Finite Group Databasebeta

← C3×#10592659 #13162998

Subgroups — C3×#10592659 #13162998

Order 1728.

2460 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#638C212yesmaximal, complemented
2C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×C33×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×C3×#3752C212yesmaximal, complemented
5C3×#63658C211yesmaximal, characteristic, complemented
6#10592659C311yesmaximal, characteristic, direct product factor
7C3×#4960—34nomaximal
8C24×C33C2211yescharacteristic
9C23×C3×C3⋊D3C2212yes—
10C22×C33×C4C2211yescharacteristic
11C22×C33×C4C2211yescharacteristic
12C2×C3×#638—24no—
13C2×C3×#638—24no—
14C2×C3×#638—24no—
15C22×C3×C3⋊Dic3C2212yes—
16C2×C3×C4⋊(C3⋊D3)—24no—
17C2×C3×C4⋊(C3⋊D3)—22no—
18C33×C22⋊C4—22no—
19C3×#3752—24no—
20C22×#638C2×C312yescomplemented
21C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
22C22×C3×#67—38no—
23C22×C3×D12—34no—
24C2×C32×C22⋊C4—24no—
25C2×C32×C22⋊C4D314yescomplemented
26C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
27C2×#3752C2×C312yescomplemented
28C2×C3×(C2×D3)⋊C4—38no—
29#63658C2×C311yescharacteristic, complemented
30C3×C3⋊#95—34no—
31C23×C33—21no—
32C23×C33—22no—
33C23×C33—22no—
34C23×C33—21no—
35C23×C33D412yescomplemented
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—22no—
38C22×C3×C3⋊D3—22no—
39C22×C3×C3⋊D3—22no—
40C22×C3×C3⋊D3—22no—
41C22×C3×C3⋊D3—22no—
42C22×C3×C3⋊D3—22no—
43C22×C3×C3⋊D3D414yescomplemented
44C2×C33×C4—22no—
45C2×C33×C4—21no—
46C2×C33×C4—22no—
47C2×C33×C4D412yescomplemented
48C3×#638—216no—
49C2×C3×C3⋊Dic3—22no—
50C2×C3×C3⋊Dic3—22no—
51C2×C3×C3⋊Dic3D414yescomplemented
52C3×C4⋊(C3⋊D3)—28no—
53C3×#392—91no—
54#4960—34no—
55C24×C32—24no—
56C24×C32C2×D314yes—
57C24×C32C22×C311yescharacteristic
58C23×C3⋊D3C22×C312yes—
59C23×C3×D3—38no—
60C22×C32×C4—24no—
61C22×C32×C4—24no—
62C22×C32×C4C2×D314yes—
63C22×C32×C4C2×D314yes—
64C22×C32×C4C22×C311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C2×#638—24no—
67C2×#638—24no—
68C2×#638—24no—
69C22×C3⋊Dic3C22×C312yes—
70C2×C4⋊(C3⋊D3)—24no—
71C2×C4⋊(C3⋊D3)—22no—
72C2×C3×#67—616no—
73C2×C3×#67—616no—
74C2×C3×#67—616no—
75C22×C3×Dic3—38no—
76C2×C3×D12—616no—
77C2×C3×D12—68no—
78C32×C22⋊C4—216no—
79C32×C22⋊C4—28no—
80C32×C22⋊C4—22no—
81#3752—24no—
82C3×(C2×D3)⋊C4—616no—
83C22×C33—24no—
84C22×C33—22no—
85C22×C33—41no—
86C22×C33—42no—
87C22×C33—41no—
88C22×C33C2×D411yescharacteristic
89C22×C33C2×D412yes—
90C22×C33C2×D412yes—
91C22×C33C2×D411yescharacteristic
92C22×C33#4111yescharacteristic
93C2×C3×C3⋊D3—24no—
94C2×C3×C3⋊D3—24no—
95C2×C3×C3⋊D3—24no—
96C2×C3×C3⋊D3—42no—
97C2×C3×C3⋊D3—42no—
98C2×C3×C3⋊D3—42no—
99C2×C3×C3⋊D3—42no—
100C33×C4—22no—
101C33×C4—41no—
102C3×C3⋊Dic3—24no—
103C22×C3×D4—92no—
104C22×C3×D4—91no—
105C22×#67—38no—
106C22×D12—34no—
107C2×C3×C22⋊C4—24no—
108C2×C3×C22⋊C4—92no—
109C2×C3×C22⋊C4C3×D314yescomplemented
110C2×C3×C22⋊C4C3⋊D311yescharacteristic, complemented
111C2×(C2×D3)⋊C4—38no—
112C3×#95—91no—
113C3⋊#95—34no—
114C23×C32—28no—
115C23×C32—28no—
116C23×C32—24no—
117C23×C32—24no—
118C23×C32—21no—
119C23×C32—22no—
120C23×C32—22no—
121C23×C32—21no—
122C23×C32—28no—
123C23×C32—24no—
124C23×C32—28no—
125C23×C32—28no—
126C23×C32—48no—
127C23×C32—48no—
128C23×C32D1218yescomplemented
129C23×C32C3×D412yescomplemented
130C23×C32C22×D314yes—
131C23×C32C23×C311yescharacteristic
132C22×C3⋊D3—22no—
133C22×C3⋊D3—22no—
134C22×C3⋊D3—22no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3—22no—
138C22×C3⋊D3C3×D414yescomplemented
139C22×C3×D3—316no—
140C22×C3×D3—68no—
141C22×C3×D3—68no—
142C22×C3×D3—68no—
143C22×C3×D3—68no—
144C22×C3×D3—68no—
145C22×C3×D3—68no—
146C2×C32×C4—22no—
147C2×C32×C4—28no—
148C2×C32×C4—24no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—28no—
152C2×C32×C4—28no—
153C2×C32×C4—28no—
154C2×C32×C4—48no—
155C2×C32×C4—48no—
156C2×C32×C4#6718yescomplemented
157C2×C32×C4C3×D412yescomplemented
158#638—216no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3C3×D414yescomplemented
162C4⋊(C3⋊D3)—28no—
163C3×#67—664no—
164C2×C3×Dic3—316no—
165C2×C3×Dic3—68no—
166C2×C3×Dic3—68no—
167C3×D12—632no—
168#392—91noSylow
169C2×C33—42no—
170C2×C33#12012yes—
171C2×C33#12011yescharacteristic
172C2×C33#12112yes—
173C2×C33#12111yescharacteristic
174C2×C33#12711yescharacteristic
175C3×C3⋊D3—44no—
176C24×C3—24no—
177C24×C3—92no—
178C24×C3C2×C3×D314yes—
179C24×C3C2×C3⋊D311yescharacteristic
180C23×D3—38no—
181C2×C3×D4—184no—
182C2×C3×D4—184no—
183C2×C3×D4—184no—
184C2×C3×D4—184no—
185C2×C3×D4—182no—
186C22×C3×C4—24no—
187C22×C3×C4—24no—
188C22×C3×C4—92no—
189C22×C3×C4C2×C3×D314yes—
190C22×C3×C4C2×C3×D314yes—
191C22×C3×C4C2×C3⋊D311yescharacteristic
192C22×C3×C4C2×C3⋊D311yescharacteristic
193C2×#67—616no—
194C2×#67—616no—
195C2×#67—616no—
196C22×Dic3—38no—
197C2×D12—616no—
198C2×D12—68no—
199C3×C22⋊C4—216no—
200C3×C22⋊C4—28no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—184no—
203(C2×D3)⋊C4—616no—
204C22×C32—216no—
205C22×C32—28no—
206C22×C32—24no—
207C22×C32—22no—
208C22×C32—24no—
209C22×C32—24no—
210C22×C32—28no—
211C22×C32—28no—
212C22×C32—24no—
213C22×C32—216no—
214C22×C32—416no—
215C22×C32—416no—
216C22×C32—48no—
217C22×C32—48no—
218C22×C32—48no—
219C22×C32—44no—
220C22×C32—44no—
221C22×C32—41no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32C2×D1214yes—
225C22×C32#23514yes—
226C22×C32D3×D418yes—
227C22×C32D3×D418yes—
228C22×C32C2×#6714yes—
229C22×C32C2×C3×D411yescharacteristic
230C22×C32C2×C3×D412yes—
231C22×C32C2×C3×D412yes—
232C22×C32C2×C3×D411yescharacteristic
233C22×C32C3×#4111yescharacteristic
234C2×C3⋊D3—24no—
235C2×C3⋊D3—24no—
236C2×C3⋊D3—24no—
237C2×C3⋊D3—42no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3—42no—
241C2×C3×D3—616no—
242C2×C3×D3—616no—
243C2×C3×D3—616no—
244C2×C3×D3—128no—
245C2×C3×D3—128no—
246C2×C3×D3—128no—
247C2×C3×D3—128no—
248C32×C4—22no—
249C32×C4—28no—
250C32×C4—48no—
251C32×C4—48no—
252C32×C4—44no—
253C32×C4—41no—
254C3⋊Dic3—24no—
255C3×Dic3—616no—
256C22×D4—92no—
257C22×D4—91no—
258C2×C22⋊C4—92no—
259C2×C22⋊C4C3×C3⋊D311yescharacteristic, complemented
260#95—91no—
261C33#39211yescharacteristic, complemented, Sylow
262C23×C3—28no—
263C23×C3—28no—
264C23×C3—24no—
265C23×C3—28no—
266C23×C3—28no—
267C23×C3—24no—
268C23×C3—21no—
269C23×C3—22no—
270C23×C3—22no—
271C23×C3—21no—
272C23×C3—28no—
273C23×C3—24no—
274C23×C3—48no—
275C23×C3—48no—
276C23×C3—94no—
277C23×C3—182no—
278C23×C3—182no—
279C23×C3—182no—
280C23×C3—182no—
281C23×C3—182no—
282C23×C3—182no—
283C23×C3C3×D1218yescomplemented
284C23×C3C4⋊(C3⋊D3)12yescomplemented
285C23×C3C22×C3×D314yes—
286C23×C3C22×C3⋊D311yescharacteristic
287C22×D3—316no—
288C22×D3—68no—
289C22×D3—68no—
290C22×D3—68no—
291C22×D3—68no—
292C22×D3—68no—
293C22×D3—68no—
294C3×D4—188no—
295C3×D4—1816no—
296C2×C3×C4—28no—
297C2×C3×C4—22no—
298C2×C3×C4—28no—
299C2×C3×C4—28no—
300C2×C3×C4—24no—
301C2×C3×C4—22no—
302C2×C3×C4—21no—
303C2×C3×C4—28no—
304C2×C3×C4—48no—
305C2×C3×C4—48no—
306C2×C3×C4—94no—
307C2×C3×C4—182no—
308C2×C3×C4—182no—
309C2×C3×C4C3×#6718yescomplemented
310C2×C3×C4#63812yescomplemented
311#67—664no—
312C2×Dic3—316no—
313C2×Dic3—68no—
314C2×Dic3—68no—
315D12—632no—
316C2×C32—28no—
317C2×C32—28no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—416no—
322C2×C32—48no—
323C2×C32—42no—
324C2×C32#88218yes—
325C2×C32#88418yes—
326C2×C32#93014yes—
327C2×C32#93714yes—
328C2×C32#94014yes—
329C2×C32C3×#12012yes—
330C2×C32C3×#12011yescharacteristic
331C2×C32C3×#12112yes—
332C2×C32C3×#12111yescharacteristic
333C2×C32C3×#12711yescharacteristic
334C3⋊D3—44no—
335C3×D3—1216no—
336C24—92no—
337C24C2×C3×C3⋊D311yescharacteristic
338C2×D4—184no—
339C2×D4—184no—
340C2×D4—184no—
341C2×D4—184no—
342C2×D4—182no—
343C22×C4—92no—
344C22×C4C2×C3×C3⋊D311yescharacteristic
345C22×C4C2×C3×C3⋊D311yescharacteristic
346C22⋊C4—22no—
347C22⋊C4—184no—
348C22×C3—216no—
349C22×C3—216no—
350C22×C3—28no—
351C22×C3—24no—
352C22×C3—22no—
353C22×C3—24no—
354C22×C3—24no—
355C22×C3—28no—
356C22×C3—28no—
357C22×C3—24no—
358C22×C3—416no—
359C22×C3—416no—
360C22×C3—48no—
361C22×C3—48no—
362C22×C3—48no—
363C22×C3—44no—
364C22×C3—44no—
365C22×C3—42no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—184no—
369C22×C3—184no—
370C22×C3—184no—
371C22×C3—362no—
372C22×C3—362no—
373C22×C3—362no—
374C22×C3—362no—
375C22×C3C2×C3×D1214yes—
376C22×C3C3×#23514yes—
377C22×C3C3×D3×D418yes—
378C22×C3C3×D3×D418yes—
379C22×C3C2×C3×#6714yes—
380C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
381C22×C3#382811yescharacteristic
382C22×C3D4×C3⋊D312yes—
383C22×C3D4×C3⋊D312yes—
384C22×C3C2×#63811yescharacteristic
385C2×D3—616no—
386C2×D3—616no—
387C2×D3—616no—
388C2×D3—128no—
389C2×D3—128no—
390C2×D3—128no—
391C2×D3—128no—
392C3×C4—28no—
393C3×C4—22no—
394C3×C4—48no—
395C3×C4—48no—
396C3×C4—44no—
397C3×C4—41no—
398C3×C4—184no—
399Dic3—616no—
400C32—24no—
401C32#496014yescomplemented
402C32C3×#39211yescharacteristic, complemented
403C23—21no—
404C23—22no—
405C23—22no—
406C23—21no—
407C23—94no—
408C23—182no—
409C23—182no—
410C23—182no—
411C23—182no—
412C23—182no—
413C23—182no—
414C23C3×C4⋊(C3⋊D3)12yescomplemented
415C23C22×C3×C3⋊D311yescharacteristic
416D4—188no—
417D4—1816no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—94no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4C3×#63812yescomplemented
425C2×C3—28no—
426C2×C3—28no—
427C2×C3—24no—
428C2×C3—24no—
429C2×C3—24no—
430C2×C3—416no—
431C2×C3—48no—
432C2×C3—42no—
433C2×C3—364no—
434C2×C3C3×#88218yes—
435C2×C3C3×#88418yes—
436C2×C3C3×#93014yes—
437C2×C3C3×#93714yes—
438C2×C3C3×#94014yes—
439C2×C3#6409112yes—
440C2×C3#6409312yes—
441C2×C3#6413911yescharacteristic
442C2×C3#6414611yescharacteristic
443C2×C3#6414911yescharacteristic
444D3—1216no—
445C22—24no—
446C22—22no—
447C22—42no—
448C22—41no—
449C22—41no—
450C22—184no—
451C22—184no—
452C22—184no—
453C22—362no—
454C22—362no—
455C22—362no—
456C22—362no—
457C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
458C22C3×#382811yescharacteristic
459C22C3×D4×C3⋊D312yes—
460C22C3×D4×C3⋊D312yes—
461C22C2×C3×#63811yescharacteristic
462C4—22no—
463C4—41no—
464C4—184no—
465C3—24no—
466C3C3×#496014yescomplemented
467C3#1059265911yescharacteristic, direct product factor
468C2—42no—
469C2—364no—
470C2C3×#6409112yes—
471C2C3×#6409312yes—
472C2C3×#6413911yescharacteristic
473C2C3×#6414611yescharacteristic
474C2C3×#6414911yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutator, cosocleC23×C32C23×C3
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C22⋊C4C2
socleC23×C33C23
hypercenterC2×C3×C22⋊C4C3⋊D3
hypocenterC32C3×#392
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×#638C2semidirect
2. C22×C3×C4⋊(C3⋊D3)C2semidirect
3. C2×C33×C22⋊C4C2semidirect
4. C2×C3×#3752C2semidirect
5. C3×#63658C2semidirect
6. #10592659C3direct (N × H)
20. C22×#638C2×C3semidirect
21. C22×C4⋊(C3⋊D3)C2×C3semidirect
25. C2×C32×C22⋊C4D3semidirect
26. C2×C32×C22⋊C4C2×C3semidirect
27. C2×#3752C2×C3semidirect
29. #63658C2×C3semidirect
35. C23×C33D4semidirect
43. C22×C3×C3⋊D3D4semidirect
47. C2×C33×C4D4semidirect
51. C2×C3×C3⋊Dic3D4semidirect
109. C2×C3×C22⋊C4C3×D3semidirect
110. C2×C3×C22⋊C4C3⋊D3semidirect
128. C23×C32D12semidirect
129. C23×C32C3×D4semidirect
138. C22×C3⋊D3C3×D4semidirect
156. C2×C32×C4#67semidirect
157. C2×C32×C4C3×D4semidirect
161. C2×C3⋊Dic3C3×D4semidirect
259. C2×C22⋊C4C3×C3⋊D3semidirect
261. C33#392semidirect
283. C23×C3C3×D12semidirect
284. C23×C3C4⋊(C3⋊D3)semidirect
309. C2×C3×C4C3×#67semidirect
310. C2×C3×C4#638semidirect
401. C32#4960semidirect
402. C32C3×#392semidirect
414. C23C3×C4⋊(C3⋊D3)semidirect
424. C2×C4C3×#638semidirect
466. C3C3×#4960semidirect
467. C3#10592659direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6382
C22×C3×C4⋊(C3⋊D3)2
C2×C33×C22⋊C42
C2×C3×#37522
C3×#636582
#105926593
C3×#49603