C2×C3×C4×D312yes—
488C2×C4C3×#23514yes—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—21no—
494C2×C3—122no—
495C2×C3C4×#23511yescharacteristic
496C2×C3#550411yescharacteristic
497C2×C3D3×#11711yescharacteristic
498C2×C3#554012yes—
499C2×C3#556812yes—
500C2×C3C2×C3×#11711yescharacteristic
501C2×C3C3×C4×#4111yescharacteristic
502C2×C3C3×#52011yescharacteristic
503C2×C3C3×#55312yes—
504C2×C3C3×#55612yes—
505D3—42no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—122no—
510C22—122no—
511C22C22×C3×C4×D311yescharacteristic
512C22C2×C3×#23512yes—
513C22C3×#100411yescharacteristic
514C22C3×#100711yescharacteristic
515C22C3×D3×#4112yes—
516C4—22no—
517C4—22no—
518C4—24no—
519C4—22no—
520C4—34no—
521C4—42no—
522C4—62no—
523C4—62no—
524C4—122no—
525C4—122no—
526C3—21no—
527C3#7363511yescharacteristic, direct product factor
528C3C3×#234711yescharacteristic, complemented
529C2—122no—
530C2C3×C4×#23511yescharacteristic
531C2C3×#550411yescharacteristic
532C2C3×D3×#11711yescharacteristic
533C2C3×#554012yes—
534C2C3×#556812yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socleC23×C32C24
prime radicalC23×C3×D3C23
cosocleC23×C3C24×C3
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2347
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C32×C4×C4⋊C4C2semidirect
4. C3×#4976C2semidirect
6. C3×#4937C2semidirect
7. C3×C4×Dic3⋊C4C2semidirect
10. C3×#4651C2semidirect
11. C3×#4649C2semidirect
13. #73635C3direct (N × H)
29. C3×#872C4semidirect
49. C3×C4×C4⋊C4D3semidirect
50. C3×C4×C4⋊C4C2×C3semidirect
53. #4976C2×C3semidirect
55. #4937C2×C3semidirect
56. C4×Dic3⋊C4C2×C3semidirect
59. #4651C2×C3semidirect
60. #4649C2×C3semidirect
123. #872C3×C4semidirect
184. C4×C4⋊C4C3×D3semidirect
455. C32#2347semidirect
527. C3#73635direct (N × H)
528. C3C3×#2347semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C32×C4×C4⋊C42
C3×#49902
C3×#49762
C3×#49402
C3×#49372
C3×C4×Dic3⋊C42
C3×#46672
C3×#46662
C3×#46512
C3×#46492
C3×#23473
#736353