Finite Group Databasebeta

← C2×C3×Dic3⋊(C3⋊C8) #13158145

Subgroups — C2×C3×Dic3⋊(C3⋊C8) #13158145

Order 1728.

866 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
2C2×C32×C4×Dic3C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C214yesmaximal, direct product factor
5C2×C3×Dic3⋊C8—31nomaximal
6C2×C3×C4⋊(C3⋊C8)—31nomaximal
7C2×Dic3⋊(C3⋊C8)C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C32×Dic3C411yescharacteristic
10C22×C32×Dic3C411yescharacteristic
11C2×C3×C3⋊(C3⋊C8)—22no—
12C2×C3×C3⋊(C3⋊C8)C2212yes—
13C32×C4×Dic3C412yes—
14C32×C4×Dic3C2212yes—
15C2×C32×C3⋊C8—22no—
16C2×C32×C3⋊C8C2212yes—
17C22×C32×C8—31no—
18C2×C32×C42—31no—
19C22×C3⋊(C3⋊C8)C2×C311yescharacteristic
20C2×C3×C4×Dic3—21no—
21C2×C3×C4×Dic3D311yescharacteristic
22C2×C3×C4×Dic3C2×C311yescharacteristic
23C22×C3×C3⋊C8—21no—
24C22×C3×C3⋊C8—31no—
25C22×C3×C3⋊C8—31no—
26C22×C3×C3⋊C8—61no—
27C22×C3×C3⋊C8D311yescharacteristic
28C22×C3×C3⋊C8C2×C311yescharacteristic
29C3×Dic3⋊C8—34no—
30C3×C4⋊(C3⋊C8)—34no—
31Dic3⋊(C3⋊C8)C2×C314yesdirect product factor
32C23×C33C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4D412yes—
35C2×C33×C4Q812yes—
36C2×C33×C4C2311yescharacteristic
37C2×C32×Dic3—22no—
38C2×C32×Dic3C814yescomplemented
39C2×C32×Dic3C2×C412yes—
40C2×C32×Dic3C2×C412yes—
41C3×C3⋊(C3⋊C8)—22no—
42C32×C3⋊C8—22no—
43C2×C3×C4⋊C8—91no—
44C2×Dic3⋊C8—31no—
45C2×C4⋊(C3⋊C8)—31no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C22×C32×C4—21no—
49C22×C32×C4—31no—
50C22×C32×C4—31no—
51C22×C32×C4—41no—
52C22×C32×C4C2×D311yescharacteristic
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C22×C3×Dic3—21no—
56C22×C3×Dic3—21no—
57C22×C3×Dic3Dic311yescharacteristic
58C22×C3×Dic3Dic311yescharacteristic
59C22×C3×Dic3C3×C411yescharacteristic
60C22×C3×Dic3C3×C411yescharacteristic
61C2×C32×C8—32no—
62C2×C32×C8—62no—
63C32×C42—32no—
64C32×C42—32no—
65C2×C3⋊(C3⋊C8)—22no—
66C2×C3⋊(C3⋊C8)C22×C312yes—
67C3×C4×Dic3—22no—
68C3×C4×Dic3—22no—
69C3×C4×Dic3Dic312yes—
70C3×C4×Dic3C3×C412yes—
71C3×C4×Dic3C2×D312yes—
72C3×C4×Dic3C22×C312yes—
73C2×C3×C3⋊C8—24no—
74C2×C3×C3⋊C8—22no—
75C2×C3×C3⋊C8—22no—
76C2×C3×C3⋊C8—22no—
77C2×C3×C3⋊C8—32no—
78C2×C3×C3⋊C8—32no—
79C2×C3×C3⋊C8—62no—
80C2×C3×C3⋊C8—64no—
81C2×C3×C3⋊C8—62no—
82C2×C3×C3⋊C8—62no—
83C2×C3×C3⋊C8C2×D312yes—
84C2×C3×C3⋊C8C22×C312yes—
85C22×C33C4⋊C412yes—
86C22×C33C2×C812yes—
87C22×C33M1612yes—
88C22×C33C22×C411yescharacteristic
89C33×C4C4⋊C412yes—
90C33×C4C2×D411yescharacteristic
91C33×C4C2×Q811yescharacteristic
92C32×Dic3—22no—
93C32×Dic3C2×C814yescomplemented
94C22×C3×C8—31no—
95C22×C3×C8—31no—
96C22×C3×C8—61no—
97C22×C3×C8—91no—
98C2×C3×C42—31no—
99C2×C3×C42—31no—
100C2×C3×C42—61no—
101C2×C4×Dic3C3×D311yescharacteristic
102C22×C3⋊C8—31no—
103C22×C3⋊C8—31no—
104C22×C3⋊C8—61no—
105C22×C3⋊C8C3×D311yescharacteristic
106C3×C4⋊C8—94no—
107Dic3⋊C8—34no—
108C4⋊(C3⋊C8)—34no—
109C23×C32—21no—
110C23×C32—21no—
111C23×C32—21no—
112C23×C32—41no—
113C23×C32C4×D311yescharacteristic
114C23×C32C2×Dic311yescharacteristic
115C23×C32C2×C3×C411yescharacteristic
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—22no—
123C2×C32×C4—21no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—34no—
129C2×C32×C4—32no—
130C2×C32×C4—32no—
131C2×C32×C4—41no—
132C2×C32×C4—42no—
133C2×C32×C4—42no—
134C2×C32×C4—41no—
135C2×C32×C4—62no—
136C2×C32×C4Dic612yes—
137C2×C32×C4Dic612yes—
138C2×C32×C4C4×D311yescharacteristic
139C2×C32×C4D1212yes—
140C2×C32×C4C2×Dic311yescharacteristic
141C2×C32×C4#6712yes—
142C2×C32×C4C2×C3×C411yescharacteristic
143C2×C32×C4C3×D412yes—
144C2×C32×C4C3×Q812yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C23×C311yescharacteristic
148C2×C3×Dic3—24no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—24no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3C3⋊C814yescomplemented
155C2×C3×Dic3C3×C814yescomplemented
156C2×C3×Dic3C2×Dic312yes—
157C2×C3×Dic3C2×Dic312yes—
158C2×C3×Dic3C2×C3×C412yes—
159C2×C3×Dic3C2×C3×C412yes—
160C32×C8—62no—
161C3⋊(C3⋊C8)—22no—
162C3×C3⋊C8—24no—
163C3×C3⋊C8—22no—
164C3×C3⋊C8—22no—
165C3×C3⋊C8—64no—
166C3×C3⋊C8—62no—
167C3×C3⋊C8—62no—
168C2×C4⋊C8—91noSylow
169C2×C33C4⋊C814yescomplemented
170C2×C33C2×C4⋊C411yescharacteristic
171C2×C33C22×C811yescharacteristic
172C2×C33C2×M1611yescharacteristic
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—31no—
177C22×C3×C4—31no—
178C22×C3×C4—31no—
179C22×C3×C4—31no—
180C22×C3×C4—41no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4D3211yescharacteristic
184C22×C3×C4C2×C3×D311yescharacteristic
185C22×C3×C4C2×C3×D311yescharacteristic
186C22×Dic3C3×Dic311yescharacteristic
187C22×Dic3C3×Dic311yescharacteristic
188C2×C3×C8—32no—
189C2×C3×C8—32no—
190C2×C3×C8—62no—
191C2×C3×C8—64no—
192C2×C3×C8—62no—
193C2×C3×C8—62no—
194C2×C3×C8—92no—
195C2×C3×C8—182no—
196C3×C42—32no—
197C3×C42—32no—
198C3×C42—32no—
199C3×C42—32no—
200C3×C42—62no—
201C3×C42—62no—
202C4×Dic3C3×Dic312yes—
203C4×Dic3C2×C3×D312yes—
204C2×C3⋊C8—22no—
205C2×C3⋊C8—32no—
206C2×C3⋊C8—32no—
207C2×C3⋊C8—62no—
208C2×C3⋊C8—64no—
209C2×C3⋊C8—62no—
210C2×C3⋊C8—62no—
211C2×C3⋊C8C2×C3×D312yes—
212C22×C32—22no—
213C22×C32—22no—
214C22×C32—22no—
215C22×C32—22no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—42no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—41no—
228C22×C32D3×C812yes—
229C22×C32ΓL(1,𝔽25)12yes—
230C22×C32C2×C3⋊C812yes—
231C22×C32#20812yes—
232C22×C32Dic3⋊C412yes—
233C22×C32C4⋊Dic312yes—
234C22×C32C3×C4⋊C412yes—
235C22×C32C2×C3×C812yes—
236C22×C32C3×M1612yes—
237C22×C32C2×C4×D311yescharacteristic
238C22×C32C22×Dic311yescharacteristic
239C22×C32C22×C3×C411yescharacteristic
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—34no—
250C32×C4—42no—
251C32×C4—41no—
252C32×C4—41no—
253C32×C4—62no—
254C32×C4Dic3⋊C412yes—
255C32×C4C4⋊Dic312yes—
256C32×C4C3×C4⋊C412yes—
257C32×C4C2×Dic611yescharacteristic
258C32×C4C2×Dic611yescharacteristic
259C32×C4C2×D1211yescharacteristic
260C32×C4C2×#6711yescharacteristic
261C32×C4C2×C3×D411yescharacteristic
262C32×C4C2×C3×Q811yescharacteristic
263C3×Dic3—24no—
264C3×Dic3—22no—
265C3×Dic3—22no—
266C3×Dic3—24no—
267C3×Dic3C2×C3⋊C814yescomplemented
268C3×Dic3C2×C3×C814yescomplemented
269C22×C8—31no—
270C22×C8—91no—
271C2×C42—31no—
272C4⋊C8—94no—
273C33C2×C4⋊C811yescharacteristic, complemented, Sylow
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—41no—
278C23×C3D3×Dic311yescharacteristic
279C23×C3C3×C4×D311yescharacteristic
280C23×C3C2×C3×Dic311yescharacteristic
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—34no—
294C2×C3×C4—34no—
295C2×C3×C4—32no—
296C2×C3×C4—32no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—41no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—64no—
306C2×C3×C4—64no—
307C2×C3×C4—62no—
308C2×C3×C4—62no—
309C2×C3×C4D3×Dic311yescharacteristic
310C2×C3×C4#62612yes—
311C2×C3×C4#62712yes—
312C2×C3×C4C3×Dic612yes—
313C2×C3×C4C3×Dic612yes—
314C2×C3×C4C3×C4×D311yescharacteristic
315C2×C3×C4C3×D1212yes—
316C2×C3×C4C2×C3×Dic311yescharacteristic
317C2×C3×C4C3×#6712yes—
318C2×C3×C4C2×D3211yescharacteristic
319C2×C3×C4C22×C3×D311yescharacteristic
320C2×C3×C4C22×C3×D311yescharacteristic
321C2×Dic3—22no—
322C2×Dic3C3×C3⋊C814yescomplemented
323C2×Dic3C2×C3×Dic312yes—
324C2×Dic3C2×C3×Dic312yes—
325C3×C8—64no—
326C3×C8—62no—
327C3×C8—62no—
328C3×C8—182no—
329C3⋊C8—22no—
330C3⋊C8—64no—
331C3⋊C8—62no—
332C3⋊C8—62no—
333C2×C32—21no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—24no—
342C2×C32—24no—
343C2×C32—24no—
344C2×C32—21no—
345C2×C32—44no—
346C2×C32—41no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32C4⋊(C3⋊C8)14yescomplemented
350C2×C32Dic3⋊C814yescomplemented
351C2×C32C3×C4⋊C814yescomplemented
352C2×C32C2×D3×C811yescharacteristic
353C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
354C2×C32C22×C3⋊C811yescharacteristic
355C2×C32C2×#20811yescharacteristic
356C2×C32C2×Dic3⋊C411yescharacteristic
357C2×C32C2×C4⋊Dic311yescharacteristic
358C2×C32C2×C3×C4⋊C411yescharacteristic
359C2×C32C22×C3×C811yescharacteristic
360C2×C32C2×C3×M1611yescharacteristic
361C22×C4—31no—
362C22×C4—31no—
363C22×C4C3×D3211yescharacteristic
364C2×C8—32no—
365C2×C8—62no—
366C2×C8—92no—
367C2×C8—182no—
368C42—32no—
369C42—32no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—22no—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3D3×C3⋊C812yes—
387C22×C3#371112yes—
388C22×C3Dic3⋊Dic312yes—
389C22×C3C3×D3×C812yes—
390C22×C3C3×ΓL(1,𝔽25)12yes—
391C22×C3C2×C3×C3⋊C812yes—
392C22×C3C3×#20812yes—
393C22×C3C3×Dic3⋊C412yes—
394C22×C3C3×C4⋊Dic312yes—
395C22×C3C2×D3×Dic311yescharacteristic
396C22×C3C2×C3×C4×D311yescharacteristic
397C22×C3C22×C3×Dic311yescharacteristic
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—34no—
408C3×C4—34no—
409C3×C4—41no—
410C3×C4—41no—
411C3×C4—42no—
412C3×C4—64no—
413C3×C4—64no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4Dic3⋊Dic312yes—
417C3×C4C3×Dic3⋊C412yes—
418C3×C4C3×C4⋊Dic312yes—
419C3×C4C2×#62611yescharacteristic
420C3×C4C2×#62711yescharacteristic
421C3×C4C2×C3×Dic611yescharacteristic
422C3×C4C2×C3×Dic611yescharacteristic
423C3×C4C2×C3×D1211yescharacteristic
424C3×C4C2×C3×#6711yescharacteristic
425Dic3—22no—
426Dic3C2×C3×C3⋊C814yescomplemented
427C32—21no—
428C32—21no—
429C32—21no—
430C32—41no—
431C32C2×C4⋊(C3⋊C8)11yescharacteristic, complemented
432C32C2×Dic3⋊C811yescharacteristic, complemented
433C32C2×C3×C4⋊C811yescharacteristic, complemented
434C23C3×D3×Dic311yescharacteristic
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—32no—
438C2×C4—62no—
439C2×C4C3×D3×Dic311yescharacteristic
440C2×C4C3×#62612yes—
441C2×C4C3×#62712yes—
442C2×C4C2×C3×D3211yescharacteristic
443C8—62no—
444C8—182no—
445C2×C3—24no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—44no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3Dic3⋊(C3⋊C8)14yesdirect product factor
462C2×C3C3×C4⋊(C3⋊C8)14yescomplemented
463C2×C3C3×Dic3⋊C814yescomplemented
464C2×C3C2×D3×C3⋊C811yescharacteristic
465C2×C3C2×#371111yescharacteristic
466C2×C3C2×Dic3⋊Dic311yescharacteristic
467C2×C3C2×C3×D3×C811yescharacteristic
468C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
469C2×C3C22×C3×C3⋊C811yescharacteristic
470C2×C3C2×C3×#20811yescharacteristic
471C2×C3C2×C3×Dic3⋊C411yescharacteristic
472C2×C3C2×C3×C4⋊Dic311yescharacteristic
473C22C3×D3×C3⋊C812yes—
474C22C3×#371112yes—
475C22C3×Dic3⋊Dic312yes—
476C22C2×C3×D3×Dic311yescharacteristic
477C4—34no—
478C4—62no—
479C4C3×Dic3⋊Dic312yes—
480C4C2×C3×#62611yescharacteristic
481C4C2×C3×#62711yescharacteristic
482C3—21no—
483C3—21no—
484C3—21no—
485C3—41no—
486C3C2×Dic3⋊(C3⋊C8)11yescharacteristic, direct product factor
487C3C2×C3×C4⋊(C3⋊C8)11yescharacteristic, complemented
488C3C2×C3×Dic3⋊C811yescharacteristic, complemented
489C2C3×Dic3⋊(C3⋊C8)14yesdirect product factor
490C2C2×C3×D3×C3⋊C811yescharacteristic
491C2C2×C3×#371111yescharacteristic
492C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutatorC2×C32C22×C3×C8
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypocenterC32C2×C3×C4⋊C8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×Dic3⋊(C3⋊C8)C2direct (N × H)
7. C2×Dic3⋊(C3⋊C8)C3direct (N × H)
31. Dic3⋊(C3⋊C8)C2×C3direct (N × H)
38. C2×C32×Dic3C8semidirect
93. C32×Dic3C2×C8semidirect
154. C2×C3×Dic3C3⋊C8semidirect
155. C2×C3×Dic3C3×C8semidirect
169. C2×C33C4⋊C8semidirect
267. C3×Dic3C2×C3⋊C8semidirect
268. C3×Dic3C2×C3×C8semidirect
273. C33C2×C4⋊C8semidirect
322. C2×Dic3C3×C3⋊C8semidirect
349. C2×C32C4⋊(C3⋊C8)semidirect
350. C2×C32Dic3⋊C8semidirect
351. C2×C32C3×C4⋊C8semidirect
426. Dic3C2×C3×C3⋊C8semidirect
431. C32C2×C4⋊(C3⋊C8)semidirect
432. C32C2×Dic3⋊C8semidirect
433. C32C2×C3×C4⋊C8semidirect
461. C2×C3Dic3⋊(C3⋊C8)direct (N × H)
462. C2×C3C3×C4⋊(C3⋊C8)semidirect
463. C2×C3C3×Dic3⋊C8semidirect
486. C3C2×Dic3⋊(C3⋊C8)direct (N × H)
487. C3C2×C3×C4⋊(C3⋊C8)semidirect
488. C3C2×C3×Dic3⋊C8semidirect
489. C2C3×Dic3⋊(C3⋊C8)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊(C3⋊C8)2
C2×C32×C4×Dic32
C22×C32×C3⋊C82
C3×Dic3⋊(C3⋊C8)2
C2×C3×Dic3⋊C83
C2×C3×C4⋊(C3⋊C8)3
C2×Dic3⋊(C3⋊C8)3