Finite Group Databasebeta

← C3×#10590695 #13157695

Subgroups — C3×#10590695 #13157695

Order 1728.

622 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic
2C3×#63855C211yesmaximal, characteristic
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic
4C3×#63641C211yesmaximal, characteristic, complemented
5C32×Dic3⋊C8C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C3×#5119—31nomaximal
9C3×#4839—31nomaximal
10#10590695C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C32×C4×Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C32×ΓL(1,𝔽25)—22no—
19C3×Dic3⋊Dic3C411yescharacteristic
20C3×Dic3⋊Dic3C411yescharacteristic
21C3×(C2×D3)⋊Dic3C411yescharacteristic
22C3×(C2×D3)⋊Dic3C411yescharacteristic
23C3×D3×C3⋊C8—22no—
24C2×C32×M16—31no—
25C3×#926—31no—
26C22×C3×C3⋊C8—31no—
27C2×C3×ΓL(1,𝔽25)—21no—
28C2×C3×ΓL(1,𝔽25)D311yescharacteristic
29C2×C3×ΓL(1,𝔽25)C2×C311yescharacteristic
30C2×C3×D3×C8—31no—
31C3×#872—31no—
32#63855C2×C311yescharacteristic
33C2×D3×C3⋊C8C2×C311yescharacteristic
34C32×C4⋊C8—31no—
35#63641C2×C311yescharacteristic, complemented
36C3×Dic3⋊C8—21no—
37C3×Dic3⋊C8—31no—
38C3×Dic3⋊C8—31no—
39C3×Dic3⋊C8—31no—
40C3×Dic3⋊C8—31no—
41C3×Dic3⋊C8—31no—
42C3×Dic3⋊C8—61no—
43C3×Dic3⋊C8D311yescharacteristic, complemented
44C3×Dic3⋊C8C2×C311yescharacteristic, complemented
45C3×C4⋊(C3⋊C8)—31no—
46#63572C2×C311yescharacteristic, complemented
47Dic3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
48C22×C32×D3C2×C411yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C32×C4×D3D412yes—
56C32×C4×D3Q812yes—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60C32×C3⋊C8—21no—
61C3×#424—91no—
62#5119—31no—
63#4839—31no—
64C22×C32×C4—31no—
65C2×C3×C4×D3—21no—
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×C4×D3C22×C311yescharacteristic
68C32×M16—62no—
69C2×C32×C8—21no—
70C2×C32×C8—21no—
71C2×C32×C8—21no—
72C2×C32×C8—31no—
73C2×C32×C8—31no—
74C2×C32×C8—41no—
75C2×C32×C8C2×D311yescharacteristic
76C2×C32×C8C2×D311yescharacteristic
77C2×C32×C8C22×C311yescharacteristic
78C32×C42—31no—
79C4×C3⋊Dic3C22×C311yescharacteristic
80C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
81C3×#217—31no—
82C3×#217—31no—
83C3×(C2×D3)⋊C4—31no—
84C3×(C2×D3)⋊C4—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×Dic3⋊C4—31no—
88C3×Dic3⋊C4—31no—
89C3×C4×Dic3—21no—
90C3×C4×Dic3—31no—
91C3×C4×Dic3—31no—
92C3×C4×Dic3—61no—
93C3×C4×Dic3C2×D311yescharacteristic
94C3×C4×Dic3C22×C311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—62no—
100C2×C3×C3⋊C8—61no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C22×C311yescharacteristic
104C2×C3×C3⋊C8C22×C311yescharacteristic
105C3×ΓL(1,𝔽25)—24no—
106C3×ΓL(1,𝔽25)—22no—
107C3×ΓL(1,𝔽25)—22no—
108C3×D3×C8—62no—
109Dic3⋊Dic3C3×C411yescharacteristic
110Dic3⋊Dic3C3×C411yescharacteristic
111(C2×D3)⋊Dic3C3×C411yescharacteristic
112(C2×D3)⋊Dic3C3×C411yescharacteristic
113D3×C3⋊C8—22no—
114C22×C33C22×C411yescharacteristic
115C2×C32×D3—21no—
116C2×C32×D3—21no—
117C2×C32×D3C4⋊C412yes—
118C33×C4C2×D411yescharacteristic
119C33×C4C2×Q811yescharacteristic
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3—21no—
122C32×Dic3—21no—
123C32×Dic3—21no—
124C32×Dic3C4⋊C412yes—
125C2×C3×M16—31no—
126C2×C3×M16—31no—
127C2×C3×M16—61no—
128C22×C3×C8—91no—
129C3×#117—91no—
130#926—31no—
131C22×C3⋊C8—31no—
132C2×ΓL(1,𝔽25)C3×D311yescharacteristic
133C2×D3×C8—31no—
134#872—31no—
135C3×C4⋊C8—31no—
136C3×C4⋊C8—31no—
137C3×C4⋊C8—61no—
138C3×C4⋊C8—91no—
139C3×C4⋊C8—91no—
140C3×C4⋊C8—91no—
141Dic3⋊C8—31no—
142Dic3⋊C8—31no—
143Dic3⋊C8—31no—
144Dic3⋊C8—31no—
145Dic3⋊C8—31no—
146Dic3⋊C8—61no—
147Dic3⋊C8C3×D311yescharacteristic, complemented
148C4⋊(C3⋊C8)—31no—
149C23×C32—31no—
150C22×C3×D3—21no—
151C22×C3×D3C4×D311yescharacteristic
152C22×C3×D3C2×C3×C411yescharacteristic
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—31no—
157C2×C32×C4—32no—
158C2×C32×C4—32no—
159C2×C32×C4—31no—
160C2×C32×C4—31no—
161C2×C32×C4—41no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C23×C311yescharacteristic
165C2×C3⋊Dic3C2×C3×C411yescharacteristic
166C2×C3⋊Dic3C2×C3×C411yescharacteristic
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3C4×D311yescharacteristic
177C2×C3×Dic3C4×D311yescharacteristic
178C2×C3×Dic3C4×D311yescharacteristic
179C2×C3×Dic3C2×C3×C411yescharacteristic
180C2×C3×Dic3C2×C3×C411yescharacteristic
181C2×C3×Dic3C2×C3×C411yescharacteristic
182C3×C4×D3—22no—
183C3×C4×D3—22no—
184C3×C4×D3Dic612yes—
185C3×C4×D3#6712yes—
186C3×C4×D3C3×D412yes—
187C3×C4×D3C3×Q812yes—
188C32×C8—21no—
189C32×C8—22no—
190C32×C8—21no—
191C32×C8—21no—
192C32×C8—41no—
193C32×C8—41no—
194C32×C8—42no—
195C32×C8—61no—
196C32×C8—61no—
197C3⋊(C3⋊C8)—21no—
198C3×C3⋊C8—22no—
199C3×C3⋊C8—21no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—121no—
207#424—91noSylow
208C2×C33C2×C4⋊C411yescharacteristic
209C2×C33#13111yescharacteristic
210C2×C33#13111yescharacteristic
211C32×D3—22no—
212C22×C3×C4—31no—
213C22×C3×C4—31no—
214C22×C3×C4—61no—
215C2×C4×D3C2×C3×D311yescharacteristic
216C3×M16—62no—
217C3×M16—62no—
218C3×M16—64no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—41no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8—91no—
230C2×C3×C8—182no—
231C2×C3×C8D3211yescharacteristic
232C2×C3×C8C2×C3×D311yescharacteristic
233C2×C3×C8C2×C3×D311yescharacteristic
234C3×C4⋊C4—91no—
235C3×C4⋊C4—91no—
236C3×C22⋊C4—91no—
237C3×C22⋊C4—91no—
238C3×C42—31no—
239C3×C42—31no—
240C3×C42—61no—
241C3×C42—91no—
242#217—31no—
243#217—31no—
244(C2×D3)⋊C4—31no—
245(C2×D3)⋊C4—31no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248Dic3⋊C4—31no—
249Dic3⋊C4—31no—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—61no—
253C4×Dic3C2×C3×D311yescharacteristic
254C2×C3⋊C8—31no—
255C2×C3⋊C8—31no—
256C2×C3⋊C8—61no—
257C2×C3⋊C8—62no—
258C2×C3⋊C8C2×C3×D311yescharacteristic
259C2×C3⋊C8C2×C3×D311yescharacteristic
260ΓL(1,𝔽25)—22no—
261D3×C8—62no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—32no—
266C22×C32—41no—
267C22×C32—61no—
268C22×C32—61no—
269C22×C32C2×C4×D311yescharacteristic
270C22×C32C2×C4×D311yescharacteristic
271C22×C32C22×C3×C411yescharacteristic
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3—21no—
275C2×C3×D3—21no—
276C2×C3×D3—22no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3Dic3⋊C412yes—
280C2×C3×D3C3×C4⋊C412yes—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—32no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—61no—
291C32×C4—61no—
292C32×C4C2×Dic611yescharacteristic
293C32×C4D3×D411yescharacteristic
294C32×C4D3×Q811yescharacteristic
295C32×C4C2×#6711yescharacteristic
296C32×C4C2×C3×D411yescharacteristic
297C32×C4C2×C3×Q811yescharacteristic
298C3⋊Dic3—21no—
299C3⋊Dic3—21no—
300C3×Dic3—22no—
301C3×Dic3—21no—
302C3×Dic3—21no—
303C3×Dic3—21no—
304C3×Dic3—21no—
305C3×Dic3—22no—
306C3×Dic3—41no—
307C3×Dic3—62no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—121no—
313C3×Dic3Dic3⋊C412yes—
314C3×Dic3C3×C4⋊C412yes—
315C2×M16—31no—
316C22×C8—91no—
317#117—91no—
318C4⋊C8—31no—
319C4⋊C8—91no—
320C4⋊C8—91no—
321C4⋊C8—91no—
322C33#42411yescharacteristic, complemented, Sylow
323C23×C3—31no—
324C23×C3—31no—
325C23×C3—61no—
326C22×D3C3×C4×D311yescharacteristic
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—32no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—31no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—62no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—61no—
346C2×C3×C4—91no—
347C2×C3×C4—91no—
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C22×C3×D311yescharacteristic
350C2×C3×C4C22×C3×D311yescharacteristic
351C2×Dic3—31no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3C3×C4×D311yescharacteristic
358C2×Dic3C3×C4×D311yescharacteristic
359C2×Dic3C3×C4×D311yescharacteristic
360C4×D3C3×Dic612yes—
361C4×D3C3×#6712yes—
362C3×C8—22no—
363C3×C8—21no—
364C3×C8—21no—
365C3×C8—21no—
366C3×C8—41no—
367C3×C8—41no—
368C3×C8—42no—
369C3×C8—62no—
370C3×C8—61no—
371C3×C8—61no—
372C3×C8—61no—
373C3×C8—61no—
374C3×C8—121no—
375C3×C8—181no—
376C3⋊C8—21no—
377C3⋊C8—21no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—62no—
394C2×C32D3×C4⋊C411yescharacteristic
395C2×C32#90111yescharacteristic
396C2×C32#90711yescharacteristic
397C2×C32#90711yescharacteristic
398C2×C32#90711yescharacteristic
399C2×C32C2×Dic3⋊C411yescharacteristic
400C2×C32C2×C3×C4⋊C411yescharacteristic
401C2×C32C3×#13111yescharacteristic
402C2×C32C3×#13111yescharacteristic
403C3×D3—22no—
404C3×D3—22no—
405C3×D3—42no—
406C22×C4—31no—
407M16—62no—
408C2×C8—31no—
409C2×C8—31no—
410C2×C8—91no—
411C2×C8—182no—
412C2×C8C3×D3211yescharacteristic
413C4⋊C4—91no—
414C4⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—91no—
417C42—31no—
418C42—91no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—32no—
423C22×C3—32no—
424C22×C3—41no—
425C22×C3—62no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3C4×D3211yescharacteristic
433C22×C3C2×C3×C4×D311yescharacteristic
434C22×C3C2×C3×C4×D311yescharacteristic
435C2×D3—21no—
436C2×D3—21no—
437C2×D3C3×Dic3⋊C412yes—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—32no—
445C3×C4—32no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—61no—
453C3×C4—61no—
454C3×C4—121no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4D3×Dic611yescharacteristic
458C3×C4D3×#6711yescharacteristic
459C3×C4C2×C3×Dic611yescharacteristic
460C3×C4C3×D3×D411yescharacteristic
461C3×C4C3×D3×Q811yescharacteristic
462C3×C4C2×C3×#6711yescharacteristic
463Dic3—21no—
464Dic3—21no—
465Dic3—62no—
466Dic3—61no—
467Dic3—61no—
468Dic3—61no—
469Dic3—61no—
470Dic3—121no—
471Dic3C3×Dic3⋊C412yes—
472C32—21no—
473C32—21no—
474C32—21no—
475C32—41no—
476C32#483911yescharacteristic, complemented
477C32#511911yescharacteristic, complemented
478C32C3×#42411yescharacteristic, complemented
479C23—31no—
480C2×C4—31no—
481C2×C4—32no—
482C2×C4—32no—
483C2×C4—31no—
484C2×C4—31no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4C2×C3×D3211yescharacteristic
488C8—21no—
489C8—61no—
490C8—61no—
491C8—181no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—41no—
502C2×C3—41no—
503C2×C3—41no—
504C2×C3—62no—
505C2×C3—62no—
506C2×C3—122no—
507C2×C3#6380411yescharacteristic
508C2×C3#6380511yescharacteristic
509C2×C3D3×Dic3⋊C411yescharacteristic
510C2×C3C3×D3×C4⋊C411yescharacteristic
511C2×C3C3×#90111yescharacteristic
512C2×C3C3×#90711yescharacteristic
513C2×C3C3×#90711yescharacteristic
514C2×C3C3×#90711yescharacteristic
515C2×C3C2×C3×Dic3⋊C411yescharacteristic
516D3—22no—
517C22—32no—
518C22—61no—
519C22—61no—
520C22C3×C4×D3211yescharacteristic
521C4—32no—
522C4—61no—
523C4—61no—
524C4—181no—
525C4—181no—
526C4C3×D3×Dic611yescharacteristic
527C4C3×D3×#6711yescharacteristic
528C3—21no—
529C3—21no—
530C3—21no—
531C3—41no—
532C3#1059069511yescharacteristic, direct product factor
533C3C3×#483911yescharacteristic, complemented
534C3C3×#511911yescharacteristic, complemented
535C2—62no—
536C2C3×#6380411yescharacteristic
537C2C3×#6380511yescharacteristic
538C2C3×D3×Dic3⋊C411yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC22×C32×D3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63641C2semidirect
5. C32×Dic3⋊C8C2semidirect
6. C3×#63572C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
10. #10590695C3direct (N × H)
35. #63641C2×C3semidirect
43. C3×Dic3⋊C8D3semidirect
44. C3×Dic3⋊C8C2×C3semidirect
46. #63572C2×C3semidirect
47. Dic3⋊(C3⋊C8)C2×C3semidirect
147. Dic3⋊C8C3×D3semidirect
322. C33#424semidirect
476. C32#4839semidirect
477. C32#5119semidirect
478. C32C3×#424semidirect
532. C3#10590695direct (N × H)
533. C3C3×#4839semidirect
534. C3C3×#5119semidirect

Maximal subgroups

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

subgroupindex
C2×C32×ΓL(1,𝔽25)2
C3×#638552
C2×C3×D3×C3⋊C82
C3×#636412
C32×Dic3⋊C82
C3×#635722
C3×Dic3⋊(C3⋊C8)2
C3×#51193
C3×#48393
#105906953