Finite Group Databasebeta

← C2×C3×#64226 #13181022

Subgroups — C2×C3×#64226 #13181022

Order 1728.

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

normal subgroups only

This group has 928 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
1C2×C3×#3798C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×C3×#3788C211yesmaximal, characteristic, complemented
4C2×C3×#3776C212yesmaximal, complemented
5C2×C3×#3775C212yesmaximal, complemented
6C3×#64226C218yesmaximal, direct product factor
7C2×#64226C311yesmaximal, characteristic, direct product factor
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×C4×D3—21no—
10C2×C32×Dic6—21no—
11C2×C3×#627C2212yes—
12C2×C3×#627C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×#625C2211yescharacteristic
15C2×C3×#624—21no—
16C3×#3798—22no—
17C3×#3798C2214yes—
18C3×#3797—22no—
19C3×#3797C2214yescomplemented
20C2×C3×#622C2212yes—
21C3×#3788C2214yescomplemented
22C3×#3776C2218yescomplemented
23C3×#3775C2218yescomplemented
24C2×C3×D3×Q8—61no—
25C2×C3×#235—61no—
26C2×#3798C2×C311yescharacteristic
27C2×#3797C2×C311yescharacteristic, complemented
28C2×#3788C2×C311yescharacteristic, complemented
29C2×#3776C2×C312yescomplemented
30C2×#3775C2×C312yescomplemented
31#64226C2×C318yesdirect product factor
32C22×C3×C3⋊D3D411yescharacteristic
33C22×C32×D3—21no—
34C2×C33×C4D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C3×C4×C3⋊D3D412yes—
37C3×C4×C3⋊D3C2312yes—
38C2×C32×Dic3—22no—
39C2×C32×Dic3—21no—
40C32×C4×D3—24no—
41C32×Dic6—24no—
42C3×#627—22no—
43C3×#627—21no—
44C3×#627C2314yes—
45C3×#627C2312yes—
46C3×#626—24no—
47C3×#625—21no—
48C3×#625C2312yes—
49C3×#624—24no—
50C3×#622C2314yes—
51C2×C3×#137—91nomaximal
52C2×C32×Q8—61no—
53C22×C32×C4—61no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×C3×#67—61no—
56C2×C3×#67—61no—
57C3×D3×Q8—64no—
58C3×D3×Q8—64no—
59C3×#235—64no—
60C3×#235—64no—
61C2×C3×D12—61no—
62C2×C3×C4×D3—21no—
63C2×C3×C4×D3—21no—
64C2×C3×C4×D3—41no—
65C2×C3×C4×D3—62no—
66C2×C3×C4×D3—61no—
67C2×C3×C4×D3—61no—
68C2×C3×Dic6—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—41no—
71C2×C3×Dic6—61no—
72C2×C3×Dic6—62no—
73C2×#627C22×C312yes—
74C2×#627C22×C311yescharacteristic
75C2×#626—21no—
76C2×#625C22×C311yescharacteristic
77C2×#624—21no—
78#3798—22no—
79#3798C22×C314yes—
80#3797—22no—
81#3797C22×C314yescomplemented
82C2×#622C22×C312yes—
83#3788C22×C314yescomplemented
84#3776C22×C318yescomplemented
85#3775C22×C318yescomplemented
86C22×C33C2×D411yescharacteristic
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3C2×D412yes—
89C2×C32×D3—22no—
90C2×C32×D3—42no—
91C33×C4C2×D412yes—
92C3×C3⋊Dic3C2×D411yescharacteristic
93C3×C3⋊Dic3C2411yescharacteristic
94C32×Dic3—24no—
95C32×Dic3—22no—
96C2×C3×#41—91no—
97C22×C3×Q8—91no—
98C2×D3×Q8—61no—
99C2×#235—61no—
100C3×#137—98no—
101C2×C3×Q16—92no—
102C2×C3×SD16—92no—
103C2×C3×M16—91no—
104C23×C32—61no—
105C22×C3⋊D3C3×D411yescharacteristic
106C22×C3×D3—21no—
107C22×C3×D3—21no—
108C22×C3×D3—41no—
109C22×C3×D3—61no—
110C22×C3×D3—61no—
111C32×Q8—64no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—64no—
117C2×C32×C4—62no—
118C2×C32×C4—61no—
119C2×C32×C4C3×D411yescharacteristic
120C2×C3⋊Dic3C23×C311yescharacteristic
121C4×C3⋊D3C3×D412yes—
122C4×C3⋊D3C23×C312yes—
123C3×#67—62no—
124C3×#67—64no—
125C3×#67—62no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—42no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—61no—
133C2×C3×Dic3—61no—
134C3×D12—64no—
135C3×C4×D3—24no—
136C3×C4×D3—24no—
137C3×C4×D3—44no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140C3×C4×D3—62no—
141C3×C4×D3—62no—
142C3×C4×D3—68no—
143C3×Dic6—24no—
144C3×Dic6—24no—
145C3×Dic6—44no—
146C3×Dic6—62no—
147C3×Dic6—64no—
148C3×Dic6—62no—
149C3×Dic6—64no—
150#627—22no—
151#627—21no—
152#627C23×C314yes—
153#627C23×C312yes—
154#626—24no—
155#625—21no—
156#625C23×C312yes—
157#624—24no—
158#622C23×C314yes—
159C2×#137—91noSylow
160C2×C33#13712yescomplemented
161C2×C33C22×D411yescharacteristic
162C3×C3⋊D3—22no—
163C32×D3—42no—
164C3×#41—94no—
165C3×#41—182no—
166C2×C3×Q8—61no—
167C2×C3×Q8—61no—
168C2×C3×Q8—92no—
169C2×C3×Q8—91no—
170C2×C3×Q8—94no—
171C2×C3×Q8—121no—
172C2×C3×Q8—182no—
173C2×C3×D4—91no—
174C2×C3×D4—181no—
175C22×C3×C4—61no—
176C22×C3×C4—61no—
177C22×C3×C4—91no—
178C22×C3×C4—121no—
179C22×C3×C4—181no—
180C2×#67—61no—
181C2×#67—61no—
182D3×Q8—64no—
183D3×Q8—64no—
184#235—64no—
185#235—64no—
186C2×D12—61no—
187C2×C4×D3—21no—
188C2×C4×D3—61no—
189C2×C4×D3—61no—
190C2×C4×D3—62no—
191C2×Dic6—21no—
192C2×Dic6—61no—
193C2×Dic6—62no—
194C3×Q16—98no—
195C3×SD16—98no—
196C3×M16—94no—
197C2×C3×C8—92no—
198C22×C32—21no—
199C22×C32—21no—
200C22×C32—41no—
201C22×C32—41no—
202C22×C32—62no—
203C22×C32—122no—
204C22×C32C2×C3×D411yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3C2×C3×D412yes—
207C2×C3×D3—22no—
208C2×C3×D3—22no—
209C2×C3×D3—42no—
210C2×C3×D3—42no—
211C2×C3×D3—44no—
212C2×C3×D3—42no—
213C2×C3×D3—62no—
214C2×C3×D3—62no—
215C2×C3×D3—64no—
216C2×C3×D3—122no—
217C32×C4—22no—
218C32×C4—22no—
219C32×C4—42no—
220C32×C4—42no—
221C32×C4—64no—
222C32×C4—62no—
223C32×C4C2×C3×D412yes—
224C3⋊Dic3C2×C3×D411yescharacteristic
225C3⋊Dic3C24×C311yescharacteristic
226C3×Dic3—24no—
227C3×Dic3—22no—
228C3×Dic3—24no—
229C3×Dic3—22no—
230C3×Dic3—44no—
231C3×Dic3—42no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C2×#41—91no—
237C22×Q8—91no—
238#137—98no—
239C2×Q16—92no—
240C2×SD16—92no—
241C2×M16—91no—
242C33C2×#13711yescharacteristic, complemented, Sylow
243C23×C3—61no—
244C23×C3—61no—
245C23×C3—91no—
246C23×C3—121no—
247C22×D3—21no—
248C22×D3—61no—
249C22×D3—61no—
250C3×Q8—64no—
251C3×Q8—64no—
252C3×Q8—94no—
253C3×Q8—92no—
254C3×Q8—124no—
255C3×Q8—182no—
256C3×Q8—181no—
257C3×D4—92no—
258C3×D4—184no—
259C3×D4—181no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—64no—
265C2×C3×C4—62no—
266C2×C3×C4—61no—
267C2×C3×C4—61no—
268C2×C3×C4—62no—
269C2×C3×C4—64no—
270C2×C3×C4—91no—
271C2×C3×C4—92no—
272C2×C3×C4—92no—
273C2×C3×C4—124no—
274C2×C3×C4—121no—
275C2×C3×C4—122no—
276C2×C3×C4—184no—
277C2×C3×C4GO+(4,𝔽2)11yescharacteristic
278#67—62no—
279#67—64no—
280#67—62no—
281C2×Dic3—22no—
282C2×Dic3—21no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285D12—64no—
286C4×D3—24no—
287C4×D3—62no—
288C4×D3—62no—
289C4×D3—62no—
290C4×D3—62no—
291C4×D3—68no—
292Dic6—24no—
293Dic6—62no—
294Dic6—64no—
295Dic6—62no—
296Dic6—64no—
297C3×C8—94no—
298C2×C32—21no—
299C2×C32—21no—
300C2×C32—22no—
301C2×C32—22no—
302C2×C32—42no—
303C2×C32—42no—
304C2×C32—41no—
305C2×C32—41no—
306C2×C32—122no—
307C2×C32C3×#13712yescomplemented
308C2×C32C22×C3×D411yescharacteristic
309C3⋊D3—22no—
310C3×D3—42no—
311C3×D3—44no—
312C3×D3—42no—
313C3×D3—64no—
314C3×D3—122no—
315#41—94no—
316#41—182no—
317C2×Q8—61no—
318C2×Q8—92no—
319C2×Q8—91no—
320C2×Q8—94no—
321C2×Q8—182no—
322C2×D4—91no—
323C2×D4—181no—
324C22×C4—61no—
325C22×C4—91no—
326C22×C4—181no—
327Q16—98no—
328SD16—98no—
329M16—94no—
330C2×C8—92no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—62no—
336C22×C3—62no—
337C22×C3—92no—
338C22×C3—122no—
339C22×C3—122no—
340C22×C3—124no—
341C22×C3—122no—
342C22×C3—182no—
343C22×C3C2×GO+(4,𝔽2)11yescharacteristic
344C2×D3—22no—
345C2×D3—42no—
346C2×D3—62no—
347C2×D3—62no—
348C2×D3—64no—
349C2×D3—122no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—64no—
355C3×C4—62no—
356C3×C4—62no—
357C3×C4—64no—
358C3×C4—91no—
359C3×C4—91no—
360C3×C4—122no—
361C3×C4—124no—
362C3×C4C2×GO+(4,𝔽2)12yes—
363Dic3—24no—
364Dic3—22no—
365Dic3—61no—
366Dic3—61no—
367Dic3—61no—
368Dic3—61no—
369C32—21no—
370C32—21no—
371C32—41no—
372C32—41no—
373C32C2×C3×#13711yescharacteristic, complemented
374C23—61no—
375C23—91no—
376Q8—64no—
377Q8—94no—
378Q8—92no—
379Q8—182no—
380Q8—181no—
381D4—92no—
382D4—184no—
383D4—181no—
384C2×C4—61no—
385C2×C4—62no—
386C2×C4—64no—
387C2×C4—91no—
388C2×C4—92no—
389C2×C4—92no—
390C2×C4—184no—
391C2×C4C3×GO+(4,𝔽2)11yescharacteristic
392C8—94no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—22no—
396C2×C3—22no—
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—42no—
400C2×C3—42no—
401C2×C3—122no—
402C2×C3—124no—
403C2×C3—122no—
404C2×C3—182no—
405C2×C3#6422612yesdirect product factor
406C2×C3C22×GO+(4,𝔽2)11yescharacteristic
407D3—42no—
408D3—64no—
409D3—122no—
410C22—62no—
411C22—92no—
412C22—122no—
413C22—182no—
414C22C2×C3×GO+(4,𝔽2)11yescharacteristic
415C4—62no—
416C4—64no—
417C4—91no—
418C4—91no—
419C4C2×C3×GO+(4,𝔽2)12yes—
420C3—21no—
421C3—21no—
422C3—41no—
423C3—41no—
424C3C2×#6422611yescharacteristic, direct product factor
425C2—122no—
426C2—182no—
427C2C3×#6422612yesdirect product factor
428C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#3798C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#137
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C2×C3×#3788C2semidirect
4. C2×C3×#3776C2semidirect
5. C2×C3×#3775C2semidirect
6. C3×#64226C2direct (N × H)
7. C2×#64226C3direct (N × H)
19. C3×#3797C22semidirect
21. C3×#3788C22semidirect
22. C3×#3776C22semidirect
23. C3×#3775C22semidirect
27. C2×#3797C2×C3semidirect
28. C2×#3788C2×C3semidirect
29. C2×#3776C2×C3semidirect
30. C2×#3775C2×C3semidirect
31. #64226C2×C3direct (N × H)
81. #3797C22×C3semidirect
83. #3788C22×C3semidirect
84. #3776C22×C3semidirect
85. #3775C22×C3semidirect
160. C2×C33#137semidirect
242. C33C2×#137semidirect
307. C2×C32C3×#137semidirect
373. C32C2×C3×#137semidirect
405. C2×C3#64226direct (N × H)
424. C3C2×#64226direct (N × H)
427. C2C3×#64226direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37982
C2×C3×#37972
C2×C3×#37882
C2×C3×#37762
C2×C3×#37752
C3×#642262
C2×#642263
C2×C3×#1379