Finite Group Databasebeta

← C2×C3⋊D3×Dic3⋊C4 #13176901

Subgroups — C2×C3⋊D3×Dic3⋊C4 #13176901

Order 1728.

2876 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×Dic3×C3⋊D3C211yesmaximal, characteristic
3C22×Dic3×C3⋊D3C211yesmaximal, characteristic
4C2×#88857C211yesmaximal, characteristic, complemented
5C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
6C2×#88806C211yesmaximal, characteristic, complemented
7C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C3⋊D3×Dic3⋊C4C218yesmaximal, direct product factor
9C2×C4⋊C4×C3⋊D3—31nomaximal
10C2×D3×Dic3⋊C4—34nomaximal
11C23×C3×C3⋊D3C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3C2211yescharacteristic
16C2×C3×C4×C3⋊D3—24no—
17C2×C3×C4×C3⋊D3C2214yes—
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×Dic3×C3⋊D3—24no—
21C2×Dic3×C3⋊D3C418yescomplemented
22C2×Dic3×C3⋊D3C2214yes—
23C2×Dic3×C3⋊D3C2214yes—
24#88857C2214yescomplemented
25C32×Dic3⋊C4C2214yescomplemented
26#88806C2214yescomplemented
27Dic3⋊(C3⋊Dic3)C2214yescomplemented
28C22×C4×C3⋊D3—31no—
29C22×C4×C3⋊D3—31no—
30C22×C4×C3⋊D3D311yescharacteristic
31C22×C3×C4×D3—34no—
32C22×D3×Dic3—34no—
33C22×D3×Dic3—34no—
34C2×C32×C4⋊C4—31no—
35C2×C4⋊(C3⋊Dic3)—31no—
36C2×#3750—31no—
37C2×#3750—31no—
38C2×#3750—34no—
39C2×#3750—64no—
40C4⋊C4×C3⋊D3—38no—
41C2×C3×Dic3⋊C4D314yescomplemented
42C2×#3724—34no—
43C2×Dic3⋊Dic3—34no—
44D3×Dic3⋊C4—332no—
45C23×C33C2311yescharacteristic
46C22×C3×C3⋊D3C2×C414yes—
47C22×C3×C3⋊D3D414yes—
48C22×C3×C3⋊D3Q814yes—
49C22×C3×C3⋊D3C2312yes—
50C2×C33×C4—22no—
51C2×C33×C4C2312yes—
52C2×C3⋊(C3⋊Dic3)—22no—
53C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56C2×C3×C3⋊Dic3—22no—
57C2×C3×C3⋊Dic3C2312yes—
58C3×C4×C3⋊D3—28no—
59C2×C32×Dic3—22no—
60C2×C32×Dic3C2×C414yescomplemented
61C2×C32×Dic3C2312yes—
62C2×C32×Dic3C2312yes—
63Dic3×C3⋊D3—28no—
64Dic3×C3⋊D3C2×C4116yescomplemented
65C22×Dic3⋊C4—91no—
66C2×D3×C4⋊C4—94no—
67C23×C3⋊D3C2×D311yescharacteristic
68C23×C3×D3—34no—
69C22×C32×C4—24no—
70C22×C32×C4—31no—
71C22×C32×C4—31no—
72C22×C32×C4C2×D314yes—
73C22×C32×C4C2×D311yescharacteristic
74C22×C3⋊Dic3—31no—
75C22×C3⋊Dic3—34no—
76C22×C3⋊Dic3—31no—
77C22×C3⋊Dic3—34no—
78C22×C3⋊Dic3—64no—
79C22×C3⋊Dic3—64no—
80C22×C3⋊Dic3C2×D311yescharacteristic
81C2×C4×C3⋊D3—24no—
82C2×C4×C3⋊D3—38no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—34no—
85C2×C4×C3⋊D3—64no—
86C2×C4×C3⋊D3C2×D314yes—
87C22×C3×Dic3—34no—
88C22×C3×Dic3C2×D314yes—
89C22×C3×Dic3C2×D314yes—
90C2×C3×C4×D3—316no—
91C2×C3×C4×D3—616no—
92C2×D3×Dic3—332no—
93C2×D3×Dic3—316no—
94C2×D3×Dic3—316no—
95C2×D3×Dic3—616no—
96C32×C4⋊C4—34no—
97C4⋊(C3⋊Dic3)—34no—
98#3750—34no—
99#3750—316no—
100#3750—34no—
101#3750—616no—
102C3×Dic3⋊C4C2×D3116yescomplemented
103#3724—316no—
104Dic3⋊Dic3—316no—
105C22×C33C22×C412yes—
106C22×C33C2×D412yes—
107C22×C33C2×Q812yes—
108C22×C33C2411yescharacteristic
109C2×C3×C3⋊D3C4⋊C4116yescomplemented
110C2×C3×C3⋊D3C22×C414yes—
111C2×C3×C3⋊D3C2×D414yes—
112C2×C3×C3⋊D3C2×Q814yes—
113C33×C4—22no—
114C3⋊(C3⋊Dic3)—22no—
115C3⋊(C3⋊Dic3)C22×C414yescomplemented
116C3×C3⋊Dic3—22no—
117C32×Dic3—22no—
118C32×Dic3C22×C414yescomplemented
119C23×C3×C4—91no—
120C23×Dic3—91no—
121C23×Dic3—91no—
122C22×C4×D3—34no—
123C22×C4×D3—94no—
124C22×C4×D3—94no—
125C2×C3×C4⋊C4—34no—
126C2×C4⋊Dic3—94no—
127C2×Dic3⋊C4—94no—
128C2×Dic3⋊C4—94no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4—91no—
131C2×Dic3⋊C4—91no—
132C2×Dic3⋊C4—98no—
133C2×Dic3⋊C4—184no—
134C2×Dic3⋊C4C3⋊D311yescharacteristic, direct product factor
135D3×C4⋊C4—932no—
136C23×C32—24no—
137C23×C32C22×D314yes—
138C23×C32C22×D311yescharacteristic
139C22×C3⋊D3Dic614yes—
140C22×C3⋊D3C4×D314yes—
141C22×C3⋊D3#6714yes—
142C22×C3⋊D3C22×D312yes—
143C22×C3×D3—316no—
144C22×C3×D3—316no—
145C22×C3×D3—316no—
146C22×C3×D3—38no—
147C2×C32×C4—22no—
148C2×C32×C4—28no—
149C2×C32×C4—28no—
150C2×C32×C4—34no—
151C2×C32×C4—32no—
152C2×C32×C4—32no—
153C2×C32×C4—48no—
154C2×C32×C4—62no—
155C2×C32×C4C22×D318yes—
156C2×C32×C4C22×D312yes—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—34no—
159C2×C3⋊Dic3—32no—
160C2×C3⋊Dic3—32no—
161C2×C3⋊Dic3—316no—
162C2×C3⋊Dic3—38no—
163C2×C3⋊Dic3—38no—
164C2×C3⋊Dic3—68no—
165C2×C3⋊Dic3—616no—
166C2×C3⋊Dic3—68no—
167C2×C3⋊Dic3—62no—
168C2×C3⋊Dic3—68no—
169C2×C3⋊Dic3—128no—
170C2×C3⋊Dic3C22×D312yes—
171C4×C3⋊D3—28no—
172C4×C3⋊D3—316no—
173C4×C3⋊D3—68no—
174C2×C3×Dic3—28no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3—68no—
177C2×C3×Dic3C4×D3116yescomplemented
178C2×C3×Dic3C22×D318yes—
179C2×C3×Dic3C22×D318yes—
180C3×C4×D3—632no—
181D3×Dic3—364no—
182D3×Dic3—632no—
183C22×C4⋊C4—271noSylow
184C2×C33C2×C4⋊C414yescomplemented
185C2×C33C23×C411yescharacteristic
186C2×C33C22×D411yescharacteristic
187C2×C33C22×Q811yescharacteristic
188C3×C3⋊D3C2×C4⋊C418yescomplemented
189C24×C3—91no—
190C23×D3—34no—
191C22×C3×C4—24no—
192C22×C3×C4—34no—
193C22×C3×C4—34no—
194C22×C3×C4—91no—
195C22×C3×C4—94no—
196C22×C3×C4—184no—
197C22×C3×C4D3214yes—
198C22×C3×C4C2×C3⋊D311yescharacteristic
199C22×Dic3—34no—
200C22×Dic3—94no—
201C22×Dic3—91no—
202C22×Dic3—98no—
203C22×Dic3—94no—
204C22×Dic3—94no—
205C22×Dic3—91no—
206C22×Dic3—94no—
207C22×Dic3—184no—
208C22×Dic3—184no—
209C22×Dic3—184no—
210C22×Dic3C2×C3⋊D311yescharacteristic
211C22×Dic3C2×C3⋊D311yescharacteristic
212C2×C4×D3—316no—
213C2×C4×D3—616no—
214C2×C4×D3—916no—
215C2×C4×D3—932no—
216C2×C4×D3—916no—
217C2×C4×D3—1816no—
218C3×C4⋊C4—316no—
219C4⋊Dic3—916no—
220Dic3⋊C4—916no—
221Dic3⋊C4—916no—
222Dic3⋊C4—94no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—1816no—
226Dic3⋊C4C2×C3⋊D314yesdirect product factor
227C22×C32—24no—
228C22×C32—28no—
229C22×C32—28no—
230C22×C32—28no—
231C22×C32C2×Dic612yes—
232C22×C32C2×C4×D318yes—
233C22×C32C2×C4×D312yes—
234C22×C32D3×D418yes—
235C22×C32D3×Q818yes—
236C22×C32C2×#6712yes—
237C22×C32C23×D314yes—
238C22×C32C23×D311yescharacteristic
239C2×C3⋊D3Dic3⋊C4116yesdirect product factor
240C2×C3⋊D3C2×Dic614yes—
241C2×C3⋊D3C2×C4×D314yes—
242C2×C3⋊D3C2×#6714yes—
243C2×C3×D3—364no—
244C2×C3×D3—316no—
245C2×C3×D3—316no—
246C2×C3×D3—316no—
247C32×C4—22no—
248C32×C4—28no—
249C32×C4—34no—
250C32×C4—48no—
251C32×C4—62no—
252C3⋊Dic3—22no—
253C3⋊Dic3—34no—
254C3⋊Dic3—316no—
255C3⋊Dic3—616no—
256C3⋊Dic3—62no—
257C3⋊Dic3—68no—
258C3⋊Dic3—128no—
259C3×Dic3—28no—
260C3×Dic3—68no—
261C3×Dic3C2×C4×D3116yescomplemented
262C23×C4—91no—
263C23×C4—271no—
264C23×C4—271no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—278no—
267C2×C4⋊C4—271no—
268C2×C4⋊C4—271no—
269C2×C4⋊C4—271no—
270C33C22×C4⋊C411yescharacteristic, complemented, Sylow
271C23×C3—24no—
272C23×C3—92no—
273C23×C3—94no—
274C23×C3—94no—
275C23×C3—94no—
276C23×C3C2×D3214yes—
277C23×C3C22×C3⋊D311yescharacteristic
278C22×D3—38no—
279C22×D3—316no—
280C22×D3—316no—
281C22×D3—316no—
282C2×C3×C4—28no—
283C2×C3×C4—22no—
284C2×C3×C4—28no—
285C2×C3×C4—38no—
286C2×C3×C4—316no—
287C2×C3×C4—38no—
288C2×C3×C4—48no—
289C2×C3×C4—68no—
290C2×C3×C4—92no—
291C2×C3×C4—188no—
292C2×C3×C4—182no—
293C2×C3×C4C2×D3218yes—
294C2×C3×C4C22×C3⋊D312yes—
295C2×Dic3—22no—
296C2×Dic3—38no—
297C2×Dic3—68no—
298C2×Dic3—98no—
299C2×Dic3—92no—
300C2×Dic3—916no—
301C2×Dic3—94no—
302C2×Dic3—916no—
303C2×Dic3—98no—
304C2×Dic3—92no—
305C2×Dic3—188no—
306C2×Dic3—1816no—
307C2×Dic3—188no—
308C2×Dic3—188no—
309C2×Dic3—188no—
310C2×Dic3—182no—
311C2×Dic3—368no—
312C2×Dic3C4×C3⋊D314yescomplemented
313C2×Dic3C22×C3⋊D312yes—
314C2×Dic3C22×C3⋊D312yes—
315C4×D3—632no—
316C4×D3—964no—
317C4×D3—1832no—
318C2×C32—24no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—216no—
322C2×C32D3×C4⋊C4116yescomplemented
323C2×C32C2×Dic3⋊C414yescomplemented
324C2×C32C22×Dic611yescharacteristic
325C2×C32C22×C4×D314yes—
326C2×C32C22×C4×D311yescharacteristic
327C2×C32C2×D3×D414yes—
328C2×C32C2×D3×Q814yes—
329C2×C32C22×#6711yescharacteristic
330C3⋊D3C2×Dic3⋊C418yesdirect product factor
331C3×D3—332no—
332C24—91no—
333C22×C4—31no—
334C22×C4—31no—
335C22×C4—91no—
336C22×C4—94no—
337C22×C4—184no—
338C22×C4—274no—
339C22×C4—278no—
340C22×C4—274no—
341C22×C4—271no—
342C22×C4—271no—
343C22×C4—544no—
344C22×C4D3×C3⋊D311yescharacteristic
345C4⋊C4—34no—
346C4⋊C4—274no—
347C4⋊C4—274no—
348C4⋊C4—274no—
349C22×C3—24no—
350C22×C3—28no—
351C22×C3—28no—
352C22×C3—28no—
353C22×C3—916no—
354C22×C3—94no—
355C22×C3—94no—
356C22×C3—94no—
357C22×C3D3×Dic618yes—
358C22×C3C4×D3218yes—
359C22×C3D3×#6718yes—
360C22×C3C2×C4×C3⋊D312yes—
361C22×C3D4×C3⋊D312yes—
362C22×C3Q8×C3⋊D312yes—
363C22×C3C22×D3214yes—
364C22×C3C23×C3⋊D311yescharacteristic
365C2×D3—316no—
366C2×D3—316no—
367C2×D3—316no—
368C2×D3—364no—
369C3×C4—28no—
370C3×C4—22no—
371C3×C4—316no—
372C3×C4—48no—
373C3×C4—68no—
374C3×C4—182no—
375Dic3—22no—
376Dic3—68no—
377Dic3—916no—
378Dic3—94no—
379Dic3—1816no—
380Dic3—188no—
381Dic3—182no—
382Dic3—368no—
383Dic3C2×C4×C3⋊D314yescomplemented
384C32—24no—
385C32C2×D3×C4⋊C414yescomplemented
386C32C22×Dic3⋊C411yescharacteristic, complemented
387C23—92no—
388C23—94no—
389C23—94no—
390C23—94no—
391C23C2×D3×C3⋊D311yescharacteristic
392C2×C4—22no—
393C2×C4—32no—
394C2×C4—34no—
395C2×C4—32no—
396C2×C4—62no—
397C2×C4—92no—
398C2×C4—188no—
399C2×C4—182no—
400C2×C4—2716no—
401C2×C4—272no—
402C2×C4—274no—
403C2×C4—272no—
404C2×C4—548no—
405C2×C4—542no—
406C2×C4C2×D3×C3⋊D312yes—
407C2×C3—24no—
408C2×C3—24no—
409C2×C3—24no—
410C2×C3—216no—
411C2×C3—98no—
412C2×C3D3×Dic3⋊C4116yescomplemented
413C2×C3C4⋊C4×C3⋊D314yescomplemented
414C2×C3C2×D3×Dic614yes—
415C2×C3C2×C4×D3214yes—
416C2×C3C2×D3×#6714yes—
417C2×C3C22×C4×C3⋊D311yescharacteristic
418C2×C3C2×D4×C3⋊D311yescharacteristic
419C2×C3C2×Q8×C3⋊D311yescharacteristic
420D3—332no—
421C22—94no—
422C22—94no—
423C22—94no—
424C22—916no—
425C22C3⋊D3×Dic612yes—
426C22C4×D3×C3⋊D312yes—
427C22C3⋊D3×#6712yes—
428C22C22×D3×C3⋊D311yescharacteristic
429C4—22no—
430C4—34no—
431C4—62no—
432C4—182no—
433C4—274no—
434C4—542no—
435C3—24no—
436C3C2×D3×Dic3⋊C414yescomplemented
437C3C2×C4⋊C4×C3⋊D311yescharacteristic, complemented
438C2—98no—
439C2C3⋊D3×Dic3⋊C414yesdirect product factor
440C2C2×C3⋊D3×Dic611yescharacteristic
441C2C2×C4×D3×C3⋊D311yescharacteristic
442C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC22×C33C24
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C22×C4⋊C4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. C2×#88857C2semidirect
5. C2×C32×Dic3⋊C4C2semidirect
6. C2×#88806C2semidirect
7. C2×Dic3⋊(C3⋊Dic3)C2semidirect
8. C3⋊D3×Dic3⋊C4C2direct (N × H)
21. C2×Dic3×C3⋊D3C4semidirect
24. #88857C22semidirect
25. C32×Dic3⋊C4C22semidirect
26. #88806C22semidirect
27. Dic3⋊(C3⋊Dic3)C22semidirect
41. C2×C3×Dic3⋊C4D3semidirect
53. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
60. C2×C32×Dic3C2×C4semidirect
64. Dic3×C3⋊D3C2×C4semidirect
102. C3×Dic3⋊C4C2×D3semidirect
109. C2×C3×C3⋊D3C4⋊C4semidirect
115. C3⋊(C3⋊Dic3)C22×C4semidirect
118. C32×Dic3C22×C4semidirect
134. C2×Dic3⋊C4C3⋊D3direct (N × H)
177. C2×C3×Dic3C4×D3semidirect
184. C2×C33C2×C4⋊C4semidirect
188. C3×C3⋊D3C2×C4⋊C4semidirect
226. Dic3⋊C4C2×C3⋊D3direct (N × H)
239. C2×C3⋊D3Dic3⋊C4direct (N × H)
261. C3×Dic3C2×C4×D3semidirect
270. C33C22×C4⋊C4semidirect
312. C2×Dic3C4×C3⋊D3semidirect
322. C2×C32D3×C4⋊C4semidirect
323. C2×C32C2×Dic3⋊C4semidirect
330. C3⋊D3C2×Dic3⋊C4direct (N × H)
383. Dic3C2×C4×C3⋊D3semidirect
385. C32C2×D3×C4⋊C4semidirect
386. C32C22×Dic3⋊C4semidirect
412. C2×C3D3×Dic3⋊C4semidirect
413. C2×C3C4⋊C4×C3⋊D3semidirect
436. C3C2×D3×Dic3⋊C4semidirect
437. C3C2×C4⋊C4×C3⋊D3semidirect
439. C2C3⋊D3×Dic3⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×Dic3×C3⋊D32
C2×#888572
C2×C32×Dic3⋊C42
C2×#888062
C2×Dic3⋊(C3⋊Dic3)2
C3⋊D3×Dic3⋊C42
C2×C4⋊C4×C3⋊D33
C2×D3×Dic3⋊C43