Finite Group Databasebeta

← C3×#10608034 #14327997

Subgroups — C3×#10608034 #14327997

Order 1920.

1120 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
1C2×C3×D4×Dic5C211yesmaximal, characteristic
2C3×C5×#390C211yesmaximal, characteristic
3C3×#65510C211yesmaximal, characteristic
4C3×#65266C211yesmaximal, characteristic, complemented
5C3×C4⋊C4×Dic5C211yesmaximal, characteristic, complemented
6C3×#65234C212yesmaximal, complemented
7C3×#65233C212yesmaximal, complemented
8C3×#65232C212yesmaximal, complemented
9C3×C22⋊C4×Dic5C212yesmaximal, complemented
10C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
11C3×C4×#683C211yesmaximal, characteristic, complemented
12#10608034C311yesmaximal, characteristic, direct product factor, Hall
13C22×C3×C5×D4C2211yescharacteristic
14C23×C3×Dic5C2212yes—
15C2×C3×C5×C4⋊C4C2211yescharacteristic
16C2×C3×C5×C22⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2212yes—
18C2×C3×C42×C5C2211yescharacteristic
19C2×C3×C22⋊Dic5C2212yes—
20C2×C3×C22⋊Dic5C2212yes—
21C2×C3×C22⋊Dic5C2212yes—
22C3×D4×Dic5C418yescomplemented
23C2×C3×C4⋊Dic5C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×#683C2211yescharacteristic
26C2×C3×#683C2211yescharacteristic
27C2×C3×C4×Dic5C2211yescharacteristic
28C2×C3×C4×Dic5C2212yes—
29C2×C3×C4×Dic5C2212yes—
30C2×C3×C4×Dic5C2211yescharacteristic
31C3×#3963C2211yescharacteristic
32C3×#3963C2211yescharacteristic
33C3×#3963C2212yes—
34C3×#3963C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#2357—51nomaximal, Hall
38C2×D4×Dic5C2×C311yescharacteristic
39C5×#390C2×C311yescharacteristic
40#65510C2×C311yescharacteristic
41#65266C2×C311yescharacteristic, complemented
42C4⋊C4×Dic5C2×C311yescharacteristic, complemented
43#65234C2×C312yescomplemented
44#65233C2×C312yescomplemented
45#65232C2×C312yescomplemented
46C22⋊C4×Dic5C2×C312yescomplemented
47C4×C4⋊Dic5C2×C311yescharacteristic, complemented
48C4×#683C2×C311yescharacteristic, complemented
49C24×C3×C5C2312yes—
50C2×C3×C5×D4—22no—
51C2×C3×C5×D4—22no—
52C2×C3×C5×D4C2×C414yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×Dic5—42no—
58C22×C3×Dic5—42no—
59C22×C3×Dic5C2×C418yes—
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2311yescharacteristic
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2311yescharacteristic
65C22×C3×Dic5C2311yescharacteristic
66C3×C5×C4⋊C4—22no—
67C3×C5×C22⋊C4—24no—
68C3×C5×C22⋊C4—42no—
69C3×C42×C5—22no—
70C3×C22⋊Dic5—24no—
71C3×C22⋊Dic5—42no—
72C3×C22⋊Dic5C2×C418yes—
73C3×C4⋊Dic5—22no—
74C3×C4⋊Dic5C2×C414yes—
75C3×#683—22no—
76C3×#683—22no—
77C3×C4×Dic5—22no—
78C3×C4×Dic5—42no—
79C3×C4×Dic5—42no—
80C3×C4×Dic5C2×C414yes—
81C2×C3×C4×D4—51no—
82C3×#390—51no—
83C3×#390D511yescharacteristic
84C3×#388—52no—
85C3×#388—52no—
86C3×#382—51no—
87C3×#381—52no—
88C3×C4×C4⋊C4—51no—
89C3×C4×C4⋊C4—51no—
90C3×C4×C4⋊C4—51no—
91C3×C4×C22⋊C4—52no—
92C22×C5×D4C22×C311yescharacteristic
93C23×Dic5C22×C312yes—
94C2×C5×C4⋊C4C22×C311yescharacteristic
95C2×C5×C22⋊C4C22×C312yes—
96C2×C5×C22⋊C4C22×C312yes—
97C2×C42×C5C22×C311yescharacteristic
98C2×C22⋊Dic5C22×C312yes—
99C2×C22⋊Dic5C22×C312yes—
100C2×C22⋊Dic5C22×C312yes—
101D4×Dic5C3×C418yescomplemented
102C2×C4⋊Dic5C22×C311yescharacteristic
103C2×C4⋊Dic5C22×C311yescharacteristic
104C2×#683C22×C311yescharacteristic
105C2×#683C22×C311yescharacteristic
106C2×C4×Dic5C22×C311yescharacteristic
107C2×C4×Dic5C22×C312yes—
108C2×C4×Dic5C22×C312yes—
109C2×C4×Dic5C22×C311yescharacteristic
110#3963C22×C311yescharacteristic
111#3963C22×C311yescharacteristic
112#3963C22×C312yes—
113#3963C22×C312yes—
114#3963C22×C312yes—
115#3963C22×C312yes—
116#2357—51noSylow
117C23×C3×C5—22no—
118C23×C3×C5—22no—
119C23×C3×C5—22no—
120C23×C3×C5—22no—
121C23×C3×C5—22no—
122C23×C3×C5—22no—
123C23×C3×C5C22×C414yes—
124C23×C3×C5C2411yescharacteristic
125C3×C5×D4—28no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5—21no—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5C22×C412yes—
135C2×C3×C4×C5#4112yes—
136C2×C3×C4×C5#4112yes—
137C2×C3×C4×C5#4112yes—
138C2×C3×Dic5—21no—
139C2×C3×Dic5—21no—
140C2×C3×Dic5—21no—
141C2×C3×Dic5—21no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—21no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—48no—
153C2×C3×Dic5C22×C412yes—
154C2×C3×Dic5C22×C412yes—
155C2×C3×Dic5C22×C414yes—
156C2×C3×Dic5#4112yes—
157C22×C3×D4C2×D511yescharacteristic
158C23×C3×C4—52no—
159C3×C4×D4—58no—
160C2×C3×C4⋊C4—51no—
161C2×C3×C4⋊C4—51no—
162C2×C3×C4⋊C4—51no—
163C2×C3×C4⋊C4—51no—
164C2×C3×C4⋊C4C2×D511yescharacteristic
165C2×C3×C22⋊C4—52no—
166C2×C3×C22⋊C4—52no—
167C2×C3×C22⋊C4—52no—
168C2×C3×C22⋊C4C2×D512yes—
169C2×C3×C22⋊C4C2×D512yes—
170C2×C3×C42—51no—
171C2×C3×C42—51no—
172C2×C3×C42—52no—
173C2×C3×C42—52no—
174C2×C3×C42C2×D511yescharacteristic
175C3×#95—51no—
176C3×#95—52no—
177C3×#95—51no—
178C3×#95—52no—
179C3×#95—52no—
180C3×#95—52no—
181C24×C5C23×C312yes—
182C2×C5×D4—22no—
183C2×C5×D4—22no—
184C2×C5×D4C2×C3×C414yes—
185C22×C4×C5C23×C311yescharacteristic
186C22×C4×C5C23×C312yes—
187C22×C4×C5C23×C311yescharacteristic
188C22×C4×C5C23×C311yescharacteristic
189C22×Dic5—42no—
190C22×Dic5—42no—
191C22×Dic5C2×C3×C418yes—
192C22×Dic5C23×C311yescharacteristic
193C22×Dic5C23×C312yes—
194C22×Dic5C23×C311yescharacteristic
195C22×Dic5C23×C312yes—
196C22×Dic5C23×C311yescharacteristic
197C22×Dic5C23×C311yescharacteristic
198C5×C4⋊C4—22no—
199C5×C22⋊C4—24no—
200C5×C22⋊C4—42no—
201C42×C5—22no—
202C22⋊Dic5—24no—
203C22⋊Dic5—42no—
204C22⋊Dic5C2×C3×C418yes—
205C4⋊Dic5—22no—
206C4⋊Dic5C2×C3×C414yes—
207#683—22no—
208#683—22no—
209C4×Dic5—22no—
210C4×Dic5—42no—
211C4×Dic5—42no—
212C4×Dic5C2×C3×C414yes—
213C2×C4×D4—51no—
214#390—51no—
215#390C3×D511yescharacteristic
216#388—52no—
217#388—52no—
218#382—51no—
219#381—52no—
220C4×C4⋊C4—51no—
221C4×C4⋊C4—51no—
222C4×C4⋊C4—51no—
223C4×C22⋊C4—52no—
224C22×C3×C5—24no—
225C22×C3×C5—24no—
226C22×C3×C5—24no—
227C22×C3×C5—42no—
228C22×C3×C5—42no—
229C22×C3×C5—42no—
230C22×C3×C5—42no—
231C22×C3×C5C23×C411yescharacteristic
232C22×C3×C5C2×#4111yescharacteristic
233C22×C3×C5C2×#4111yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5C2×#4111yescharacteristic
236C22×C3×C5H(5,𝔽2)11yescharacteristic
237C22×C3×C5H(5,𝔽2)11yescharacteristic
238C3×C4×C5—22no—
239C3×C4×C5—41no—
240C3×C4×C5—42no—
241C3×C4×C5#11714yes—
242C3×Dic5—22no—
243C3×Dic5—22no—
244C3×Dic5—24no—
245C3×Dic5—22no—
246C3×Dic5—41no—
247C3×Dic5—42no—
248C24×C3C22×D512yes—
249C2×C3×D4—22no—
250C2×C3×D4—22no—
251C2×C3×D4C4×D514yes—
252C22×C3×C4—51no—
253C22×C3×C4—52no—
254C22×C3×C4—51no—
255C22×C3×C4—51no—
256C22×C3×C4—58no—
257C22×C3×C4—51no—
258C22×C3×C4—52no—
259C22×C3×C4—202no—
260C22×C3×C4—202no—
261C22×C3×C4C22×D512yes—
262C22×C3×C4C22×D511yescharacteristic
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D511yescharacteristic
265C3×C4⋊C4—22no—
266C3×C4⋊C4—54no—
267C3×C4⋊C4—102no—
268C3×C4⋊C4—102no—
269C3×C4⋊C4—102no—
270C3×C22⋊C4—24no—
271C3×C22⋊C4—42no—
272C3×C22⋊C4—58no—
273C3×C22⋊C4—104no—
274C3×C22⋊C4—202no—
275C3×C42—22no—
276C3×C42—54no—
277C3×C42—102no—
278C3×C42—202no—
279C3×C42—202no—
280C23×C5—22no—
281C23×C5—22no—
282C23×C5—22no—
283C23×C5—22no—
284C23×C5—22no—
285C23×C5—22no—
286C23×C5C22×C3×C414yes—
287C23×C5C24×C311yescharacteristic
288C5×D4—28no—
289C2×C4×C5—21no—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—22no—
294C2×C4×C5—22no—
295C2×C4×C5—22no—
296C2×C4×C5—21no—
297C2×C4×C5C22×C3×C412yes—
298C2×C4×C5C3×#4112yes—
299C2×C4×C5C3×#4112yes—
300C2×C4×C5C3×#4112yes—
301C2×Dic5—21no—
302C2×Dic5—21no—
303C2×Dic5—21no—
304C2×Dic5—21no—
305C2×Dic5—22no—
306C2×Dic5—22no—
307C2×Dic5—21no—
308C2×Dic5—21no—
309C2×Dic5—21no—
310C2×Dic5—21no—
311C2×Dic5—22no—
312C2×Dic5—22no—
313C2×Dic5—22no—
314C2×Dic5—21no—
315C2×Dic5—48no—
316C2×Dic5C22×C3×C412yes—
317C2×Dic5C22×C3×C412yes—
318C2×Dic5C22×C3×C414yes—
319C2×Dic5C3×#4112yes—
320C22×D4C2×C3×D511yescharacteristic
321C23×C4—52no—
322C4×D4—58no—
323C2×C4⋊C4—51no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—51no—
327C2×C4⋊C4C2×C3×D511yescharacteristic
328C2×C22⋊C4—52no—
329C2×C22⋊C4—52no—
330C2×C22⋊C4—52no—
331C2×C22⋊C4C2×C3×D512yes—
332C2×C22⋊C4C2×C3×D512yes—
333C2×C42—51no—
334C2×C42—51no—
335C2×C42—52no—
336C2×C42—52no—
337C2×C42C2×C3×D511yescharacteristic
338#95—51no—
339#95—52no—
340#95—51no—
341#95—52no—
342#95—52no—
343#95—52no—
344C2×C3×C5—44no—
345C2×C3×C5C2×#11711yescharacteristic
346C2×C3×C5C4×#4111yescharacteristic
347C2×C3×C5#51811yescharacteristic
348C2×C3×C5#55311yescharacteristic
349C2×C3×C5#55311yescharacteristic
350C2×C3×C5#55911yescharacteristic
351C2×C3×C5#55911yescharacteristic
352C23×C3—22no—
353C23×C3—22no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3C2×C4×D514yes—
359C23×C3C23×D511yescharacteristic
360C3×D4—28no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—52no—
370C2×C3×C4—52no—
371C2×C3×C4—54no—
372C2×C3×C4—52no—
373C2×C3×C4—101no—
374C2×C3×C4—101no—
375C2×C3×C4—101no—
376C2×C3×C4—102no—
377C2×C3×C4—101no—
378C2×C3×C4—101no—
379C2×C3×C4—102no—
380C2×C3×C4—101no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—102no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—101no—
387C2×C3×C4—208no—
388C2×C3×C4C2×C4×D512yes—
389C2×C3×C4#71112yes—
390C2×C3×C4#71112yes—
391C2×C3×C4#71112yes—
392C22×C5—24no—
393C22×C5—24no—
394C22×C5—24no—
395C22×C5—42no—
396C22×C5—42no—
397C22×C5—42no—
398C22×C5—42no—
399C22×C5C23×C3×C411yescharacteristic
400C22×C5C2×C3×#4111yescharacteristic
401C22×C5C2×C3×#4111yescharacteristic
402C22×C5C2×C3×#4111yescharacteristic
403C22×C5C2×C3×#4111yescharacteristic
404C22×C5C3×H(5,𝔽2)11yescharacteristic
405C22×C5C3×H(5,𝔽2)11yescharacteristic
406C4×C5—22no—
407C4×C5—42no—
408C4×C5—41no—
409C4×C5C3×#11714yes—
410Dic5—22no—
411Dic5—22no—
412Dic5—24no—
413Dic5—22no—
414Dic5—42no—
415Dic5—41no—
416C24C22×C3×D512yes—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4C3×C4×D514yes—
420C22×C4—51no—
421C22×C4—52no—
422C22×C4—51no—
423C22×C4—51no—
424C22×C4—58no—
425C22×C4—51no—
426C22×C4—52no—
427C22×C4—202no—
428C22×C4—202no—
429C22×C4C22×C3×D512yes—
430C22×C4C22×C3×D511yescharacteristic
431C22×C4C22×C3×D511yescharacteristic
432C22×C4C22×C3×D511yescharacteristic
433C4⋊C4—22no—
434C4⋊C4—54no—
435C4⋊C4—102no—
436C4⋊C4—102no—
437C4⋊C4—102no—
438C22⋊C4—24no—
439C22⋊C4—42no—
440C22⋊C4—58no—
441C22⋊C4—104no—
442C22⋊C4—202no—
443C42—22no—
444C42—54no—
445C42—102no—
446C42—202no—
447C42—202no—
448C3×C5#235711yescharacteristic, complemented, Hall
449C22×C3—24no—
450C22×C3—24no—
451C22×C3—24no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3C22×C4×D511yescharacteristic
457C22×C3C2×#71111yescharacteristic
458C22×C3C2×#71111yescharacteristic
459C22×C3C2×#71111yescharacteristic
460C22×C3#414411yescharacteristic
461C22×C3D5×#4111yescharacteristic
462C22×C3#414911yescharacteristic
463C3×C4—22no—
464C3×C4—41no—
465C3×C4—42no—
466C3×C4—102no—
467C3×C4—104no—
468C3×C4—102no—
469C3×C4—102no—
470C3×C4—201no—
471C3×C4—202no—
472C3×C4#402314yes—
473C2×C5—44no—
474C2×C5C2×C3×#11711yescharacteristic
475C2×C5C3×C4×#4111yescharacteristic
476C2×C5C3×#51811yescharacteristic
477C2×C5C3×#55311yescharacteristic
478C2×C5C3×#55311yescharacteristic
479C2×C5C3×#55911yescharacteristic
480C2×C5C3×#55911yescharacteristic
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23C2×C3×C4×D514yes—
488C23C23×C3×D511yescharacteristic
489D4—28no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—52no—
499C2×C4—52no—
500C2×C4—54no—
501C2×C4—52no—
502C2×C4—101no—
503C2×C4—101no—
504C2×C4—101no—
505C2×C4—102no—
506C2×C4—101no—
507C2×C4—101no—
508C2×C4—102no—
509C2×C4—101no—
510C2×C4—101no—
511C2×C4—101no—
512C2×C4—102no—
513C2×C4—102no—
514C2×C4—102no—
515C2×C4—101no—
516C2×C4—208no—
517C2×C4C2×C3×C4×D512yes—
518C2×C4C3×#71112yes—
519C2×C4C3×#71112yes—
520C2×C4C3×#71112yes—
521C2×C3—44no—
522C2×C3C2×#402311yescharacteristic
523C2×C3C4×#71111yescharacteristic
524C2×C3#6587911yescharacteristic
525C2×C3#6594811yescharacteristic
526C2×C3#6595211yescharacteristic
527C2×C3#6601511yescharacteristic
528C2×C3#6604711yescharacteristic
529C5C3×#235711yescharacteristic, complemented, Sylow
530C22—24no—
531C22—24no—
532C22—24no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22C22×C3×C4×D511yescharacteristic
538C22C2×C3×#71111yescharacteristic
539C22C2×C3×#71111yescharacteristic
540C22C2×C3×#71111yescharacteristic
541C22C3×#414411yescharacteristic
542C22C3×D5×#4111yescharacteristic
543C22C3×#414911yescharacteristic
544C4—22no—
545C4—41no—
546C4—42no—
547C4—102no—
548C4—102no—
549C4—104no—
550C4—102no—
551C4—201no—
552C4—202no—
553C4C3×#402314yes—
554C3#1060803411yescharacteristic, direct product factor, Sylow
555C2—44no—
556C2C2×C3×#402311yescharacteristic
557C2C3×C4×#71111yescharacteristic
558C2C3×#6587911yescharacteristic
559C2C3×#6594811yescharacteristic
560C2C3×#6595211yescharacteristic
561C2C3×#6601511yescharacteristic
562C2C3×#6604711yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#390C2
socleC23×C3×C5C24
prime radicalC22×C3×C5×D4C22
cosocleC23×C5C24×C3
hypercenterC3×#390D5
hypocenterC5C3×#2357
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#390

Complements

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

NH ≅ G/Ntype
4. C3×#65266C2semidirect
5. C3×C4⋊C4×Dic5C2semidirect
6. C3×#65234C2semidirect
7. C3×#65233C2semidirect
8. C3×#65232C2semidirect
9. C3×C22⋊C4×Dic5C2semidirect
10. C3×C4×C4⋊Dic5C2semidirect
11. C3×C4×#683C2semidirect
12. #10608034C3direct (N × H)
22. C3×D4×Dic5C4semidirect
41. #65266C2×C3semidirect
42. C4⋊C4×Dic5C2×C3semidirect
43. #65234C2×C3semidirect
44. #65233C2×C3semidirect
45. #65232C2×C3semidirect
46. C22⋊C4×Dic5C2×C3semidirect
47. C4×C4⋊Dic5C2×C3semidirect
48. C4×#683C2×C3semidirect
101. D4×Dic5C3×C4semidirect
448. C3×C5#2357semidirect
529. C5C3×#2357semidirect
554. C3#10608034direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Dic52
C3×C5×#3902
C3×#655102
C3×#652662
C3×C4⋊C4×Dic52
C3×#652342
C3×#652332
C3×#652322
C3×C22⋊C4×Dic52
C3×C4×C4⋊Dic52
C3×C4×#6832
#106080343
C3×#23575