Finite Group Databasebeta

← Dic3×#805 #11783876

Subgroups — Dic3×#805 #11783876

Order 1152.

775 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#235C211yesmaximal, characteristic
2C3×C42×Dic3C211yesmaximal, characteristic, complemented
3C2×C3×#805C211yesmaximal, characteristic
4C2×#63569C211yesmaximal, characteristic
5Dic3×#208C211yesmaximal, characteristic, complemented
6#10589389C211yesmaximal, characteristic, complemented
7#10589162C211yesmaximal, characteristic, complemented
8Dic3×#104—31nomaximal
9C4×#805—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×C42C2211yescharacteristic
12#64135C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×C4×Dic3—22no—
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18Dic3×#67—22no—
19Dic3×D12C412yescomplemented
20C4×D3×Dic3—21no—
21#63855—21no—
22Dic3×Dic6C412yescomplemented
23C3×#805C414yescomplemented
24#63569C414yescomplemented
25#63554—21no—
26Dic3×C3⋊C8—21no—
27Dic3×#41—31no—
28C4×#235—31no—
29C2×C3×#104—31no—
30C3×C43—31no—
31C2×#837—31no—
32Dic3×M16—31no—
33C42×Dic3D311yescharacteristic, complemented
34C2×#805—31no—
35C2×#805D311yescharacteristic
36C4×#208—31no—
37#4555—31no—
38#4523—31no—
39#4523—31no—
40#4523—31no—
41#4523—61no—
42C22×C32×C4—21no—
43C22×C32×C4C2311yescharacteristic
44C2×C3×#67—21no—
45C22×C3×Dic3—21no—
46C22×C3×Dic3D412yes—
47C3×#235—21no—
48C3×#235—21no—
49C3×#235C2×C412yes—
50C3×#235C2×C412yes—
51C2×C3×D12C2×C411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×Dic6C2×C411yescharacteristic
54C2×D3×Dic3—22no—
55C32×C42—21no—
56C32×C42C2×C412yescomplemented
57#3757—21no—
58C4⋊(C3⋊Dic3)C2×C412yes—
59C4×C3⋊Dic3—21no—
60#3748—21no—
61#3748C2×C412yescomplemented
62C2×C3⋊(C3⋊C8)—21no—
63C3×C4×Dic3—28no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3C2×C412yes—
66C3×C4×Dic3C2×C412yescomplemented
67C3×C4×Dic3D411yescharacteristic
68C3×C4×Dic3D411yescharacteristic
69C3×#208—21no—
70C3×#208C2×C412yescomplemented
71C2×C3×C3⋊C8—21no—
72Dic3⋊Dic3—22no—
73(C2×D3)⋊Dic3—22no—
74Dic32—22no—
75C4×#104—91noSylow
76C2×C3×#41—31no—
77C2×#235C2×D311yescharacteristic
78C2×C3×M16—31no—
79C2×C3×C42—21no—
80C2×C3×C42—31no—
81C2×C3×C42—31no—
82C2×C3×C42—62no—
83C2×C3×C42C2×D311yescharacteristic
84C2×C3×C42C2×D311yescharacteristic
85Q8×Dic3—32no—
86D4×Dic3—32no—
87D4×Dic3—62no—
88C4×#67—62no—
89#926—31no—
90#926—31no—
91#926—61no—
92#926—61no—
93C2×C4×Dic3—22no—
94C2×C4×Dic3—61no—
95C2×C4×Dic3C2×D311yescharacteristic
96C2×C4×Dic3C2×D311yescharacteristic
97C2×#208—31no—
98C2×#208—31no—
99C2×#208—61no—
100C2×#208C2×D311yescharacteristic
101C4×D12—32no—
102#872—61no—
103C42×D3—61no—
104C4×Dic6—32no—
105C3×#104—34no—
106#837—34no—
107#815—61no—
108C8×Dic3—61no—
109#805—34no—
110#805Dic314yesdirect product factor
111#803—61no—
112C4×C3⋊C8—61no—
113C23×C32C2×D411yescharacteristic
114C22×C3×D3—21no—
115C2×C32×C4—24no—
116C2×C32×C4—22no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C22×C411yescharacteristic
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C3⋊Dic3—22no—
124C3×#67—22no—
125C3×#67—22no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3C22⋊C412yes—
131C2×C3×Dic3C22⋊C412yes—
132C3×D12—21no—
133C3×D12C4212yescomplemented
134C3×C4×D3—22no—
135C3×C4×D3—22no—
136C3×Dic6—21no—
137C3×Dic6C4212yescomplemented
138D3×Dic3—44no—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—22no—
141C4×#41—91no—
142C2×#104—31no—
143C2×#104—91no—
144C4×M16—91no—
145C43—31no—
146#339—91no—
147#339—91no—
148C3×#41—32no—
149C3×#41—32no—
150C3×#41—61no—
151C3×#41—61no—
152C2×C3×Q8—31no—
153C2×C3×D4—31no—
154C2×C3×D4—61no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—32no—
159C22×C3×C4—41no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C22×D311yescharacteristic
164C2×#67—21no—
165C22×Dic3—21no—
166C22×Dic3—62no—
167C22×Dic3#6712yes—
168#235—21no—
169#235—21no—
170#235C4×D312yes—
171#235C2×Dic312yes—
172C2×D12C4×D311yescharacteristic
173C2×C4×D3—21no—
174C2×C4×D3—62no—
175C2×Dic6C4×D311yescharacteristic
176C3×M16—32no—
177C3×M16—61no—
178C2×C3×C8—61no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—22no—
182C3×C42—32no—
183C3×C42—31no—
184C3×C42—31no—
185C3×C42—32no—
186C3×C42—41no—
187C3×C42—68no—
188C3×C42—61no—
189C3×C42C4×D312yescomplemented
190C3×C42C2×Dic312yescomplemented
191#217—61no—
192#217—61no—
193#217—62no—
194#217—121no—
195(C2×D3)⋊C4—62no—
196C4⋊Dic3—32no—
197C4⋊Dic3—32no—
198C4⋊Dic3—62no—
199C4⋊Dic3—62no—
200Dic3⋊C4—62no—
201C4×Dic3—28no—
202C4×Dic3—21no—
203C4×Dic3—62no—
204C4×Dic3—61no—
205C4×Dic3—61no—
206C4×Dic3—62no—
207C4×Dic3—121no—
208C4×Dic3C4×D312yes—
209C4×Dic3C4×D312yescomplemented
210C4×Dic3D1211yescharacteristic
211C4×Dic3D1211yescharacteristic
212#208—21no—
213#208—32no—
214#208—32no—
215#208—62no—
216#208—61no—
217#208—61no—
218#208—121no—
219#208C2×Dic312yescomplemented
220C2×C3⋊C8—21no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—121no—
224C22×C32—22no—
225C22×C32C2×C22⋊C411yescharacteristic
226C22×C32C4×D411yescharacteristic
227C22×C32C4×D411yescharacteristic
228C2×C3×D3—22no—
229C2×C3×D3—42no—
230C32×C4—24no—
231C32×C4C2×C4211yescharacteristic
232C32×C4#11711yescharacteristic
233C32×C4C4×D411yescharacteristic
234C32×C4C4×D411yescharacteristic
235C3⋊Dic3—42no—
236C3×Dic3—24no—
237C3×Dic3—22no—
238C3×Dic3—22no—
239C3×Dic3#10414yescomplemented
240C2×#41—31no—
241C2×M16—31no—
242C2×M16—91no—
243C4×Q8—92no—
244C4×D4—92no—
245C4×D4—182no—
246#117—91no—
247#117—181no—
248C2×C42—31no—
249C2×C42—31no—
250C2×C42—62no—
251C2×C42—181no—
252C2×C42D3211yescharacteristic
253#104—34no—
254#104—94no—
255#97—181no—
256C4×C8—181no—
257C23×C3—21no—
258C23×C3—61no—
259C23×C3D3×D411yescharacteristic
260C23×C3C2×#6711yescharacteristic
261C22×D3—21no—
262C3×Q8—32no—
263C3×Q8—61no—
264C3×D4—32no—
265C3×D4—62no—
266C3×D4—62no—
267C3×D4—61no—
268C2×C3×C4—24no—
269C2×C3×C4—24no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—62no—
285C2×C3×C4—61no—
286C2×C3×C4—62no—
287C2×C3×C4—62no—
288C2×C3×C4C2×C4×D311yescharacteristic
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4C2×C4×D311yescharacteristic
291C2×C3×C4C2×C4×D311yescharacteristic
292C2×C3×C4C2×C4×D311yescharacteristic
293C2×C3×C4C2×D1211yescharacteristic
294C2×C3×C4#23511yescharacteristic
295C2×C3×C4#23511yescharacteristic
296C2×C3×C4D3×D411yescharacteristic
297C2×C3×C4#23711yescharacteristic
298C2×C3×C4#23711yescharacteristic
299C2×C3×C4C22×Dic311yescharacteristic
300#67—22no—
301#67—22no—
302C2×Dic3—24no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—24no—
306C2×Dic3—62no—
307C2×Dic3—62no—
308C2×Dic3—122no—
309C2×Dic3—124no—
310C2×Dic3(C2×D3)⋊C412yes—
311C2×Dic3(C2×D3)⋊C412yes—
312D12—21no—
313D12C4×Dic312yescomplemented
314C4×D3—22no—
315C4×D3—22no—
316C4×D3—124no—
317Dic6—21no—
318Dic6C4×Dic312yescomplemented
319C3×C8—62no—
320C3⋊C8—22no—
321C3⋊C8—62no—
322C3⋊C8—62no—
323C3⋊C8—122no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32C4×C22⋊C411yescharacteristic
327C2×C32C2×#10412yes—
328C3×D3—42no—
329#41—32no—
330#41—32no—
331#41—61no—
332#41—61no—
333C2×Q8—31no—
334C2×D4—31no—
335C2×D4—61no—
336C22×C4—21no—
337C22×C4—32no—
338C22×C4—61no—
339C22×C4—61no—
340C22×C4—182no—
341C22×C4C2×D3211yescharacteristic
342M16—32no—
343M16—61no—
344M16—92no—
345M16—181no—
346C2×C8—61no—
347C2×C8—181no—
348C4⋊C4—92no—
349C4⋊C4—182no—
350C22⋊C4—182no—
351C22⋊C4—181no—
352C42—21no—
353C42—32no—
354C42—31no—
355C42—31no—
356C42—32no—
357C42—68no—
358C42—61no—
359C42—182no—
360C42—181no—
361C42D3×Dic312yescomplemented
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—42no—
368C22×C3—62no—
369C22×C3—122no—
370C22×C3D3×C22⋊C411yescharacteristic
371C22×C3#67⋊C411yescharacteristic
372C22×C3C2×(C2×D3)⋊C411yescharacteristic
373C22×C3C4×#6711yescharacteristic
374C22×C3C4×#6711yescharacteristic
375C22×C3D4×Dic311yescharacteristic
376C2×D3—22no—
377C2×D3—42no—
378C3×C4—24no—
379C3×C4—24no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—34no—
385C3×C4—44no—
386C3×C4—64no—
387C3×C4—62no—
388C3×C4—62no—
389C3×C4C42×D311yescharacteristic
390C3×C4#87211yescharacteristic
391C3×C4C4×D1211yescharacteristic
392C3×C4C4×D1211yescharacteristic
393C3×C4#87711yescharacteristic
394C3×C4#67⋊C411yescharacteristic
395C3×C4C2×C4×Dic311yescharacteristic
396C3×C4D4×Dic311yescharacteristic
397Dic3—24no—
398Dic3—22no—
399Dic3—22no—
400Dic3—122no—
401Dic3—122no—
402Dic3—242no—
403Dic3#80514yesdirect product factor
404C32C4×#10411yescharacteristic, complemented, Sylow
405C23—61no—
406C23D3×#6711yescharacteristic
407Q8—32no—
408Q8—61no—
409D4—32no—
410D4—62no—
411D4—62no—
412D4—61no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—64no—
418C2×C4—64no—
419C2×C4—62no—
420C2×C4—61no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—182no—
424C2×C4—364no—
425C2×C4#379511yescharacteristic
426C2×C4C4×D3211yescharacteristic
427C2×C4C4×D3211yescharacteristic
428C2×C4D3×D1211yescharacteristic
429C2×C4C2×D3×Dic311yescharacteristic
430C2×C4#380411yescharacteristic
431C8—62no—
432C8—182no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—22no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—122no—
442C2×C3D3×#10412yes—
443C2×C3C2×#80512yes—
444C2×C3C4×(C2×D3)⋊C411yescharacteristic
445C2×C3Dic3×C22⋊C411yescharacteristic
446D3—42no—
447C22—22no—
448C22—62no—
449C22—122no—
450C22#6387911yescharacteristic
451C22D3×(C2×D3)⋊C411yescharacteristic
452C22Dic3×#6711yescharacteristic
453C4—24no—
454C4—34no—
455C4—64no—
456C4—62no—
457C4—62no—
458C4—362no—
459C4C4×D3×Dic311yescharacteristic
460C4#6387711yescharacteristic
461C4#6388011yescharacteristic
462C4Dic3×D1211yescharacteristic
463C3—21no—
464C3C4×#80511yescharacteristic, complemented
465C3Dic3×#10411yescharacteristic, complemented
466C2—21no—
467C2—21no—
468C2—122no—
469C2D3×#80512yes—
470C2Dic3×(C2×D3)⋊C411yescharacteristic
subgroupG/H
centerC2×C4D3×D12
commutatorC32×C4C2×C42
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC22×C32C2×C22⋊C4
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32C4×#104
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C3×C42×Dic3C2semidirect
5. Dic3×#208C2semidirect
6. #10589389C2semidirect
7. #10589162C2semidirect
19. Dic3×D12C4semidirect
22. Dic3×Dic6C4semidirect
23. C3×#805C4semidirect
24. #63569C4semidirect
33. C42×Dic3D3semidirect
56. C32×C42C2×C4semidirect
61. #3748C2×C4semidirect
66. C3×C4×Dic3C2×C4semidirect
70. C3×#208C2×C4semidirect
110. #805Dic3direct (N × H)
133. C3×D12C42semidirect
137. C3×Dic6C42semidirect
189. C3×C42C4×D3semidirect
190. C3×C42C2×Dic3semidirect
209. C4×Dic3C4×D3semidirect
219. #208C2×Dic3semidirect
239. C3×Dic3#104semidirect
313. D12C4×Dic3semidirect
318. Dic6C4×Dic3semidirect
361. C42D3×Dic3semidirect
403. Dic3#805direct (N × H)
404. C32C4×#104semidirect
464. C3C4×#805semidirect
465. C3Dic3×#104semidirect

Maximal subgroups

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

subgroupindex
Dic3×#2352
C3×C42×Dic32
C2×C3×#8052
C2×#635692
Dic3×#2082
#105893892
#105891622
Dic3×#1043
C4×#8053