Finite Group Databasebeta

← C3×#10596019 #13179302

Subgroups — C3×#10596019 #13179302

Order 1728.

2236 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3828C212yesmaximal, complemented
2C33×C4×D4C211yesmaximal, characteristic, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C3×#64147C211yesmaximal, characteristic, complemented
5C3×C4×#638C212yesmaximal, complemented
6C3×#64104C211yesmaximal, characteristic, complemented
7C3×#64097C211yesmaximal, characteristic, complemented
8C3×#64094C212yesmaximal, complemented
9C3×#64093C212yesmaximal, complemented
10C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C3×C42×C3⋊D3C211yesmaximal, characteristic, complemented
12#10596019C311yesmaximal, characteristic, direct product factor
13C3×#5548—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×#638C2212yes—
19C3×#3828—28no—
20C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
21C2×C3×C4⋊(C3⋊D3)C2212yes—
22C2×C3×C4×C3⋊D3C2211yescharacteristic
23C2×C3×C4×C3⋊D3C2211yescharacteristic
24C2×C3×C4×C3⋊D3C2212yes—
25C2×C3×C4×C3⋊D3C2211yescharacteristic
26C2×C3×#634C2212yes—
27C33×C4⋊C4C2211yescharacteristic
28C33×C22⋊C4C2212yes—
29C33×C42C2211yescharacteristic
30C3×#3757C2212yes—
31C3×#3752C2212yes—
32C3×#3752C2212yes—
33C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2211yescharacteristic
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C3×C4×C3⋊Dic3C2211yescharacteristic
38C2×#3828C2×C312yescomplemented
39C2×C3×#235—38no—
40C32×C4×D4—24no—
41C32×C4×D4D314yescomplemented
42C32×C4×D4C2×C311yescharacteristic, complemented
43#64149C2×C311yescharacteristic, complemented
44#64147C2×C311yescharacteristic, complemented
45C4×#638C2×C312yescomplemented
46#64104C2×C311yescharacteristic, complemented
47#64097C2×C311yescharacteristic, complemented
48#64094C2×C312yescomplemented
49#64093C2×C312yescomplemented
50C4×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
51C42×C3⋊D3C2×C311yescharacteristic, complemented
52C3×#940—34no—
53C3×#938—34no—
54C3×C4×#67—38no—
55C3×#895—34no—
56C3×#888—34no—
57C3×#885—38no—
58C3×#884—38no—
59C3×C4×D12—34no—
60C3×C42×D3—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2312yes—
63C22×C3×C3⋊D3C2311yescharacteristic
64C33×D4—22no—
65C2×C33×C4—24no—
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2312yes—
68C2×C33×C4C2311yescharacteristic
69C2×C33×C4C2311yescharacteristic
70C3×#638—24no—
71C3×#638—24no—
72C3×#638—24no—
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2311yescharacteristic
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4⋊(C3⋊D3)—24no—
78C3×C4⋊(C3⋊D3)—22no—
79C3×C4×C3⋊D3—24no—
80C3×C4×C3⋊D3—22no—
81C3×C4×C3⋊D3—22no—
82C3×C4×C3⋊D3D414yescomplemented
83C3×#634—24no—
84C3×#532—91no—
85#5548—34no—
86C2×C32×D4—24no—
87C2×C32×D4C2×D314yes—
88C2×C32×D4C22×C311yescharacteristic
89C22×C32×C4—28no—
90C22×C32×C4C2×D318yes—
91C22×C32×C4C22×C312yes—
92C2×#638C22×C312yes—
93C2×#638C22×C312yes—
94C2×#638C22×C312yes—
95#3828—28no—
96C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
97C2×C4⋊(C3⋊D3)C22×C312yes—
98C2×C4×C3⋊D3C22×C311yescharacteristic
99C2×C4×C3⋊D3C22×C311yescharacteristic
100C2×C4×C3⋊D3C22×C312yes—
101C2×C4×C3⋊D3C22×C311yescharacteristic
102C2×#634C22×C312yes—
103C2×C3×#67—38no—
104C2×C3×#67—38no—
105C2×C3×#67—38no—
106C3×#235—632no—
107C2×C3×D12—38no—
108C2×C3×D12—34no—
109C2×C3×C4×D3—34no—
110C2×C3×C4×D3—34no—
111C2×C3×C4×D3—34no—
112C2×C3×C4×D3—38no—
113C2×C3×Dic6—38no—
114C32×C4⋊C4—24no—
115C32×C4⋊C4C2×D314yes—
116C32×C4⋊C4C22×C311yescharacteristic
117C32×C22⋊C4—28no—
118C32×C22⋊C4C2×D318yes—
119C32×C22⋊C4C22×C312yes—
120C32×C42—24no—
121C32×C42C2×D314yes—
122C32×C42C22×C311yescharacteristic
123#3757C22×C312yes—
124#3752C22×C312yes—
125#3752C22×C312yes—
126C4⋊(C3⋊Dic3)C22×C311yescharacteristic
127#3750C22×C312yes—
128C4×C3⋊Dic3C22×C311yescharacteristic
129C4×C3⋊Dic3C22×C311yescharacteristic
130C4×C3⋊Dic3C22×C311yescharacteristic
131C3×#217—38no—
132C3×(C2×D3)⋊C4—38no—
133C3×(C2×D3)⋊C4—38no—
134C3×C4⋊Dic3—34no—
135C3×Dic3⋊C4—38no—
136C3×C4×Dic3—34no—
137C3×C4×Dic3—34no—
138C3×C4×Dic3—34no—
139C22×C33—22no—
140C22×C33—22no—
141C22×C33—22no—
142C22×C33C2411yescharacteristic
143C2×C3×C3⋊D3—21no—
144C2×C3×C3⋊D3—21no—
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3—22no—
147C2×C3×C3⋊D3—22no—
148C2×C3×C3⋊D3C2×D412yes—
149C33×C4—22no—
150C33×C4—21no—
151C33×C4C2×D412yes—
152C33×C4#4112yescomplemented
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—21no—
155C3×C3⋊Dic3C2×D412yes—
156C3×C3⋊Dic3#4112yescomplemented
157C2×C3×#41—92no—
158C2×#235—38no—
159C3×C4⋊Q8—91no—
160C3×#127—91no—
161C3×#124—92no—
162C3×#121—91no—
163C3×#121—91no—
164C3×#121—92no—
165C3×C4×D4—24no—
166C3×C4×D4—92no—
167C3×C4×D4—91no—
168C3×C4×D4C3×D314yescomplemented
169C3×C4×D4C3⋊D311yescharacteristic, complemented
170C2×C3×C42—91no—
171#940—34no—
172#938—34no—
173C4×#67—38no—
174#895—34no—
175#888—34no—
176#885—38no—
177#884—38no—
178C4×D12—34no—
179C42×D3—34no—
180C23×C32—28no—
181C23×C32C22×D318yes—
182C23×C32C23×C312yes—
183C22×C3⋊D3C23×C312yes—
184C22×C3⋊D3C23×C311yescharacteristic
185C22×C3×D3—34no—
186C22×C3×D3—38no—
187C32×D4—216no—
188C32×D4—28no—
189C32×D4—22no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—24no—
193C2×C32×C4—24no—
194C2×C32×C4—28no—
195C2×C32×C4—216no—
196C2×C32×C4—416no—
197C2×C32×C4C22×D314yes—
198C2×C32×C4C22×D318yes—
199C2×C32×C4C22×D314yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C23×C311yescharacteristic
202C2×C32×C4C23×C312yes—
203C2×C32×C4C23×C311yescharacteristic
204C2×C32×C4C23×C311yescharacteristic
205#638—24no—
206#638—24no—
207#638—24no—
208C2×C3⋊Dic3C23×C311yescharacteristic
209C2×C3⋊Dic3C23×C312yes—
210C2×C3⋊Dic3C23×C311yescharacteristic
211C2×C3⋊Dic3C23×C311yescharacteristic
212C4⋊(C3⋊D3)—24no—
213C4⋊(C3⋊D3)—22no—
214C4×C3⋊D3—24no—
215C4×C3⋊D3—22no—
216C4×C3⋊D3—22no—
217C4×C3⋊D3C3×D414yescomplemented
218#634—24no—
219C3×#67—616no—
220C3×#67—616no—
221C3×#67—616no—
222C2×C3×Dic3—34no—
223C2×C3×Dic3—34no—
224C2×C3×Dic3—34no—
225C2×C3×Dic3—38no—
226C3×D12—616no—
227C3×D12—68no—
228C3×C4×D3—316no—
229C3×C4×D3—68no—
230C3×C4×D3—68no—
231C3×C4×D3—616no—
232C3×Dic6—616no—
233#532—91noSylow
234C2×C33—42no—
235C2×C33C22×D411yescharacteristic
236C2×C33C2×#4111yescharacteristic
237C2×C33C2×#4111yescharacteristic
238C3×C3⋊D3—22no—
239C3×C3⋊D3—42no—
240C3×#41—188no—
241C2×C3×Q8—92no—
242C2×C3×D4—24no—
243C2×C3×D4—92no—
244C2×C3×D4—92no—
245C2×C3×D4—92no—
246C2×C3×D4—91no—
247C2×C3×D4—92no—
248C2×C3×D4C2×C3×D314yes—
249C2×C3×D4C2×C3⋊D311yescharacteristic
250C22×C3×C4—28no—
251C22×C3×C4—91no—
252C22×C3×C4—91no—
253C22×C3×C4—91no—
254C22×C3×C4—92no—
255C22×C3×C4C2×C3×D318yes—
256C22×C3×C4C2×C3⋊D312yes—
257C2×#67—38no—
258C2×#67—38no—
259C2×#67—38no—
260#235—632no—
261C2×D12—34no—
262C2×D12—38no—
263C2×C4×D3—34no—
264C2×C4×D3—34no—
265C2×C4×D3—34no—
266C2×C4×D3—38no—
267C2×Dic6—38no—
268C3×C4⋊C4—24no—
269C3×C4⋊C4—92no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4C2×C3×D314yes—
272C3×C4⋊C4C2×C3⋊D311yescharacteristic
273C3×C22⋊C4—28no—
274C3×C22⋊C4—92no—
275C3×C22⋊C4—92no—
276C3×C22⋊C4—92no—
277C3×C22⋊C4C2×C3×D318yes—
278C3×C22⋊C4C2×C3⋊D312yes—
279C3×C42—24no—
280C3×C42—91no—
281C3×C42—91no—
282C3×C42—91no—
283C3×C42C2×C3×D314yes—
284C3×C42C2×C3⋊D311yescharacteristic
285#217—38no—
286(C2×D3)⋊C4—38no—
287(C2×D3)⋊C4—38no—
288C4⋊Dic3—34no—
289Dic3⋊C4—38no—
290C4×Dic3—34no—
291C4×Dic3—34no—
292C4×Dic3—34no—
293C22×C32—216no—
294C22×C32—28no—
295C22×C32—22no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—24no—
299C22×C32—28no—
300C22×C32—28no—
301C22×C32—48no—
302C22×C32—48no—
303C22×C32C23×D314yes—
304C22×C32C24×C311yescharacteristic
305C2×C3⋊D3—21no—
306C2×C3⋊D3—21no—
307C2×C3⋊D3—22no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3C2×C3×D412yes—
311C2×C3×D3—38no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—68no—
315C2×C3×D3—68no—
316C2×C3×D3—68no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—28no—
320C32×C4—28no—
321C32×C4—28no—
322C32×C4—24no—
323C32×C4—44no—
324C32×C4—48no—
325C32×C4#23518yescomplemented
326C32×C4D3×D418yes—
327C32×C4C2×C3×D412yes—
328C32×C4C3×#4112yescomplemented
329C3⋊Dic3—22no—
330C3⋊Dic3—21no—
331C3⋊Dic3C2×C3×D412yes—
332C3⋊Dic3C3×#4112yescomplemented
333C3×Dic3—38no—
334C3×Dic3—38no—
335C3×Dic3—64no—
336C3×Dic3—68no—
337C2×#41—92no—
338C4⋊Q8—91no—
339#127—91no—
340#124—92no—
341#121—91no—
342#121—91no—
343#121—92no—
344C4×D4—92no—
345C4×D4—91no—
346C4×D4C3×C3⋊D311yescharacteristic, complemented
347C2×C42—91no—
348C33#53211yescharacteristic, complemented, Sylow
349C23×C3—28no—
350C23×C3—91no—
351C23×C3—92no—
352C23×C3C22×C3×D318yes—
353C23×C3C22×C3⋊D312yes—
354C22×D3—34no—
355C22×D3—38no—
356C3×Q8—184no—
357C3×D4—216no—
358C3×D4—28no—
359C3×D4—22no—
360C3×D4—184no—
361C3×D4—184no—
362C3×D4—184no—
363C3×D4—184no—
364C3×D4—182no—
365C2×C3×C4—216no—
366C2×C3×C4—24no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—24no—
370C2×C3×C4—28no—
371C2×C3×C4—416no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—94no—
375C2×C3×C4—91no—
376C2×C3×C4—92no—
377C2×C3×C4—182no—
378C2×C3×C4—182no—
379C2×C3×C4—184no—
380C2×C3×C4C22×C3×D314yes—
381C2×C3×C4C22×C3×D318yes—
382C2×C3×C4C22×C3×D314yes—
383C2×C3×C4C22×C3×D314yes—
384C2×C3×C4C22×C3⋊D311yescharacteristic
385C2×C3×C4C22×C3⋊D312yes—
386C2×C3×C4C22×C3⋊D311yescharacteristic
387C2×C3×C4C22×C3⋊D311yescharacteristic
388#67—616no—
389#67—616no—
390#67—616no—
391C2×Dic3—34no—
392C2×Dic3—34no—
393C2×Dic3—34no—
394C2×Dic3—38no—
395D12—616no—
396D12—68no—
397C4×D3—316no—
398C4×D3—68no—
399C4×D3—68no—
400C4×D3—616no—
401Dic6—616no—
402C2×C32—24no—
403C2×C32—24no—
404C2×C32—24no—
405C2×C32—416no—
406C2×C32—48no—
407C2×C32—42no—
408C2×C32C2×#23514yes—
409C2×C32C2×D3×D414yes—
410C2×C32D3×#4114yes—
411C2×C32C22×C3×D411yescharacteristic
412C2×C32C2×C3×#4111yescharacteristic
413C2×C32C2×C3×#4111yescharacteristic
414C3⋊D3—22no—
415C3⋊D3—42no—
416C3×D3—68no—
417C3×D3—128no—
418#41—188no—
419C2×Q8—92no—
420C2×D4—92no—
421C2×D4—92no—
422C2×D4—92no—
423C2×D4—91no—
424C2×D4—92no—
425C2×D4C2×C3×C3⋊D311yescharacteristic
426C22×C4—91no—
427C22×C4—91no—
428C22×C4—91no—
429C22×C4—92no—
430C22×C4C2×C3×C3⋊D312yes—
431C4⋊C4—92no—
432C4⋊C4—91no—
433C4⋊C4C2×C3×C3⋊D311yescharacteristic
434C22⋊C4—92no—
435C22⋊C4—92no—
436C22⋊C4—92no—
437C22⋊C4C2×C3×C3⋊D312yes—
438C42—91no—
439C42—91no—
440C42—91no—
441C42C2×C3×C3⋊D311yescharacteristic
442C22×C3—216no—
443C22×C3—28no—
444C22×C3—28no—
445C22×C3—28no—
446C22×C3—22no—
447C22×C3—22no—
448C22×C3—22no—
449C22×C3—24no—
450C22×C3—48no—
451C22×C3—48no—
452C22×C3—92no—
453C22×C3—181no—
454C22×C3—181no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3—182no—
458C22×C3C23×C3×D314yes—
459C22×C3C23×C3⋊D311yescharacteristic
460C2×D3—38no—
461C2×D3—64no—
462C2×D3—64no—
463C2×D3—68no—
464C2×D3—68no—
465C2×D3—68no—
466C3×C4—28no—
467C3×C4—24no—
468C3×C4—22no—
469C3×C4—21no—
470C3×C4—28no—
471C3×C4—28no—
472C3×C4—48no—
473C3×C4—44no—
474C3×C4—92no—
475C3×C4—92no—
476C3×C4—181no—
477C3×C4—182no—
478C3×C4C3×#23518yescomplemented
479C3×C4C3×D3×D418yes—
480C3×C4#382812yescomplemented
481C3×C4D4×C3⋊D312yes—
482Dic3—38no—
483Dic3—38no—
484Dic3—64no—
485Dic3—68no—
486C32—24no—
487C32#554814yescomplemented
488C32C3×#53211yescharacteristic, complemented
489C23—91no—
490C23—92no—
491C23C22×C3×C3⋊D312yes—
492Q8—184no—
493D4—22no—
494D4—184no—
495D4—184no—
496D4—184no—
497D4—184no—
498D4—182no—
499C2×C4—24no—
500C2×C4—91no—
501C2×C4—91no—
502C2×C4—94no—
503C2×C4—91no—
504C2×C4—92no—
505C2×C4—182no—
506C2×C4—182no—
507C2×C4—184no—
508C2×C4C22×C3×C3⋊D311yescharacteristic
509C2×C4C22×C3×C3⋊D312yes—
510C2×C4C22×C3×C3⋊D311yescharacteristic
511C2×C4C22×C3×C3⋊D311yescharacteristic
512C2×C3—24no—
513C2×C3—24no—
514C2×C3—24no—
515C2×C3—416no—
516C2×C3—48no—
517C2×C3—42no—
518C2×C3—182no—
519C2×C3—362no—
520C2×C3C2×C3×#23514yes—
521C2×C3C2×C3×D3×D414yes—
522C2×C3C3×D3×#4114yes—
523C2×C3C2×#382811yescharacteristic
524C2×C3C2×D4×C3⋊D311yescharacteristic
525C2×C3#41×C3⋊D311yescharacteristic
526D3—68no—
527D3—128no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—92no—
532C22—181no—
533C22—181no—
534C22—182no—
535C22—182no—
536C22—182no—
537C22C23×C3×C3⋊D311yescharacteristic
538C4—22no—
539C4—21no—
540C4—92no—
541C4—92no—
542C4—181no—
543C4—182no—
544C4C3×#382812yescomplemented
545C4C3×D4×C3⋊D312yes—
546C3—24no—
547C3C3×#554814yescomplemented
548C3#1059601911yescharacteristic, direct product factor
549C2—42no—
550C2—182no—
551C2—362no—
552C2C2×C3×#382811yescharacteristic
553C2C2×C3×D4×C3⋊D311yescharacteristic
554C2C3×#41×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C3×C4C22×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3828C2semidirect
2. C33×C4×D4C2semidirect
3. C3×#64149C2semidirect
4. C3×#64147C2semidirect
5. C3×C4×#638C2semidirect
6. C3×#64104C2semidirect
7. C3×#64097C2semidirect
8. C3×#64094C2semidirect
9. C3×#64093C2semidirect
10. C3×C4×C4⋊(C3⋊D3)C2semidirect
11. C3×C42×C3⋊D3C2semidirect
12. #10596019C3direct (N × H)
38. C2×#3828C2×C3semidirect
41. C32×C4×D4D3semidirect
42. C32×C4×D4C2×C3semidirect
43. #64149C2×C3semidirect
44. #64147C2×C3semidirect
45. C4×#638C2×C3semidirect
46. #64104C2×C3semidirect
47. #64097C2×C3semidirect
48. #64094C2×C3semidirect
49. #64093C2×C3semidirect
50. C4×C4⋊(C3⋊D3)C2×C3semidirect
51. C42×C3⋊D3C2×C3semidirect
82. C3×C4×C3⋊D3D4semidirect
152. C33×C4#41semidirect
156. C3×C3⋊Dic3#41semidirect
168. C3×C4×D4C3×D3semidirect
169. C3×C4×D4C3⋊D3semidirect
217. C4×C3⋊D3C3×D4semidirect
325. C32×C4#235semidirect
328. C32×C4C3×#41semidirect
332. C3⋊Dic3C3×#41semidirect
346. C4×D4C3×C3⋊D3semidirect
348. C33#532semidirect
478. C3×C4C3×#235semidirect
480. C3×C4#3828semidirect
487. C32#5548semidirect
488. C32C3×#532semidirect
544. C4C3×#3828semidirect
547. C3C3×#5548semidirect
548. C3#10596019direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38282
C33×C4×D42
C3×#641492
C3×#641472
C3×C4×#6382
C3×#641042
C3×#640972
C3×#640942
C3×#640932
C3×C4×C4⋊(C3⋊D3)2
C3×C42×C3⋊D32
#105960193
C3×#55483