Finite Group Databasebeta

← C3×#73458 #11805226

Subgroups — C3×#73458 #11805226

Order 1152.

1225 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Dic3⋊C4C211yesmaximal, characteristic
2C32×C4×C22⋊C4C211yesmaximal, characteristic
3C3×#5212C211yesmaximal, characteristic
4C3×#4942C211yesmaximal, characteristic
5C3×#4942C211yesmaximal, characteristic
6C3×Dic3×C22⋊C4C211yesmaximal, characteristic
7C3×Dic3×C22⋊C4C211yesmaximal, characteristic
8C3×#4933C212yesmaximal, complemented
9C3×C4×Dic3⋊C4C212yesmaximal, complemented
10C3×#4655C212yesmaximal, complemented
11C3×#4647C212yesmaximal, complemented
12C3×#2345—31nomaximal
13#73458C311yesmaximal, characteristic, direct product factor
14C23×C32×C4C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C2×C32×C22⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4C2211yescharacteristic
19C2×C32×C42C2212yes—
20C2×C3×#217C2211yescharacteristic
21C2×C3×#217C2211yescharacteristic
22C2×C3×C4⋊Dic3C2212yes—
23C2×C3×Dic3⋊C4C418yescomplemented
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×Dic3⋊C4C2212yes—
26C2×C3×Dic3⋊C4C2211yescharacteristic
27C2×C3×Dic3⋊C4C2211yescharacteristic
28C2×C3×Dic3⋊C4C2211yescharacteristic
29C2×C3×Dic3⋊C4C2211yescharacteristic
30C2×C3×C4×Dic3C2212yes—
31C2×C3×C4×Dic3C2212yes—
32C32×#95C2212yes—
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2212yes—
36C3×C3⋊#95C2212yes—
37C3×C3⋊#95C2212yes—
38C22×C3×C4⋊C4—31no—
39C22×Dic3⋊C4C2×C311yescharacteristic
40C3×#389—32no—
41C3×#387—32no—
42C3×#387—32no—
43C3×#380—31no—
44C3×#380—31no—
45C3×#380—31no—
46C3×C4×C4⋊C4—32no—
47C3×C4×C22⋊C4—21no—
48C3×C4×C22⋊C4—31no—
49C3×C4×C22⋊C4—31no—
50C3×C4×C22⋊C4D311yescharacteristic
51C3×C4×C22⋊C4C2×C311yescharacteristic
52#5212C2×C311yescharacteristic
53#4942C2×C311yescharacteristic
54#4942C2×C311yescharacteristic
55Dic3×C22⋊C4C2×C311yescharacteristic
56Dic3×C22⋊C4C2×C311yescharacteristic
57#4933C2×C312yescomplemented
58C4×Dic3⋊C4C2×C312yescomplemented
59#4655C2×C312yescomplemented
60#4647C2×C312yescomplemented
61C24×C32C2311yescharacteristic
62C22×C32×C4—42no—
63C22×C32×C4C2×C414yes—
64C22×C32×C4C2312yes—
65C22×C32×C4C2311yescharacteristic
66C22×C32×C4C2311yescharacteristic
67C22×C32×C4C2312yes—
68C22×C3×Dic3—24no—
69C22×C3×Dic3—42no—
70C22×C3×Dic3C2×C414yes—
71C22×C3×Dic3C2×C414yes—
72C22×C3×Dic3C2312yes—
73C22×C3×Dic3C2312yes—
74C22×C3×Dic3C2311yescharacteristic
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2311yescharacteristic
77C22×C3×Dic3C2311yescharacteristic
78C32×C22⋊C4—22no—
79C32×C22⋊C4Q814yes—
80C32×C42—42no—
81C3×#217—22no—
82C3×#217—22no—
83C3×C4⋊Dic3—42no—
84C3×Dic3⋊C4—24no—
85C3×Dic3⋊C4—42no—
86C3×Dic3⋊C4C2×C414yes—
87C3×Dic3⋊C4C2×C414yes—
88C3×Dic3⋊C4C2×C414yes—
89C3×Dic3⋊C4C2×C414yes—
90C3×C4×Dic3—24no—
91C3×C4×Dic3—42no—
92#2345—31noSylow
93C23×C3×C4—21no—
94C23×C3×C4—31no—
95C23×C3×C4—31no—
96C23×C3×C4C2×D311yescharacteristic
97C23×C3×C4C22×C311yescharacteristic
98C23×Dic3C22×C311yescharacteristic
99C23×Dic3C22×C311yescharacteristic
100C2×C3×C4⋊C4—38no—
101C2×C3×C4⋊C4—31no—
102C2×C3×C4⋊C4—31no—
103C2×C3×C4⋊C4—31no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C22⋊C4—21no—
109C2×C3×C22⋊C4—21no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C22⋊C4—31no—
112C2×C3×C22⋊C4C2×D311yescharacteristic
113C2×C3×C22⋊C4C2×D311yescharacteristic
114C2×C3×C22⋊C4C22×C311yescharacteristic
115C2×C3×C22⋊C4C22×C311yescharacteristic
116C2×C3×C42—22no—
117C2×C3×C42—32no—
118C2×C3×C42—32no—
119C2×C3×C42C2×D312yes—
120C2×C3×C42C22×C312yes—
121C2×#217C22×C311yescharacteristic
122C2×#217C22×C311yescharacteristic
123C2×C4⋊Dic3C22×C312yes—
124C2×Dic3⋊C4C3×C418yescomplemented
125C2×Dic3⋊C4C22×C312yes—
126C2×Dic3⋊C4C22×C312yes—
127C2×Dic3⋊C4C22×C311yescharacteristic
128C2×Dic3⋊C4C22×C311yescharacteristic
129C2×Dic3⋊C4C22×C311yescharacteristic
130C2×Dic3⋊C4C22×C311yescharacteristic
131C2×C4×Dic3C22×C312yes—
132C2×C4×Dic3C22×C312yes—
133C3×#95—22no—
134C3×#95—32no—
135C3×#95—32no—
136C3×#95—32no—
137C3×#95—32no—
138C3×#95—32no—
139C3×#95C2×D312yes—
140C3×#95C22×C312yes—
141C3⋊#95C22×C312yes—
142C3⋊#95C22×C312yes—
143C3⋊#95C22×C312yes—
144C3⋊#95C22×C312yes—
145C3⋊#95C22×C312yes—
146C23×C32—22no—
147C23×C32—22no—
148C23×C32C22×C412yes—
149C23×C32C2×Q812yes—
150C23×C32#4112yes—
151C23×C32C2411yescharacteristic
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—22no—
159C2×C32×C4—44no—
160C2×C32×C4C22×C412yes—
161C2×C32×C4C22×C412yes—
162C2×C32×C4C2×Q814yes—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—28no—
167C2×C3×Dic3—22no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—44no—
174C2×C3×Dic3C22×C412yes—
175C2×C3×Dic3C22×C412yes—
176C2×C3×Dic3C22×C412yes—
177C2×C3×Dic3C22×C412yes—
178C2×C3×Dic3#4114yes—
179C22×C4⋊C4—31no—
180#389—32no—
181#387—32no—
182#387—32no—
183#380—31no—
184#380—31no—
185#380—31no—
186C4×C4⋊C4—32no—
187C4×C22⋊C4—31no—
188C4×C22⋊C4—31no—
189C4×C22⋊C4C3×D311yescharacteristic
190C24×C3—21no—
191C24×C3C22×D311yescharacteristic
192C24×C3C23×C311yescharacteristic
193C22×C3×C4—24no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—22no—
198C22×C3×C4—31no—
199C22×C3×C4—34no—
200C22×C3×C4—31no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—34no—
204C22×C3×C4—32no—
205C22×C3×C4—32no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4—44no—
209C22×C3×C4—64no—
210C22×C3×C4—122no—
211C22×C3×C4C4×D314yes—
212C22×C3×C4C2×C3×C414yes—
213C22×C3×C4C22×D312yes—
214C22×C3×C4C22×D312yes—
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C23×C312yes—
218C22×C3×C4C23×C311yescharacteristic
219C22×C3×C4C23×C311yescharacteristic
220C22×C3×C4C23×C312yes—
221C22×Dic3—24no—
222C22×Dic3—42no—
223C22×Dic3C2×C3×C414yes—
224C22×Dic3C2×C3×C414yes—
225C22×Dic3C23×C312yes—
226C22×Dic3C23×C312yes—
227C22×Dic3C23×C311yescharacteristic
228C22×Dic3C23×C311yescharacteristic
229C22×Dic3C23×C311yescharacteristic
230C22×Dic3C23×C311yescharacteristic
231C3×C4⋊C4—34no—
232C3×C4⋊C4—34no—
233C3×C4⋊C4—34no—
234C3×C4⋊C4—34no—
235C3×C4⋊C4—64no—
236C3×C4⋊C4—122no—
237C3×C4⋊C4—122no—
238C3×C22⋊C4—24no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—24no—
242C3×C22⋊C4—62no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4Dic614yes—
245C3×C22⋊C4C3×Q814yes—
246C3×C42—42no—
247C3×C42—42no—
248C3×C42—44no—
249C3×C42—64no—
250C3×C42—122no—
251#217—22no—
252#217—22no—
253C4⋊Dic3—42no—
254Dic3⋊C4—24no—
255Dic3⋊C4—42no—
256Dic3⋊C4C2×C3×C414yes—
257Dic3⋊C4C2×C3×C414yes—
258Dic3⋊C4C2×C3×C414yes—
259Dic3⋊C4C2×C3×C414yes—
260C4×Dic3—24no—
261C4×Dic3—42no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32C4×Q814yes—
267C22×C32C23×C411yescharacteristic
268C22×C32C22×Q811yescharacteristic
269C22×C32C2×#4111yescharacteristic
270C22×C32C2×#4112yes—
271C22×C32H(5,𝔽2)12yes—
272C32×C4—22no—
273C32×C4—22no—
274C32×C4—24no—
275C32×C4—42no—
276C3×Dic3—22no—
277C3×Dic3—22no—
278C3×Dic3—22no—
279C3×Dic3—42no—
280C3×Dic3—42no—
281C3×Dic3#11714yes—
282C23×C4—31no—
283C23×C4—31no—
284C23×C4C2×C3×D311yescharacteristic
285C2×C4⋊C4—38no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—32no—
291C2×C4⋊C4—32no—
292C2×C4⋊C4—32no—
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4C2×C3×D311yescharacteristic
296C2×C22⋊C4C2×C3×D311yescharacteristic
297C2×C42—32no—
298C2×C42—32no—
299C2×C42C2×C3×D312yes—
300#95—32no—
301#95—32no—
302#95—32no—
303#95—32no—
304#95—32no—
305#95C2×C3×D312yes—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—22no—
309C23×C3—22no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—42no—
315C23×C3—42no—
316C23×C3C2×Dic612yes—
317C23×C3C2×C4×D312yes—
318C23×C3#23512yes—
319C23×C3C22×C3×C412yes—
320C23×C3C2×C3×Q812yes—
321C23×C3C3×#4112yes—
322C23×C3C23×D311yescharacteristic
323C23×C3C24×C311yescharacteristic
324C2×C3×C4—24no—
325C2×C3×C4—22no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—32no—
344C2×C3×C4—34no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—44no—
354C2×C3×C4—44no—
355C2×C3×C4—48no—
356C2×C3×C4—68no—
357C2×C3×C4—62no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—124no—
367C2×C3×C4C2×Dic614yes—
368C2×C3×C4C2×C4×D312yes—
369C2×C3×C4C2×C4×D312yes—
370C2×C3×C4C22×C3×C412yes—
371C2×C3×C4C22×C3×C412yes—
372C2×C3×C4C2×C3×Q814yes—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—28no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—44no—
384C2×Dic3C22×C3×C412yes—
385C2×Dic3C22×C3×C412yes—
386C2×Dic3C22×C3×C412yes—
387C2×Dic3C22×C3×C412yes—
388C2×Dic3C3×#4114yes—
389C2×C32—24no—
390C2×C32C2×#11711yescharacteristic
391C2×C32C2×C4×Q811yescharacteristic
392C2×C32#51811yescharacteristic
393C2×C32#55312yes—
394C2×C32#55412yes—
395C24C22×C3×D311yescharacteristic
396C22×C4—31no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—34no—
401C22×C4—34no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—42no—
405C22×C4—64no—
406C22×C4—122no—
407C22×C4C3×C4×D314yes—
408C22×C4C22×C3×D312yes—
409C22×C4C22×C3×D312yes—
410C22×C4C22×C3×D311yescharacteristic
411C22×C4C22×C3×D311yescharacteristic
412C4⋊C4—34no—
413C4⋊C4—34no—
414C4⋊C4—34no—
415C4⋊C4—34no—
416C4⋊C4—64no—
417C4⋊C4—122no—
418C4⋊C4—122no—
419C22⋊C4—22no—
420C22⋊C4—62no—
421C22⋊C4—62no—
422C22⋊C4C3×Dic614yes—
423C42—42no—
424C42—64no—
425C42—122no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—24no—
429C22×C3—24no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—24no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—24no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3C4×Dic614yes—
445C22×C3C3×C4×Q814yes—
446C22×C3C22×Dic611yescharacteristic
447C22×C3C22×C4×D311yescharacteristic
448C22×C3C2×#23511yescharacteristic
449C22×C3#100412yes—
450C22×C3D3×#4112yes—
451C22×C3C23×C3×C411yescharacteristic
452C22×C3C22×C3×Q811yescharacteristic
453C22×C3C2×C3×#4111yescharacteristic
454C22×C3C2×C3×#4112yes—
455C22×C3C3×H(5,𝔽2)12yes—
456C3×C4—24no—
457C3×C4—22no—
458C3×C4—22no—
459C3×C4—24no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—24no—
463C3×C4—34no—
464C3×C4—42no—
465C3×C4—44no—
466C3×C4—44no—
467C3×C4—42no—
468C3×C4—42no—
469C3×C4—62no—
470C3×C4—62no—
471C3×C4—62no—
472C3×C4—122no—
473C3×C4—122no—
474Dic3—22no—
475Dic3—22no—
476Dic3—22no—
477Dic3—42no—
478Dic3—42no—
479Dic3C3×#11714yes—
480C32#234511yescharacteristic, complemented, Sylow
481C23—22no—
482C23—22no—
483C23C2×C3×Dic612yes—
484C23C2×C3×C4×D312yes—
485C23C3×#23512yes—
486C23C23×C3×D311yescharacteristic
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—32no—
495C2×C4—34no—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—32no—
499C2×C4—44no—
500C2×C4—68no—
501C2×C4—62no—
502C2×C4—62no—
503C2×C4—62no—
504C2×C4—62no—
505C2×C4—62no—
506C2×C4—62no—
507C2×C4—62no—
508C2×C4—62no—
509C2×C4—62no—
510C2×C4—124no—
511C2×C4C2×C3×Dic614yes—
512C2×C4C2×C3×C4×D312yes—
513C2×C4C2×C3×C4×D312yes—
514C2×C3—24no—
515C2×C3—24no—
516C2×C3—21no—
517C2×C3—22no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—44no—
522C2×C3C2×C4×Dic611yescharacteristic
523C2×C3#548611yescharacteristic
524C2×C3D3×#11711yescharacteristic
525C2×C3#553912yes—
526C2×C3#554012yes—
527C2×C3C2×C3×#11711yescharacteristic
528C2×C3C2×C3×C4×Q811yescharacteristic
529C2×C3C3×#51811yescharacteristic
530C2×C3C3×#55312yes—
531C2×C3C3×#55412yes—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22C3×C4×Dic614yes—
537C22C22×C3×Dic611yescharacteristic
538C22C22×C3×C4×D311yescharacteristic
539C22C2×C3×#23511yescharacteristic
540C22C3×#100412yes—
541C22C3×D3×#4112yes—
542C4—24no—
543C4—22no—
544C4—22no—
545C4—34no—
546C4—42no—
547C4—62no—
548C4—62no—
549C4—62no—
550C4—122no—
551C4—122no—
552C3—21no—
553C3#7345811yescharacteristic, direct product factor
554C3C3×#234511yescharacteristic, complemented
555C2—24no—
556C2C2×C3×C4×Dic611yescharacteristic
557C2C3×#548611yescharacteristic
558C2C3×D3×#11711yescharacteristic
559C2C3×#553912yes—
560C2C3×#554012yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C22⋊C4C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC3×C4×C22⋊C4D3
hypocenterC3C3×#2345
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
8. C3×#4933C2semidirect
9. C3×C4×Dic3⋊C4C2semidirect
10. C3×#4655C2semidirect
11. C3×#4647C2semidirect
13. #73458C3direct (N × H)
23. C2×C3×Dic3⋊C4C4semidirect
57. #4933C2×C3semidirect
58. C4×Dic3⋊C4C2×C3semidirect
59. #4655C2×C3semidirect
60. #4647C2×C3semidirect
124. C2×Dic3⋊C4C3×C4semidirect
480. C32#2345semidirect
553. C3#73458direct (N × H)
554. C3C3×#2345semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic3⋊C42
C32×C4×C22⋊C42
C3×#52122
C3×#49422
C3×Dic3×C22⋊C42
C3×#49332
C3×C4×Dic3⋊C42
C3×#46552
C3×#46472
C3×#23453
#734583