Finite Group Databasebeta

← C3×#10596017 #13179300

Subgroups — C3×#10596017 #13179300

Order 1728.

2140 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C33×C4×D4C211yesmaximal, characteristic, complemented
3C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C3×C4×#638C212yesmaximal, complemented
5C2×C3×#3750C212yesmaximal, complemented
6C3×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
7C3×#64096C211yesmaximal, characteristic, complemented
8C3×#64090C212yesmaximal, complemented
9C3×#64086C212yesmaximal, complemented
10C3×#64081C211yesmaximal, characteristic, complemented
11C3×C4×#634C211yesmaximal, characteristic, complemented
12#10596017C311yesmaximal, characteristic, direct product factor
13C3×#5546—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C33×C4C2212yes—
16C2×C3×#638C2212yes—
17C22×C3×C3⋊Dic3C2212yes—
18C22×C3×C3⋊Dic3C2212yes—
19C3×#3830C418yescomplemented
20C2×C3×C4×C3⋊D3C2212yes—
21C2×C3×C4×C3⋊D3C2211yescharacteristic
22C2×C3×#634C2211yescharacteristic
23C33×C4⋊C4C2211yescharacteristic
24C33×C22⋊C4C2212yes—
25C33×C42C2211yescharacteristic
26C3×#3757C2212yes—
27C3×#3752C2211yescharacteristic
28C3×#3752C2211yescharacteristic
29C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
30C3×#3750C2212yes—
31C3×#3750C2211yescharacteristic
32C3×#3750C2211yescharacteristic
33C3×#3750C2212yes—
34C3×#3750C2212yes—
35C3×C4×C3⋊Dic3C2212yes—
36C3×C4×C3⋊Dic3C2211yescharacteristic
37C3×C4×C3⋊Dic3C2212yes—
38C2×#3830C2×C311yescharacteristic
39C2×C3×#237—34no—
40C32×C4×D4—24no—
41C32×C4×D4D314yescomplemented
42C32×C4×D4C2×C311yescharacteristic, complemented
43D4×C3⋊Dic3C2×C311yescharacteristic, complemented
44C4×#638C2×C312yescomplemented
45C2×#3750C2×C312yescomplemented
46C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
47#64096C2×C311yescharacteristic, complemented
48#64090C2×C312yescomplemented
49#64086C2×C312yescomplemented
50#64081C2×C311yescharacteristic, complemented
51C4×#634C2×C311yescharacteristic, complemented
52C3×D4×Dic3—34no—
53C3×C4×#67—38no—
54C2×C3×Dic3⋊C4—38no—
55C3×D3×C4⋊C4—34no—
56C3×#887—34no—
57C3×#67⋊C4—38no—
58C3×#877—38no—
59C3×#872—34no—
60C3×C4×Dic6—34no—
61C23×C33C2312yes—
62C22×C3×C3⋊D3C2311yescharacteristic
63C33×D4C2×C414yescomplemented
64C2×C33×C4—42no—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2312yes—
67C2×C33×C4C2311yescharacteristic
68C2×C33×C4C2311yescharacteristic
69C3×#638C2×C418yescomplemented
70C2×C3×C3⋊Dic3—42no—
71C2×C3×C3⋊Dic3C2×C418yescomplemented
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C2×C3×C3⋊Dic3C2312yes—
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77C3×C4×C3⋊D3—42no—
78C3×C4×C3⋊D3C2×C414yescomplemented
79C3×#634C2×C414yescomplemented
80C3×#520—91no—
81#5546—34no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C22×C311yescharacteristic
85C22×C32×C4—28no—
86C22×C32×C4C2×D318yes—
87C22×C32×C4C22×C312yes—
88C2×#638C22×C312yes—
89C22×C3⋊Dic3C22×C312yes—
90C22×C3⋊Dic3C22×C312yes—
91#3830C3×C418yescomplemented
92C2×C4×C3⋊D3C22×C312yes—
93C2×C4×C3⋊D3C22×C311yescharacteristic
94C2×#634C22×C311yescharacteristic
95C2×C3×#67—38no—
96C22×C3×Dic3—38no—
97C22×C3×Dic3—38no—
98C3×#237—332no—
99C2×C3×C4×D3—34no—
100C2×C3×C4×D3—38no—
101C2×C3×Dic6—34no—
102C32×C4⋊C4—24no—
103C32×C4⋊C4C2×D314yes—
104C32×C4⋊C4C22×C311yescharacteristic
105C32×C22⋊C4—28no—
106C32×C22⋊C4C2×D318yes—
107C32×C22⋊C4C22×C312yes—
108C32×C42—24no—
109C32×C42C2×D314yes—
110C32×C42C22×C311yescharacteristic
111#3757C22×C312yes—
112#3752C22×C311yescharacteristic
113#3752C22×C311yescharacteristic
114C4⋊(C3⋊Dic3)C22×C311yescharacteristic
115#3750C22×C312yes—
116#3750C22×C311yescharacteristic
117#3750C22×C311yescharacteristic
118#3750C22×C312yes—
119#3750C22×C312yes—
120C4×C3⋊Dic3C22×C312yes—
121C4×C3⋊Dic3C22×C311yescharacteristic
122C4×C3⋊Dic3C22×C312yes—
123C3×#217—38no—
124C3×(C2×D3)⋊C4—34no—
125C3×(C2×D3)⋊C4—34no—
126C3×C4⋊Dic3—34no—
127C3×Dic3⋊C4—34no—
128C3×Dic3⋊C4—38no—
129C3×Dic3⋊C4—38no—
130C3×Dic3⋊C4—34no—
131C3×Dic3⋊C4—38no—
132C3×C4×Dic3—38no—
133C3×C4×Dic3—38no—
134C3×C4×Dic3—34no—
135C22×C33—22no—
136C22×C33—22no—
137C22×C33C22×C414yes—
138C22×C33C2411yescharacteristic
139C2×C3×C3⋊D3—21no—
140C2×C3×C3⋊D3—21no—
141C2×C3×C3⋊D3C22×C412yes—
142C33×C4—21no—
143C33×C4—22no—
144C33×C4—21no—
145C33×C4C22×C412yes—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—21no—
149C3×C3⋊Dic3C22×C414yes—
150C3×C3⋊Dic3C22×C412yes—
151C2×C3×#41—91no—
152C2×#237—34no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C3×C4×D4—24no—
156C3×C4×D4—91no—
157C3×C4×D4—92no—
158C3×C4×D4—92no—
159C3×C4×D4C3×D314yescomplemented
160C3×C4×D4C3⋊D311yescharacteristic, complemented
161C3×#117—92no—
162C3×#117—91no—
163C2×C3×C4⋊C4—92no—
164C2×C3×C4⋊C4—91no—
165D4×Dic3—34no—
166C4×#67—38no—
167C2×Dic3⋊C4—38no—
168D3×C4⋊C4—34no—
169#887—34no—
170#67⋊C4—38no—
171#877—38no—
172#872—34no—
173C4×Dic6—34no—
174C23×C32—28no—
175C23×C32C22×D318yes—
176C23×C32C23×C312yes—
177C22×C3⋊D3C23×C311yescharacteristic
178C22×C3×D3—34no—
179C32×D4—216no—
180C32×D4C4×D3116yescomplemented
181C32×D4C2×C3×C414yescomplemented
182C2×C32×C4—24no—
183C2×C32×C4—24no—
184C2×C32×C4—24no—
185C2×C32×C4—28no—
186C2×C32×C4—48no—
187C2×C32×C4—42no—
188C2×C32×C4—416no—
189C2×C32×C4C22×D314yes—
190C2×C32×C4C22×D318yes—
191C2×C32×C4C22×D314yes—
192C2×C32×C4C22×D314yes—
193C2×C32×C4C23×C311yescharacteristic
194C2×C32×C4C23×C312yes—
195C2×C32×C4C23×C311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197#638C2×C3×C418yescomplemented
198C2×C3⋊Dic3—42no—
199C2×C3⋊Dic3C2×C3×C418yescomplemented
200C2×C3⋊Dic3C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C311yescharacteristic
202C2×C3⋊Dic3C23×C312yes—
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C311yescharacteristic
205C4×C3⋊D3—42no—
206C4×C3⋊D3C2×C3×C414yescomplemented
207#634C2×C3×C414yescomplemented
208C3×#67—332no—
209C2×C3×Dic3—332no—
210C2×C3×Dic3—38no—
211C2×C3×Dic3—34no—
212C2×C3×Dic3—34no—
213C2×C3×Dic3—34no—
214C2×C3×Dic3—38no—
215C2×C3×Dic3—128no—
216C3×C4×D3—316no—
217C3×C4×D3—128no—
218C3×Dic6—316no—
219#520—91noSylow
220C2×C33—24no—
221C2×C33C23×C411yescharacteristic
222C2×C33H(5,𝔽2)11yescharacteristic
223C2×C332−1+411yescharacteristic
224C3×C3⋊D3—22no—
225C3×#41—98no—
226C2×C3×Q8—91no—
227C2×C3×D4—24no—
228C2×C3×D4—92no—
229C2×C3×D4C2×C3×D314yes—
230C2×C3×D4C2×C3⋊D311yescharacteristic
231C22×C3×C4—28no—
232C22×C3×C4—92no—
233C22×C3×C4—91no—
234C22×C3×C4—92no—
235C22×C3×C4—92no—
236C22×C3×C4C2×C3×D318yes—
237C22×C3×C4C2×C3⋊D312yes—
238C2×#67—38no—
239C22×Dic3—38no—
240C22×Dic3—38no—
241#237—332no—
242C2×C4×D3—34no—
243C2×C4×D3—38no—
244C2×Dic6—34no—
245C3×C4⋊C4—24no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4—92no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—91no—
252C3×C4⋊C4C2×C3×D314yes—
253C3×C4⋊C4C2×C3⋊D311yescharacteristic
254C3×C22⋊C4—28no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4—91no—
257C3×C22⋊C4—91no—
258C3×C22⋊C4C2×C3×D318yes—
259C3×C22⋊C4C2×C3⋊D312yes—
260C3×C42—24no—
261C3×C42—92no—
262C3×C42—92no—
263C3×C42—91no—
264C3×C42C2×C3×D314yes—
265C3×C42C2×C3⋊D311yescharacteristic
266#217—38no—
267(C2×D3)⋊C4—34no—
268(C2×D3)⋊C4—34no—
269C4⋊Dic3—34no—
270Dic3⋊C4—38no—
271Dic3⋊C4—34no—
272Dic3⋊C4—34no—
273Dic3⋊C4—38no—
274Dic3⋊C4—38no—
275C4×Dic3—38no—
276C4×Dic3—34no—
277C4×Dic3—38no—
278C22×C32—22no—
279C22×C32—22no—
280C22×C32—24no—
281C22×C32—216no—
282C22×C32—28no—
283C22×C32—28no—
284C22×C32—48no—
285C22×C32—48no—
286C22×C32C2×C4×D3116yes—
287C22×C32C22×C3×C414yes—
288C22×C32C23×D314yes—
289C22×C32C24×C311yescharacteristic
290C2×C3⋊D3—21no—
291C2×C3⋊D3—21no—
292C2×C3⋊D3C22×C3×C412yes—
293C2×C3×D3—38no—
294C2×C3×D3—64no—
295C2×C3×D3—64no—
296C32×C4—28no—
297C32×C4—24no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—28no—
302C32×C4—28no—
303C32×C4—24no—
304C32×C4—44no—
305C32×C4—48no—
306C32×C4C2×C4×D318yes—
307C32×C4C22×C3×C412yes—
308C3⋊Dic3—22no—
309C3⋊Dic3—21no—
310C3⋊Dic3—21no—
311C3⋊Dic3C22×C3×C414yes—
312C3⋊Dic3C22×C3×C412yes—
313C3×Dic3—316no—
314C3×Dic3—38no—
315C3×Dic3—68no—
316C3×Dic3—64no—
317C3×Dic3—64no—
318C2×#41—91no—
319C4×Q8—91no—
320C4×Q8—91no—
321C4×D4—91no—
322C4×D4—92no—
323C4×D4—92no—
324C4×D4C3×C3⋊D311yescharacteristic, complemented
325#117—92no—
326#117—91no—
327C2×C4⋊C4—92no—
328C2×C4⋊C4—91no—
329C33#52011yescharacteristic, complemented, Sylow
330C23×C3—28no—
331C23×C3—91no—
332C23×C3C22×C3×D318yes—
333C23×C3C22×C3⋊D312yes—
334C22×D3—34no—
335C3×Q8—94no—
336C3×D4—216no—
337C3×D4—98no—
338C3×D4C3×C4×D3116yescomplemented
339C3×D4C4×C3⋊D314yescomplemented
340C2×C3×C4—24no—
341C2×C3×C4—24no—
342C2×C3×C4—24no—
343C2×C3×C4—28no—
344C2×C3×C4—48no—
345C2×C3×C4—42no—
346C2×C3×C4—416no—
347C2×C3×C4—92no—
348C2×C3×C4—91no—
349C2×C3×C4—98no—
350C2×C3×C4—94no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—362no—
355C2×C3×C4—362no—
356C2×C3×C4C22×C3×D314yes—
357C2×C3×C4C22×C3×D318yes—
358C2×C3×C4C22×C3×D314yes—
359C2×C3×C4C22×C3×D314yes—
360C2×C3×C4C22×C3⋊D311yescharacteristic
361C2×C3×C4C22×C3⋊D312yes—
362C2×C3×C4C22×C3⋊D311yescharacteristic
363C2×C3×C4C22×C3⋊D311yescharacteristic
364#67—332no—
365C2×Dic3—38no—
366C2×Dic3—34no—
367C2×Dic3—332no—
368C2×Dic3—34no—
369C2×Dic3—34no—
370C2×Dic3—38no—
371C2×Dic3—128no—
372C4×D3—316no—
373C4×D3—128no—
374Dic6—316no—
375C2×C32—24no—
376C2×C32—24no—
377C2×C32—24no—
378C2×C32—24no—
379C2×C32—216no—
380C2×C32—416no—
381C2×C32C22×C4×D314yes—
382C2×C32#100414yes—
383C2×C32#101014yes—
384C2×C32C23×C3×C411yescharacteristic
385C2×C32C3×H(5,𝔽2)11yescharacteristic
386C2×C32C3×2−1+411yescharacteristic
387C3⋊D3—22no—
388C3×D3—68no—
389#41—98no—
390C2×Q8—91no—
391C2×D4—92no—
392C2×D4C2×C3×C3⋊D311yescharacteristic
393C22×C4—92no—
394C22×C4—91no—
395C22×C4—92no—
396C22×C4—92no—
397C22×C4C2×C3×C3⋊D312yes—
398C4⋊C4—92no—
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—92no—
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4C2×C3×C3⋊D311yescharacteristic
405C22⋊C4—92no—
406C22⋊C4—91no—
407C22⋊C4—91no—
408C22⋊C4C2×C3×C3⋊D312yes—
409C42—92no—
410C42—92no—
411C42—91no—
412C42C2×C3×C3⋊D311yescharacteristic
413C22×C3—28no—
414C22×C3—28no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—24no—
418C22×C3—216no—
419C22×C3—48no—
420C22×C3—48no—
421C22×C3—92no—
422C22×C3—181no—
423C22×C3—181no—
424C22×C3C2×C3×C4×D3116yes—
425C22×C3C2×C4×C3⋊D314yes—
426C22×C3C23×C3×D314yes—
427C22×C3C23×C3⋊D311yescharacteristic
428C2×D3—38no—
429C2×D3—64no—
430C2×D3—64no—
431C3×C4—28no—
432C3×C4—28no—
433C3×C4—24no—
434C3×C4—24no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—28no—
439C3×C4—48no—
440C3×C4—44no—
441C3×C4—92no—
442C3×C4—94no—
443C3×C4—182no—
444C3×C4—181no—
445C3×C4—181no—
446C3×C4C2×C3×C4×D318yes—
447C3×C4C2×C4×C3⋊D312yes—
448Dic3—316no—
449Dic3—38no—
450Dic3—68no—
451Dic3—64no—
452Dic3—64no—
453C32—24no—
454C32#554614yescomplemented
455C32C3×#52011yescharacteristic, complemented
456C23—91no—
457C23C22×C3×C3⋊D312yes—
458Q8—94no—
459D4—98no—
460D4C3×C4×C3⋊D314yescomplemented
461C2×C4—42no—
462C2×C4—92no—
463C2×C4—91no—
464C2×C4—98no—
465C2×C4—94no—
466C2×C4—91no—
467C2×C4—92no—
468C2×C4—91no—
469C2×C4—362no—
470C2×C4—362no—
471C2×C4C22×C3×C3⋊D311yescharacteristic
472C2×C4C22×C3×C3⋊D312yes—
473C2×C4C22×C3×C3⋊D311yescharacteristic
474C2×C4C22×C3×C3⋊D311yescharacteristic
475C2×C3—216no—
476C2×C3—24no—
477C2×C3—24no—
478C2×C3—24no—
479C2×C3—24no—
480C2×C3—416no—
481C2×C3—182no—
482C2×C3C22×C3×C4×D314yes—
483C2×C3C3×#100414yes—
484C2×C3C3×#101014yes—
485C2×C3C22×C4×C3⋊D311yescharacteristic
486C2×C3#6436111yescharacteristic
487C2×C3#6436711yescharacteristic
488D3—68no—
489C22—22no—
490C22—22no—
491C22—92no—
492C22—181no—
493C22—181no—
494C22C2×C3×C4×C3⋊D314yes—
495C22C23×C3×C3⋊D311yescharacteristic
496C4—21no—
497C4—22no—
498C4—21no—
499C4—92no—
500C4—94no—
501C4—182no—
502C4—181no—
503C4—181no—
504C4C2×C3×C4×C3⋊D312yes—
505C3—24no—
506C3C3×#554614yescomplemented
507C3#1059601711yescharacteristic, direct product factor
508C2—24no—
509C2—182no—
510C2C22×C3×C4×C3⋊D311yescharacteristic
511C2C3×#6436111yescharacteristic
512C2C3×#6436711yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4×D4C2
socleC22×C33C24
prime radicalC2×C3×#3830C2
cosocleC22×C32C24×C3
hypercenterC3×C4×D4C3⋊D3
hypocenterC32C3×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
2. C33×C4×D4C2semidirect
3. C3×D4×C3⋊Dic3C2semidirect
4. C3×C4×#638C2semidirect
5. C2×C3×#3750C2semidirect
6. C3×C4⋊C4×C3⋊D3C2semidirect
7. C3×#64096C2semidirect
8. C3×#64090C2semidirect
9. C3×#64086C2semidirect
10. C3×#64081C2semidirect
11. C3×C4×#634C2semidirect
12. #10596017C3direct (N × H)
19. C3×#3830C4semidirect
41. C32×C4×D4D3semidirect
42. C32×C4×D4C2×C3semidirect
43. D4×C3⋊Dic3C2×C3semidirect
44. C4×#638C2×C3semidirect
45. C2×#3750C2×C3semidirect
46. C4⋊C4×C3⋊D3C2×C3semidirect
47. #64096C2×C3semidirect
48. #64090C2×C3semidirect
49. #64086C2×C3semidirect
50. #64081C2×C3semidirect
51. C4×#634C2×C3semidirect
63. C33×D4C2×C4semidirect
69. C3×#638C2×C4semidirect
71. C2×C3×C3⋊Dic3C2×C4semidirect
78. C3×C4×C3⋊D3C2×C4semidirect
79. C3×#634C2×C4semidirect
91. #3830C3×C4semidirect
159. C3×C4×D4C3×D3semidirect
160. C3×C4×D4C3⋊D3semidirect
180. C32×D4C4×D3semidirect
181. C32×D4C2×C3×C4semidirect
197. #638C2×C3×C4semidirect
199. C2×C3⋊Dic3C2×C3×C4semidirect
206. C4×C3⋊D3C2×C3×C4semidirect
207. #634C2×C3×C4semidirect
324. C4×D4C3×C3⋊D3semidirect
329. C33#520semidirect
338. C3×D4C3×C4×D3semidirect
339. C3×D4C4×C3⋊D3semidirect
454. C32#5546semidirect
455. C32C3×#520semidirect
460. D4C3×C4×C3⋊D3semidirect
506. C3C3×#5546semidirect
507. C3#10596017direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C33×C4×D42
C3×D4×C3⋊Dic32
C3×C4×#6382
C2×C3×#37502
C3×C4⋊C4×C3⋊D32
C3×#640962
C3×#640902
C3×#640862
C3×#640812
C3×C4×#6342
#105960173
C3×#55463