Finite Group Databasebeta

← C2×C3×#64239 #13181053

Subgroups — C2×C3×#64239 #13181053

Order 1728.

944 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 178 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×#627C211yesmaximal, characteristic, complemented
2C2×C3×#3805C211yesmaximal, characteristic
3C2×C3×#3792C211yesmaximal, characteristic, complemented
4C2×C3×#3776C212yesmaximal, complemented
5C3×#64239C218yesmaximal, direct product factor
6C2×C3×#3775C212yesmaximal, complemented
7C2×#64239C311yesmaximal, characteristic, direct product factor
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×C32×#67—21no—
10C22×C32×Dic3—21no—
11C2×C3×#627—22no—
12C2×C3×#627—21no—
13C2×C3×#627C2212yes—
14C2×C3×#627C2211yescharacteristic
15C2×C3×#627C2214yescomplemented
16C2×C3×#625C2211yescharacteristic
17C3×#3805—22no—
18C3×#3805C2214yes—
19C2×C3×D3×Dic3—21no—
20C3×#3792C2214yescomplemented
21C2×C3×#622C2212yes—
22C3×#3776C2218yescomplemented
23C3×#3775C2218yescomplemented
24C2×C3×#237—61no—
25C22×C3×Dic6—61no—
26C22×#627C2×C311yescharacteristic, complemented
27C2×#3805C2×C311yescharacteristic
28C2×#3792C2×C311yescharacteristic, complemented
29C2×#3776C2×C312yescomplemented
30#64239C2×C318yesdirect product factor
31C2×#3775C2×C312yescomplemented
32C23×C33D411yescharacteristic
33C22×C32×D3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3D412yes—
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C3×C3⋊Dic3C2312yes—
38C32×#67—24no—
39C2×C32×Dic3—21no—
40C2×C32×Dic3—22no—
41C2×C32×Dic3—24no—
42C3×#627—21no—
43C3×#627—22no—
44C3×#627—24no—
45C3×#627C2314yes—
46C3×#627C2312yes—
47C3×#625—21no—
48C3×#625C2312yes—
49C3×D3×Dic3—24no—
50C3×#622C2314yes—
51C2×C3×#137—91nomaximal
52C2×C32×D4—61no—
53C22×C32×C4—61no—
54C22×C3⋊Dic3C22×C311yescharacteristic
55C2×C3×#67—21no—
56C2×C3×#67—21no—
57C2×C3×#67—41no—
58C2×C3×#67—61no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—21no—
61C22×C3×Dic3—41no—
62C22×C3×Dic3—61no—
63C22×C3×Dic3—61no—
64C22×C3×Dic3—61no—
65C3×#237—64no—
66C3×#237—64no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—62no—
69C2×C3×Dic6—64no—
70C2×C3×Dic6—61no—
71C2×C3×Dic6—64no—
72C2×C3×Dic6—62no—
73C2×#627—22no—
74C2×#627—21no—
75C2×#627C22×C311yescharacteristic
76C2×#627C22×C312yes—
77C2×#627C22×C314yescomplemented
78C2×#625C22×C311yescharacteristic
79#3805—22no—
80#3805C22×C314yes—
81C2×D3×Dic3—21no—
82#3792C22×C314yescomplemented
83C2×#622C22×C312yes—
84#3776C22×C318yescomplemented
85#3775C22×C318yescomplemented
86C22×C33—22no—
87C22×C33C2×D412yes—
88C22×C33C2×D411yescharacteristic
89C2×C32×D3—22no—
90C2×C32×D3—42no—
91C3×C3⋊Dic3C2×D412yes—
92C3×C3⋊Dic3C2×D411yescharacteristic
93C3×C3⋊Dic3C2411yescharacteristic
94C32×Dic3—22no—
95C32×Dic3—24no—
96C2×C3×#41—91no—
97C22×C3×Q8—91no—
98C2×#237—61no—
99C22×Dic6—61no—
100C3×#137—98no—
101C2×C3×Q16—92no—
102C2×C3×SD16—92no—
103C2×C3×M16—91no—
104C23×C32—21no—
105C23×C32—21no—
106C23×C32—41no—
107C23×C32—41no—
108C23×C32—61no—
109C23×C32C3×D411yescharacteristic
110C22×C3×D3—21no—
111C22×C3×D3—21no—
112C22×C3×D3—41no—
113C32×D4—64no—
114C2×C32×C4—64no—
115C2×C32×C4—62no—
116C2×C32×C4—61no—
117C2×C3⋊Dic3C3×D411yescharacteristic
118C2×C3⋊Dic3C3×D412yes—
119C2×C3⋊Dic3C23×C311yescharacteristic
120C2×C3⋊Dic3C23×C312yes—
121C3×#67—24no—
122C3×#67—24no—
123C3×#67—44no—
124C3×#67—62no—
125C3×#67—62no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—42no—
133C2×C3×Dic3—44no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—62no—
136C2×C3×Dic3—61no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—62no—
139C2×C3×Dic3—62no—
140C2×C3×Dic3—61no—
141C2×C3×Dic3—61no—
142C2×C3×Dic3—62no—
143C2×C3×Dic3—64no—
144C3×C4×D3—64no—
145C3×Dic6—64no—
146C3×Dic6—62no—
147C3×Dic6—64no—
148C3×Dic6—68no—
149C3×Dic6—62no—
150#627—21no—
151#627—22no—
152#627—24no—
153#627C23×C314yes—
154#627C23×C312yes—
155#625—21no—
156#625C23×C312yes—
157D3×Dic3—24no—
158#622C23×C314yes—
159C2×#137—91noSylow
160C2×C33—22no—
161C2×C33#13712yescomplemented
162C2×C33C22×D411yescharacteristic
163C32×D3—42no—
164C3×#41—94no—
165C3×#41—182no—
166C2×C3×Q8—92no—
167C2×C3×Q8—91no—
168C2×C3×Q8—94no—
169C2×C3×Q8—182no—
170C2×C3×Q8—181no—
171C2×C3×D4—61no—
172C2×C3×D4—61no—
173C2×C3×D4—91no—
174C2×C3×D4—121no—
175C22×C3×C4—61no—
176C22×C3×C4—61no—
177C22×C3×C4—91no—
178C22×C3×C4—121no—
179C22×C3×C4—181no—
180C2×#67—21no—
181C2×#67—61no—
182C22×Dic3—21no—
183C22×Dic3—61no—
184C22×Dic3—61no—
185C22×Dic3—61no—
186#237—64no—
187#237—64no—
188C2×C4×D3—61no—
189C2×Dic6—62no—
190C2×Dic6—61no—
191C2×Dic6—64no—
192C2×Dic6—62no—
193C2×Dic6—64no—
194C3×Q16—98no—
195C3×SD16—98no—
196C3×M16—94no—
197C2×C3×C8—92no—
198C22×C32—24no—
199C22×C32—22no—
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—41no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—44no—
210C22×C32—62no—
211C22×C32—82no—
212C22×C32—122no—
213C22×C32C2×C3×D412yes—
214C22×C32C2×C3×D411yescharacteristic
215C2×C3×D3—22no—
216C2×C3×D3—22no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—82no—
221C32×C4—64no—
222C32×C4—62no—
223C3⋊Dic3C2×C3×D412yes—
224C3⋊Dic3C2×C3×D411yescharacteristic
225C3⋊Dic3C24×C311yescharacteristic
226C3×Dic3—24no—
227C3×Dic3—22no—
228C3×Dic3—22no—
229C3×Dic3—24no—
230C3×Dic3—44no—
231C3×Dic3—42no—
232C3×Dic3—62no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—62no—
236C3×Dic3—61no—
237C3×Dic3—61no—
238C2×#41—91no—
239C22×Q8—91no—
240#137—98no—
241C2×Q16—92no—
242C2×SD16—92no—
243C2×M16—91no—
244C33C2×#13711yescharacteristic, complemented, Sylow
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—61no—
250C23×C3—61no—
251C23×C3—121no—
252C23×C3GO+(4,𝔽2)11yescharacteristic
253C22×D3—21no—
254C3×Q8—94no—
255C3×Q8—92no—
256C3×Q8—184no—
257C3×Q8—182no—
258C3×Q8—181no—
259C3×D4—64no—
260C3×D4—64no—
261C3×D4—92no—
262C3×D4—124no—
263C3×D4—181no—
264C2×C3×C4—62no—
265C2×C3×C4—64no—
266C2×C3×C4—61no—
267C2×C3×C4—64no—
268C2×C3×C4—62no—
269C2×C3×C4—61no—
270C2×C3×C4—92no—
271C2×C3×C4—92no—
272C2×C3×C4—91no—
273C2×C3×C4—91no—
274C2×C3×C4—124no—
275C2×C3×C4—122no—
276C2×C3×C4—121no—
277C2×C3×C4—184no—
278#67—24no—
279#67—62no—
280#67—62no—
281C2×Dic3—22no—
282C2×Dic3—24no—
283C2×Dic3—21no—
284C2×Dic3—61no—
285C2×Dic3—62no—
286C2×Dic3—61no—
287C2×Dic3—62no—
288C2×Dic3—61no—
289C2×Dic3—62no—
290C2×Dic3—61no—
291C2×Dic3—62no—
292C2×Dic3—64no—
293C4×D3—64no—
294Dic6—64no—
295Dic6—62no—
296Dic6—64no—
297Dic6—68no—
298Dic6—62no—
299C3×C8—94no—
300C2×C32—24no—
301C2×C32—22no—
302C2×C32—21no—
303C2×C32—21no—
304C2×C32—22no—
305C2×C32—22no—
306C2×C32—42no—
307C2×C32—42no—
308C2×C32—41no—
309C2×C32—41no—
310C2×C32—42no—
311C2×C32—44no—
312C2×C32—82no—
313C2×C32—122no—
314C2×C32C3×#13712yescomplemented
315C2×C32C22×C3×D411yescharacteristic
316C3×D3—42no—
317C3×D3—42no—
318C3×D3—82no—
319#41—94no—
320#41—182no—
321C2×Q8—92no—
322C2×Q8—91no—
323C2×Q8—94no—
324C2×Q8—181no—
325C2×Q8—182no—
326C2×D4—61no—
327C2×D4—91no—
328C22×C4—61no—
329C22×C4—91no—
330C22×C4—181no—
331Q16—98no—
332SD16—98no—
333M16—94no—
334C2×C8—92no—
335C22×C3—24no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—42no—
342C22×C3—44no—
343C22×C3—42no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—42no—
347C22×C3—62no—
348C22×C3—62no—
349C22×C3—82no—
350C22×C3—122no—
351C22×C3—122no—
352C22×C3—122no—
353C22×C3—242no—
354C22×C3C2×GO+(4,𝔽2)12yes—
355C22×C3C2×GO+(4,𝔽2)11yescharacteristic
356C2×D3—22no—
357C2×D3—42no—
358C3×C4—64no—
359C3×C4—62no—
360C3×C4—64no—
361C3×C4—62no—
362C3×C4—91no—
363C3×C4—92no—
364C3×C4—91no—
365C3×C4—124no—
366C3×C4—122no—
367Dic3—24no—
368Dic3—22no—
369Dic3—62no—
370Dic3—61no—
371Dic3—61no—
372Dic3—61no—
373Dic3—61no—
374Dic3—62no—
375C32—21no—
376C32—21no—
377C32—41no—
378C32—41no—
379C32C2×C3×#13711yescharacteristic, complemented
380C23—61no—
381C23C3×GO+(4,𝔽2)11yescharacteristic
382Q8—94no—
383Q8—92no—
384Q8—184no—
385Q8—182no—
386Q8—181no—
387D4—64no—
388D4—92no—
389D4—181no—
390C2×C4—64no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—91no—
394C2×C4—92no—
395C2×C4—91no—
396C2×C4—92no—
397C2×C4—184no—
398C8—94no—
399C2×C3—24no—
400C2×C3—22no—
401C2×C3—22no—
402C2×C3—22no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—42no—
406C2×C3—44no—
407C2×C3—42no—
408C2×C3—42no—
409C2×C3—41no—
410C2×C3—41no—
411C2×C3—82no—
412C2×C3—122no—
413C2×C3—122no—
414C2×C3—242no—
415C2×C3#6423912yesdirect product factor
416C2×C3C22×GO+(4,𝔽2)11yescharacteristic
417D3—42no—
418C22—22no—
419C22—62no—
420C22—122no—
421C22C2×C3×GO+(4,𝔽2)12yes—
422C22C2×C3×GO+(4,𝔽2)11yescharacteristic
423C4—64no—
424C4—62no—
425C4—91no—
426C4—92no—
427C4—91no—
428C3—21no—
429C3—21no—
430C3—41no—
431C3—41no—
432C3C2×#6423911yescharacteristic, direct product factor
433C2—22no—
434C2—122no—
435C2C3×#6423912yesdirect product factor
436C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
prime radicalC2×C3×#3805C2
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C2×C3×#137
LCSG ▷ 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×#627C2semidirect
3. C2×C3×#3792C2semidirect
4. C2×C3×#3776C2semidirect
5. C3×#64239C2direct (N × H)
6. C2×C3×#3775C2semidirect
7. C2×#64239C3direct (N × H)
15. C2×C3×#627C22semidirect
20. C3×#3792C22semidirect
22. C3×#3776C22semidirect
23. C3×#3775C22semidirect
26. C22×#627C2×C3semidirect
28. C2×#3792C2×C3semidirect
29. C2×#3776C2×C3semidirect
30. #64239C2×C3direct (N × H)
31. C2×#3775C2×C3semidirect
77. C2×#627C22×C3semidirect
82. #3792C22×C3semidirect
84. #3776C22×C3semidirect
85. #3775C22×C3semidirect
161. C2×C33#137semidirect
244. C33C2×#137semidirect
314. C2×C32C3×#137semidirect
379. C32C2×C3×#137semidirect
415. C2×C3#64239direct (N × H)
432. C3C2×#64239direct (N × H)
435. C2C3×#64239direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×#38052
C2×C3×#37922
C2×C3×#37762
C3×#642392
C2×C3×#37752
C2×#642393
C2×C3×#1379