Finite Group Databasebeta

← C3⋊D3×#1007 #13181630

Subgroups — C3⋊D3×#1007 #13181630

Order 1728.

5348 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724682C211yesmaximal, characteristic, complemented
2C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C32×#1007C211yesmaximal, characteristic, complemented
4C3⋊D3×#239C214yesmaximal, complemented
5D3×Q8×C3⋊D3C214yesmaximal, complemented
6#11724591C214yesmaximal, complemented
7#11724590C214yesmaximal, complemented
8C3⋊D3×#235C216yesmaximal, complemented
9#11724562C216yesmaximal, complemented
10C3⋊D3×2−1+4—31nomaximal
11D3×#1007—34nomaximal
12C2×C33×Q8C2211yescharacteristic, complemented
13#89080C2214yescomplemented
14Q8×C3⋊(C3⋊D3)C2214yescomplemented
15#89076C2216yescomplemented
16C3×Q8×C3⋊D3C2214yes—
17C3×Q8×C3⋊D3C2214yescomplemented
18C2×C3×C4×C3⋊D3C2213yes—
19C2×C3×#634C2213yescomplemented
20C32×#239C2214yescomplemented
21C32×D3×Q8C2214yescomplemented
22C32×#235C2216yescomplemented
23C3⋊D3×#67C2214yes—
24#89028C22112yescomplemented
25C3⋊D3×D12C2216yes—
26C4×D3×C3⋊D3C22112yes—
27#89022C22112yescomplemented
28#89020C22112yescomplemented
29#89019C22112yescomplemented
30#89018C2216yescomplemented
31C3⋊D3×Dic6C2216yes—
32#89015C22112yescomplemented
33#89014C2216yescomplemented
34D3×#634C22112yescomplemented
35C32×2−1+4—31no—
36#64367—34no—
37#64367—36no—
38#41×C3⋊D3—34no—
39#41×C3⋊D3—36no—
40#64364—31no—
41#64364—34no—
42#64364—34no—
43#64364—64no—
44C2×Q8×C3⋊D3—34no—
45C2×Q8×C3⋊D3D311yescharacteristic, complemented
46C3×#1007D314yescomplemented
47C2×C3×D3×Q8—34no—
48D3×#239—316no—
49D32×Q8—316no—
50#64316—316no—
51#64315—316no—
52D3×#235—324no—
53#64297—324no—
54C22×C3×C3⋊D3C2311yescharacteristic
55C2×D3×C3⋊D3C2314yes—
56C33×Q8C2314yes—
57C2×C33×C4C2313yes—
58#6312C2314yes—
59C4⋊(C3⋊(C3⋊D3))C2316yes—
60C4×C3⋊(C3⋊D3)C23112yes—
61#6308C2316yes—
62C2×C3×C3⋊Dic3C2313yes—
63C3×C4×C3⋊D3C2316yes—
64C3×C4×C3⋊D3C2316yes—
65C3×#634C23112yes—
66C32×#67C2314yes—
67C32×D12C2316yes—
68C32×C4×D3C23112yes—
69C32×Dic6C2316yes—
70#6293C23112yes—
71#6292C2314yes—
72#6291C23112yes—
73#6290C2314yes—
74#6289C23112yes—
75Dic3×C3⋊D3C2314yes—
76D3×C3⋊Dic3C23112yes—
77D3×2−1+4—94no—
78C2×#1007—91no—
79C32×#41—34no—
80C32×#41—36no—
81C2×C32×Q8—24no—
82C2×C32×Q8—34no—
83C2×C32×Q8C2×D314yescomplemented
84C2×C32×Q8C2×D311yescharacteristic, complemented
85#3832—34no—
86#3832—316no—
87#3832—36no—
88#3832—616no—
89Q8×C3⋊D3—34no—
90Q8×C3⋊D3—316no—
91Q8×C3⋊D3—36no—
92Q8×C3⋊D3—312no—
93Q8×C3⋊D3—616no—
94Q8×C3⋊D3C2×D314yes—
95Q8×C3⋊D3C2×D314yescomplemented
96#3830—312no—
97#3830—36no—
98#3830—312no—
99D4×C3⋊D3—34no—
100D4×C3⋊D3—36no—
101#3828—36no—
102#3828—324no—
103#3828—312no—
104#3828—312no—
105#3828—624no—
106C2×C4×C3⋊D3—312no—
107C2×C4×C3⋊D3C2×D313yes—
108C2×#634—312no—
109C2×#634C2×D313yescomplemented
110C3×#239C2×D3116yescomplemented
111C3×D3×Q8—316no—
112C3×D3×Q8—316no—
113C3×D3×Q8C2×D3116yescomplemented
114C3×#235C2×D3124yescomplemented
115C2×C3×C4×D3—312no—
116C2×C3×Dic6—312no—
117D3×#67—316no—
118#3804—348no—
119D3×D12—324no—
120C4×D32—348no—
121#3799—348no—
122#3799—324no—
123#3798—348no—
124#3797—348no—
125#3796—348no—
126#3795—324no—
127D3×Dic6—348no—
128D3×Dic6—324no—
129C22×C33C2411yescharacteristic
130C2×C3⋊(C3⋊D3)C2414yes—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3C2411yescharacteristic
133C2×C3×C3⋊D3C2411yescharacteristic
134C2×C32×D3C2414yes—
135D3×C3⋊D3—28no—
136C33×C4C2416yes—
137C3⋊(C3⋊Dic3)C2414yes—
138C3×C3⋊Dic3C2416yes—
139C32×Dic3C2414yes—
140C3×2−1+4—34no—
141C22×C3×Q8—91no—
142#1010—924no—
143#1010—916no—
144D3×#41—924no—
145D3×#41—916no—
146#1007—916no—
147#1007—94no—
148#1007—91no—
149#1007—94no—
150#1007—94no—
151#1007—96no—
152#1007—184no—
153#1007C3⋊D311yescharacteristic, direct product factor
154C2×#239—94no—
155C2×D3×Q8—34no—
156C2×D3×Q8—916no—
157C2×D3×Q8—94no—
158C2×#235—96no—
159C22×C3⋊D3—34no—
160C22×C3⋊D3C22×D311yescharacteristic
161C22×C3×D3—34no—
162C2×D32—316no—
163C32×Q8—216no—
164C32×Q8—36no—
165C32×Q8C22×D3116yes—
166C32×Q8C22×D314yes—
167C32×D4—34no—
168C32×D4—36no—
169C2×C32×C4—212no—
170C2×C32×C4—312no—
171C2×C32×C4C22×D3112yes—
172C2×C32×C4C22×D313yes—
173#638—34no—
174#638—34no—
175#638—312no—
176#638—316no—
177#638—616no—
178C2×C3⋊Dic3—312no—
179C2×C3⋊Dic3C22×D313yes—
180C4⋊(C3⋊D3)—34no—
181C4⋊(C3⋊D3)—36no—
182C4⋊(C3⋊D3)—324no—
183C4⋊(C3⋊D3)—624no—
184C4×C3⋊D3—34no—
185C4×C3⋊D3—312no—
186C4×C3⋊D3—312no—
187C4×C3⋊D3—348no—
188C4×C3⋊D3—648no—
189C4×C3⋊D3C22×D316yes—
190C4×C3⋊D3C22×D316yes—
191#634—36no—
192#634—312no—
193#634—324no—
194#634—624no—
195#634C22×D3112yes—
196C3×#67C22×D3116yes—
197C2×C3×Dic3—312no—
198C3×D12C22×D3124yes—
199C3×C4×D3—324no—
200C3×C4×D3—324no—
201C3×C4×D3C22×D3148yes—
202C3×Dic6—348no—
203C3×Dic6C22×D3124yes—
204#627—348no—
205#626—348no—
206#626—316no—
207#625—316no—
208#624—348no—
209D3×Dic3—348no—
210D3×Dic3—316no—
211C2×2−1+4—271noSylow
212C2×C33—21no—
213C2×C33C2511yescharacteristic
214C3⋊(C3⋊D3)—24no—
215C3×C3⋊D3—21no—
216C3×C3⋊D32−1+412yescomplemented
217C32×D3—24no—
218C3×#41—316no—
219C3×#41—324no—
220C2×C3×Q8—24no—
221C2×C3×Q8—316no—
222C2×C3×Q8—94no—
223C2×C3×Q8—94no—
224C2×C3×Q8—93no—
225C2×C3×Q8D3214yescomplemented
226C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
227C22×C3×C4—93no—
228C2×#67—94no—
229#239—924no—
230#239—916no—
231#239—94no—
232#239—912no—
233#239—912no—
234#239—1816no—
235#239C2×C3⋊D314yescomplemented
236D3×Q8—316no—
237D3×Q8—316no—
238D3×Q8—948no—
239D3×Q8—924no—
240D3×Q8—916no—
241D3×Q8—94no—
242D3×Q8—912no—
243D3×Q8—912no—
244D3×Q8—1816no—
245D3×Q8C2×C3⋊D314yescomplemented
246#237—948no—
247#237—924no—
248#237—948no—
249D3×D4—924no—
250D3×D4—916no—
251#235—948no—
252#235—948no—
253#235—924no—
254#235—96no—
255#235—912no—
256#235—96no—
257#235—912no—
258#235—96no—
259#235—1824no—
260#235C2×C3⋊D316yescomplemented
261C2×D12—96no—
262C2×C4×D3—312no—
263C2×C4×D3—948no—
264C2×C4×D3—912no—
265C2×Dic6—312no—
266C2×Dic6—948no—
267C2×Dic6—96no—
268C22×C32—24no—
269C22×C32—34no—
270C22×C32C23×D314yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—22no—
273C2×C3⋊D3—34no—
274C2×C3⋊D3—316no—
275C2×C3⋊D3—616no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3C23×D311yescharacteristic
278C2×C3⋊D3C23×D311yescharacteristic
279C2×C3×D3—34no—
280C2×C3×D3—34no—
281C2×C3×D3—68no—
282C2×C3×D3C23×D3116yes—
283D32—632no—
284C32×C4—224no—
285C32×C4—34no—
286C32×C4C23×D3124yes—
287C32×C4C23×D316yes—
288C3⋊Dic3—34no—
289C3⋊Dic3—316no—
290C3⋊Dic3—616no—
291C3⋊Dic3C23×D316yes—
292C3×Dic3—324no—
293C3×Dic3C23×D3116yes—
2942−1+4—31no—
2952−1+4—274no—
2962−1+4—274no—
2972−1+4—276no—
2982−1+4—271no—
299C2×#41—276no—
300C2×#41—274no—
301C22×Q8—91no—
302C22×Q8—274no—
303C33C2×2−1+411yescharacteristic, complemented, Sylow
304C23×C3—91no—
305C22×D3—34no—
306C22×D3—916no—
307C22×D3—94no—
308C3×Q8—216no—
309C3×Q8—324no—
310C3×Q8—912no—
311C3×Q8C2×D32116yes—
312C3×Q8C22×C3⋊D314yes—
313C3×D4—316no—
314C3×D4—324no—
315C2×C3×C4—212no—
316C2×C3×C4—348no—
317C2×C3×C4—93no—
318C2×C3×C4—96no—
319C2×C3×C4—96no—
320C2×C3×C4C2×D32112yes—
321C2×C3×C4C22×C3⋊D313yes—
322#67—948no—
323#67—916no—
324#67—916no—
325#67—94no—
326#67—94no—
327#67—94no—
328#67—1816no—
329#67C22×C3⋊D314yes—
330C2×Dic3—312no—
331C2×Dic3—948no—
332C2×Dic3—94no—
333D12—916no—
334D12—912no—
335D12—924no—
336D12—96no—
337D12—1824no—
338D12C22×C3⋊D316yes—
339C4×D3—324no—
340C4×D3—324no—
341C4×D3—948no—
342C4×D3—916no—
343C4×D3—912no—
344C4×D3—948no—
345C4×D3—912no—
346C4×D3—912no—
347C4×D3—1848no—
348C4×D3C22×C3⋊D3112yes—
349Dic6—348no—
350Dic6—948no—
351Dic6—924no—
352Dic6—96no—
353Dic6—912no—
354Dic6—1824no—
355Dic6C22×C3⋊D316yes—
356C2×C32—21no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—44no—
360C2×C32—64no—
361C2×C32C24×D314yes—
362C2×C32C24×D311yescharacteristic
363C3⋊D3—21no—
364C3⋊D3—64no—
365C3⋊D3—616no—
366C3⋊D3—1216no—
367C3⋊D3#100712yesdirect product factor
368C3×D3—216no—
369C3×D3—38no—
370C3×D3—64no—
371#41—34no—
372#41—36no—
373#41—2712no—
374#41—2712no—
375#41—2712no—
376#41—276no—
377#41—276no—
378#41—2712no—
379#41—274no—
380#41—276no—
381C2×Q8—34no—
382C2×Q8—94no—
383C2×Q8—94no—
384C2×Q8—93no—
385C2×Q8—2712no—
386C2×Q8—276no—
387C2×Q8—2712no—
388C2×Q8—274no—
389C2×Q8D3×C3⋊D311yescharacteristic, complemented
390C2×D4—276no—
391C2×D4—274no—
392C22×C4—93no—
393C22×C4—2712no—
394C22×C3—24no—
395C22×C3—316no—
396C22×C3—91no—
397C22×C3—91no—
398C22×C3—182no—
399C22×C3C22×D3214yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—34no—
402C2×D3—34no—
403C2×D3—68no—
404C2×D3—916no—
405C2×D3—94no—
406C2×D3—1816no—
407C2×D3—188no—
408C2×D3—1832no—
409C2×D3C23×C3⋊D314yes—
410C3×C4—224no—
411C3×C4—316no—
412C3×C4—96no—
413C3×C4C22×D32124yes—
414C3×C4C23×C3⋊D316yes—
415Dic3—324no—
416Dic3—916no—
417Dic3—94no—
418Dic3—1816no—
419Dic3C23×C3⋊D314yes—
420C32—24no—
421C32C2×#100711yescharacteristic, complemented
422C32D3×2−1+414yescomplemented
423C23—91no—
424C23—274no—
425Q8—36no—
426Q8—912no—
427Q8—2712no—
428Q8—276no—
429Q8C2×D3×C3⋊D314yes—
430D4—34no—
431D4—36no—
432D4—274no—
433D4—2712no—
434D4—274no—
435D4—274no—
436D4—276no—
437C2×C4—312no—
438C2×C4—93no—
439C2×C4—96no—
440C2×C4—96no—
441C2×C4—2712no—
442C2×C4—274no—
443C2×C4—2712no—
444C2×C4—2712no—
445C2×C4C2×D3×C3⋊D313yes—
446C2×C3—24no—
447C2×C3—21no—
448C2×C3—24no—
449C2×C3—44no—
450C2×C3—616no—
451C2×C3—92no—
452C2×C3—181no—
453C2×C3C23×D3214yes—
454C2×C3C24×C3⋊D311yescharacteristic
455D3—24no—
456D3—38no—
457D3—64no—
458D3—1816no—
459D3—184no—
460D3—3616no—
461C22—34no—
462C22—91no—
463C22—91no—
464C22—182no—
465C22—274no—
466C22—548no—
467C22C22×D3×C3⋊D311yescharacteristic
468C4—34no—
469C4—96no—
470C4—274no—
471C4C22×D3×C3⋊D316yes—
472C3—24no—
473C3D3×#100714yescomplemented
474C3C3⋊D3×2−1+411yescharacteristic, complemented
475C2—21no—
476C2—64no—
477C2—92no—
478C2—181no—
479C2—544no—
480C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×C3⋊D3
commutator, socle, cosocleC2×C33C25
FittingC2×C33×Q8C22
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #11724682C2semidirect
2. C2×C3×Q8×C3⋊D3C2semidirect
3. C32×#1007C2semidirect
4. C3⋊D3×#239C2semidirect
5. D3×Q8×C3⋊D3C2semidirect
6. #11724591C2semidirect
7. #11724590C2semidirect
8. C3⋊D3×#235C2semidirect
9. #11724562C2semidirect
12. C2×C33×Q8C22semidirect
13. #89080C22semidirect
14. Q8×C3⋊(C3⋊D3)C22semidirect
15. #89076C22semidirect
17. C3×Q8×C3⋊D3C22semidirect
19. C2×C3×#634C22semidirect
20. C32×#239C22semidirect
21. C32×D3×Q8C22semidirect
22. C32×#235C22semidirect
24. #89028C22semidirect
27. #89022C22semidirect
28. #89020C22semidirect
29. #89019C22semidirect
30. #89018C22semidirect
32. #89015C22semidirect
33. #89014C22semidirect
34. D3×#634C22semidirect
45. C2×Q8×C3⋊D3D3semidirect
46. C3×#1007D3semidirect
83. C2×C32×Q8C2×D3semidirect
84. C2×C32×Q8C2×D3semidirect
95. Q8×C3⋊D3C2×D3semidirect
109. C2×#634C2×D3semidirect
110. C3×#239C2×D3semidirect
113. C3×D3×Q8C2×D3semidirect
114. C3×#235C2×D3semidirect
153. #1007C3⋊D3direct (N × H)
216. C3×C3⋊D32−1+4semidirect
225. C2×C3×Q8D32semidirect
226. C2×C3×Q8C2×C3⋊D3semidirect
235. #239C2×C3⋊D3semidirect
245. D3×Q8C2×C3⋊D3semidirect
260. #235C2×C3⋊D3semidirect
303. C33C2×2−1+4semidirect
367. C3⋊D3#1007direct (N × H)
389. C2×Q8D3×C3⋊D3semidirect
421. C32C2×#1007semidirect
422. C32D3×2−1+4semidirect
473. C3D3×#1007semidirect
474. C3C3⋊D3×2−1+4semidirect

Maximal subgroups

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

subgroupindex
#117246822
C2×C3×Q8×C3⋊D32
C32×#10072
C3⋊D3×#2392
D3×Q8×C3⋊D32
#117245912
#117245902
C3⋊D3×#2352
#117245622
C3⋊D3×2−1+43
D3×#10073