Finite Group Databasebeta

← C3×#10590725 #13157726

Subgroups — C3×#10590725 #13157726

Order 1728.

658 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#625C211yesmaximal, characteristic
2C2×C3×#3711C211yesmaximal, characteristic, complemented
3C2×C3×D3×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#63642C211yesmaximal, characteristic, complemented
5C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7C3×#63572C211yesmaximal, characteristic, complemented
8C3×#4730—31nomaximal
9C3×#4728—31nomaximal
10#10590725C311yesmaximal, characteristic, direct product factor
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×C3×#625C411yescharacteristic
14C2×C33×C8C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19C3×#3724C411yescharacteristic
20C3×(C2×D3)⋊Dic3C411yescharacteristic
21C3×(C2×D3)⋊Dic3C411yescharacteristic
22C3×#3711—22no—
23C3×D3×C3⋊C8—22no—
24C3×C4×#67—31no—
25C3×C4×#67—31no—
26C2×C3×#208—31no—
27C22×C3×C3⋊C8—31no—
28C2×C3×ΓL(1,𝔽25)—31no—
29C2×C3×D3×C8—31no—
30C4×#625C2×C311yescharacteristic
31C2×#3711C2×C311yescharacteristic, complemented
32C2×D3×C3⋊C8C2×C311yescharacteristic, complemented
33C32×C22⋊C8—31no—
34C32×C22⋊C8—31no—
35#63642C2×C311yescharacteristic, complemented
36C3×(C2×D3)⋊C8—21no—
37C3×(C2×D3)⋊C8—21no—
38C3×(C2×D3)⋊C8D311yescharacteristic, complemented
39C3×(C2×D3)⋊C8D311yescharacteristic, complemented
40C3×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
41C3×(C2×D3)⋊C8C2×C311yescharacteristic, complemented
42C3×#815—31no—
43C3×#815—31no—
44C3×#815—61no—
45C3×Dic3⋊C8—31no—
46C3×Dic3⋊C8—31no—
47#63572C2×C311yescharacteristic, complemented
48C22×C32×D3C2×C411yescharacteristic
49C22×C32×D3C2×C411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C2×C32×Dic3C2×C411yescharacteristic
54C2×C32×Dic3C2×C411yescharacteristic
55C32×C4×D3—21no—
56C32×C4×D3—21no—
57C32×C4×D3—21no—
58C32×C4×D3—21no—
59C3×#625—22no—
60C33×C8—21no—
61C3×C3⋊(C3⋊C8)D412yescomplemented
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64C3×#435—91no—
65#4730—31no—
66#4728—31no—
67C22×C32×C4—31no—
68C22×C32×C4—31no—
69C2×C3×#67—31no—
70C2×C3×#67—31no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—21no—
73C2×C3×C4×D3C2×D311yescharacteristic
74C2×C3×C4×D3C2×D311yescharacteristic
75C2×C3×C4×D3C22×C311yescharacteristic
76C2×C3×C4×D3C22×C311yescharacteristic
77C2×#625C3×C411yescharacteristic
78C2×C32×C8—21no—
79C2×C32×C8—21no—
80C2×C32×C8—21no—
81C2×C32×C8—31no—
82C2×C32×C8—31no—
83C2×C32×C8—41no—
84C2×C32×C8C2×D311yescharacteristic
85C2×C32×C8C2×D311yescharacteristic
86C2×C32×C8C22×C311yescharacteristic
87C4×C3⋊Dic3C22×C311yescharacteristic
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89C3×#217—31no—
90C3×#217—31no—
91C3×(C2×D3)⋊C4—31no—
92C3×(C2×D3)⋊C4—31no—
93C3×Dic3⋊C4—31no—
94C3×Dic3⋊C4—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—61no—
98C3×#208—62no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—62no—
104C2×C3×C3⋊C8—61no—
105C2×C3×C3⋊C8C2×D311yescharacteristic
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C2×C3×C3⋊C8C22×C311yescharacteristic
109C3×ΓL(1,𝔽25)—62no—
110C3×D3×C8—62no—
111#3724C3×C411yescharacteristic
112(C2×D3)⋊Dic3C3×C411yescharacteristic
113(C2×D3)⋊Dic3C3×C411yescharacteristic
114#3711—22no—
115D3×C3⋊C8—22no—
116C22×C33C22×C411yescharacteristic
117C2×C32×D3—21no—
118C2×C32×D3—21no—
119C2×C32×D3—21no—
120C2×C32×D3—21no—
121C2×C32×D3—21no—
122C2×C32×D3M1612yescomplemented
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3×C3⋊Dic3—21no—
126C3×C3⋊Dic3—21no—
127C32×Dic3—21no—
128C32×Dic3—21no—
129C2×C3×M16—91no—
130C22×C3×C8—91no—
131C3×C4×D4—91no—
132C4×#67—31no—
133C4×#67—31no—
134C2×#208—31no—
135C22×C3⋊C8—31no—
136C2×ΓL(1,𝔽25)—31no—
137C2×D3×C8—31no—
138C3×C4⋊C8—91no—
139C3×C22⋊C8—31no—
140C3×C22⋊C8—31no—
141C3×C22⋊C8—31no—
142C3×C22⋊C8—31no—
143C3×C22⋊C8—61no—
144C3×C22⋊C8—61no—
145C3×#97—91no—
146(C2×D3)⋊C8C3×D311yescharacteristic, complemented
147(C2×D3)⋊C8C3×D311yescharacteristic, complemented
148#815—31no—
149#815—31no—
150#815—61no—
151Dic3⋊C8—31no—
152Dic3⋊C8—31no—
153C23×C32—31no—
154C23×C32—31no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3C4×D311yescharacteristic
158C22×C3×D3C4×D311yescharacteristic
159C22×C3×D3C2×C3×C411yescharacteristic
160C22×C3×D3C2×C3×C411yescharacteristic
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—31no—
165C2×C32×C4—31no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C23×C311yescharacteristic
174C2×C3⋊Dic3C2×C3×C411yescharacteristic
175C2×C3⋊Dic3C2×C3×C411yescharacteristic
176C3×#67—62no—
177C3×#67—62no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—31no—
183C2×C3×Dic3—31no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3C4×D311yescharacteristic
187C2×C3×Dic3C4×D311yescharacteristic
188C2×C3×Dic3C2×C3×C411yescharacteristic
189C2×C3×Dic3C2×C3×C411yescharacteristic
190C3×C4×D3—21no—
191C3×C4×D3—21no—
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—21no—
197C3×C4×D3—21no—
198C3×C4×D3—41no—
199C3×C4×D3—41no—
200C3×C4×D3—41no—
201C3×C4×D3—41no—
202#625—22no—
203C32×C8—21no—
204C32×C8—22no—
205C32×C8—21no—
206C32×C8—21no—
207C32×C8—41no—
208C32×C8—41no—
209C32×C8—42no—
210C32×C8—61no—
211C32×C8—61no—
212C3⋊(C3⋊C8)C3×D412yescomplemented
213C3×C3⋊C8—21no—
214C3×C3⋊C8—21no—
215C3×C3⋊C8—21no—
216C3×C3⋊C8—21no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—41no—
221C3×C3⋊C8—62no—
222#435—91noSylow
223C2×C33C4×D411yescharacteristic
224C2×C33C2×M1611yescharacteristic
225C2×C33#13111yescharacteristic
226C32×D3—22no—
227C32×D3—41no—
228C2×C3×D4—91no—
229C22×C3×C4—31no—
230C22×C3×C4—31no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—61no—
234C22×C3×C4—61no—
235C2×#67—31no—
236C2×#67—31no—
237C2×C4×D3C2×C3×D311yescharacteristic
238C2×C4×D3C2×C3×D311yescharacteristic
239C3×M16—182no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—31no—
244C2×C3×C8—31no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—41no—
248C2×C3×C8—61no—
249C2×C3×C8—61no—
250C2×C3×C8—91no—
251C2×C3×C8—182no—
252C2×C3×C8D3211yescharacteristic
253C2×C3×C8C2×C3×D311yescharacteristic
254C2×C3×C8C2×C3×D311yescharacteristic
255C3×C4⋊C4—91no—
256C3×C22⋊C4—91no—
257C3×C22⋊C4—91no—
258C3×C42—91no—
259#217—31no—
260#217—31no—
261(C2×D3)⋊C4—31no—
262(C2×D3)⋊C4—31no—
263Dic3⋊C4—31no—
264Dic3⋊C4—31no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—61no—
268#208—62no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8—62no—
273C2×C3⋊C8C2×C3×D311yescharacteristic
274C2×C3⋊C8C2×C3×D311yescharacteristic
275ΓL(1,𝔽25)—62no—
276D3×C8—62no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—32no—
281C22×C32—41no—
282C22×C32—61no—
283C22×C32—61no—
284C22×C32—61no—
285C22×C32—61no—
286C22×C32—61no—
287C22×C32C2×C4×D311yescharacteristic
288C22×C32C2×C4×D311yescharacteristic
289C22×C32C22×C3×C411yescharacteristic
290C2×C3×D3—21no—
291C2×C3×D3—21no—
292C2×C3×D3—21no—
293C2×C3×D3—21no—
294C2×C3×D3—22no—
295C2×C3×D3—21no—
296C2×C3×D3—21no—
297C2×C3×D3—21no—
298C2×C3×D3—21no—
299C2×C3×D3—21no—
300C2×C3×D3—21no—
301C2×C3×D3—22no—
302C2×C3×D3—41no—
303C2×C3×D3—41no—
304C2×C3×D3—41no—
305C2×C3×D3—41no—
306C2×C3×D3ΓL(1,𝔽25)12yescomplemented
307C2×C3×D3C3×M1612yescomplemented
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4—61no—
317C32×C4—61no—
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4#23711yescharacteristic
322C32×C4C2×C3×D411yescharacteristic
323C32×C4C3×#4111yescharacteristic
324C3⋊Dic3—21no—
325C3⋊Dic3—21no—
326C3×Dic3—21no—
327C3×Dic3—21no—
328C3×Dic3—21no—
329C3×Dic3—21no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—61no—
335C3×Dic3—61no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C2×M16—91no—
339C22×C8—91no—
340C4×D4—91no—
341C4⋊C8—91no—
342C22⋊C8—31no—
343C22⋊C8—31no—
344#97—91no—
345C33#43511yescharacteristic, complemented, Sylow
346C23×C3—31no—
347C23×C3—31no—
348C23×C3—31no—
349C23×C3—31no—
350C23×C3—61no—
351C23×C3—61no—
352C22×D3C3×C4×D311yescharacteristic
353C22×D3C3×C4×D311yescharacteristic
354C3×D4—182no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—41no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—91no—
374C2×C3×C4—91no—
375C2×C3×C4—121no—
376C2×C3×C4—121no—
377C2×C3×C4—121no—
378C2×C3×C4—121no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C22×C3×D311yescharacteristic
381C2×C3×C4C22×C3×D311yescharacteristic
382#67—62no—
383#67—62no—
384C2×Dic3—31no—
385C2×Dic3—31no—
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3C3×C4×D311yescharacteristic
391C2×Dic3C3×C4×D311yescharacteristic
392C4×D3—21no—
393C4×D3—21no—
394C4×D3—21no—
395C4×D3—21no—
396C3×C8—22no—
397C3×C8—21no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8—42no—
403C3×C8—61no—
404C3×C8—61no—
405C3×C8—61no—
406C3×C8—61no—
407C3×C8—92no—
408C3×C8—121no—
409C3×C8—121no—
410C3⋊C8—21no—
411C3⋊C8—21no—
412C3⋊C8—32no—
413C3⋊C8—32no—
414C3⋊C8—62no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32—41no—
427C2×C32—62no—
428C2×C32—121no—
429C2×C32#67⋊C411yescharacteristic
430C2×C32#67⋊C411yescharacteristic
431C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
432C2×C32#90111yescharacteristic
433C2×C32D3×M1611yescharacteristic
434C2×C32#90711yescharacteristic
435C2×C32C3×C4×D411yescharacteristic
436C2×C32C2×C3×M1611yescharacteristic
437C2×C32C3×#13111yescharacteristic
438C3×D3—22no—
439C3×D3—22no—
440C3×D3—42no—
441C3×D3—42no—
442C3×D3—41no—
443C3×D3—41no—
444C2×D4—91no—
445C22×C4—31no—
446C22×C4—31no—
447M16—182no—
448C2×C8—31no—
449C2×C8—31no—
450C2×C8—91no—
451C2×C8—182no—
452C2×C8C3×D3211yescharacteristic
453C4⋊C4—91no—
454C22⋊C4—91no—
455C22⋊C4—91no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—32no—
461C22×C3—32no—
462C22×C3—41no—
463C22×C3—62no—
464C22×C3—62no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3—61no—
468C22×C3—61no—
469C22×C3—61no—
470C22×C3—61no—
471C22×C3—61no—
472C22×C3—61no—
473C22×C3—61no—
474C22×C3—61no—
475C22×C3—121no—
476C22×C3—121no—
477C22×C3—121no—
478C22×C3—121no—
479C22×C3C4×D3211yescharacteristic
480C22×C3C2×C3×C4×D311yescharacteristic
481C22×C3C2×C3×C4×D311yescharacteristic
482C2×D3—21no—
483C2×D3—21no—
484C2×D3—21no—
485C2×D3—21no—
486C2×D3—21no—
487C2×D3C3×ΓL(1,𝔽25)12yescomplemented
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—41no—
495C3×C4—41no—
496C3×C4—61no—
497C3×C4—61no—
498C3×C4—61no—
499C3×C4—61no—
500C3×C4—121no—
501C3×C4—121no—
502C3×C4—181no—
503C3×C4—181no—
504C3×C4#380211yescharacteristic
505C3×C4#380511yescharacteristic
506C3×C4C3×D3×D411yescharacteristic
507C3×C4C3×D3×D411yescharacteristic
508C3×C4C3×#23711yescharacteristic
509C3×C4C3×#23711yescharacteristic
510Dic3—21no—
511Dic3—21no—
512Dic3—61no—
513Dic3—61no—
514Dic3—61no—
515Dic3—61no—
516Dic3—121no—
517Dic3—121no—
518C32—21no—
519C32—21no—
520C32—21no—
521C32—41no—
522C32#472811yescharacteristic, complemented
523C32#473011yescharacteristic, complemented
524C32C3×#43511yescharacteristic, complemented
525C23—31no—
526C23—31no—
527D4—182no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4C2×C3×D3211yescharacteristic
537C8—21no—
538C8—61no—
539C8—61no—
540C8—92no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—41no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—62no—
554C2×C3—62no—
555C2×C3—122no—
556C2×C3—122no—
557C2×C3—121no—
558C2×C3—121no—
559C2×C3D3×ΓL(1,𝔽25)11yescharacteristic
560C2×C3#6380411yescharacteristic
561C2×C3#6390211yescharacteristic
562C2×C3C3×#67⋊C411yescharacteristic
563C2×C3C3×#67⋊C411yescharacteristic
564C2×C3C2×C3×ΓL(1,𝔽25)11yescharacteristic
565C2×C3C3×#90111yescharacteristic
566C2×C3C3×D3×M1611yescharacteristic
567C2×C3C3×#90711yescharacteristic
568D3—22no—
569D3—41no—
570C22—32no—
571C22—61no—
572C22—61no—
573C22—61no—
574C22—61no—
575C22—61no—
576C22C3×C4×D3211yescharacteristic
577C4—61no—
578C4—61no—
579C4—181no—
580C4—181no—
581C4C3×#380211yescharacteristic
582C4C3×#380511yescharacteristic
583C3—21no—
584C3—21no—
585C3—21no—
586C3—41no—
587C3#1059072511yescharacteristic, direct product factor
588C3C3×#472811yescharacteristic, complemented
589C3C3×#473011yescharacteristic, complemented
590C2—62no—
591C2—121no—
592C2C3×D3×ΓL(1,𝔽25)11yescharacteristic
593C2C3×#6380411yescharacteristic
594C2C3×#6390211yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC2×C3×#625C4
cosocleC2×C32×C4C23×C3
hypercenterC2×C3×C8D32
hypocenterC32C3×#435
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
2. C2×C3×#3711C2semidirect
3. C2×C3×D3×C3⋊C8C2semidirect
4. C3×#63642C2semidirect
5. C32×(C2×D3)⋊C8C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. C3×#63572C2semidirect
10. #10590725C3direct (N × H)
31. C2×#3711C2×C3semidirect
32. C2×D3×C3⋊C8C2×C3semidirect
35. #63642C2×C3semidirect
38. C3×(C2×D3)⋊C8D3semidirect
39. C3×(C2×D3)⋊C8D3semidirect
40. C3×(C2×D3)⋊C8C2×C3semidirect
41. C3×(C2×D3)⋊C8C2×C3semidirect
47. #63572C2×C3semidirect
61. C3×C3⋊(C3⋊C8)D4semidirect
122. C2×C32×D3M16semidirect
146. (C2×D3)⋊C8C3×D3semidirect
147. (C2×D3)⋊C8C3×D3semidirect
212. C3⋊(C3⋊C8)C3×D4semidirect
306. C2×C3×D3ΓL(1,𝔽25)semidirect
307. C2×C3×D3C3×M16semidirect
345. C33#435semidirect
487. C2×D3C3×ΓL(1,𝔽25)semidirect
522. C32#4728semidirect
523. C32#4730semidirect
524. C32C3×#435semidirect
587. C3#10590725direct (N × H)
588. C3C3×#4728semidirect
589. C3C3×#4730semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#6252
C2×C3×#37112
C2×C3×D3×C3⋊C82
C3×#636422
C32×(C2×D3)⋊C82
C3×#635722
C3×#47303
C3×#47283
#105907253