Finite Group Databasebeta

← C3×D3×C4⋊(C3⋊C8) #13157338

Subgroups — C3×D3×C4⋊(C3⋊C8) #13157338

Order 1728.

766 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C42×D3C211yesmaximal, characteristic
2C2×C3×D3×C3⋊C8C212yesmaximal
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×Dic3⋊(C3⋊C8)C212yesmaximal, complemented
6C2×C3×C4⋊(C3⋊C8)—31nomaximal
7C3×D3×C4⋊C8—31nomaximal
8D3×C4⋊(C3⋊C8)C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3C411yescharacteristic
10C2×C32×C4×D3C411yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2212yes—
14C32×C4×Dic3C411yescharacteristic
15C32×C4×Dic3C411yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2212yes—
18C3×D3×C3⋊C8—24no—
19C2×C32×C42—31no—
20C22×C3×C3⋊C8—32no—
21C2×C3×D3×C8—32no—
22C3×C42×D3—21no—
23C3×C42×D3D311yescharacteristic
24C3×C42×D3C2×C311yescharacteristic
25C2×D3×C3⋊C8C2×C312yes—
26C32×C4⋊C8—31no—
27C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
28C3×Dic3⋊C8—32no—
29C3×C4⋊(C3⋊C8)—21no—
30C3×C4⋊(C3⋊C8)—31no—
31C3×C4⋊(C3⋊C8)—31no—
32C3×C4⋊(C3⋊C8)—32no—
33C3×C4⋊(C3⋊C8)—61no—
34C3×C4⋊(C3⋊C8)D311yescharacteristic, direct product factor
35C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
36Dic3⋊(C3⋊C8)C2×C312yescomplemented
37C22×C32×D3C2×C411yescharacteristic
38C2×C33×C4C2×C411yescharacteristic
39C2×C33×C4C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C32×Dic3C2×C411yescharacteristic
42C2×C32×Dic3C2×C411yescharacteristic
43C2×C32×Dic3C2×C411yescharacteristic
44C32×C4×D3—22no—
45C32×C4×D3C814yescomplemented
46C32×C4×D3D412yes—
47C32×C4×D3Q812yes—
48C3×C3⋊(C3⋊C8)—22no—
49C32×C3⋊C8—22no—
50C2×C3×C4⋊C8—91no—
51C2×C4⋊(C3⋊C8)—31no—
52D3×C4⋊C8—31no—
53C22×C32×C4—31no—
54C22×C32×C4—31no—
55C22×C32×C4—31no—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3Dic311yescharacteristic
60C2×C3×C4×D3Dic311yescharacteristic
61C2×C3×C4×D3C3×C411yescharacteristic
62C2×C3×C4×D3C3×C411yescharacteristic
63C2×C3×C4×D3C2×D311yescharacteristic
64C2×C3×C4×D3C22×C311yescharacteristic
65C2×C32×C8—32no—
66C32×C42—21no—
67C32×C42—21no—
68C32×C42—21no—
69C32×C42—31no—
70C32×C42—31no—
71C32×C42—31no—
72C32×C42—41no—
73C32×C42C2×D311yescharacteristic
74C32×C42C2×D311yescharacteristic
75C32×C42C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C312yes—
77C3×C4×Dic3—21no—
78C3×C4×Dic3—21no—
79C3×C4×Dic3—21no—
80C3×C4×Dic3Dic311yescharacteristic
81C3×C4×Dic3Dic311yescharacteristic
82C3×C4×Dic3C3×C411yescharacteristic
83C3×C4×Dic3C3×C411yescharacteristic
84C3×C4×Dic3C2×D311yescharacteristic
85C3×C4×Dic3C22×C311yescharacteristic
86C2×C3×C3⋊C8—22no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—32no—
89C2×C3×C3⋊C8—64no—
90C2×C3×C3⋊C8—62no—
91C2×C3×C3⋊C8C2×D312yes—
92C2×C3×C3⋊C8C22×C312yes—
93C3×D3×C8—64no—
94D3×C3⋊C8—24no—
95C22×C33C22×C411yescharacteristic
96C2×C32×D3C4⋊C412yes—
97C2×C32×D3C2×C812yes—
98C2×C32×D3M1612yes—
99C33×C4—21no—
100C33×C4C2×C812yescomplemented
101C33×C4C2×D411yescharacteristic
102C33×C4C2×Q811yescharacteristic
103C32×Dic3—21no—
104C32×Dic3C4⋊C412yes—
105C32×Dic3C2×C812yescomplemented
106C22×C3×C8—92no—
107C2×C3×C42—31no—
108C2×C3×C42—31no—
109C2×C3×C42—61no—
110C22×C3⋊C8—32no—
111C2×D3×C8—32no—
112C42×D3C3×D311yescharacteristic
113C3×C4⋊C8—31no—
114C3×C4⋊C8—31no—
115C3×C4⋊C8—61no—
116C3×C4⋊C8—92no—
117C3×C4⋊C8—91no—
118Dic3⋊C8—32no—
119C4⋊(C3⋊C8)—31no—
120C4⋊(C3⋊C8)—31no—
121C4⋊(C3⋊C8)—32no—
122C4⋊(C3⋊C8)—61no—
123C4⋊(C3⋊C8)C3×D311yescharacteristic, direct product factor
124C23×C32—31no—
125C22×C3×D3—21no—
126C22×C3×D3C2×Dic311yescharacteristic
127C22×C3×D3C2×C3×C411yescharacteristic
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—31no—
138C2×C32×C4—31no—
139C2×C32×C4—34no—
140C2×C32×C4—32no—
141C2×C32×C4—32no—
142C2×C32×C4—31no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4—62no—
147C2×C32×C4C4×D311yescharacteristic
148C2×C32×C4C4×D311yescharacteristic
149C2×C32×C4C2×Dic311yescharacteristic
150C2×C32×C4C2×Dic311yescharacteristic
151C2×C32×C4C2×C3×C411yescharacteristic
152C2×C32×C4C2×C3×C411yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C23×C311yescharacteristic
156C2×C3×Dic3—21no—
157C2×C3×Dic3—21no—
158C2×C3×Dic3—21no—
159C2×C3×Dic3C2×Dic311yescharacteristic
160C2×C3×Dic3C2×Dic311yescharacteristic
161C2×C3×Dic3C2×Dic311yescharacteristic
162C2×C3×Dic3C2×C3×C411yescharacteristic
163C2×C3×Dic3C2×C3×C411yescharacteristic
164C2×C3×Dic3C2×C3×C411yescharacteristic
165C3×C4×D3—24no—
166C3×C4×D3—22no—
167C3×C4×D3—22no—
168C3×C4×D3—24no—
169C3×C4×D3—22no—
170C3×C4×D3—22no—
171C3×C4×D3C3⋊C814yescomplemented
172C3×C4×D3C3×C814yescomplemented
173C3×C4×D3Dic612yes—
174C3×C4×D3D1212yes—
175C3×C4×D3C3×D412yes—
176C3×C4×D3C3×Q812yes—
177C32×C8—62no—
178C3⋊(C3⋊C8)—22no—
179C3×C3⋊C8—22no—
180C3×C3⋊C8—22no—
181C3×C3⋊C8—42no—
182C3×C3⋊C8—62no—
183C3×C3⋊C8—62no—
184C3×C3⋊C8—122no—
185C2×C4⋊C8—91noSylow
186C2×C33C2×C4⋊C411yescharacteristic
187C2×C33C22×C811yescharacteristic
188C2×C33C2×M1611yescharacteristic
189C32×D3C4⋊C814yescomplemented
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—61no—
197C22×C3×C4—61no—
198C22×C3×C4—61no—
199C2×C4×D3C3×Dic311yescharacteristic
200C2×C4×D3C3×Dic311yescharacteristic
201C2×C4×D3C2×C3×D311yescharacteristic
202C2×C3×C8—32no—
203C2×C3×C8—32no—
204C2×C3×C8—62no—
205C2×C3×C8—92no—
206C2×C3×C8—184no—
207C3×C42—21no—
208C3×C42—21no—
209C3×C42—21no—
210C3×C42—31no—
211C3×C42—31no—
212C3×C42—31no—
213C3×C42—31no—
214C3×C42—31no—
215C3×C42—31no—
216C3×C42—41no—
217C3×C42—61no—
218C3×C42—61no—
219C3×C42—61no—
220C3×C42D3211yescharacteristic
221C3×C42C2×C3×D311yescharacteristic
222C3×C42C2×C3×D311yescharacteristic
223C4×Dic3C3×Dic311yescharacteristic
224C4×Dic3C3×Dic311yescharacteristic
225C4×Dic3C2×C3×D311yescharacteristic
226C2×C3⋊C8—32no—
227C2×C3⋊C8—32no—
228C2×C3⋊C8—62no—
229C2×C3⋊C8—64no—
230C2×C3⋊C8C2×C3×D312yes—
231D3×C8—64no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—32no—
236C22×C32—32no—
237C22×C32—32no—
238C22×C32—41no—
239C22×C32C2×C4×D311yescharacteristic
240C22×C32C22×Dic311yescharacteristic
241C22×C32C22×C3×C411yescharacteristic
242C2×C3×D3—22no—
243C2×C3×D3—22no—
244C2×C3×D3—22no—
245C2×C3×D3C2×C3⋊C812yes—
246C2×C3×D3#20812yes—
247C2×C3×D3C4⋊Dic312yes—
248C2×C3×D3C3×C4⋊C412yes—
249C2×C3×D3C2×C3×C812yes—
250C2×C3×D3C3×M1612yes—
251C32×C4—21no—
252C32×C4—22no—
253C32×C4—21no—
254C32×C4—21no—
255C32×C4—22no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—21no—
259C32×C4—21no—
260C32×C4—21no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—21no—
264C32×C4—32no—
265C32×C4—32no—
266C32×C4—42no—
267C32×C4—41no—
268C32×C4—41no—
269C32×C4—41no—
270C32×C4—41no—
271C32×C4—42no—
272C32×C4—61no—
273C32×C4D3×C812yescomplemented
274C32×C4C2×C3⋊C812yescomplemented
275C32×C4C2×C3×C812yescomplemented
276C32×C4C2×Dic611yescharacteristic
277C32×C4C2×D1211yescharacteristic
278C32×C4D3×D411yescharacteristic
279C32×C4D3×Q811yescharacteristic
280C32×C4C2×C3×D411yescharacteristic
281C32×C4C2×C3×Q811yescharacteristic
282C3×Dic3—22no—
283C3×Dic3—21no—
284C3×Dic3—21no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3C2×C3⋊C812yescomplemented
288C3×Dic3C4⋊Dic312yes—
289C3×Dic3C3×C4⋊C412yes—
290C3×Dic3C2×C3×C812yescomplemented
291C22×C8—92no—
292C2×C42—31no—
293C4⋊C8—31no—
294C4⋊C8—92no—
295C4⋊C8—91no—
296C33C2×C4⋊C811yescharacteristic, complemented, Sylow
297C23×C3—31no—
298C23×C3—31no—
299C23×C3—61no—
300C22×D3C2×C3×Dic311yescharacteristic
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—34no—
315C2×C3×C4—34no—
316C2×C3×C4—32no—
317C2×C3×C4—32no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—64no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—64no—
334C2×C3×C4D3×Dic311yescharacteristic
335C2×C3×C4D3×Dic311yescharacteristic
336C2×C3×C4C3×C4×D311yescharacteristic
337C2×C3×C4C3×C4×D311yescharacteristic
338C2×C3×C4C2×C3×Dic311yescharacteristic
339C2×C3×C4C2×C3×Dic311yescharacteristic
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C22×C3×D311yescharacteristic
342C2×C3×C4C22×C3×D311yescharacteristic
343C2×Dic3C2×C3×Dic311yescharacteristic
344C2×Dic3C2×C3×Dic311yescharacteristic
345C2×Dic3C2×C3×Dic311yescharacteristic
346C4×D3—22no—
347C4×D3C3×C3⋊C814yescomplemented
348C4×D3C3×Dic612yes—
349C4×D3C3×D1212yes—
350C3×C8—62no—
351C3×C8—62no—
352C3×C8—122no—
353C3×C8—182no—
354C3⋊C8—22no—
355C3⋊C8—62no—
356C3⋊C8—62no—
357C3⋊C8—122no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—21no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—34no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32—41no—
371C2×C32D3×C4⋊C411yescharacteristic
372C2×C32C2×D3×C811yescharacteristic
373C2×C32D3×M1611yescharacteristic
374C2×C32C22×C3⋊C811yescharacteristic
375C2×C32C2×#20811yescharacteristic
376C2×C32C2×C4⋊Dic311yescharacteristic
377C2×C32C2×C3×C4⋊C411yescharacteristic
378C2×C32C22×C3×C811yescharacteristic
379C2×C32C2×C3×M1611yescharacteristic
380C3×D3—24no—
381C3×D3C4⋊(C3⋊C8)14yesdirect product factor
382C3×D3C3×C4⋊C814yescomplemented
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C2×C8—32no—
387C2×C8—92no—
388C2×C8—184no—
389C42—31no—
390C42—31no—
391C42—31no—
392C42C3×D3211yescharacteristic
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—32no—
397C22×C3—32no—
398C22×C3—32no—
399C22×C3—32no—
400C22×C3—32no—
401C22×C3—32no—
402C22×C3—41no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—62no—
406C22×C3C2×D3×Dic311yescharacteristic
407C22×C3C2×C3×C4×D311yescharacteristic
408C22×C3C22×C3×Dic311yescharacteristic
409C2×D3C2×C3×C3⋊C812yes—
410C2×D3C3×#20812yes—
411C2×D3C3×C4⋊Dic312yes—
412C3×C4—22no—
413C3×C4—21no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—22no—
425C3×C4—32no—
426C3×C4—32no—
427C3×C4—32no—
428C3×C4—32no—
429C3×C4—41no—
430C3×C4—41no—
431C3×C4—42no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—42no—
435C3×C4—62no—
436C3×C4—62no—
437C3×C4—62no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4D3×C3⋊C812yescomplemented
441C3×C4C3×D3×C812yescomplemented
442C3×C4C2×C3×C3⋊C812yescomplemented
443C3×C4D3×Dic611yescharacteristic
444C3×C4D3×D1211yescharacteristic
445C3×C4C2×C3×Dic611yescharacteristic
446C3×C4C2×C3×D1211yescharacteristic
447C3×C4C3×D3×D411yescharacteristic
448C3×C4C3×D3×Q811yescharacteristic
449Dic3—21no—
450Dic3C2×C3×C3⋊C812yescomplemented
451Dic3C3×C4⋊Dic312yes—
452C32—21no—
453C32—21no—
454C32—21no—
455C32—41no—
456C32D3×C4⋊C811yescharacteristic, complemented
457C32C2×C4⋊(C3⋊C8)11yescharacteristic, complemented
458C32C2×C3×C4⋊C811yescharacteristic, complemented
459C23—31no—
460C2×C4—31no—
461C2×C4—31no—
462C2×C4—32no—
463C2×C4—32no—
464C2×C4—34no—
465C2×C4—31no—
466C2×C4—62no—
467C2×C4C3×D3×Dic311yescharacteristic
468C2×C4C3×D3×Dic311yescharacteristic
469C2×C4C2×C3×D3211yescharacteristic
470C8—62no—
471C8—182no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—34no—
482C2×C3—34no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—64no—
487C2×C3C2×D3×C3⋊C811yescharacteristic
488C2×C3D3×#20811yescharacteristic
489C2×C3D3×C4⋊Dic311yescharacteristic
490C2×C3C3×D3×C4⋊C411yescharacteristic
491C2×C3C2×C3×D3×C811yescharacteristic
492C2×C3C3×D3×M1611yescharacteristic
493C2×C3C22×C3×C3⋊C811yescharacteristic
494C2×C3C2×C3×#20811yescharacteristic
495C2×C3C2×C3×C4⋊Dic311yescharacteristic
496D3C3×C4⋊(C3⋊C8)14yesdirect product factor
497C22—32no—
498C22—32no—
499C22—32no—
500C22C2×C3×D3×Dic311yescharacteristic
501C4—21no—
502C4—32no—
503C4—32no—
504C4—61no—
505C4C3×D3×C3⋊C812yescomplemented
506C4C3×D3×Dic611yescharacteristic
507C4C3×D3×D1211yescharacteristic
508C3—21no—
509C3—21no—
510C3—21no—
511C3—41no—
512C3D3×C4⋊(C3⋊C8)11yescharacteristic, direct product factor
513C3C3×D3×C4⋊C811yescharacteristic, complemented
514C3C2×C3×C4⋊(C3⋊C8)11yescharacteristic, complemented
515C2—34no—
516C2C2×C3×D3×C3⋊C811yescharacteristic
517C2C3×D3×#20811yescharacteristic
518C2C3×D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
3. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. C32×C4⋊(C3⋊C8)C2semidirect
5. C3×Dic3⋊(C3⋊C8)C2semidirect
8. D3×C4⋊(C3⋊C8)C3direct (N × H)
27. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
34. C3×C4⋊(C3⋊C8)D3direct (N × H)
35. C3×C4⋊(C3⋊C8)C2×C3semidirect
36. Dic3⋊(C3⋊C8)C2×C3semidirect
45. C32×C4×D3C8semidirect
100. C33×C4C2×C8semidirect
105. C32×Dic3C2×C8semidirect
123. C4⋊(C3⋊C8)C3×D3direct (N × H)
171. C3×C4×D3C3⋊C8semidirect
172. C3×C4×D3C3×C8semidirect
189. C32×D3C4⋊C8semidirect
273. C32×C4D3×C8semidirect
274. C32×C4C2×C3⋊C8semidirect
275. C32×C4C2×C3×C8semidirect
287. C3×Dic3C2×C3⋊C8semidirect
290. C3×Dic3C2×C3×C8semidirect
296. C33C2×C4⋊C8semidirect
347. C4×D3C3×C3⋊C8semidirect
381. C3×D3C4⋊(C3⋊C8)direct (N × H)
382. C3×D3C3×C4⋊C8semidirect
440. C3×C4D3×C3⋊C8semidirect
441. C3×C4C3×D3×C8semidirect
442. C3×C4C2×C3×C3⋊C8semidirect
450. Dic3C2×C3×C3⋊C8semidirect
456. C32D3×C4⋊C8semidirect
457. C32C2×C4⋊(C3⋊C8)semidirect
458. C32C2×C3×C4⋊C8semidirect
496. D3C3×C4⋊(C3⋊C8)direct (N × H)
505. C4C3×D3×C3⋊C8semidirect
512. C3D3×C4⋊(C3⋊C8)direct (N × H)
513. C3C3×D3×C4⋊C8semidirect
514. C3C2×C3×C4⋊(C3⋊C8)semidirect

Maximal subgroups

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

subgroupindex
C32×C42×D32
C2×C3×D3×C3⋊C82
C3×C4⋊(C3⋊(C3⋊C8))2
C32×C4⋊(C3⋊C8)2
C3×Dic3⋊(C3⋊C8)2
C2×C3×C4⋊(C3⋊C8)3
C3×D3×C4⋊C83
D3×C4⋊(C3⋊C8)3