Finite Group Databasebeta

← C3×#74360 #11805419

Subgroups — C3×#74360 #11805419

Order 1152.

1645 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#893C212yesmaximal
2C2×C3×D3×C4⋊C4C212yesmaximal, complemented
3C2×C3×C4×D12C211yesmaximal, characteristic
4C32×#384C211yesmaximal, characteristic, complemented
5C3×#4989C214yesmaximal, complemented
6C3×#4987C212yesmaximal, complemented
7C3×#4972C212yesmaximal, complemented
8C3×C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
9C3×D4×C4⋊C4—31nomaximal
10#74360C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2214yes—
13C22×C3×C4×D3C2212yes—
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C4⋊C4C2212yes—
16C2×C32×C4⋊C4C2212yes—
17C2×C32×C42C2211yescharacteristic
18C2×C3×(C2×D3)⋊C4C2214yes—
19C2×C3×(C2×D3)⋊C4C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4⋊Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×Dic3⋊C4C2214yes—
24C2×C3×C4×Dic3C2212yes—
25C3×#893—28no—
26C3×D3×C4⋊C4—28no—
27C3×C4×D12C418yescomplemented
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2214yes—
30C2×C3×C4×D4—32no—
31C2×C3×C4×D4—31no—
32C22×C3×C4⋊C4—32no—
33C2×#893C2×C312yes—
34C2×D3×C4⋊C4C2×C312yescomplemented
35C2×C4×D12C2×C311yescharacteristic
36C3×#389—32no—
37C3×#385—34no—
38C3×#384—21no—
39C3×#384D311yescharacteristic, complemented
40C3×#384C2×C311yescharacteristic, complemented
41C3×#380—32no—
42C3×C4×C4⋊C4—31no—
43#4989C2×C314yescomplemented
44#4987C2×C312yescomplemented
45#4972C2×C312yescomplemented
46C4×C4⋊Dic3C2×C311yescharacteristic, complemented
47C23×C3×D3C2312yes—
48C22×C32×C4C2314yes—
49C22×C32×C4C2311yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3C2312yes—
53C22×C3×Dic3C2312yes—
54C22×C3×Dic3C2312yes—
55C2×C3×D12C2×C414yes—
56C2×C3×D12D414yes—
57C2×C3×D12Q814yes—
58C2×C3×C4×D3—24no—
59C2×C3×C4×D3—24no—
60C2×C3×C4×D3—28no—
61C2×C3×C4×D3—28no—
62C2×C3×C4×D3—44no—
63C2×C3×C4×D3C2×C418yescomplemented
64C32×C4⋊C4—24no—
65C32×C4⋊C4—24no—
66C32×C4⋊C4—42no—
67C32×C42C2×C414yescomplemented
68C3×(C2×D3)⋊C4—28no—
69C3×(C2×D3)⋊C4C2×C418yescomplemented
70C3×C4⋊Dic3—22no—
71C3×C4⋊Dic3C2×C414yescomplemented
72C3×C4⋊Dic3D414yescomplemented
73C3×Dic3⋊C4—28no—
74C3×C4×Dic3—24no—
75D4×C4⋊C4—31noSylow
76C22×C3×D4—31no—
77C23×C3×C4—32no—
78C23×C3×C4—34no—
79C22×D12C22×C311yescharacteristic
80C22×C4×D3C22×C314yes—
81C22×C4×D3C22×C312yes—
82C3×C4×D4—38no—
83C3×C4×D4—68no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—34no—
90C2×C3×C4⋊C4—31no—
91C2×C3×C4⋊C4—68no—
92C2×C3×C4⋊C4C2×D312yes—
93C2×C3×C4⋊C4C2×D312yes—
94C2×C3×C4⋊C4C2×D312yes—
95C2×C3×C4⋊C4C22×C312yes—
96C2×C3×C4⋊C4C22×C312yes—
97C2×C3×C4⋊C4C22×C312yes—
98C2×C3×C22⋊C4—32no—
99C2×C3×C22⋊C4—34no—
100C2×C3×C42—21no—
101C2×C3×C42—32no—
102C2×C3×C42C2×D311yescharacteristic
103C2×C3×C42C22×C311yescharacteristic
104C2×(C2×D3)⋊C4C22×C314yes—
105C2×(C2×D3)⋊C4C22×C312yes—
106C2×C4⋊Dic3C22×C311yescharacteristic
107C2×C4⋊Dic3C22×C311yescharacteristic
108C2×C4⋊Dic3C22×C311yescharacteristic
109C2×Dic3⋊C4C22×C314yes—
110C2×C4×Dic3C22×C312yes—
111#893—28no—
112D3×C4⋊C4—28no—
113C4×D12C3×C418yescomplemented
114C3×#95—32no—
115C3×#95—34no—
116C3⋊#95C22×C312yes—
117C3⋊#95C22×C314yes—
118C23×C32C2411yescharacteristic
119C22×C3×D3—22no—
120C22×C3×D3—22no—
121C22×C3×D3—22no—
122C22×C3×D3—22no—
123C22×C3×D3C22×C414yes—
124C22×C3×D3C2×D414yes—
125C22×C3×D3C2×Q814yes—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4C22×C412yes—
132C2×C32×C4C22×C412yes—
133C2×C32×C4C22×C412yes—
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×Q812yes—
137C2×C32×C4#4112yes—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3C22×C414yes—
146C2×C3×Dic3C2×D414yes—
147C3×D12C4⋊C4116yescomplemented
148C3×C4×D3—216no—
149C3×C4×D3—416no—
150C2×C4×D4—32no—
151C2×C4×D4—31no—
152C22×C4⋊C4—32no—
153#389—32no—
154#385—34no—
155#384C3×D311yescharacteristic, complemented
156#380—32no—
157C4×C4⋊C4—31no—
158C24×C3—32no—
159C23×D3C23×C312yes—
160C2×C3×D4—34no—
161C2×C3×D4—34no—
162C2×C3×D4—34no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—24no—
167C22×C3×C4—32no—
168C22×C3×C4—32no—
169C22×C3×C4—38no—
170C22×C3×C4—32no—
171C22×C3×C4—68no—
172C22×C3×C4—68no—
173C22×C3×C4—64no—
174C22×C3×C4—64no—
175C22×C3×C4—124no—
176C22×C3×C4C22×D314yes—
177C22×C3×C4C22×D311yescharacteristic
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C22×D311yescharacteristic
180C22×C3×C4C23×C314yes—
181C22×C3×C4C23×C311yescharacteristic
182C22×C3×C4C23×C311yescharacteristic
183C22×C3×C4C23×C311yescharacteristic
184C22×Dic3C23×C312yes—
185C22×Dic3C23×C312yes—
186C22×Dic3C23×C312yes—
187C2×D12C2×C3×C414yes—
188C2×D12C3×D414yes—
189C2×D12C3×Q814yes—
190C2×C4×D3—24no—
191C2×C4×D3—24no—
192C2×C4×D3—28no—
193C2×C4×D3—28no—
194C2×C4×D3—44no—
195C2×C4×D3C2×C3×C418yescomplemented
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—34no—
201C3×C4⋊C4—34no—
202C3×C4⋊C4—44no—
203C3×C4⋊C4—44no—
204C3×C4⋊C4—42no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4—44no—
207C3×C4⋊C4—68no—
208C3×C4⋊C4—62no—
209C3×C22⋊C4—38no—
210C3×C22⋊C4—68no—
211C3×C42—24no—
212C3×C42—64no—
213C3×C42C4×D314yescomplemented
214C3×C42C2×C3×C414yescomplemented
215(C2×D3)⋊C4—28no—
216(C2×D3)⋊C4C2×C3×C418yescomplemented
217C4⋊Dic3—22no—
218C4⋊Dic3C2×C3×C414yescomplemented
219C4⋊Dic3C3×D414yescomplemented
220Dic3⋊C4—28no—
221C4×Dic3—24no—
222C22×C32C23×C411yescharacteristic
223C22×C32C22×D411yescharacteristic
224C22×C32C22×D411yescharacteristic
225C22×C32C22×Q811yescharacteristic
226C22×C32C2×#4111yescharacteristic
227C22×C32H(5,𝔽2)11yescharacteristic
228C22×C322−1+411yescharacteristic
229C2×C3×D3—24no—
230C2×C3×D3—24no—
231C2×C3×D3—24no—
232C2×C3×D3—24no—
233C2×C3×D3—24no—
234C2×C3×D3—24no—
235C2×C3×D3C2×C4⋊C418yescomplemented
236C32×C4—22no—
237C32×C4—44no—
238C32×C4C2×C4⋊C414yescomplemented
239C32×C4C4×D414yescomplemented
240C3×Dic3—24no—
241C3×Dic3—24no—
242C3×Dic3—42no—
243C22×D4—31no—
244C23×C4—32no—
245C23×C4—34no—
246C4×D4—38no—
247C4×D4—68no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—68no—
253C2×C4⋊C4C2×C3×D312yes—
254C2×C4⋊C4C2×C3×D312yes—
255C2×C4⋊C4C2×C3×D312yes—
256C2×C22⋊C4—32no—
257C2×C22⋊C4—34no—
258C2×C42—32no—
259C2×C42C2×C3×D311yescharacteristic
260#95—32no—
261#95—34no—
262C23×C3—21no—
263C23×C3—34no—
264C23×C3—34no—
265C23×C3—34no—
266C23×C3—62no—
267C23×C3—62no—
268C23×C3—62no—
269C23×C3—62no—
270C23×C3C23×D311yescharacteristic
271C23×C3C24×C311yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3C22×C3×C414yes—
277C22×D3C2×C3×D414yes—
278C22×D3C2×C3×Q814yes—
279C3×D4—316no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—24no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—34no—
300C2×C3×C4—34no—
301C2×C3×C4—44no—
302C2×C3×C4—44no—
303C2×C3×C4—44no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—616no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—1216no—
313C2×C3×C4C2×C4×D312yes—
314C2×C3×C4C2×C4×D312yes—
315C2×C3×C4C2×C4×D312yes—
316C2×C3×C4D3×D412yes—
317C2×C3×C4D3×D412yes—
318C2×C3×C4D3×Q812yes—
319C2×C3×C4#23912yes—
320C2×C3×C4C22×C3×C412yes—
321C2×C3×C4C22×C3×C412yes—
322C2×C3×C4C22×C3×C412yes—
323C2×C3×C4C2×C3×D412yes—
324C2×C3×C4C2×C3×D412yes—
325C2×C3×C4C2×C3×Q812yes—
326C2×C3×C4C3×#4112yes—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—22no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3C22×C3×C414yes—
335C2×Dic3C2×C3×D414yes—
336D12C3×C4⋊C4116yescomplemented
337C4×D3—216no—
338C4×D3—416no—
339C2×C32C22×C4⋊C411yescharacteristic
340C2×C32C2×C4×D411yescharacteristic
341C2×C32#52011yescharacteristic
342C2×C32D4211yescharacteristic
343C2×C32#54711yescharacteristic
344C2×C32D4×Q811yescharacteristic
345C2×C32#55411yescharacteristic
346C3×D3—28no—
347C24—32no—
348C2×D4—34no—
349C2×D4—34no—
350C2×D4—34no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—38no—
354C22×C4—32no—
355C22×C4—68no—
356C22×C4—68no—
357C22×C4—64no—
358C22×C4—64no—
359C22×C4—124no—
360C22×C4C22×C3×D314yes—
361C22×C4C22×C3×D311yescharacteristic
362C22×C4C22×C3×D311yescharacteristic
363C22×C4C22×C3×D311yescharacteristic
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—34no—
367C4⋊C4—34no—
368C4⋊C4—42no—
369C4⋊C4—68no—
370C4⋊C4—62no—
371C22⋊C4—38no—
372C22⋊C4—68no—
373C42—64no—
374C42C3×C4×D314yescomplemented
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—38no—
383C22×C3—64no—
384C22×C3—64no—
385C22×C3—64no—
386C22×C3—64no—
387C22×C3—64no—
388C22×C3—64no—
389C22×C3C22×C4×D311yescharacteristic
390C22×C3C2×D3×D411yescharacteristic
391C22×C3C2×D3×D411yescharacteristic
392C22×C3#100411yescharacteristic
393C22×C3C2×D3×Q811yescharacteristic
394C22×C3C2×#23911yescharacteristic
395C22×C3#100711yescharacteristic
396C22×C3C23×C3×C411yescharacteristic
397C22×C3C22×C3×D411yescharacteristic
398C22×C3C22×C3×D411yescharacteristic
399C22×C3C22×C3×Q811yescharacteristic
400C22×C3C2×C3×#4111yescharacteristic
401C22×C3C3×H(5,𝔽2)11yescharacteristic
402C22×C3C3×2−1+411yescharacteristic
403C2×D3—24no—
404C2×D3—24no—
405C2×D3—24no—
406C2×D3—24no—
407C2×D3—24no—
408C2×D3—24no—
409C2×D3C2×C3×C4⋊C418yescomplemented
410C3×C4—24no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—44no—
416C3×C4—44no—
417C3×C4—64no—
418C3×C4—64no—
419C3×C4—84no—
420C3×C4—122no—
421C3×C4D3×C4⋊C414yescomplemented
422C3×C4#89314yescomplemented
423C3×C4C2×C3×C4⋊C414yescomplemented
424C3×C4C3×C4×D414yescomplemented
425Dic3—24no—
426Dic3—24no—
427Dic3—42no—
428C32D4×C4⋊C411yescharacteristic, complemented, Sylow
429C23—34no—
430C23—34no—
431C23—34no—
432C23—62no—
433C23—62no—
434C23—62no—
435C23—62no—
436C23C23×C3×D311yescharacteristic
437D4—316no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—34no—
444C2×C4—34no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—616no—
450C2×C4—62no—
451C2×C4—62no—
452C2×C4—62no—
453C2×C4—1216no—
454C2×C4C2×C3×C4×D312yes—
455C2×C4C2×C3×C4×D312yes—
456C2×C4C2×C3×C4×D312yes—
457C2×C4C3×D3×D412yes—
458C2×C4C3×D3×D412yes—
459C2×C4C3×D3×Q812yes—
460C2×C4C3×#23912yes—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—68no—
469C2×C3C2×D3×C4⋊C411yescharacteristic
470C2×C3C2×#89311yescharacteristic
471C2×C3#550411yescharacteristic
472C2×C3#571711yescharacteristic
473C2×C3#572611yescharacteristic
474C2×C3#572711yescharacteristic
475C2×C3#573011yescharacteristic
476C2×C3C22×C3×C4⋊C411yescharacteristic
477C2×C3C2×C3×C4×D411yescharacteristic
478C2×C3C3×#52011yescharacteristic
479C2×C3C3×D4211yescharacteristic
480C2×C3C3×#54711yescharacteristic
481C2×C3C3×D4×Q811yescharacteristic
482C2×C3C3×#55411yescharacteristic
483D3—28no—
484C22—38no—
485C22—64no—
486C22—64no—
487C22—64no—
488C22—64no—
489C22—64no—
490C22—64no—
491C22C22×C3×C4×D311yescharacteristic
492C22C2×C3×D3×D411yescharacteristic
493C22C2×C3×D3×D411yescharacteristic
494C22C3×#100411yescharacteristic
495C22C2×C3×D3×Q811yescharacteristic
496C22C2×C3×#23911yescharacteristic
497C22C3×#100711yescharacteristic
498C4—22no—
499C4—44no—
500C4—64no—
501C4—64no—
502C4—122no—
503C4C3×D3×C4⋊C414yescomplemented
504C4C3×#89314yescomplemented
505C3—21no—
506C3#7436011yescharacteristic, direct product factor
507C3C3×D4×C4⋊C411yescharacteristic, complemented
508C2—68no—
509C2C2×C3×D3×C4⋊C411yescharacteristic
510C2C2×C3×#89311yescharacteristic
511C2C3×#550411yescharacteristic
512C2C3×#571711yescharacteristic
513C2C3×#572611yescharacteristic
514C2C3×#572711yescharacteristic
515C2C3×#573011yescharacteristic
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×#384C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×#384D3
hypocenterC3C3×D4×C4⋊C4
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×#384

Complements

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

NH ≅ G/Ntype
2. C2×C3×D3×C4⋊C4C2semidirect
4. C32×#384C2semidirect
5. C3×#4989C2semidirect
6. C3×#4987C2semidirect
7. C3×#4972C2semidirect
8. C3×C4×C4⋊Dic3C2semidirect
10. #74360C3direct (N × H)
27. C3×C4×D12C4semidirect
34. C2×D3×C4⋊C4C2×C3semidirect
39. C3×#384D3semidirect
40. C3×#384C2×C3semidirect
43. #4989C2×C3semidirect
44. #4987C2×C3semidirect
45. #4972C2×C3semidirect
46. C4×C4⋊Dic3C2×C3semidirect
63. C2×C3×C4×D3C2×C4semidirect
67. C32×C42C2×C4semidirect
69. C3×(C2×D3)⋊C4C2×C4semidirect
71. C3×C4⋊Dic3C2×C4semidirect
72. C3×C4⋊Dic3D4semidirect
113. C4×D12C3×C4semidirect
147. C3×D12C4⋊C4semidirect
155. #384C3×D3semidirect
195. C2×C4×D3C2×C3×C4semidirect
213. C3×C42C4×D3semidirect
214. C3×C42C2×C3×C4semidirect
216. (C2×D3)⋊C4C2×C3×C4semidirect
218. C4⋊Dic3C2×C3×C4semidirect
219. C4⋊Dic3C3×D4semidirect
235. C2×C3×D3C2×C4⋊C4semidirect
238. C32×C4C2×C4⋊C4semidirect
239. C32×C4C4×D4semidirect
336. D12C3×C4⋊C4semidirect
374. C42C3×C4×D3semidirect
409. C2×D3C2×C3×C4⋊C4semidirect
421. C3×C4D3×C4⋊C4semidirect
422. C3×C4#893semidirect
423. C3×C4C2×C3×C4⋊C4semidirect
424. C3×C4C3×C4×D4semidirect
428. C32D4×C4⋊C4semidirect
503. C4C3×D3×C4⋊C4semidirect
504. C4C3×#893semidirect
506. C3#74360direct (N × H)
507. C3C3×D4×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8932
C2×C3×D3×C4⋊C42
C2×C3×C4×D122
C32×#3842
C3×#49892
C3×#49872
C3×#49722
C3×C4×C4⋊Dic32
C3×D4×C4⋊C43
#743603