Finite Group Databasebeta

← C2×C3×#64227 #13181023

Subgroups — C2×C3×#64227 #13181023

Order 1728.

1080 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3802C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C2×C3×#3787C211yesmaximal, characteristic, complemented
4C2×C3×#3775C214yesmaximal, complemented
5C3×#64227C218yesmaximal, direct product factor
6C2×#64227C311yesmaximal, characteristic, direct product factor
7C22×C3×D32—21no—
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12—21no—
10C2×C32×Dic6—21no—
11C2×C3×#627C2212yescomplemented
12C2×C3×#625C2212yes—
13C2×C3×#624—21no—
14C3×#3802—22no—
15C3×#3802C2214yes—
16C3×#3797—22no—
17C3×#3797C2214yescomplemented
18C2×C3×#622C2212yescomplemented
19C3×#3787C2214yescomplemented
20C3×#3775C22116yescomplemented
21C2×C3×D3×Q8—61no—
22C2×C3×D3×D4—61no—
23C2×#3802C2×C311yescharacteristic
24C2×#3797C2×C311yescharacteristic, complemented
25C2×#3787C2×C311yescharacteristic, complemented
26C2×#3775C2×C314yescomplemented
27#64227C2×C318yesdirect product factor
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C32×D3—22no—
30C2×C3×D32—24no—
31C2×C3×D32—44no—
32C2×C33×C4D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C3×C4×C3⋊D3D412yes—
35C3×C4×C3⋊D3C2312yes—
36C2×C32×Dic3—22no—
37C32×D12—24no—
38C32×Dic6—24no—
39C3×#627—22no—
40C3×#627C2314yescomplemented
41C3×#625—22no—
42C3×#625C2314yes—
43C3×#624—24no—
44C3×#622C2314yescomplemented
45C22×C3×SD16—91nomaximal
46C23×C3×D3—62no—
47C22×D32—21no—
48C2×C32×Q8—61no—
49C2×C32×D4—61no—
50C2×C4×C3⋊D3C22×C311yescharacteristic
51C2×C3×#67—62no—
52C3×D3×Q8—64no—
53C3×D3×Q8—64no—
54C3×D3×D4—64no—
55C3×D3×D4—64no—
56C2×C3×D12—21no—
57C2×C3×D12—21no—
58C2×C3×D12—41no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—61no—
61C2×C3×C4×D3—61no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—41no—
65C2×C3×Dic6—62no—
66C2×#627C22×C312yescomplemented
67C2×#625C22×C312yes—
68C2×#624—21no—
69#3802—22no—
70#3802C22×C314yes—
71#3797—22no—
72#3797C22×C314yescomplemented
73C2×#622C22×C312yescomplemented
74#3787C22×C314yescomplemented
75#3775C22×C3116yescomplemented
76C22×C33C2×D411yescharacteristic
77C2×C3×C3⋊D3SD1614yescomplemented
78C2×C3×C3⋊D3C2×D412yes—
79C2×C32×D3—24no—
80C2×C32×D3—44no—
81C3×D32—48no—
82C33×C4C2×D412yes—
83C3×C3⋊Dic3C2×D411yescharacteristic
84C3×C3⋊Dic3C2411yescharacteristic
85C32×Dic3—24no—
86C22×C3×Q8—91no—
87C22×C3×D4—91no—
88C2×D3×Q8—61no—
89C2×D3×D4—61no—
90C2×C3×SD16—94no—
91C2×C3×SD16—98no—
92C22×C3×C8—91no—
93C23×C32—62no—
94C22×C3⋊D3C3×D411yescharacteristic
95C22×C3×D3—22no—
96C22×C3×D3—22no—
97C22×C3×D3—42no—
98C22×C3×D3—61no—
99C22×C3×D3—61no—
100C22×C3×D3—68no—
101C22×C3×D3—128no—
102C2×D32—24no—
103C2×D32—44no—
104C32×Q8—64no—
105C32×D4—64no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4—41no—
109C2×C32×C4—41no—
110C2×C32×C4—62no—
111C2×C32×C4C3×D411yescharacteristic
112C2×C3⋊Dic3C23×C311yescharacteristic
113C4×C3⋊D3C3×D412yes—
114C4×C3⋊D3C23×C312yes—
115C3×#67—64no—
116C3×#67—64no—
117C2×C3×Dic3—22no—
118C2×C3×Dic3—22no—
119C2×C3×Dic3—42no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C3×D12—24no—
123C3×D12—24no—
124C3×D12—44no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—68no—
130C3×Dic6—24no—
131C3×Dic6—24no—
132C3×Dic6—44no—
133C3×Dic6—64no—
134C3×Dic6—64no—
135#627—22no—
136#627C23×C314yescomplemented
137#625—22no—
138#625C23×C314yes—
139#624—24no—
140#622C23×C314yescomplemented
141C22×SD16—91noSylow
142C2×C33C2×SD1612yescomplemented
143C2×C33C22×D411yescharacteristic
144C3×C3⋊D3C2×SD1614yescomplemented
145C32×D3—44no—
146C24×C3—181no—
147C23×D3—62no—
148C2×C3×Q8—61no—
149C2×C3×Q8—61no—
150C2×C3×Q8—92no—
151C2×C3×Q8—94no—
152C2×C3×Q8—121no—
153C2×C3×Q8—182no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—94no—
157C2×C3×D4—92no—
158C2×C3×D4—121no—
159C2×C3×D4—182no—
160C22×C3×C4—91no—
161C22×C3×C4—181no—
162C2×#67—62no—
163D3×Q8—64no—
164D3×Q8—64no—
165D3×D4—64no—
166D3×D4—64no—
167C2×D12—21no—
168C2×C4×D3—61no—
169C2×C4×D3—61no—
170C2×C4×D3—62no—
171C2×Dic6—21no—
172C2×Dic6—62no—
173C3×SD16—916no—
174C2×C3×C8—94no—
175C2×C3×C8—92no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—41no—
179C22×C32—41no—
180C22×C32—64no—
181C22×C32—124no—
182C22×C32C2×C3×D411yescharacteristic
183C2×C3⋊D3C3×SD1614yescomplemented
184C2×C3⋊D3C2×C3×D412yes—
185C2×C3×D3—24no—
186C2×C3×D3—24no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C2×C3×D3—44no—
190C2×C3×D3—64no—
191C2×C3×D3—64no—
192C2×C3×D3—62no—
193C2×C3×D3—62no—
194C2×C3×D3—84no—
195C2×C3×D3—1216no—
196D32—48no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4—64no—
202C32×C4C2×C3×D412yes—
203C3⋊Dic3C2×C3×D411yescharacteristic
204C3⋊Dic3C24×C311yescharacteristic
205C3×Dic3—24no—
206C3×Dic3—24no—
207C3×Dic3—44no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—61no—
211C3×Dic3—61no—
212C22×Q8—91no—
213C22×D4—91no—
214C2×SD16—98no—
215C2×SD16—94no—
216C22×C8—91no—
217C33C22×SD1611yescharacteristic, complemented, Sylow
218C23×C3—62no—
219C23×C3—62no—
220C23×C3—91no—
221C23×C3—122no—
222C23×C3—184no—
223C23×C3—364no—
224C22×D3—22no—
225C22×D3—61no—
226C22×D3—61no—
227C22×D3—68no—
228C22×D3—128no—
229C3×Q8—64no—
230C3×Q8—64no—
231C3×Q8—94no—
232C3×Q8—124no—
233C3×Q8—182no—
234C3×D4—64no—
235C3×D4—64no—
236C3×D4—94no—
237C3×D4—124no—
238C3×D4—182no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—91no—
246C2×C3×C4—92no—
247C2×C3×C4—92no—
248C2×C3×C4—122no—
249C2×C3×C4—184no—
250C2×C3×C4GO+(4,𝔽2)11yescharacteristic
251#67—64no—
252#67—64no—
253C2×Dic3—22no—
254C2×Dic3—61no—
255C2×Dic3—61no—
256D12—24no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261C4×D3—68no—
262Dic6—24no—
263Dic6—64no—
264Dic6—64no—
265C3×C8—94no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—22no—
269C2×C32—22no—
270C2×C32—42no—
271C2×C32—42no—
272C2×C32—41no—
273C2×C32—41no—
274C2×C32—124no—
275C2×C32C2×C3×SD1612yescomplemented
276C2×C32C22×C3×D411yescharacteristic
277C3⋊D3C2×C3×SD1614yescomplemented
278C3×D3—44no—
279C3×D3—44no—
280C3×D3—64no—
281C3×D3—64no—
282C3×D3—84no—
283C24—181no—
284C2×Q8—61no—
285C2×Q8—92no—
286C2×Q8—94no—
287C2×Q8—182no—
288C2×D4—61no—
289C2×D4—94no—
290C2×D4—92no—
291C2×D4—182no—
292C22×C4—91no—
293C22×C4—181no—
294SD16—916no—
295C2×C8—94no—
296C2×C8—92no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—64no—
302C22×C3—64no—
303C22×C3—92no—
304C22×C3—94no—
305C22×C3—124no—
306C22×C3—124no—
307C22×C3—124no—
308C22×C3—244no—
309C22×C3—368no—
310C22×C3C2×GO+(4,𝔽2)11yescharacteristic
311C2×D3—24no—
312C2×D3—44no—
313C2×D3—62no—
314C2×D3—62no—
315C2×D3—64no—
316C2×D3—64no—
317C2×D3—1216no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—42no—
321C3×C4—42no—
322C3×C4—64no—
323C3×C4—64no—
324C3×C4—91no—
325C3×C4—91no—
326C3×C4—124no—
327C3×C4C2×GO+(4,𝔽2)12yes—
328Dic3—24no—
329Dic3—61no—
330Dic3—61no—
331Dic3—61no—
332Dic3—61no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32C22×C3×SD1611yescharacteristic, complemented
338C23—62no—
339C23—91no—
340C23—184no—
341C23—364no—
342Q8—64no—
343Q8—94no—
344Q8—182no—
345D4—64no—
346D4—94no—
347D4—182no—
348C2×C4—62no—
349C2×C4—91no—
350C2×C4—92no—
351C2×C4—92no—
352C2×C4—184no—
353C2×C4C3×GO+(4,𝔽2)11yescharacteristic
354C8—94no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—22no—
358C2×C3—22no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—42no—
362C2×C3—42no—
363C2×C3—94no—
364C2×C3—124no—
365C2×C3—124no—
366C2×C3—244no—
367C2×C3#6422712yesdirect product factor
368C2×C3C22×GO+(4,𝔽2)11yescharacteristic
369D3—44no—
370D3—64no—
371D3—64no—
372C22—64no—
373C22—92no—
374C22—94no—
375C22—124no—
376C22—368no—
377C22C2×C3×GO+(4,𝔽2)11yescharacteristic
378C4—64no—
379C4—91no—
380C4—91no—
381C4C2×C3×GO+(4,𝔽2)12yes—
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3C2×#6422711yescharacteristic, direct product factor
387C2—94no—
388C2—124no—
389C2C3×#6422712yesdirect product factor
390C2C22×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×#3802C2
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C22×C3×SD16
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×#3787C2semidirect
4. C2×C3×#3775C2semidirect
5. C3×#64227C2direct (N × H)
6. C2×#64227C3direct (N × H)
11. C2×C3×#627C22semidirect
17. C3×#3797C22semidirect
18. C2×C3×#622C22semidirect
19. C3×#3787C22semidirect
20. C3×#3775C22semidirect
24. C2×#3797C2×C3semidirect
25. C2×#3787C2×C3semidirect
26. C2×#3775C2×C3semidirect
27. #64227C2×C3direct (N × H)
40. C3×#627C23semidirect
44. C3×#622C23semidirect
66. C2×#627C22×C3semidirect
72. #3797C22×C3semidirect
73. C2×#622C22×C3semidirect
74. #3787C22×C3semidirect
75. #3775C22×C3semidirect
77. C2×C3×C3⋊D3SD16semidirect
136. #627C23×C3semidirect
140. #622C23×C3semidirect
142. C2×C33C2×SD16semidirect
144. C3×C3⋊D3C2×SD16semidirect
183. C2×C3⋊D3C3×SD16semidirect
217. C33C22×SD16semidirect
275. C2×C32C2×C3×SD16semidirect
277. C3⋊D3C2×C3×SD16semidirect
337. C32C22×C3×SD16semidirect
367. C2×C3#64227direct (N × H)
386. C3C2×#64227direct (N × H)
389. C2C3×#64227direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37972
C2×C3×#37872
C2×C3×#37752
C3×#642272
C2×#642273
C22×C3×SD169