Finite Group Databasebeta

← C2×D4×#1204 #14482953

Subgroups — C2×D4×#1204 #14482953

Order 1920.

3136 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1204C212yesmaximal, complemented
2C2×D4×Dic15C211yesmaximal, characteristic
3C2×C5×D4×Dic3C211yesmaximal, characteristic
4C2×C3×D4×Dic5C211yesmaximal, characteristic
5C2×#91576C212yesmaximal, complemented
6C2×#91575C212yesmaximal, complemented
7C2×#91574C212yesmaximal, complemented
8D4×#1204C2116yesmaximal, direct product factor
9C2×#91470C211yesmaximal, characteristic, complemented
10C2×#91469C211yesmaximal, characteristic, complemented
11C2×#91468C211yesmaximal, characteristic, complemented
12C2×C4×#1204C211yesmaximal, characteristic, complemented
13C2×D4×Dic10—31nomaximal, Hall
14C22×C3×C5×D4C2211yescharacteristic
15C23×Dic15C2212yes—
16C23×C5×Dic3C2212yes—
17C23×C3×Dic5C2212yes—
18C22×#1204—216no—
19C22×#1204C2212yes—
20C22×#1204C2212yes—
21C22×#1204C2212yes—
22C22×#1204C2211yescharacteristic
23C22×#1204C22116yescomplemented
24C2×#6719C2212yes—
25D4×Dic15C2218yes—
26C2×C4⋊Dic15C2211yescharacteristic
27C2×C4×Dic15C2211yescharacteristic
28C2×C5×#217C2212yes—
29C5×D4×Dic3C2218yes—
30C2×C5×C4⋊Dic3C2211yescharacteristic
31C2×C4×C5×Dic3C2211yescharacteristic
32C2×C3×C22⋊Dic5C2212yes—
33C3×D4×Dic5C2218yes—
34C2×C3×C4⋊Dic5C2211yescharacteristic
35C2×C3×C4×Dic5C2211yescharacteristic
36#91576C22116yescomplemented
37#91575C22116yescomplemented
38#91574C22116yescomplemented
39C2×Dic3⋊Dic5C2211yescharacteristic
40C2×Dic3⋊Dic5C2212yes—
41C2×#6669C2211yescharacteristic
42C2×#6669C2212yes—
43C2×#6668C2211yescharacteristic
44C2×#6668C2212yes—
45#91470C2218yescomplemented
46#91469C2218yescomplemented
47#91468C2218yescomplemented
48C4×#1204C2218yescomplemented
49C2×D4×Dic6—51nomaximal, Hall
50C23×Dic10—32no—
51C2×C4×C5×D4—31no—
52C2×D4×Dic5—31no—
53C2×D4×Dic5D311yescharacteristic
54C2×#4070—32no—
55D4×Dic10—316no—
56C2×#4034—31no—
57C2×#4034—31no—
58C2×#4024—32no—
59C2×#4024—32no—
60C2×#4015—31no—
61C2×C4×Dic10—31no—
62C24×C3×C5C2312yes—
63C2×C3×C5×D4Q818yescomplemented
64C2×C3×C5×D4C2314yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15—28no—
67C22×Dic15C2312yes—
68C22×Dic15C2318yes—
69C22×Dic15C2311yescharacteristic
70C22×Dic15C2311yescharacteristic
71C22×C5×Dic3—28no—
72C22×C5×Dic3C2311yescharacteristic
73C22×C5×Dic3C2312yes—
74C22×C5×Dic3C2318yes—
75C22×C5×Dic3C2311yescharacteristic
76C22×C3×Dic5—28no—
77C22×C3×Dic5C2318yes—
78C22×C3×Dic5C2311yescharacteristic
79C22×C3×Dic5C2312yes—
80C22×C3×Dic5C2311yescharacteristic
81C2×#1204—28no—
82C2×#1204—28no—
83C2×#1204—28no—
84C2×#1204—264no—
85C2×#1204D418yesdirect product factor
86C2×#1204C2318yes—
87C2×#1204C2318yes—
88C2×#1204C2318yes—
89C2×#1204C2314yes—
90#6719C2318yes—
91C4⋊Dic15C2314yes—
92C4×Dic15C2314yes—
93C5×#217C2318yes—
94C5×C4⋊Dic3C2314yes—
95C4×C5×Dic3C2314yes—
96C3×C22⋊Dic5C2318yes—
97C3×C4⋊Dic5C2314yes—
98C3×C4×Dic5C2314yes—
99Dic3⋊Dic5C2314yes—
100Dic3⋊Dic5C2318yes—
101#6669C2314yes—
102#6669C2318yes—
103#6668C2314yes—
104#6668C2318yes—
105C23×Dic6—52no—
106C2×C3×C4×D4—51no—
107C2×D4×Dic3—51no—
108C2×D4×Dic3D511yescharacteristic
109C2×#924—52no—
110D4×Dic6—516no—
111C2×#888—51no—
112C2×#888—51no—
113C2×#878—52no—
114C2×#878—52no—
115C2×#869—51no—
116C2×C4×Dic6—51no—
117C22×C5×D4C2×D311yescharacteristic
118C23×C4×C5—32no—
119C23×Dic5—32no—
120C23×Dic5C2×D312yes—
121C22×Dic10—31no—
122C22×Dic10—32no—
123C22×Dic10—32no—
124C22×Dic10—32no—
125C22×Dic10—316no—
126C22×Dic10—616no—
127C4×C5×D4—38no—
128C2×C5×C4⋊C4—31no—
129C2×C5×C22⋊C4—32no—
130C2×C42×C5—31no—
131C2×C22⋊Dic5—32no—
132C2×C22⋊Dic5C2×D312yes—
133D4×Dic5—38no—
134D4×Dic5C2×D318yes—
135C2×C4⋊Dic5—31no—
136C2×C4⋊Dic5—31no—
137C2×C4⋊Dic5—32no—
138C2×C4⋊Dic5C2×D311yescharacteristic
139#4070—316no—
140C2×#683—31no—
141C2×#683—31no—
142C2×#683—32no—
143C2×#683—32no—
144C2×C4×Dic5—31no—
145C2×C4×Dic5C2×D311yescharacteristic
146#4034—38no—
147#4034—38no—
148#4024—316no—
149#4024—316no—
150#4015—38no—
151C4×Dic10—38no—
152C2×D4×Q8—151noSylow
153C23×C3×C5—24no—
154C23×C3×C5—24no—
155C23×C3×C5C2×Q818yescomplemented
156C23×C3×C5C2414yes—
157C23×C3×C5C2411yescharacteristic
158C3×C5×D4C2×Q8116yescomplemented
159C2×C3×C4×C5C2×Q814yescomplemented
160C2×C3×C4×C5C2412yes—
161C2×Dic15—24no—
162C2×Dic15—22no—
163C2×Dic15—216no—
164C2×Dic15C2×D414yes—
165C2×Dic15C2412yes—
166C2×Dic15C2414yes—
167C2×Dic15C2412yes—
168C2×C5×Dic3—24no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3—216no—
171C2×C5×Dic3C2×D414yes—
172C2×C5×Dic3C2412yes—
173C2×C5×Dic3C2414yes—
174C2×C5×Dic3C2412yes—
175C2×C3×Dic5—24no—
176C2×C3×Dic5—22no—
177C2×C3×Dic5—216no—
178C2×C3×Dic5C2×D414yes—
179C2×C3×Dic5C2412yes—
180C2×C3×Dic5C2414yes—
181C2×C3×Dic5C2412yes—
182#1204—216no—
183#1204—216no—
184#1204—216no—
185#1204C2×D4116yesdirect product factor
186C22×C3×D4C2×D511yescharacteristic
187C23×C3×C4—52no—
188C23×Dic3—52no—
189C23×Dic3C2×D512yes—
190C22×Dic6—51no—
191C22×Dic6—52no—
192C22×Dic6—52no—
193C22×Dic6—52no—
194C22×Dic6—516no—
195C22×Dic6—1016no—
196C3×C4×D4—58no—
197C2×C3×C4⋊C4—51no—
198C2×C3×C22⋊C4—52no—
199C2×C3×C42—51no—
200C2×#217—52no—
201C2×#217C2×D512yes—
202D4×Dic3—58no—
203D4×Dic3C2×D518yes—
204C2×C4⋊Dic3—51no—
205C2×C4⋊Dic3—51no—
206C2×C4⋊Dic3—52no—
207C2×C4⋊Dic3C2×D511yescharacteristic
208#924—516no—
209C2×Dic3⋊C4—51no—
210C2×Dic3⋊C4—51no—
211C2×Dic3⋊C4—52no—
212C2×Dic3⋊C4—52no—
213C2×C4×Dic3—51no—
214C2×C4×Dic3C2×D511yescharacteristic
215#888—58no—
216#888—58no—
217#878—516no—
218#878—516no—
219#869—58no—
220C4×Dic6—58no—
221C24×C5C22×D312yes—
222C2×C5×D4Dic618yescomplemented
223C2×C5×D4C22×D314yes—
224C22×C4×C5—31no—
225C22×C4×C5—31no—
226C22×C4×C5—32no—
227C22×C4×C5—38no—
228C22×C4×C5—68no—
229C22×C4×C5C22×D311yescharacteristic
230C22×Dic5—28no—
231C22×Dic5—31no—
232C22×Dic5—31no—
233C22×Dic5—32no—
234C22×Dic5—38no—
235C22×Dic5—68no—
236C22×Dic5C22×D311yescharacteristic
237C22×Dic5C22×D312yes—
238C22×Dic5C22×D318yes—
239C22×Dic5C22×D311yescharacteristic
240C2×Dic10—38no—
241C2×Dic10—34no—
242C2×Dic10—38no—
243C2×Dic10—38no—
244C2×Dic10—38no—
245C2×Dic10—664no—
246C2×Dic10—68no—
247C2×Dic10—68no—
248C2×Dic10—68no—
249C5×C4⋊C4—34no—
250C5×C22⋊C4—38no—
251C42×C5—34no—
252C22⋊Dic5—38no—
253C22⋊Dic5C22×D318yes—
254C4⋊Dic5—34no—
255C4⋊Dic5—34no—
256C4⋊Dic5—38no—
257C4⋊Dic5C22×D314yes—
258#683—34no—
259#683—38no—
260#683—34no—
261#683—38no—
262C4×Dic5—34no—
263C4×Dic5C22×D314yes—
264C23×Q8—152no—
265D4×Q8—1516no—
266C2×C4⋊Q8—151no—
267C2×C4⋊Q8—151no—
268C2×C4⋊Q8—151no—
269C2×C22⋊Q8—152no—
270C2×C22⋊Q8—152no—
271C2×C22⋊Q8—152no—
272C2×C4×Q8—151no—
273C2×C4×D4—31no—
274C2×C4×D4—51no—
275C2×C4×D4—151no—
276C22×C3×C5—216no—
277C22×C3×C5—24no—
278C22×C3×C5—24no—
279C22×C3×C5C22×D412yes—
280C22×C3×C5C22×Q812yes—
281C22×C3×C5C22×Q818yescomplemented
282C22×C3×C52−1+412yes—
283C22×C3×C5C2511yescharacteristic
284C3×C4×C5C22×Q814yescomplemented
285Dic15—24no—
286Dic15—22no—
287Dic15C22×D414yes—
288C3×Dic5—24no—
289C3×Dic5—22no—
290C3×Dic5C22×D414yes—
291C5×Dic3—24no—
292C5×Dic3—22no—
293C5×Dic3C22×D414yes—
294C24×C3C22×D512yes—
295C2×C3×D4Dic1018yescomplemented
296C2×C3×D4C22×D514yes—
297C22×C3×C4—51no—
298C22×C3×C4—51no—
299C22×C3×C4—52no—
300C22×C3×C4—58no—
301C22×C3×C4—108no—
302C22×C3×C4C22×D511yescharacteristic
303C22×Dic3—28no—
304C22×Dic3—51no—
305C22×Dic3—58no—
306C22×Dic3—51no—
307C22×Dic3—52no—
308C22×Dic3—108no—
309C22×Dic3C22×D518yes—
310C22×Dic3C22×D511yescharacteristic
311C22×Dic3C22×D512yes—
312C22×Dic3C22×D511yescharacteristic
313C2×Dic6—58no—
314C2×Dic6—54no—
315C2×Dic6—58no—
316C2×Dic6—58no—
317C2×Dic6—58no—
318C2×Dic6—1064no—
319C2×Dic6—108no—
320C2×Dic6—108no—
321C2×Dic6—108no—
322C3×C4⋊C4—54no—
323C3×C22⋊C4—58no—
324C3×C42—54no—
325#217—58no—
326#217C22×D518yes—
327C4⋊Dic3—54no—
328C4⋊Dic3—54no—
329C4⋊Dic3—58no—
330C4⋊Dic3C22×D514yes—
331Dic3⋊C4—54no—
332Dic3⋊C4—58no—
333Dic3⋊C4—54no—
334Dic3⋊C4—58no—
335C4×Dic3—54no—
336C4×Dic3C22×D514yes—
337C23×C5—24no—
338C23×C5—24no—
339C23×C5C2×Dic618yescomplemented
340C23×C5C23×D314yes—
341C23×C5C23×D311yescharacteristic
342C5×D4C2×Dic6116yescomplemented
343C2×C4×C5—34no—
344C2×C4×C5—32no—
345C2×C4×C5—32no—
346C2×C4×C5—34no—
347C2×C4×C5—616no—
348C2×C4×C5—64no—
349C2×C4×C5—62no—
350C2×C4×C5C2×Dic614yescomplemented
351C2×C4×C5C23×D312yes—
352C2×Dic5—24no—
353C2×Dic5—22no—
354C2×Dic5—216no—
355C2×Dic5—34no—
356C2×Dic5—32no—
357C2×Dic5—32no—
358C2×Dic5—34no—
359C2×Dic5—616no—
360C2×Dic5—64no—
361C2×Dic5—62no—
362C2×Dic5D3×D414yes—
363C2×Dic5C23×D312yes—
364C2×Dic5C23×D314yes—
365C2×Dic5C23×D312yes—
366Dic10—316no—
367Dic10—616no—
368Dic10—616no—
369Dic10—616no—
370C22×Q8—151no—
371C22×Q8—152no—
372C22×Q8—152no—
373C22×Q8—152no—
374C22×Q8—1516no—
375C22×Q8—3016no—
376C22×D4D3×D511yescharacteristic
377C23×C4—32no—
378C23×C4—52no—
379C23×C4—152no—
380C4⋊Q8—158no—
381C4⋊Q8—158no—
382C4⋊Q8—158no—
383C22⋊Q8—1516no—
384C22⋊Q8—1516no—
385C22⋊Q8—1516no—
386C4×Q8—158no—
387C4×D4—38no—
388C4×D4—58no—
389C4×D4—158no—
390C2×C4⋊C4—31no—
391C2×C4⋊C4—51no—
392C2×C4⋊C4—151no—
393C2×C4⋊C4—151no—
394C2×C4⋊C4—151no—
395C2×C4⋊C4—152no—
396C2×C4⋊C4—152no—
397C2×C4⋊C4—152no—
398C2×C4⋊C4—151no—
399C2×C22⋊C4—32no—
400C2×C22⋊C4—52no—
401C2×C22⋊C4—152no—
402C2×C42—31no—
403C2×C42—51no—
404C2×C42—151no—
405C2×C3×C5—28no—
406C2×C3×C5D4×Q814yescomplemented
407C2×C3×C5C23×D411yescharacteristic
408C2×C3×C5C23×Q811yescharacteristic
409C2×C3×C5C2×2−1+411yescharacteristic
410C23×C3—24no—
411C23×C3—24no—
412C23×C3C2×Dic1018yescomplemented
413C23×C3C23×D514yes—
414C23×C3C23×D511yescharacteristic
415C3×D4C2×Dic10116yescomplemented
416C2×C3×C4—54no—
417C2×C3×C4—52no—
418C2×C3×C4—52no—
419C2×C3×C4—54no—
420C2×C3×C4—1016no—
421C2×C3×C4—104no—
422C2×C3×C4—102no—
423C2×C3×C4C2×Dic1014yescomplemented
424C2×C3×C4C23×D512yes—
425C2×Dic3—24no—
426C2×Dic3—22no—
427C2×Dic3—216no—
428C2×Dic3—54no—
429C2×Dic3—52no—
430C2×Dic3—52no—
431C2×Dic3—54no—
432C2×Dic3—1016no—
433C2×Dic3—104no—
434C2×Dic3—102no—
435C2×Dic3D4×D514yes—
436C2×Dic3C23×D512yes—
437C2×Dic3C23×D514yes—
438C2×Dic3C23×D512yes—
439Dic6—516no—
440Dic6—1016no—
441Dic6—1016no—
442Dic6—1016no—
443C22×C5—24no—
444C22×C5—24no—
445C22×C5—216no—
446C22×C5C22×Dic612yes—
447C22×C5C22×Dic618yescomplemented
448C22×C5C2×D3×D412yes—
449C22×C5#101012yes—
450C22×C5C24×D311yescharacteristic
451C4×C5—34no—
452C4×C5—64no—
453C4×C5—62no—
454C4×C5C22×Dic614yescomplemented
455Dic5—24no—
456Dic5—22no—
457Dic5—34no—
458Dic5—64no—
459Dic5—62no—
460Dic5C2×D3×D414yes—
461C24C2×D3×D512yes—
462C2×Q8—154no—
463C2×Q8—158no—
464C2×Q8—158no—
465C2×Q8—158no—
466C2×Q8—158no—
467C2×Q8—308no—
468C2×Q8—308no—
469C2×Q8—308no—
470C2×Q8—3064no—
471C2×D4#120418yesdirect product factor
472C2×D4C2×D3×D514yes—
473C22×C4—31no—
474C22×C4—38no—
475C22×C4—31no—
476C22×C4—32no—
477C22×C4—51no—
478C22×C4—58no—
479C22×C4—51no—
480C22×C4—52no—
481C22×C4—68no—
482C22×C4—108no—
483C22×C4—151no—
484C22×C4—158no—
485C22×C4—151no—
486C22×C4—152no—
487C22×C4—308no—
488C22×C4C2×D3×D511yescharacteristic
489C4⋊C4—34no—
490C4⋊C4—54no—
491C4⋊C4—154no—
492C4⋊C4—154no—
493C4⋊C4—154no—
494C4⋊C4—158no—
495C4⋊C4—158no—
496C4⋊C4—158no—
497C4⋊C4—154no—
498C22⋊C4—38no—
499C22⋊C4—58no—
500C22⋊C4—158no—
501C42—34no—
502C42—54no—
503C42—154no—
504C3×C5C2×D4×Q811yescharacteristic, complemented, Hall
505C22×C3—216no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3C22×Dic1012yes—
509C22×C3C22×Dic1018yescomplemented
510C22×C3C2×D4×D512yes—
511C22×C3#415012yes—
512C22×C3C24×D511yescharacteristic
513C3×C4—54no—
514C3×C4—104no—
515C3×C4—102no—
516C3×C4C22×Dic1014yescomplemented
517Dic3—24no—
518Dic3—22no—
519Dic3—54no—
520Dic3—104no—
521Dic3—102no—
522Dic3C2×D4×D514yes—
523C2×C5—28no—
524C2×C5D4×Dic614yescomplemented
525C2×C5C23×Dic611yescharacteristic
526C2×C5C22×D3×D411yescharacteristic
527C2×C5C2×#101011yescharacteristic
528C23—24no—
529C23—24no—
530C23C2×#120418yescomplemented
531C23C22×D3×D514yes—
532C23C22×D3×D511yescharacteristic
533Q8—1516no—
534Q8—3016no—
535Q8—3016no—
536Q8—3016no—
537D4C2×#1204116yesdirect product factor
538C2×C4—34no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—34no—
542C2×C4—54no—
543C2×C4—52no—
544C2×C4—52no—
545C2×C4—54no—
546C2×C4—616no—
547C2×C4—62no—
548C2×C4—64no—
549C2×C4—1016no—
550C2×C4—102no—
551C2×C4—104no—
552C2×C4—154no—
553C2×C4—152no—
554C2×C4—152no—
555C2×C4—154no—
556C2×C4—3016no—
557C2×C4—302no—
558C2×C4—304no—
559C2×C4C2×#120414yescomplemented
560C2×C4C22×D3×D512yes—
561C2×C3—28no—
562C2×C3D4×Dic1014yescomplemented
563C2×C3C23×Dic1011yescharacteristic
564C2×C3C22×D4×D511yescharacteristic
565C2×C3C2×#415011yescharacteristic
566C5C2×D4×Dic611yescharacteristic, complemented, Sylow
567C22—24no—
568C22—24no—
569C22—216no—
570C22#9202012yes—
571C22D3×D4×D512yes—
572C22C22×#120412yes—
573C22C22×#120418yescomplemented
574C22C23×D3×D511yescharacteristic
575C4—34no—
576C4—54no—
577C4—62no—
578C4—64no—
579C4—102no—
580C4—104no—
581C4—154no—
582C4—302no—
583C4—304no—
584C4C22×#120414yescomplemented
585C3C2×D4×Dic1011yescharacteristic, complemented, Sylow
586C2—28no—
587C2D4×#120414yesdirect product factor
588C2C2×#9202011yescharacteristic
589C2C2×D3×D4×D511yescharacteristic
590C2C23×#120411yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
Fitting, prime radicalC22×C3×C5×D4C22
socleC23×C3×C5C24
hypercenterC22×D4D3×D5
hypocenterC3×C5C2×D4×Q8
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C23×#1204C2semidirect
5. C2×#91576C2semidirect
6. C2×#91575C2semidirect
7. C2×#91574C2semidirect
8. D4×#1204C2direct (N × H)
9. C2×#91470C2semidirect
10. C2×#91469C2semidirect
11. C2×#91468C2semidirect
12. C2×C4×#1204C2semidirect
23. C22×#1204C22semidirect
36. #91576C22semidirect
37. #91575C22semidirect
38. #91574C22semidirect
45. #91470C22semidirect
46. #91469C22semidirect
47. #91468C22semidirect
48. C4×#1204C22semidirect
63. C2×C3×C5×D4Q8semidirect
85. C2×#1204D4direct (N × H)
155. C23×C3×C5C2×Q8semidirect
158. C3×C5×D4C2×Q8semidirect
159. C2×C3×C4×C5C2×Q8semidirect
185. #1204C2×D4direct (N × H)
222. C2×C5×D4Dic6semidirect
281. C22×C3×C5C22×Q8semidirect
284. C3×C4×C5C22×Q8semidirect
295. C2×C3×D4Dic10semidirect
339. C23×C5C2×Dic6semidirect
342. C5×D4C2×Dic6semidirect
350. C2×C4×C5C2×Dic6semidirect
406. C2×C3×C5D4×Q8semidirect
412. C23×C3C2×Dic10semidirect
415. C3×D4C2×Dic10semidirect
423. C2×C3×C4C2×Dic10semidirect
447. C22×C5C22×Dic6semidirect
454. C4×C5C22×Dic6semidirect
471. C2×D4#1204direct (N × H)
504. C3×C5C2×D4×Q8semidirect
509. C22×C3C22×Dic10semidirect
516. C3×C4C22×Dic10semidirect
524. C2×C5D4×Dic6semidirect
530. C23C2×#1204semidirect
537. D4C2×#1204direct (N × H)
559. C2×C4C2×#1204semidirect
562. C2×C3D4×Dic10semidirect
566. C5C2×D4×Dic6semidirect
573. C22C22×#1204semidirect
584. C4C22×#1204semidirect
585. C3C2×D4×Dic10semidirect
587. C2D4×#1204direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#12042
C2×D4×Dic152
C2×C5×D4×Dic32
C2×C3×D4×Dic52
C2×#915762
C2×#915752
C2×#915742
D4×#12042
C2×#914702
C2×#914692
C2×#914682
C2×C4×#12042
C2×D4×Dic103
C2×D4×Dic65