Finite Group Databasebeta

← C2×Dic3×#627 #13172206

Subgroups — C2×Dic3×#627 #13172206

Order 1728.

1538 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic
3C2×Dic3⋊(C3⋊Dic3)C212yesmaximal
4C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
5C2×C3×Dic32C212yesmaximal
6Dic3×#627C218yesmaximal, direct product factor
7C2×C4×#627—31nomaximal
8C2×Dic3×Dic6—32nomaximal
9C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C22×C32×Dic3C2211yescharacteristic
14C2×C3×#627C418yescomplemented
15C2×C3×#627C2214yes—
16#88815C2214yes—
17Dic3⋊(C3⋊Dic3)C2218yes—
18Dic3×C3⋊Dic3C2214yes—
19C3×Dic32C2218yes—
20C22×C3×Dic6—32no—
21C22×#627D311yescharacteristic
22C2×C4⋊(C3⋊Dic3)—32no—
23C2×C4×C3⋊Dic3—31no—
24C2×C3×C4×Dic3—32no—
25C2×C3×C4×Dic3—32no—
26C2×#3724—31no—
27C2×Dic3⋊Dic3—32no—
28C2×Dic3⋊Dic3—32no—
29C2×Dic3⋊Dic3—32no—
30C2×Dic3⋊Dic3—62no—
31C2×Dic32—32no—
32C2×Dic32—61no—
33C2×Dic32D312yes—
34C4×#627—38no—
35Dic3×Dic6—316no—
36C23×C33C2311yescharacteristic
37C2×C3⋊(C3⋊Dic3)—22no—
38C2×C3⋊(C3⋊Dic3)C2312yes—
39C2×C3×C3⋊Dic3—24no—
40C2×C3×C3⋊Dic3C2×C414yes—
41C2×C3×C3⋊Dic3C2314yes—
42C2×C3×C3⋊Dic3C2312yes—
43C2×C32×Dic3C2×C418yes—
44C2×C32×Dic3Q814yescomplemented
45C2×C32×Dic3C2314yes—
46C2×C32×Dic3C2312yes—
47C3×#627C2×C4116yescomplemented
48C2×Q8×Dic3—91no—
49C2×C4×Dic6—92no—
50C22×C32×C4—31no—
51C22×C32×C4—32no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—32no—
54C22×C3⋊Dic3—61no—
55C22×C3⋊Dic3—62no—
56C22×C3⋊Dic3—121no—
57C22×C3⋊Dic3C2×D312yes—
58C22×C3⋊Dic3C2×D311yescharacteristic
59C22×C3×Dic3—21no—
60C22×C3×Dic3—22no—
61C22×C3×Dic3—32no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—62no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C2×D312yes—
69C2×C3×Dic6—316no—
70C2×C3×Dic6—38no—
71C2×#627Dic318yesdirect product factor
72C2×#627C2×D314yes—
73C4⋊(C3⋊Dic3)—38no—
74C4×C3⋊Dic3—34no—
75C3×C4×Dic3—38no—
76C3×C4×Dic3—38no—
77#3724—34no—
78Dic3⋊Dic3—38no—
79Dic3⋊Dic3—38no—
80Dic3⋊Dic3—38no—
81Dic3⋊Dic3—68no—
82Dic32—38no—
83Dic32—64no—
84Dic32C2×D318yes—
85C22×C33C22×C412yes—
86C22×C33C2×Q812yes—
87C22×C33#4112yes—
88C22×C33C2411yescharacteristic
89C3⋊(C3⋊Dic3)—22no—
90C3×C3⋊Dic3—24no—
91C3×C3⋊Dic3C22×C414yes—
92C32×Dic3C22×C418yes—
93C32×Dic3C2×Q814yescomplemented
94C22×C3×Q8—91no—
95C22×Dic6—32no—
96C2×C3×C42—92no—
97Q8×Dic3—98no—
98C2×C4⋊Dic3—91no—
99C2×C4⋊Dic3—92no—
100C2×C4⋊Dic3—92no—
101C2×C4⋊Dic3—182no—
102C2×Dic3⋊C4—92no—
103C2×Dic3⋊C4—92no—
104C2×C4×Dic3—32no—
105C2×C4×Dic3—32no—
106C2×C4×Dic3—92no—
107C2×C4×Dic3—91no—
108C2×C4×Dic3—181no—
109C4×Dic6—916no—
110C23×C32—22no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32C22×D312yes—
114C23×C32C22×D311yescharacteristic
115C2×C32×C4—38no—
116C2×C32×C4—34no—
117C2×C32×C4—32no—
118C2×C32×C4—34no—
119C2×C3⋊Dic3—24no—
120C2×C3⋊Dic3—34no—
121C2×C3⋊Dic3—32no—
122C2×C3⋊Dic3—62no—
123C2×C3⋊Dic3—64no—
124C2×C3⋊Dic3—62no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—64no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—124no—
129C2×C3⋊Dic3—124no—
130C2×C3⋊Dic3C2×Dic314yes—
131C2×C3⋊Dic3C22×D314yes—
132C2×C3⋊Dic3C22×D312yes—
133C2×C3×Dic3—28no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3—22no—
136C2×C3×Dic3—24no—
137C2×C3×Dic3—38no—
138C2×C3×Dic3—34no—
139C2×C3×Dic3—34no—
140C2×C3×Dic3—34no—
141C2×C3×Dic3—62no—
142C2×C3×Dic3—64no—
143C2×C3×Dic3—64no—
144C2×C3×Dic3—64no—
145C2×C3×Dic3—64no—
146C2×C3×Dic3—124no—
147C2×C3×Dic3Dic618yescomplemented
148C2×C3×Dic3C4×D318yes—
149C2×C3×Dic3C2×Dic318yes—
150C2×C3×Dic3C22×D314yes—
151C2×C3×Dic3C22×D314yes—
152C2×C3×Dic3C22×D314yes—
153C3×Dic6—332no—
154#627C2×Dic3116yesdirect product factor
155C2×C4×Q8—271noSylow
156C2×C33C4×Q814yescomplemented
157C2×C33C23×C411yescharacteristic
158C2×C33C22×Q811yescharacteristic
159C2×C33C2×#4111yescharacteristic
160C2×C3×Q8—94no—
161C2×C3×Q8—98no—
162C22×C3×C4—32no—
163C22×C3×C4—32no—
164C22×C3×C4—32no—
165C22×C3×C4—61no—
166C22×C3×C4—62no—
167C22×C3×C4—91no—
168C22×C3×C4—92no—
169C22×Dic3—32no—
170C22×Dic3—32no—
171C22×Dic3—32no—
172C22×Dic3—61no—
173C22×Dic3—62no—
174C22×Dic3—92no—
175C22×Dic3—91no—
176C22×Dic3—181no—
177C22×Dic3—182no—
178C22×Dic3—361no—
179C22×Dic3D3212yes—
180C22×Dic3D3211yescharacteristic
181C2×Dic6—316no—
182C2×Dic6—38no—
183C3×C42—98no—
184C4⋊Dic3—94no—
185C4⋊Dic3—98no—
186C4⋊Dic3—98no—
187C4⋊Dic3—188no—
188Dic3⋊C4—98no—
189Dic3⋊C4—98no—
190C4×Dic3—38no—
191C4×Dic3—38no—
192C4×Dic3—98no—
193C4×Dic3—94no—
194C4×Dic3—184no—
195C22×C32—22no—
196C22×C32—21no—
197C22×C32—24no—
198C22×C32—22no—
199C22×C32—24no—
200C22×C32—22no—
201C22×C32—24no—
202C22×C32—22no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32—42no—
206C22×C32—41no—
207C22×C32C2×Dic614yes—
208C22×C32C2×C4×D314yes—
209C22×C32#23514yes—
210C22×C32D3×Q812yes—
211C22×C32#23912yes—
212C22×C32C22×Dic312yes—
213C22×C32C23×D312yes—
214C22×C32C23×D311yescharacteristic
215C32×C4—38no—
216C32×C4—34no—
217C3⋊Dic3—24no—
218C3⋊Dic3—62no—
219C3⋊Dic3—64no—
220C3⋊Dic3—64no—
221C3⋊Dic3—124no—
222C3⋊Dic3—124no—
223C3⋊Dic3C22×Dic314yes—
224C3×Dic3—28no—
225C3×Dic3—24no—
226C3×Dic3—38no—
227C3×Dic3—64no—
228C3×Dic3—64no—
229C3×Dic3—64no—
230C3×Dic3—124no—
231C3×Dic3C2×Dic618yescomplemented
232C3×Dic3C2×C4×D318yes—
233C3×Dic3C22×Dic318yes—
234C22×Q8—91no—
235C4×Q8—278no—
236C2×C4⋊C4—272no—
237C2×C4⋊C4—271no—
238C2×C42—92no—
239C2×C42—271no—
240C33C2×C4×Q811yescharacteristic, complemented, Sylow
241C23×C3—21no—
242C23×C3—22no—
243C23×C3—41no—
244C23×C3C2×D3212yes—
245C23×C3C2×D3211yescharacteristic
246C3×Q8—916no—
247C2×C3×C4—34no—
248C2×C3×C4—34no—
249C2×C3×C4—34no—
250C2×C3×C4—38no—
251C2×C3×C4—38no—
252C2×C3×C4—38no—
253C2×C3×C4—62no—
254C2×C3×C4—64no—
255C2×C3×C4—68no—
256C2×C3×C4—64no—
257C2×C3×C4—92no—
258C2×C3×C4—94no—
259C2×C3×C4—94no—
260C2×C3×C4—184no—
261C2×Dic3—34no—
262C2×Dic3—38no—
263C2×Dic3—34no—
264C2×Dic3—34no—
265C2×Dic3—64no—
266C2×Dic3—62no—
267C2×Dic3—64no—
268C2×Dic3—64no—
269C2×Dic3—64no—
270C2×Dic3—94no—
271C2×Dic3—92no—
272C2×Dic3—124no—
273C2×Dic3—182no—
274C2×Dic3—184no—
275C2×Dic3—184no—
276C2×Dic3—182no—
277C2×Dic3—184no—
278C2×Dic3—362no—
279C2×Dic3—364no—
280C2×Dic3—364no—
281C2×Dic3D3×Dic318yes—
282C2×Dic3#62714yesdirect product factor
283C2×Dic3C2×D3214yes—
284C2×Dic3C2×D3212yes—
285Dic6—332no—
286C2×C32—22no—
287C2×C32—21no—
288C2×C32—22no—
289C2×C32—21no—
290C2×C32—22no—
291C2×C32—21no—
292C2×C32—28no—
293C2×C32—24no—
294C2×C32—44no—
295C2×C32—41no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32C4×Dic618yescomplemented
299C2×C32Q8×Dic314yescomplemented
300C2×C32C22×Dic612yes—
301C2×C32C22×C4×D312yes—
302C2×C32C2×#23512yes—
303C2×C32C2×D3×Q811yescharacteristic
304C2×C32C2×#23911yescharacteristic
305C2×C32C23×Dic311yescharacteristic
306C2×Q8—94no—
307C2×Q8—98no—
308C22×C4—31no—
309C22×C4—32no—
310C22×C4—91no—
311C22×C4—92no—
312C22×C4—271no—
313C4⋊C4—278no—
314C4⋊C4—274no—
315C42—98no—
316C42—274no—
317C22×C3—21no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—24no—
321C22×C3—22no—
322C22×C3—24no—
323C22×C3—24no—
324C22×C3—22no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3D3×Dic614yes—
330C22×C3#379812yes—
331C22×C3#379914yes—
332C22×C3C4×D3212yes—
333C22×C3C2×D3×Dic314yes—
334C22×C3C2×#62712yes—
335C22×C3C22×D3212yes—
336C22×C3C22×D3211yescharacteristic
337C3×C4—38no—
338C3×C4—38no—
339C3×C4—38no—
340C3×C4—68no—
341C3×C4—64no—
342C3×C4—94no—
343C3×C4—184no—
344Dic3—38no—
345Dic3—64no—
346Dic3—64no—
347Dic3—64no—
348Dic3—124no—
349Dic3—184no—
350Dic3—182no—
351Dic3—184no—
352Dic3—364no—
353Dic3—364no—
354Dic3C2×D3×Dic318yes—
355Dic3C2×#62714yesdirect product factor
356C32—22no—
357C32—21no—
358C32—41no—
359C32C2×C4×Dic612yescomplemented
360C32C2×Q8×Dic311yescharacteristic, complemented
361C23D3311yescharacteristic
362Q8—916no—
363C2×C4—32no—
364C2×C4—34no—
365C2×C4—38no—
366C2×C4—34no—
367C2×C4—92no—
368C2×C4—94no—
369C2×C4—94no—
370C2×C4—184no—
371C2×C4—272no—
372C2×C4—542no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—22no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—28no—
380C2×C3—24no—
381C2×C3—41no—
382C2×C3—41no—
383C2×C3—41no—
384C2×C3—44no—
385C2×C3Dic3×Dic618yescomplemented
386C2×C3C4×#62714yescomplemented
387C2×C3C2×D3×Dic612yes—
388C2×C3C2×#379811yescharacteristic
389C2×C3C2×#379912yes—
390C2×C3C2×C4×D3211yescharacteristic
391C2×C3C22×D3×Dic312yes—
392C2×C3C22×#62711yescharacteristic
393C22D32×Dic312yes—
394C22D3×#62712yes—
395C22#8896512yes—
396C22C2×D3311yescharacteristic
397C4—38no—
398C4—34no—
399C4—94no—
400C4—184no—
401C4—542no—
402C3—22no—
403C3—21no—
404C3—41no—
405C3C2×Dic3×Dic612yescomplemented
406C3C2×C4×#62711yescharacteristic, complemented
407C2Dic3×#62714yesdirect product factor
408C2C2×D32×Dic311yescharacteristic
409C2C2×D3×#62711yescharacteristic
410C2C2×#8896511yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutatorC2×C33C23×C4
FrattiniC22C2×D33
Fitting, socle, prime radicalC23×C33C23
cosocleC22×C33C24
hypocenterC33C2×C4×Q8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
6. Dic3×#627C2direct (N × H)
14. C2×C3×#627C4semidirect
44. C2×C32×Dic3Q8semidirect
47. C3×#627C2×C4semidirect
71. C2×#627Dic3direct (N × H)
93. C32×Dic3C2×Q8semidirect
147. C2×C3×Dic3Dic6semidirect
154. #627C2×Dic3direct (N × H)
156. C2×C33C4×Q8semidirect
231. C3×Dic3C2×Dic6semidirect
240. C33C2×C4×Q8semidirect
282. C2×Dic3#627direct (N × H)
298. C2×C32C4×Dic6semidirect
299. C2×C32Q8×Dic3semidirect
355. Dic3C2×#627direct (N × H)
359. C32C2×C4×Dic6semidirect
360. C32C2×Q8×Dic3semidirect
385. C2×C3Dic3×Dic6semidirect
386. C2×C3C4×#627semidirect
405. C3C2×Dic3×Dic6semidirect
406. C3C2×C4×#627semidirect
407. C2Dic3×#627direct (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×#888152
C2×Dic3⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C2×C3×Dic322
Dic3×#6272
C2×C4×#6273
C2×Dic3×Dic63