Finite Group Databasebeta

← C3×#10590324 #13157339

Subgroups — C3×#10590324 #13157339

Order 1728.

638 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#872C211yesmaximal, characteristic
2C2×C3×#3711C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#4999—31nomaximal
9C3×#4839—31nomaximal
10#10590324C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×(C2×D3)⋊C4C411yescharacteristic
16C32×(C2×D3)⋊C4C411yescharacteristic
17C32×Dic3⋊C4C411yescharacteristic
18C32×Dic3⋊C4C411yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3711—22no—
23C3×D3×C3⋊C8—22no—
24C32×#117—31no—
25C2×C3×#208—31no—
26C22×C3×C3⋊C8—31no—
27C2×C3×ΓL(1,𝔽25)—31no—
28C2×C3×D3×C8—31no—
29C3×#872—21no—
30C3×#872D311yescharacteristic
31C3×#872C2×C311yescharacteristic
32C2×#3711C2×C311yescharacteristic
33C2×D3×C3⋊C8C2×C311yescharacteristic
34C32×C4⋊C8—31no—
35C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—31no—
37C3×Dic3⋊C8—31no—
38C3×C4⋊(C3⋊C8)—21no—
39C3×C4⋊(C3⋊C8)—31no—
40C3×C4⋊(C3⋊C8)—31no—
41C3×C4⋊(C3⋊C8)—31no—
42C3×C4⋊(C3⋊C8)—31no—
43C3×C4⋊(C3⋊C8)—61no—
44C3×C4⋊(C3⋊C8)D311yescharacteristic, complemented
45C3×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
46Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C22×C32×D3C2×C411yescharacteristic
49C2×C33×C4C2×C411yescharacteristic
50C2×C33×C4C2×C411yescharacteristic
51C2×C33×C4C2311yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C32×C4×D3D412yes—
56C32×C4×D3Q812yes—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8—21no—
61C3×#424—91no—
62#4999—31no—
63#4839—31no—
64C22×C32×C4—31no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×C4×D3C22×C311yescharacteristic
68C2×C32×C8—31no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4—31no—
72C32×C22⋊C4—31no—
73C32×C22⋊C4—31no—
74C32×C42—21no—
75C32×C42—21no—
76C32×C42—21no—
77C32×C42—31no—
78C32×C42—41no—
79C32×C42C2×D311yescharacteristic
80C32×C42C2×D311yescharacteristic
81C32×C42C22×C311yescharacteristic
82C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
83C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
84C3×(C2×D3)⋊C4—21no—
85C3×(C2×D3)⋊C4—21no—
86C3×(C2×D3)⋊C4Dic311yescharacteristic
87C3×(C2×D3)⋊C4Dic311yescharacteristic
88C3×(C2×D3)⋊C4C3×C411yescharacteristic
89C3×(C2×D3)⋊C4C3×C411yescharacteristic
90C3×Dic3⋊C4—21no—
91C3×Dic3⋊C4—21no—
92C3×Dic3⋊C4Dic311yescharacteristic
93C3×Dic3⋊C4Dic311yescharacteristic
94C3×Dic3⋊C4C3×C411yescharacteristic
95C3×Dic3⋊C4C3×C411yescharacteristic
96C3×C4×Dic3—21no—
97C3×C4×Dic3C2×D311yescharacteristic
98C3×C4×Dic3C22×C311yescharacteristic
99C3×#208—62no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—62no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C3×ΓL(1,𝔽25)—62no—
114C3×D3×C8—62no—
115#3711—22no—
116D3×C3⋊C8—22no—
117C22×C33C22×C411yescharacteristic
118C2×C32×D3—21no—
119C2×C32×D3—21no—
120C2×C32×D3C4⋊C412yes—
121C33×C4—21no—
122C33×C4—21no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2×Q811yescharacteristic
125C32×Dic3—21no—
126C32×Dic3—21no—
127C32×Dic3C4⋊C412yes—
128C2×C3×M16—91no—
129C22×C3×C8—91no—
130C3×#117—31no—
131C3×#117—31no—
132C3×#117—61no—
133C2×#208—31no—
134C22×C3⋊C8—31no—
135C2×ΓL(1,𝔽25)—31no—
136C2×D3×C8—31no—
137#872C3×D311yescharacteristic
138C3×C4⋊C8—31no—
139C3×C4⋊C8—31no—
140C3×C4⋊C8—61no—
141C3×C4⋊C8—91no—
142C3×C4⋊C8—91no—
143C3×C4⋊C8—91no—
144Dic3⋊C8—31no—
145Dic3⋊C8—31no—
146C4⋊(C3⋊C8)—31no—
147C4⋊(C3⋊C8)—31no—
148C4⋊(C3⋊C8)—31no—
149C4⋊(C3⋊C8)—31no—
150C4⋊(C3⋊C8)—61no—
151C4⋊(C3⋊C8)C3×D311yescharacteristic, complemented
152C23×C32—31no—
153C22×C3×D3—21no—
154C22×C3×D3C2×Dic311yescharacteristic
155C22×C3×D3C2×C3×C411yescharacteristic
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—31no—
166C2×C32×C4—32no—
167C2×C32×C4—32no—
168C2×C32×C4—31no—
169C2×C32×C4—31no—
170C2×C32×C4—41no—
171C2×C32×C4—41no—
172C2×C32×C4—41no—
173C2×C32×C4C4×D311yescharacteristic
174C2×C32×C4C4×D311yescharacteristic
175C2×C32×C4C2×Dic311yescharacteristic
176C2×C32×C4C2×Dic311yescharacteristic
177C2×C32×C4C2×C3×C411yescharacteristic
178C2×C32×C4C2×C3×C411yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3C2×Dic311yescharacteristic
186C2×C3×Dic3C2×Dic311yescharacteristic
187C2×C3×Dic3C2×Dic311yescharacteristic
188C2×C3×Dic3C2×C3×C411yescharacteristic
189C2×C3×Dic3C2×C3×C411yescharacteristic
190C2×C3×Dic3C2×C3×C411yescharacteristic
191C3×C4×D3—22no—
192C3×C4×D3—22no—
193C3×C4×D3Dic612yes—
194C3×C4×D3D1212yes—
195C3×C4×D3C3×D412yes—
196C3×C4×D3C3×Q812yes—
197C32×C8—61no—
198C32×C8—61no—
199C3⋊(C3⋊C8)—21no—
200C3⋊(C3⋊C8)—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—21no—
205C3×C3⋊C8—41no—
206C3×C3⋊C8—41no—
207C3×C3⋊C8—61no—
208C3×C3⋊C8—61no—
209C3×C3⋊C8—61no—
210C3×C3⋊C8—61no—
211C3×C3⋊C8—121no—
212C3×C3⋊C8—121no—
213#424—91noSylow
214C2×C33C2×C4⋊C411yescharacteristic
215C2×C33#13111yescharacteristic
216C2×C33#13111yescharacteristic
217C32×D3—22no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C2×C4×D3C2×C3×D311yescharacteristic
222C3×M16—182no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C2×C3×C8—91no—
231C2×C3×C8—182no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4—61no—
238C3×C22⋊C4—31no—
239C3×C22⋊C4—31no—
240C3×C22⋊C4—31no—
241C3×C22⋊C4—31no—
242C3×C22⋊C4—61no—
243C3×C22⋊C4—61no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—31no—
248C3×C42—31no—
249C3×C42—41no—
250C3×C42—61no—
251C3×C42D3211yescharacteristic
252C3×C42C2×C3×D311yescharacteristic
253C3×C42C2×C3×D311yescharacteristic
254(C2×D3)⋊C4C3×Dic311yescharacteristic
255(C2×D3)⋊C4C3×Dic311yescharacteristic
256Dic3⋊C4C3×Dic311yescharacteristic
257Dic3⋊C4C3×Dic311yescharacteristic
258C4×Dic3C2×C3×D311yescharacteristic
259#208—62no—
260C2×C3⋊C8—31no—
261C2×C3⋊C8—31no—
262C2×C3⋊C8—31no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8—61no—
266C2×C3⋊C8—62no—
267C2×C3⋊C8C2×C3×D311yescharacteristic
268C2×C3⋊C8C2×C3×D311yescharacteristic
269ΓL(1,𝔽25)—62no—
270D3×C8—62no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—21no—
274C22×C32—32no—
275C22×C32—41no—
276C22×C32—61no—
277C22×C32—61no—
278C22×C32C2×C4×D311yescharacteristic
279C22×C32C22×Dic311yescharacteristic
280C22×C32C22×C3×C411yescharacteristic
281C2×C3×D3—21no—
282C2×C3×D3—21no—
283C2×C3×D3—21no—
284C2×C3×D3—21no—
285C2×C3×D3—22no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3C4⋊Dic312yes—
289C2×C3×D3C3×C4⋊C412yes—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—21no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—41no—
310C32×C4—41no—
311C32×C4—42no—
312C32×C4—42no—
313C32×C4—61no—
314C32×C4—61no—
315C32×C4C2×Dic611yescharacteristic
316C32×C4C2×D1211yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×Q811yescharacteristic
319C32×C4C2×C3×D411yescharacteristic
320C32×C4C2×C3×Q811yescharacteristic
321C3×Dic3—21no—
322C3×Dic3—21no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—22no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C3×Dic3C4⋊Dic312yes—
329C3×Dic3C3×C4⋊C412yes—
330C2×M16—91no—
331C22×C8—91no—
332#117—31no—
333C4⋊C8—31no—
334C4⋊C8—91no—
335C4⋊C8—91no—
336C4⋊C8—91no—
337C33#42411yescharacteristic, complemented, Sylow
338C23×C3—31no—
339C23×C3—31no—
340C23×C3—61no—
341C22×D3C2×C3×Dic311yescharacteristic
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4D3×Dic311yescharacteristic
370C2×C3×C4D3×Dic311yescharacteristic
371C2×C3×C4C3×C4×D311yescharacteristic
372C2×C3×C4C3×C4×D311yescharacteristic
373C2×C3×C4C2×C3×Dic311yescharacteristic
374C2×C3×C4C2×C3×Dic311yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×C3×C4C22×C3×D311yescharacteristic
378C2×Dic3C2×C3×Dic311yescharacteristic
379C2×Dic3C2×C3×Dic311yescharacteristic
380C2×Dic3C2×C3×Dic311yescharacteristic
381C4×D3C3×Dic612yes—
382C4×D3C3×D1212yes—
383C3×C8—61no—
384C3×C8—61no—
385C3×C8—61no—
386C3×C8—61no—
387C3×C8—121no—
388C3×C8—121no—
389C3×C8—181no—
390C3×C8—181no—
391C3⋊C8—21no—
392C3⋊C8—21no—
393C3⋊C8—61no—
394C3⋊C8—61no—
395C3⋊C8—61no—
396C3⋊C8—61no—
397C3⋊C8—121no—
398C3⋊C8—121no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32—41no—
411C2×C32—62no—
412C2×C32D3×C4⋊C411yescharacteristic
413C2×C32#90111yescharacteristic
414C2×C32#90711yescharacteristic
415C2×C32C2×C4⋊Dic311yescharacteristic
416C2×C32#94811yescharacteristic
417C2×C32#94811yescharacteristic
418C2×C32C2×C3×C4⋊C411yescharacteristic
419C2×C32C3×#13111yescharacteristic
420C2×C32C3×#13111yescharacteristic
421C3×D3—22no—
422C3×D3—22no—
423C3×D3—42no—
424C22×C4—31no—
425M16—182no—
426C2×C8—31no—
427C2×C8—31no—
428C2×C8—91no—
429C2×C8—91no—
430C2×C8—182no—
431C4⋊C4—31no—
432C4⋊C4—31no—
433C22⋊C4—31no—
434C22⋊C4—31no—
435C42—31no—
436C42C3×D3211yescharacteristic
437C22×C3—21no—
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—32no—
441C22×C3—32no—
442C22×C3—41no—
443C22×C3—62no—
444C22×C3—61no—
445C22×C3—61no—
446C22×C3—61no—
447C22×C3—61no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3C2×D3×Dic311yescharacteristic
451C22×C3C2×C3×C4×D311yescharacteristic
452C22×C3C22×C3×Dic311yescharacteristic
453C2×D3—21no—
454C2×D3—21no—
455C2×D3C3×C4⋊Dic312yes—
456C3×C4—22no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—32no—
471C3×C4—32no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—42no—
477C3×C4—42no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—61no—
482C3×C4—61no—
483C3×C4—61no—
484C3×C4—61no—
485C3×C4—121no—
486C3×C4—121no—
487C3×C4D3×Dic611yescharacteristic
488C3×C4D3×D1211yescharacteristic
489C3×C4C2×C3×Dic611yescharacteristic
490C3×C4C2×C3×D1211yescharacteristic
491C3×C4C3×D3×D411yescharacteristic
492C3×C4C3×D3×Q811yescharacteristic
493Dic3—21no—
494Dic3—21no—
495Dic3C3×C4⋊Dic312yes—
496C32—21no—
497C32—21no—
498C32—21no—
499C32—41no—
500C32#483911yescharacteristic, complemented
501C32#499911yescharacteristic, complemented
502C32C3×#42411yescharacteristic, complemented
503C23—31no—
504C2×C4—31no—
505C2×C4—32no—
506C2×C4—32no—
507C2×C4—31no—
508C2×C4—31no—
509C2×C4C3×D3×Dic311yescharacteristic
510C2×C4C3×D3×Dic311yescharacteristic
511C2×C4C2×C3×D3211yescharacteristic
512C8—61no—
513C8—61no—
514C8—181no—
515C8—181no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—41no—
528C2×C3—62no—
529C2×C3—62no—
530C2×C3—122no—
531C2×C3#6381411yescharacteristic
532C2×C3#6381511yescharacteristic
533C2×C3D3×C4⋊Dic311yescharacteristic
534C2×C3C3×D3×C4⋊C411yescharacteristic
535C2×C3C3×#90111yescharacteristic
536C2×C3C3×#90711yescharacteristic
537C2×C3C2×C3×C4⋊Dic311yescharacteristic
538C2×C3C3×#94811yescharacteristic
539C2×C3C3×#94811yescharacteristic
540D3—22no—
541C22—32no—
542C22—61no—
543C22—61no—
544C22C2×C3×D3×Dic311yescharacteristic
545C4—21no—
546C4—21no—
547C4—32no—
548C4—61no—
549C4—61no—
550C4C3×D3×Dic611yescharacteristic
551C4C3×D3×D1211yescharacteristic
552C3—21no—
553C3—21no—
554C3—21no—
555C3—41no—
556C3#1059032411yescharacteristic, direct product factor
557C3C3×#483911yescharacteristic, complemented
558C3C3×#499911yescharacteristic, complemented
559C2—62no—
560C2C3×#6381411yescharacteristic
561C2C3×#6381511yescharacteristic
562C2C3×D3×C4⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
4. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590324C3direct (N × H)
35. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
44. C3×C4⋊(C3⋊C8)D3semidirect
45. C3×C4⋊(C3⋊C8)C2×C3semidirect
46. Dic3⋊(C3⋊C8)C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
151. C4⋊(C3⋊C8)C3×D3semidirect
337. C33#424semidirect
500. C32#4839semidirect
501. C32#4999semidirect
502. C32C3×#424semidirect
556. C3#10590324direct (N × H)
557. C3C3×#4839semidirect
558. C3C3×#4999semidirect

Maximal subgroups

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

subgroupindex
C32×#8722
C2×C3×#37112
C2×C3×D3×C3⋊C82
C3×C4⋊(C3⋊(C3⋊C8))2
C32×C4⋊(C3⋊C8)2
C3×Dic3⋊(C3⋊C8)2
C3×#49993
C3×#48393
#105903243