Finite Group Databasebeta

← C3×C3⋊C8×D12 #13157340

Subgroups — C3×C3⋊C8×D12 #13157340

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C211yesmaximal, characteristic
2C2×C3×D3×C3⋊C8C212yesmaximal, complemented
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×(C2×D3)⋊(C3⋊C8)C212yesmaximal, complemented
6C3×D4×C3⋊C8—31nomaximal
7C3×C8×D12—31nomaximal
8C3⋊C8×D12C311yesmaximal, characteristic, direct product factor
9C2×C32×D12C411yescharacteristic
10C2×C32×C4×D3C2212yes—
11C33×C42C2211yescharacteristic
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×(C2×D3)⋊C4C412yes—
14C32×C4⋊Dic3C411yescharacteristic
15C2×C32×C3⋊C8C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C3×D3×C3⋊C8—24no—
18C32×C4×D4—31no—
19C22×C3×C3⋊C8—32no—
20C2×C3×D3×C8—32no—
21C3×C4×D12—21no—
22C3×C4×D12D311yescharacteristic
23C3×C4×D12C2×C311yescharacteristic
24C2×D3×C3⋊C8C2×C312yescomplemented
25C32×C4×C8—31no—
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#830—32no—
28C3×(C2×D3)⋊C8—32no—
29C3×C4⋊(C3⋊C8)—31no—
30C3×C4⋊(C3⋊C8)—31no—
31C3×C4⋊(C3⋊C8)—61no—
32C3×C4×C3⋊C8—21no—
33C3×C4×C3⋊C8D311yescharacteristic, complemented
34C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
35(C2×D3)⋊(C3⋊C8)C2×C312yescomplemented
36C22×C32×D3C2×C412yes—
37C2×C33×C4C2×C411yescharacteristic
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C32×Dic3C2×C412yes—
41C32×D12C814yescomplemented
42C32×C4×D3—22no—
43C32×C4×D3—22no—
44C3×C3⋊(C3⋊C8)—22no—
45C32×C3⋊C8—21no—
46C32×C3⋊C8D412yescomplemented
47C3×C8×D4—91no—
48D4×C3⋊C8—31no—
49C8×D12—31no—
50C2×C32×D4—31no—
51C22×C32×C4—32no—
52C2×C3×D12—21no—
53C2×C3×D12Dic311yescharacteristic
54C2×C3×D12C3×C411yescharacteristic
55C2×C3×C4×D3—22no—
56C2×C3×C4×D3C2×D312yes—
57C2×C3×C4×D3C22×C312yes—
58C2×C32×C8—31no—
59C2×C32×C8—31no—
60C32×C4⋊C4—31no—
61C32×C22⋊C4—32no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42—21no—
65C32×C42—41no—
66C32×C42C2×D311yescharacteristic
67C32×C42C2×D311yescharacteristic
68C32×C42C22×C311yescharacteristic
69C2×C3⋊(C3⋊C8)C22×C312yes—
70C3×(C2×D3)⋊C4—22no—
71C3×(C2×D3)⋊C4Dic312yes—
72C3×(C2×D3)⋊C4C3×C412yes—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3Dic311yescharacteristic
75C3×C4⋊Dic3C3×C411yescharacteristic
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—32no—
80C2×C3×C3⋊C8—64no—
81C2×C3×C3⋊C8—62no—
82C2×C3×C3⋊C8C2×D311yescharacteristic
83C2×C3×C3⋊C8C2×D311yescharacteristic
84C2×C3×C3⋊C8C22×C311yescharacteristic
85C2×C3×C3⋊C8C22×C311yescharacteristic
86C3×D3×C8—64no—
87D3×C3⋊C8—24no—
88C22×C33C22×C411yescharacteristic
89C2×C32×D3—22no—
90C2×C32×D3—22no—
91C2×C32×D3C2×C814yescomplemented
92C33×C4—21no—
93C33×C4C2×C812yescomplemented
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C32×Dic3—22no—
97C22×C3×C8—92no—
98C3×C4×D4—31no—
99C3×C4×D4—31no—
100C3×C4×D4—61no—
101C22×C3⋊C8—32no—
102C2×D3×C8—32no—
103C4×D12C3×D311yescharacteristic
104C3×C4⋊C8—91no—
105C3×C22⋊C8—92no—
106C3×C4×C8—31no—
107C3×C4×C8—31no—
108C3×C4×C8—61no—
109#830—32no—
110(C2×D3)⋊C8—32no—
111C4⋊(C3⋊C8)—31no—
112C4⋊(C3⋊C8)—31no—
113C4⋊(C3⋊C8)—61no—
114C4×C3⋊C8C3×D311yescharacteristic, complemented
115C23×C32—32no—
116C22×C3×D3—22no—
117C22×C3×D3C2×Dic312yes—
118C22×C3×D3C2×C3×C412yes—
119C32×D4—34no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—32no—
130C2×C32×C4—41no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4—62no—
134C2×C32×C4—62no—
135C2×C32×C4C4×D311yescharacteristic
136C2×C32×C4C4×D311yescharacteristic
137C2×C32×C4C2×Dic311yescharacteristic
138C2×C32×C4C2×Dic311yescharacteristic
139C2×C32×C4C2×C3×C411yescharacteristic
140C2×C32×C4C2×C3×C411yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C23×C311yescharacteristic
144C2×C3×Dic3—22no—
145C2×C3×Dic3C2×Dic312yes—
146C2×C3×Dic3C2×C3×C412yes—
147C3×D12—24no—
148C3×D12C3⋊C814yesdirect product factor
149C3×D12C3×C814yescomplemented
150C3×C4×D3—22no—
151C3×C4×D3—22no—
152C3×C4×D3—22no—
153C3×C4×D3—22no—
154C3×C4×D3—42no—
155C3×C4×D3—42no—
156C32×C8—32no—
157C32×C8—61no—
158C3⋊(C3⋊C8)—22no—
159C3×C3⋊C8—22no—
160C3×C3⋊C8—21no—
161C3×C3⋊C8—21no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8—62no—
164C3×C3⋊C8—62no—
165C3×C3⋊C8—122no—
166C3×C3⋊C8D1212yesdirect product factor
167C3×C3⋊C8C3×D412yescomplemented
168C8×D4—91noSylow
169C2×C33C4×D411yescharacteristic
170C2×C33C22×C811yescharacteristic
171C2×C33#13111yescharacteristic
172C32×D3—24no—
173C2×C3×D4—31no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—32no—
177C22×C3×C4—32no—
178C22×C3×C4—62no—
179C2×D12C3×Dic311yescharacteristic
180C2×C4×D3C2×C3×D312yes—
181C2×C3×C8—31no—
182C2×C3×C8—31no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—61no—
186C2×C3×C8—61no—
187C2×C3×C8—92no—
188C2×C3×C8—184no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—31no—
191C3×C4⋊C4—61no—
192C3×C22⋊C4—32no—
193C3×C22⋊C4—32no—
194C3×C22⋊C4—62no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—41no—
199C3×C42D3211yescharacteristic
200C3×C42C2×C3×D311yescharacteristic
201C3×C42C2×C3×D311yescharacteristic
202(C2×D3)⋊C4C3×Dic312yes—
203C4⋊Dic3C3×Dic311yescharacteristic
204C2×C3⋊C8—32no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—62no—
207C2×C3⋊C8—64no—
208C2×C3⋊C8C2×C3×D311yescharacteristic
209C2×C3⋊C8C2×C3×D311yescharacteristic
210D3×C8—64no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—34no—
215C22×C32—41no—
216C22×C32—62no—
217C22×C32—62no—
218C22×C32C2×C4×D311yescharacteristic
219C22×C32C22×Dic311yescharacteristic
220C22×C32C22×C3×C411yescharacteristic
221C2×C3×D3—22no—
222C2×C3×D3—22no—
223C2×C3×D3—22no—
224C2×C3×D3—22no—
225C2×C3×D3—24no—
226C2×C3×D3—42no—
227C2×C3×D3—42no—
228C2×C3×D3C2×C3⋊C814yescomplemented
229C2×C3×D3C2×C3×C814yescomplemented
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—22no—
236C32×C4—22no—
237C32×C4—21no—
238C32×C4—21no—
239C32×C4—21no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—42no—
244C32×C4—41no—
245C32×C4—41no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—42no—
249C32×C4—62no—
250C32×C4D3×C812yescomplemented
251C32×C4C2×C3⋊C812yescomplemented
252C32×C4C2×C3×C812yescomplemented
253C32×C4C2×D1211yescharacteristic
254C32×C4#23511yescharacteristic
255C32×C4D3×D411yescharacteristic
256C32×C4#23711yescharacteristic
257C32×C4C2×C3×D411yescharacteristic
258C32×C4C3×#4111yescharacteristic
259C3×Dic3—22no—
260C3×Dic3—22no—
261C3×Dic3—42no—
262C22×C8—92no—
263C4×D4—31no—
264C4⋊C8—91no—
265C22⋊C8—92no—
266C4×C8—31no—
267C33C8×D411yescharacteristic, complemented, Sylow
268C23×C3—32no—
269C23×C3—32no—
270C23×C3—62no—
271C22×D3C2×C3×Dic312yes—
272C3×D4—34no—
273C3×D4—34no—
274C3×D4—64no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—32no—
285C2×C3×C4—32no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—122no—
295C2×C3×C4—122no—
296C2×C3×C4D3×Dic311yescharacteristic
297C2×C3×C4D3×Dic311yescharacteristic
298C2×C3×C4C3×C4×D311yescharacteristic
299C2×C3×C4C3×C4×D311yescharacteristic
300C2×C3×C4C2×C3×Dic311yescharacteristic
301C2×C3×C4C2×C3×Dic311yescharacteristic
302C2×C3×C4C2×D3211yescharacteristic
303C2×C3×C4C22×C3×D311yescharacteristic
304C2×C3×C4C22×C3×D311yescharacteristic
305C2×Dic3C2×C3×Dic312yes—
306D12C3×C3⋊C814yesdirect product factor
307C4×D3—22no—
308C4×D3—22no—
309C3×C8—32no—
310C3×C8—32no—
311C3×C8—62no—
312C3×C8—62no—
313C3×C8—61no—
314C3×C8—61no—
315C3×C8—182no—
316C3⋊C8—21no—
317C3⋊C8—62no—
318C3⋊C8—62no—
319C3⋊C8—122no—
320C3⋊C8C3×D1212yesdirect product factor
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—21no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32—64no—
334C2×C32C4×D1211yescharacteristic
335C2×C32C2×D3×C811yescharacteristic
336C2×C32#90111yescharacteristic
337C2×C32C22×C3⋊C811yescharacteristic
338C2×C32D4×Dic311yescharacteristic
339C2×C32#94811yescharacteristic
340C2×C32C3×C4×D411yescharacteristic
341C2×C32C22×C3×C811yescharacteristic
342C2×C32C3×#13111yescharacteristic
343C3×D3—24no—
344C3×D3—24no—
345C3×D3—44no—
346C2×D4—31no—
347C22×C4—32no—
348C2×C8—31no—
349C2×C8—31no—
350C2×C8—92no—
351C2×C8—184no—
352C4⋊C4—31no—
353C22⋊C4—32no—
354C42C3×D3211yescharacteristic
355C22×C3—21no—
356C22×C3—21no—
357C22×C3—21no—
358C22×C3—34no—
359C22×C3—34no—
360C22×C3—41no—
361C22×C3—64no—
362C22×C3—62no—
363C22×C3—62no—
364C22×C3—62no—
365C22×C3—62no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3C2×D3×Dic311yescharacteristic
369C22×C3C2×C3×C4×D311yescharacteristic
370C22×C3C22×C3×Dic311yescharacteristic
371C2×D3—22no—
372C2×D3—22no—
373C2×D3C2×C3×C3⋊C814yescomplemented
374C3×C4—22no—
375C3×C4—21no—
376C3×C4—21no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—42no—
390C3×C4—41no—
391C3×C4—41no—
392C3×C4—42no—
393C3×C4—62no—
394C3×C4—62no—
395C3×C4—122no—
396C3×C4D3×C3⋊C812yescomplemented
397C3×C4C3×D3×C812yescomplemented
398C3×C4C2×C3×C3⋊C812yescomplemented
399C3×C4#379511yescharacteristic
400C3×C4D3×D1211yescharacteristic
401C3×C4C2×C3×D1211yescharacteristic
402C3×C4C3×#23511yescharacteristic
403C3×C4C3×D3×D411yescharacteristic
404C3×C4C3×#23711yescharacteristic
405Dic3—22no—
406C32—21no—
407C32—21no—
408C32—21no—
409C32—41no—
410C32C8×D1211yescharacteristic, complemented
411C32D4×C3⋊C811yescharacteristic, complemented
412C32C3×C8×D411yescharacteristic, complemented
413C23—32no—
414D4—34no—
415C2×C4—32no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4C3×D3×Dic311yescharacteristic
419C2×C4C3×D3×Dic311yescharacteristic
420C2×C4C2×C3×D3211yescharacteristic
421C8—32no—
422C8—61no—
423C8—182no—
424C2×C3—21no—
425C2×C3—21no—
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—41no—
434C2×C3—41no—
435C2×C3—41no—
436C2×C3—64no—
437C2×C3—64no—
438C2×C3—124no—
439C2×C3C2×D3×C3⋊C811yescharacteristic
440C2×C3#6381411yescharacteristic
441C2×C3Dic3×D1211yescharacteristic
442C2×C3C3×C4×D1211yescharacteristic
443C2×C3C2×C3×D3×C811yescharacteristic
444C2×C3C3×#90111yescharacteristic
445C2×C3C22×C3×C3⋊C811yescharacteristic
446C2×C3C3×D4×Dic311yescharacteristic
447C2×C3C3×#94811yescharacteristic
448D3—24no—
449C22—34no—
450C22—62no—
451C22—62no—
452C22C2×C3×D3×Dic311yescharacteristic
453C4—21no—
454C4—62no—
455C4C3×D3×C3⋊C812yescomplemented
456C4C3×#379511yescharacteristic
457C4C3×D3×D1211yescharacteristic
458C3—21no—
459C3—21no—
460C3—21no—
461C3—41no—
462C3C3⋊C8×D1211yescharacteristic, direct product factor
463C3C3×C8×D1211yescharacteristic, complemented
464C3C3×D4×C3⋊C811yescharacteristic, complemented
465C2—64no—
466C2C2×C3×D3×C3⋊C811yescharacteristic
467C2C3×#6381411yescharacteristic
468C2C3×Dic3×D1211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC2×C32×D12C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×C8×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×D3×C3⋊C8C2semidirect
3. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. C32×C4×C3⋊C8C2semidirect
5. C3×(C2×D3)⋊(C3⋊C8)C2semidirect
8. C3⋊C8×D12C3direct (N × H)
24. C2×D3×C3⋊C8C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
33. C3×C4×C3⋊C8D3semidirect
34. C3×C4×C3⋊C8C2×C3semidirect
35. (C2×D3)⋊(C3⋊C8)C2×C3semidirect
41. C32×D12C8semidirect
46. C32×C3⋊C8D4semidirect
91. C2×C32×D3C2×C8semidirect
93. C33×C4C2×C8semidirect
114. C4×C3⋊C8C3×D3semidirect
148. C3×D12C3⋊C8direct (N × H)
149. C3×D12C3×C8semidirect
166. C3×C3⋊C8D12direct (N × H)
167. C3×C3⋊C8C3×D4semidirect
228. C2×C3×D3C2×C3⋊C8semidirect
229. C2×C3×D3C2×C3×C8semidirect
250. C32×C4D3×C8semidirect
251. C32×C4C2×C3⋊C8semidirect
252. C32×C4C2×C3×C8semidirect
267. C33C8×D4semidirect
306. D12C3×C3⋊C8direct (N × H)
320. C3⋊C8C3×D12direct (N × H)
373. C2×D3C2×C3×C3⋊C8semidirect
396. C3×C4D3×C3⋊C8semidirect
397. C3×C4C3×D3×C8semidirect
398. C3×C4C2×C3×C3⋊C8semidirect
410. C32C8×D12semidirect
411. C32D4×C3⋊C8semidirect
412. C32C3×C8×D4semidirect
455. C4C3×D3×C3⋊C8semidirect
462. C3C3⋊C8×D12direct (N × H)
463. C3C3×C8×D12semidirect
464. C3C3×D4×C3⋊C8semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C2×C3×D3×C3⋊C82
C3×C4⋊(C3⋊(C3⋊C8))2
C32×C4×C3⋊C82
C3×(C2×D3)⋊(C3⋊C8)2
C3×D4×C3⋊C83
C3×C8×D123
C3⋊C8×D123