Finite Group Databasebeta

← C3×#73874 #11803895

Subgroups — C3×#73874 #11803895

Order 1152.

1173 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#924C211yesmaximal, characteristic
2C32×#380C211yesmaximal, characteristic
3C2×C3×C3⋊#95C211yesmaximal, characteristic
4C3×#4978C212yesmaximal, complemented
5C3×#4976C212yesmaximal, complemented
6C3×#4946C212yesmaximal
7C3×Dic3×C22⋊C4C212yesmaximal
8C3×#4653C212yesmaximal, complemented
9C3×#4646C212yesmaximal, complemented
10C3×#2329—31nomaximal
11#73874C311yesmaximal, characteristic, direct product factor
12C23×C32×C4C2211yescharacteristic
13C23×C3×Dic3C2211yescharacteristic
14C23×C3×Dic3C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C2×C32×C4⋊C4C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C22⋊C4C2212yes—
19C2×C3×#217C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C3×#924C418yescomplemented
22C2×C3×Dic3⋊C4C2212yes—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×C4×Dic3C2212yes—
26C2×C3×C4×Dic3C2212yes—
27C32×#95C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2212yes—
33C3×C3⋊#95C2212yes—
34C2×C3×C22⋊Q8—31no—
35C2×#924C2×C311yescharacteristic
36C3×#391—32no—
37C3×#389—32no—
38C3×#387—32no—
39C3×#387—32no—
40C3×#385—32no—
41C3×#380—21no—
42C3×#380D311yescharacteristic
43C3×#380C2×C311yescharacteristic
44C3×C4×C22⋊C4—32no—
45C2×C3×#95—31no—
46C2×C3⋊#95C2×C311yescharacteristic
47#4978C2×C312yescomplemented
48#4976C2×C312yescomplemented
49#4946C2×C312yes—
50Dic3×C22⋊C4C2×C312yes—
51#4653C2×C312yescomplemented
52#4646C2×C312yescomplemented
53C24×C32C2311yescharacteristic
54C22×C32×C4—41no—
55C22×C32×C4—41no—
56C22×C32×C4C2×C414yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2312yes—
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2311yescharacteristic
61C22×C3×Dic3—22no—
62C22×C3×Dic3—41no—
63C22×C3×Dic3—41no—
64C22×C3×Dic3—41no—
65C22×C3×Dic3—41no—
66C22×C3×Dic3Q814yescomplemented
67C22×C3×Dic3C2312yes—
68C22×C3×Dic3C2312yes—
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2312yes—
71C2×C3×Dic6—41no—
72C2×C3×Dic6—41no—
73C2×C3×Dic6C2×C414yescomplemented
74C32×C4⋊C4—41no—
75C32×C4⋊C4—41no—
76C32×C22⋊C4—24no—
77C3×#217C2×C418yes—
78C3×C4⋊Dic3C2×C414yescomplemented
79C3×Dic3⋊C4—42no—
80C3×Dic3⋊C4—42no—
81C3×Dic3⋊C4C2×C418yescomplemented
82C3×C4×Dic3—42no—
83C3×C4×Dic3—42no—
84#2329—31noSylow
85C22×C3×Q8—31no—
86C23×C3×C4—21no—
87C23×C3×C4—31no—
88C23×C3×C4—31no—
89C23×C3×C4C2×D311yescharacteristic
90C23×C3×C4C22×C311yescharacteristic
91C23×Dic3C22×C311yescharacteristic
92C23×Dic3C22×C311yescharacteristic
93C22×Dic6C22×C311yescharacteristic
94C3×C22⋊Q8—38no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—32no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—32no—
100C2×C3×C4⋊C4—32no—
101C2×C3×C4⋊C4C2×D311yescharacteristic
102C2×C3×C4⋊C4C2×D311yescharacteristic
103C2×C3×C4⋊C4C22×C311yescharacteristic
104C2×C3×C4⋊C4C22×C311yescharacteristic
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4—32no—
107C2×C3×C22⋊C4C2×D312yes—
108C2×C3×C22⋊C4C22×C312yes—
109C2×C3×C42—32no—
110C2×C3×C42—32no—
111C2×#217C22×C312yes—
112C2×C4⋊Dic3C22×C311yescharacteristic
113#924C3×C418yescomplemented
114C2×Dic3⋊C4C22×C312yes—
115C2×Dic3⋊C4C22×C312yes—
116C2×Dic3⋊C4C22×C312yes—
117C2×C4×Dic3C22×C312yes—
118C2×C4×Dic3C22×C312yes—
119C3×#95—22no—
120C3×#95—32no—
121C3×#95—32no—
122C3×#95—32no—
123C3×#95—32no—
124C3×#95—32no—
125C3×#95—32no—
126C3×#95C2×D312yes—
127C3×#95C22×C312yes—
128C3⋊#95C22×C312yes—
129C3⋊#95C22×C312yes—
130C3⋊#95C22×C312yes—
131C3⋊#95C22×C312yes—
132C3⋊#95C22×C312yes—
133C3⋊#95C22×C312yes—
134C23×C32—21no—
135C23×C32—21no—
136C23×C32—21no—
137C23×C32—21no—
138C23×C32C22×C412yes—
139C23×C32C2×Q812yes—
140C23×C32#4112yes—
141C23×C32C2411yescharacteristic
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—44no—
153C2×C32×C4C22×C412yes—
154C2×C32×C4C22×C412yes—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—44no—
165C2×C3×Dic3—44no—
166C2×C3×Dic3C22×C414yes—
167C2×C3×Dic3C22×C414yes—
168C2×C3×Dic3C2×Q814yes—
169C3×Dic6—44no—
170C2×C22⋊Q8—31no—
171#391—32no—
172#389—32no—
173#387—32no—
174#387—32no—
175#385—32no—
176#380C3×D311yescharacteristic
177C4×C22⋊C4—32no—
178C2×#95—31no—
179C24×C3—21no—
180C24×C3C22×D311yescharacteristic
181C24×C3C23×C311yescharacteristic
182C2×C3×Q8—34no—
183C2×C3×Q8—121no—
184C2×C3×Q8—121no—
185C22×C3×C4—24no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—32no—
191C22×C3×C4—32no—
192C22×C3×C4—32no—
193C22×C3×C4—34no—
194C22×C3×C4—32no—
195C22×C3×C4—41no—
196C22×C3×C4—41no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—62no—
202C22×C3×C4—121no—
203C22×C3×C4—121no—
204C22×C3×C4—121no—
205C22×C3×C4—121no—
206C22×C3×C4C4×D314yes—
207C22×C3×C4C2×C3×C414yes—
208C22×C3×C4C22×D312yes—
209C22×C3×C4C22×D312yes—
210C22×C3×C4C22×D311yescharacteristic
211C22×C3×C4C22×D311yescharacteristic
212C22×C3×C4C23×C312yes—
213C22×C3×C4C23×C312yes—
214C22×C3×C4C23×C311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×Dic3—22no—
217C22×Dic3—41no—
218C22×Dic3—41no—
219C22×Dic3—41no—
220C22×Dic3—41no—
221C22×Dic3C3×Q814yescomplemented
222C22×Dic3C23×C312yes—
223C22×Dic3C23×C312yes—
224C22×Dic3C23×C312yes—
225C22×Dic3C23×C312yes—
226C2×Dic6—41no—
227C2×Dic6—41no—
228C2×Dic6C2×C3×C414yescomplemented
229C3×C4⋊C4—38no—
230C3×C4⋊C4—34no—
231C3×C4⋊C4—41no—
232C3×C4⋊C4—41no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—41no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4—42no—
237C3×C4⋊C4—122no—
238C3×C4⋊C4—122no—
239C3×C22⋊C4—24no—
240C3×C22⋊C4—24no—
241C3×C22⋊C4—38no—
242C3×C22⋊C4—44no—
243C3×C42—122no—
244C3×C42—122no—
245#217C2×C3×C418yes—
246C4⋊Dic3C2×C3×C414yescomplemented
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4C2×C3×C418yescomplemented
250C4×Dic3—42no—
251C4×Dic3—42no—
252C22×C32—22no—
253C22×C32—22no—
254C22×C32—22no—
255C22×C32—22no—
256C22×C32—22no—
257C22×C32—22no—
258C22×C32C4×Q814yes—
259C22×C32C23×C411yescharacteristic
260C22×C32C22×Q811yescharacteristic
261C22×C32C2×#4111yescharacteristic
262C22×C32H(5,𝔽2)11yescharacteristic
263C22×C32H(5,𝔽2)11yescharacteristic
264C22×C32H(5,𝔽2)11yescharacteristic
265C22×C322−1+411yescharacteristic
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—42no—
269C32×C4—42no—
270C3×Dic3—24no—
271C3×Dic3—24no—
272C3×Dic3—24no—
273C3×Dic3—42no—
274C22×Q8—31no—
275C23×C4—31no—
276C23×C4—31no—
277C23×C4C2×C3×D311yescharacteristic
278C22⋊Q8—38no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—32no—
281C2×C4⋊C4—32no—
282C2×C4⋊C4—32no—
283C2×C4⋊C4C2×C3×D311yescharacteristic
284C2×C4⋊C4C2×C3×D311yescharacteristic
285C2×C22⋊C4—32no—
286C2×C22⋊C4C2×C3×D312yes—
287C2×C42—32no—
288C2×C42—32no—
289#95—32no—
290#95—32no—
291#95—32no—
292#95—32no—
293#95—32no—
294#95—32no—
295#95C2×C3×D312yes—
296C23×C3—21no—
297C23×C3—21no—
298C23×C3—21no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3—22no—
307C23×C3—22no—
308C23×C3—41no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3C2×C4×D312yes—
313C23×C3#23712yes—
314C23×C3D3×Q812yes—
315C23×C3C22×C3×C412yes—
316C23×C3C2×C3×Q812yes—
317C23×C3C3×#4112yes—
318C23×C3C23×D311yescharacteristic
319C23×C3C24×C311yescharacteristic
320C3×Q8—124no—
321C2×C3×C4—22no—
322C2×C3×C4—22no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
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—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—34no—
346C2×C3×C4—34no—
347C2×C3×C4—34no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—42no—
356C2×C3×C4—44no—
357C2×C3×C4—44no—
358C2×C3×C4—48no—
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—62no—
367C2×C3×C4—62no—
368C2×C3×C4—124no—
369C2×C3×C4—124no—
370C2×C3×C4C2×C4×D312yes—
371C2×C3×C4C2×C4×D312yes—
372C2×C3×C4C22×C3×C412yes—
373C2×C3×C4C22×C3×C412yes—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—44no—
384C2×Dic3—44no—
385C2×Dic3C22×C3×C414yes—
386C2×Dic3C22×C3×C414yes—
387C2×Dic3C2×C3×Q814yes—
388Dic6—44no—
389C2×C32—24no—
390C2×C32C2×C4×Q811yescharacteristic
391C2×C32#51811yescharacteristic
392C2×C32#52011yescharacteristic
393C2×C32#53811yescharacteristic
394C2×C32#53911yescharacteristic
395C2×C32#54311yescharacteristic
396C2×C32#54411yescharacteristic
397C24C22×C3×D311yescharacteristic
398C2×Q8—34no—
399C2×Q8—121no—
400C2×Q8—121no—
401C22×C4—32no—
402C22×C4—32no—
403C22×C4—32no—
404C22×C4—34no—
405C22×C4—32no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—62no—
409C22×C4—121no—
410C22×C4—121no—
411C22×C4—121no—
412C22×C4—121no—
413C22×C4C3×C4×D314yes—
414C22×C4C22×C3×D312yes—
415C22×C4C22×C3×D312yes—
416C22×C4C22×C3×D311yescharacteristic
417C22×C4C22×C3×D311yescharacteristic
418C4⋊C4—38no—
419C4⋊C4—34no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—122no—
423C4⋊C4—122no—
424C22⋊C4—24no—
425C22⋊C4—38no—
426C42—122no—
427C42—122no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—22no—
440C22×C3—24no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—21no—
447C22×C3—21no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3#88714yes—
455C22×C3C3×C4×Q814yes—
456C22×C3C22×C4×D311yescharacteristic
457C22×C3C2×#23711yescharacteristic
458C22×C3#100411yescharacteristic
459C22×C3#100411yescharacteristic
460C22×C3C2×D3×Q811yescharacteristic
461C22×C3#100911yescharacteristic
462C22×C3#101011yescharacteristic
463C22×C3C23×C3×C411yescharacteristic
464C22×C3C22×C3×Q811yescharacteristic
465C22×C3C2×C3×#4111yescharacteristic
466C22×C3C3×H(5,𝔽2)11yescharacteristic
467C22×C3C3×H(5,𝔽2)11yescharacteristic
468C22×C3C3×H(5,𝔽2)11yescharacteristic
469C22×C3C3×2−1+411yescharacteristic
470C3×C4—24no—
471C3×C4—22no—
472C3×C4—22no—
473C3×C4—22no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—64no—
481C3×C4—64no—
482C3×C4—64no—
483C3×C4—82no—
484C3×C4—82no—
485C3×C4—122no—
486Dic3—24no—
487Dic3—24no—
488Dic3—24no—
489Dic3—42no—
490C32#232911yescharacteristic, complemented, Sylow
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23C2×C3×C4×D312yes—
496C23C3×#23712yes—
497C23C3×D3×Q812yes—
498C23C23×C3×D311yescharacteristic
499Q8—124no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—34no—
511C2×C4—34no—
512C2×C4—34no—
513C2×C4—44no—
514C2×C4—62no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—62no—
520C2×C4—62no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—124no—
524C2×C4—124no—
525C2×C4C2×C3×C4×D312yes—
526C2×C4C2×C3×C4×D312yes—
527C2×C3—24no—
528C2×C3—24no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—44no—
537C2×C3#548611yescharacteristic
538C2×C3C2×#88711yescharacteristic
539C2×C3#552311yescharacteristic
540C2×C3#560211yescharacteristic
541C2×C3#561711yescharacteristic
542C2×C3#562311yescharacteristic
543C2×C3#563411yescharacteristic
544C2×C3C2×C3×C4×Q811yescharacteristic
545C2×C3C3×#51811yescharacteristic
546C2×C3C3×#52011yescharacteristic
547C2×C3C3×#53811yescharacteristic
548C2×C3C3×#53911yescharacteristic
549C2×C3C3×#54311yescharacteristic
550C2×C3C3×#54411yescharacteristic
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22C3×#88714yes—
558C22C22×C3×C4×D311yescharacteristic
559C22C2×C3×#23711yescharacteristic
560C22C3×#100411yescharacteristic
561C22C3×#100411yescharacteristic
562C22C2×C3×D3×Q811yescharacteristic
563C22C3×#100911yescharacteristic
564C22C3×#101011yescharacteristic
565C4—22no—
566C4—22no—
567C4—42no—
568C4—42no—
569C4—64no—
570C4—64no—
571C4—64no—
572C4—122no—
573C3—21no—
574C3#7387411yescharacteristic, direct product factor
575C3C3×#232911yescharacteristic, complemented
576C2—24no—
577C2C3×#548611yescharacteristic
578C2C2×C3×#88711yescharacteristic
579C2C3×#552311yescharacteristic
580C2C3×#560211yescharacteristic
581C2C3×#561711yescharacteristic
582C2C3×#562311yescharacteristic
583C2C3×#563411yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#380C2
socleC23×C32C24
prime radicalC24×C32C23
cosocleC23×C3C24×C3
hypercenterC3×#380D3
hypocenterC3C3×#2329
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#380

Complements

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

NH ≅ G/Ntype
4. C3×#4978C2semidirect
5. C3×#4976C2semidirect
8. C3×#4653C2semidirect
9. C3×#4646C2semidirect
11. #73874C3direct (N × H)
21. C3×#924C4semidirect
47. #4978C2×C3semidirect
48. #4976C2×C3semidirect
51. #4653C2×C3semidirect
52. #4646C2×C3semidirect
66. C22×C3×Dic3Q8semidirect
73. C2×C3×Dic6C2×C4semidirect
78. C3×C4⋊Dic3C2×C4semidirect
81. C3×Dic3⋊C4C2×C4semidirect
113. #924C3×C4semidirect
221. C22×Dic3C3×Q8semidirect
228. C2×Dic6C2×C3×C4semidirect
246. C4⋊Dic3C2×C3×C4semidirect
249. Dic3⋊C4C2×C3×C4semidirect
490. C32#2329semidirect
574. C3#73874direct (N × H)
575. C3C3×#2329semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9242
C32×#3802
C2×C3×C3⋊#952
C3×#49782
C3×#49762
C3×#49462
C3×Dic3×C22⋊C42
C3×#46532
C3×#46462
C3×#23293
#738743