Finite Group Databasebeta

← C3×#10593519 #13166885

Subgroups — C3×#10593519 #13166885

Order 1728.

914 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#625C211yesmaximal, characteristic, complemented
2C2×C3×#3774C212yesmaximal, complemented
3C3×#63980C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593519C311yesmaximal, characteristic, direct product factor
7C23×C32×D3—21no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67—21no—
10C2×C3×#625—21no—
11C2×C3×#625—24no—
12C2×C3×#625C2212yes—
13C2×C3×#625C2211yescharacteristic, complemented
14C2×C3×#622C2212yescomplemented
15C3×#3774—24no—
16C3×#3724C2211yescharacteristic, complemented
17C3×(C2×D3)⋊Dic3—21no—
18C22×C3×#67—61no—
19C22×#625C2×C311yescharacteristic, complemented
20C2×#3774C2×C312yescomplemented
21C3×#939—61no—
22#63980C2×C311yescharacteristic, complemented
23C22⋊#622C2×C311yescharacteristic, complemented
24#63739C2×C312yescomplemented
25C23×C33D411yescharacteristic
26C22×C32×D3—24no—
27C22×C32×D3—22no—
28C22×C32×D3—21no—
29C22×C32×D3—42no—
30C22×C32×D3—42no—
31C2×C3×C3⋊Dic3—22no—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C32×#67—42no—
35C2×C32×Dic3—21no—
36C3×#625—24no—
37C3×#625—22no—
38C3×#625—22no—
39C3×#625D414yescomplemented
40C3×#622—22no—
41C3×#447—91nomaximal
42C24×C32—61no—
43C23×C3×D3—21no—
44C23×C3×D3—21no—
45C23×C3×D3—41no—
46C2×C32×D4—61no—
47C22×C3⋊Dic3C22×C311yescharacteristic
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—41no—
51C2×C3×#67—62no—
52C2×C3×#67—62no—
53C2×C3×#67—68no—
54C2×C3×#67—61no—
55C22×C3×Dic3—61no—
56C22×C3×Dic3—61no—
57C2×#625—21no—
58C2×#625—24no—
59C2×#625C22×C312yes—
60C2×#625C22×C311yescharacteristic, complemented
61C2×#622C22×C312yescomplemented
62#3774—24no—
63C3×#217—61no—
64C3×(C2×D3)⋊C4—61no—
65C3×Dic3⋊C4—61no—
66#3724C22×C311yescharacteristic, complemented
67(C2×D3)⋊Dic3—21no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33D812yescomplemented
71C22×C33C2×D411yescharacteristic
72C2×C32×D3—24no—
73C2×C32×D3—48no—
74C2×C32×D3—42no—
75C2×C32×D3—42no—
76C2×C32×D3—41no—
77C2×C32×D3—41no—
78C2×C32×D3—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3C2×D412yes—
81C32×Dic3—41no—
82C22×C3×D4—91no—
83C22×#67—61no—
84C2×C3×D8—92no—
85C3×#121—91no—
86#939—61no—
87C3×#102—92no—
88C3×C22⋊C8—91no—
89C23×C32—21no—
90C23×C32—21no—
91C23×C32—41no—
92C23×C32—41no—
93C23×C32—64no—
94C23×C32—62no—
95C23×C32—61no—
96C23×C32—122no—
97C23×C32—122no—
98C23×C32C3×D411yescharacteristic
99C22×C3×D3—24no—
100C22×C3×D3—24no—
101C22×C3×D3—22no—
102C22×C3×D3—22no—
103C22×C3×D3—21no—
104C22×C3×D3—21no—
105C22×C3×D3—42no—
106C22×C3×D3—44no—
107C22×C3×D3—41no—
108C22×C3×D3—42no—
109C22×C3×D3—42no—
110C22×C3×D3—42no—
111C22×C3×D3—44no—
112C22×C3×D3—44no—
113C22×C3×D3—42no—
114C32×D4—122no—
115C2×C32×C4—61no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3⋊Dic3C23×C311yescharacteristic
119C3×#67—42no—
120C3×#67—44no—
121C3×#67—42no—
122C3×#67—64no—
123C3×#67—64no—
124C3×#67—68no—
125C3×#67—122no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—61no—
130C2×C3×Dic3—61no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3—122no—
135#625—24no—
136#625—22no—
137#625—22no—
138#625C3×D414yescomplemented
139#622—22no—
140#447—91noSylow
141C2×C33—22no—
142C2×C33#12011yescharacteristic
143C2×C33C2×D811yescharacteristic
144C2×C33AGL(1,ℤ/8)11yescharacteristic
145C32×D3—44no—
146C32×D3—81no—
147C24×C3—61no—
148C24×C3—61no—
149C24×C3—121no—
150C23×D3—21no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C2×C3×D4—92no—
154C2×C3×D4—91no—
155C2×C3×D4—121no—
156C2×C3×D4—181no—
157C2×C3×D4—184no—
158C22×C3×C4—91no—
159C2×#67—21no—
160C2×#67—62no—
161C2×#67—61no—
162C2×#67—62no—
163C2×#67—68no—
164C22×Dic3—61no—
165C22×Dic3—61no—
166C3×D8—184no—
167C2×C3×C8—92no—
168C3×C4⋊C4—91no—
169C3×C22⋊C4—181no—
170#217—61no—
171(C2×D3)⋊C4—61no—
172Dic3⋊C4—61no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—64no—
190C22×C32—81no—
191C22×C32—81no—
192C22×C32—128no—
193C22×C32—122no—
194C22×C32—122no—
195C22×C32—121no—
196C22×C32—121no—
197C22×C32—121no—
198C22×C32C3×D812yescomplemented
199C22×C32C2×C3×D411yescharacteristic
200C2×C3×D3—24no—
201C2×C3×D3—24no—
202C2×C3×D3—44no—
203C2×C3×D3—48no—
204C2×C3×D3—48no—
205C2×C3×D3—42no—
206C2×C3×D3—42no—
207C2×C3×D3—44no—
208C2×C3×D3—44no—
209C2×C3×D3—416no—
210C2×C3×D3—42no—
211C2×C3×D3—42no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—41no—
215C2×C3×D3—42no—
216C2×C3×D3—41no—
217C2×C3×D3—41no—
218C2×C3×D3—41no—
219C2×C3×D3—81no—
220C2×C3×D3—81no—
221C32×C4—121no—
222C3⋊Dic3—21no—
223C3⋊Dic3C2×C3×D412yes—
224C3×Dic3—41no—
225C3×Dic3—42no—
226C3×Dic3—41no—
227C3×Dic3—62no—
228C3×Dic3—62no—
229C3×Dic3—62no—
230C3×Dic3—121no—
231C22×D4—91no—
232C2×D8—92no—
233#121—91no—
234#102—92no—
235C22⋊C8—91no—
236C33#44711yescharacteristic, complemented, Sylow
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—64no—
242C23×C3—62no—
243C23×C3—64no—
244C23×C3—61no—
245C23×C3—62no—
246C23×C3—61no—
247C23×C3—124no—
248C23×C3—122no—
249C23×C3—121no—
250C23×C3—122no—
251C23×C3—122no—
252C23×C3—122no—
253C23×C3—124no—
254C23×C3—124no—
255C23×C3—122no—
256C23×C3GO+(4,𝔽2)11yescharacteristic
257C22×D3—24no—
258C22×D3—22no—
259C22×D3—21no—
260C22×D3—42no—
261C22×D3—42no—
262C3×D4—94no—
263C3×D4—122no—
264C3×D4—124no—
265C3×D4—122no—
266C3×D4—182no—
267C3×D4—182no—
268C3×D4—184no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—91no—
272C2×C3×C4—91no—
273C2×C3×C4—121no—
274C2×C3×C4—182no—
275#67—42no—
276#67—64no—
277#67—64no—
278#67—68no—
279#67—122no—
280C2×Dic3—21no—
281C2×Dic3—61no—
282C2×Dic3—61no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—64no—
286C2×Dic3—122no—
287C3×C8—182no—
288C2×C32—24no—
289C2×C32—22no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—42no—
303C2×C32—44no—
304C2×C32—82no—
305C2×C32—124no—
306C2×C32—241no—
307C2×C32C3×#12011yescharacteristic
308C2×C32C2×C3×D811yescharacteristic
309C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
310C3×D3—44no—
311C3×D3—48no—
312C3×D3—44no—
313C3×D3—81no—
314C3×D3—82no—
315C3×D3—81no—
316C24—61no—
317C2×D4—61no—
318C2×D4—92no—
319C2×D4—91no—
320C2×D4—181no—
321C2×D4—184no—
322C22×C4—91no—
323D8—184no—
324C2×C8—92no—
325C4⋊C4—91no—
326C22⋊C4—181no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—64no—
344C22×C3—64no—
345C22×C3—81no—
346C22×C3—81no—
347C22×C3—124no—
348C22×C3—128no—
349C22×C3—128no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—121no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—1216no—
356C22×C3—124no—
357C22×C3—124no—
358C22×C3—122no—
359C22×C3—122no—
360C22×C3—122no—
361C22×C3—121no—
362C22×C3—121no—
363C22×C3—121no—
364C22×C3—241no—
365C22×C3—241no—
366C22×C3#377412yescomplemented
367C22×C3C2×GO+(4,𝔽2)11yescharacteristic
368C2×D3—24no—
369C2×D3—48no—
370C2×D3—42no—
371C2×D3—42no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—41no—
375C3×C4—92no—
376C3×C4—121no—
377C3×C4—122no—
378C3×C4—121no—
379C3×C4—181no—
380Dic3—41no—
381Dic3—62no—
382Dic3—62no—
383Dic3—62no—
384Dic3—121no—
385C32—21no—
386C32—21no—
387C32—41no—
388C32—41no—
389C32C3×#44711yescharacteristic, complemented
390C23—64no—
391C23—62no—
392C23—61no—
393C23—122no—
394C23—122no—
395C23C3×GO+(4,𝔽2)11yescharacteristic
396D4—94no—
397D4—122no—
398D4—182no—
399D4—182no—
400D4—184no—
401C2×C4—61no—
402C2×C4—91no—
403C2×C4—91no—
404C2×C4—182no—
405C8—182no—
406C2×C3—24no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—42no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—82no—
423C2×C3—124no—
424C2×C3—128no—
425C2×C3—124no—
426C2×C3—241no—
427C2×C3—242no—
428C2×C3—241no—
429C2×C3C2×#377411yescharacteristic
430C2×C3#6423611yescharacteristic
431C2×C3#6424111yescharacteristic
432D3—44no—
433D3—81no—
434C22—21no—
435C22—21no—
436C22—64no—
437C22—128no—
438C22—122no—
439C22—122no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22C3×#377412yescomplemented
444C22C2×C3×GO+(4,𝔽2)11yescharacteristic
445C4—92no—
446C4—121no—
447C4—181no—
448C3—21no—
449C3—21no—
450C3—41no—
451C3—41no—
452C3#1059351911yescharacteristic, direct product factor
453C2—22no—
454C2—124no—
455C2—241no—
456C2C2×C3×#377411yescharacteristic
457C2C3×#6423611yescharacteristic
458C2C3×#6424111yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#447
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#625C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63980C2semidirect
4. C3×C22⋊#622C2semidirect
5. C3×#63739C2semidirect
6. #10593519C3direct (N × H)
13. C2×C3×#625C22semidirect
14. C2×C3×#622C22semidirect
16. C3×#3724C22semidirect
19. C22×#625C2×C3semidirect
20. C2×#3774C2×C3semidirect
22. #63980C2×C3semidirect
23. C22⋊#622C2×C3semidirect
24. #63739C2×C3semidirect
39. C3×#625D4semidirect
60. C2×#625C22×C3semidirect
61. C2×#622C22×C3semidirect
66. #3724C22×C3semidirect
70. C22×C33D8semidirect
138. #625C3×D4semidirect
198. C22×C32C3×D8semidirect
236. C33#447semidirect
366. C22×C3#3774semidirect
389. C32C3×#447semidirect
443. C22C3×#3774semidirect
452. C3#10593519direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6252
C2×C3×#37742
C3×#639802
C3×C22⋊#6222
C3×#637392
#105935193
C3×#4479