Finite Group Databasebeta

← C2×#21448769 #29839363

Subgroups — C2×#21448769 #29839363

Order 15616.

3678 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D61C211yesmaximal, characteristic
2C22×#14519735C211yesmaximal, characteristic, complemented
3#21448769C2116yesmaximal, direct product factor
4C2×#16839662C212yesmaximal
5C2×#16839653C212yesmaximal, complemented
6C2×#16839650C212yesmaximal, complemented
7C2×#16839631C211yesmaximal, characteristic, complemented
8C2×#16839622C211yesmaximal, characteristic, complemented
9C2×#16839619C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic61C211yesmaximal, characteristic, complemented
11C2×C4×#11758196C212yesmaximal, complemented
12C22×#117×C61C211yesmaximal, characteristic, complemented
13C23×C22⋊D61C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C2×C4×C22⋊D61C4116yescomplemented
16C2×C4×C22⋊D61C2218yes—
17C2×#14519735—28no—
18C2×#14519735C2218yescomplemented
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×#11758196C2212yes—
22C22×#11758196C2212yes—
23C22×#11758196C2211yescharacteristic
24C22×C22⋊Dic61C2212yes—
25C22×C22⋊Dic61C2211yescharacteristic
26C22×#11758198C2212yes—
27C22×#11758198C2212yes—
28C22×#11758198C2211yescharacteristic
29C22×C4×Dic61C2212yes—
30C22×C4×Dic61C2212yes—
31C22×C4×Dic61C2211yescharacteristic
32#16839662C22116yes—
33#16839653C22116yescomplemented
34#16839650C22116yescomplemented
35#16839631C2218yescomplemented
36#16839622C2218yescomplemented
37#16839619C2218yescomplemented
38C4⋊C4×Dic61C2218yescomplemented
39C4×#11758196C22116yescomplemented
40C2×#14519684C2212yes—
41C2×#14519684C2212yes—
42C2×#14519684C2212yes—
43C24×C4×C61C2211yescharacteristic
44C2×#117×C61—28no—
45C2×#117×C61C2218yescomplemented
46C22×C4⋊C4×C61C2211yescharacteristic
47C22×C4⋊C4×C61C2211yescharacteristic
48C22×C22⋊C4×C61C2212yes—
49C22×C42×C61C2212yes—
50C24×D61C2311yescharacteristic
51C22×C22⋊D61—24no—
52C22×C22⋊D61—24no—
53C22×C22⋊D61C2×C418yes—
54C22×C22⋊D61C2314yes—
55C23×Dic61C2312yes—
56C23×Dic61C2312yes—
57C23×Dic61C2311yescharacteristic
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C22×C4×D61—24no—
61C22×C4×D61—24no—
62C22×C4×D61C2×C418yes—
63C22×C4×D61C2314yes—
64C4×C22⋊D61C2×C4164yescomplemented
65#14519735—232no—
66C2×C4⋊Dic61—24no—
67C2×C4⋊Dic61—24no—
68C2×C4⋊Dic61C2314yes—
69C2×C4⋊Dic61C2314yes—
70C2×#11758196—28no—
71C2×#11758196—44no—
72C2×#11758196C2×C418yes—
73C2×#11758196C2318yes—
74C2×#11758196C2318yes—
75C2×#11758196C2314yes—
76C2×C22⋊Dic61—44no—
77C2×C22⋊Dic61C2×C418yes—
78C2×C22⋊Dic61C2318yes—
79C2×C22⋊Dic61C2314yes—
80C2×#11758198—28no—
81C2×#11758198—44no—
82C2×#11758198C2×C418yes—
83C2×#11758198C2318yes—
84C2×#11758198C2318yes—
85C2×#11758198C2314yes—
86C2×C4×Dic61—28no—
87C2×C4×Dic61—28no—
88C2×C4×Dic61C2×C418yes—
89C2×C4×Dic61C2318yes—
90C2×C4×Dic61C2318yes—
91C2×C4×Dic61C2314yes—
92#14519684C2318yes—
93#14519684C2318yes—
94#14519684C2318yes—
95C25×C61C2311yescharacteristic
96C23×C4×C61—24no—
97C23×C4×C61—24no—
98C23×C4×C61C2×C418yes—
99C23×C4×C61C2312yes—
100C23×C4×C61C2311yescharacteristic
101C23×C4×C61C2314yes—
102C23×C4×C61C2312yes—
103C23×C4×C61C2311yescharacteristic
104#117×C61—232no—
105C2×C4⋊C4×C61—24no—
106C2×C4⋊C4×C61D418yescomplemented
107C2×C4⋊C4×C61C2314yes—
108C2×C4⋊C4×C61C2314yes—
109C2×C22⋊C4×C61—28no—
110C2×C22⋊C4×C61C2318yes—
111C2×C42×C61—28no—
112C2×C42×C61C2318yes—
113C24×C61—22no—
114C24×C61—22no—
115C24×C61—22no—
116C24×C61—22no—
117C24×C61—22no—
118C24×C61—22no—
119C24×C61C22×C414yes—
120C24×C61C2412yes—
121C24×C61C2411yescharacteristic
122C23×D61—22no—
123C23×D61—22no—
124C23×D61—22no—
125C23×D61—22no—
126C23×D61—22no—
127C23×D61—22no—
128C23×D61C22×C414yes—
129C23×D61C2412yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—24no—
134C22×C4×C61—232no—
135C22×C4×C61—28no—
136C22×C4×C61—28no—
137C22×C4×C61—24no—
138C22×C4×C61—24no—
139C22×C4×C61—24no—
140C22×C4×C61C22×C414yes—
141C22×C4×C61C22×C4116yes—
142C22×C4×C61C22×C414yes—
143C22×C4×C61C2×D418yes—
144C22×C4×C61C2×D414yes—
145C22×C4×C61#4114yes—
146C22×C4×C61C2414yes—
147C22×C4×C61C2412yes—
148C22×C4×C61C2414yes—
149C22×C4×C61C2412yes—
150C2×C22⋊D61—232no—
151C2×C22⋊D61—28no—
152C2×C22⋊D61—28no—
153C2×C22⋊D61C22×C4116yes—
154C22×Dic61—22no—
155C22×Dic61—22no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61—24no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—24no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61C22×C414yes—
165C22×Dic61C22×C414yes—
166C22×Dic61C22×C414yes—
167C22×Dic61#4114yes—
168C22×Dic61#4114yes—
169C22×Dic61C2414yes—
170C22×Dic61C2414yes—
171C22×Dic61C2412yes—
172C22×Dic61C2412yes—
173C22×Dic61C2412yes—
174C2×C4×D61—232no—
175C2×C4×D61—28no—
176C2×C4×D61—28no—
177C2×C4×D61C22×C4116yes—
178C4⋊C4×C61—28no—
179C4⋊C4×C61C2×D4116yescomplemented
180C22⋊C4×C61—216no—
181C42×C61—216no—
182C22⋊Dic61—48no—
183C22⋊Dic61C22×C4116yes—
184#11758198—216no—
185#11758198—48no—
186#11758198C22×C4116yes—
187C4⋊Dic61—28no—
188C4⋊Dic61—28no—
189#11758196—216no—
190#11758196—48no—
191#11758196C22×C4116yes—
192C4×Dic61—216no—
193C4×Dic61—216no—
194C4×Dic61C22×C4116yes—
195C23×C61—28no—
196C23×C61—28no—
197C23×C61—28no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—44no—
205C23×C61—44no—
206C23×C61—44no—
207C23×C61—44no—
208C23×C61C23×C414yes—
209C23×C61C23×C412yes—
210C23×C61C22×D412yes—
211C23×C61C2×#4112yes—
212C23×C61C2×#4112yes—
213C23×C61C2×#4112yes—
214C23×C61H(5,𝔽2)12yes—
215C23×C612−1+412yes—
216C23×C61C2511yescharacteristic
217C22×D61—28no—
218C22×D61—28no—
219C22×D61—28no—
220C22×D61—22no—
221C22×D61—22no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—44no—
227C22×D61—44no—
228C22×D61—44no—
229C22×D61—44no—
230C22×D61C23×C414yes—
231C2×C4×C61—28no—
232C2×C4×C61—216no—
233C2×C4×C61—22no—
234C2×C4×C61—24no—
235C2×C4×C61—22no—
236C2×C4×C61—24no—
237C2×C4×C61—232no—
238C2×C4×C61—24no—
239C2×C4×C61—24no—
240C2×C4×C61—24no—
241C2×C4×C61—48no—
242C2×C4×C61C4×D4116yes—
243C2×C4×C61C23×C414yes—
244C2×C4×C61C23×C414yes—
245C2×C4×C61C22×D418yes—
246C2×C4×C61C22×D414yes—
247C2×C4×C61C2×#4114yes—
248C22⋊D61—232no—
249C2×Dic61—28no—
250C2×Dic61—28no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—24no—
261C2×Dic61—48no—
262C2×Dic61—48no—
263C2×Dic61#117116yescomplemented
264C2×Dic61C23×C414yes—
265C2×Dic61C23×C414yes—
266C2×Dic61C23×C414yes—
267C2×Dic61C2×#4114yes—
268C2×Dic61C2×#4114yes—
269C4×D61—232no—
270C2×#2354—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61—416no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61C2×#11714yes—
280C22×C61C2×C4×D414yes—
281C22×C61#52014yes—
282C22×C61#54714yes—
283C22×C61#55014yes—
284C22×C61#55514yes—
285C22×C61#55614yes—
286C22×C61C24×C411yescharacteristic
287C22×C61C23×D411yescharacteristic
288C22×C61C22×#4111yescharacteristic
289C22×C61C22×#4111yescharacteristic
290C22×C61C22×#4111yescharacteristic
291C22×C61C2×H(5,𝔽2)11yescharacteristic
292C22×C61C2×2−1+411yescharacteristic
293C2×D61—24no—
294C2×D61—24no—
295C2×D61—24no—
296C2×D61—416no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—42no—
301C4×C61—24no—
302C4×C61—28no—
303C4×C61—44no—
304C4×C61C2×C4×D418yes—
305Dic61—24no—
306Dic61—24no—
307Dic61—44no—
308Dic61—44no—
309Dic61C2×#11718yescomplemented
310C22×C4×D4—611no—
311C22×#117—611no—
312C22×#117D6111yescharacteristic, complemented
313#2354—6116no—
314C2×#390—612no—
315C2×#388—612no—
316C2×#388—612no—
317C2×#384—611no—
318C2×#380—611no—
319C2×#380—611no—
320C2×C4×C4⋊C4—612no—
321C2×C4×C4⋊C4—611no—
322C2×C61—44no—
323C2×C61#235418yescomplemented
324C2×C61C22×#11711yescharacteristic
325C2×C61C22×C4×D411yescharacteristic
326C2×C61C2×#52011yescharacteristic
327C2×C61C2×#54711yescharacteristic
328C2×C61C2×#55011yescharacteristic
329C2×C61C2×#55511yescharacteristic
330C2×C61C2×#55611yescharacteristic
331D61—44no—
332C23×D4—611no—
333C24×C4—611no—
334C24×C4C2×D6111yescharacteristic
335C2×C4×D4—6116no—
336C2×C4×D4—618no—
337C2×#117—28no—
338C2×#117—618no—
339C2×#117—1228no—
340C2×#117C2×D6118yescomplemented
341C22×C4⋊C4—611no—
342C22×C4⋊C4—612no—
343C22×C4⋊C4—611no—
344C22×C4⋊C4—611no—
345C22×C4⋊C4—612no—
346C22×C4⋊C4C2×D6111yescharacteristic
347C22×C4⋊C4C2×D6111yescharacteristic
348C22×C22⋊C4—611no—
349C22×C22⋊C4—611no—
350C22×C22⋊C4—612no—
351C22×C22⋊C4—612no—
352C22×C22⋊C4—612no—
353C22×C22⋊C4C2×D6112yes—
354C22×C42—611no—
355C22×C42—612no—
356C22×C42—612no—
357C22×C42C2×D6112yes—
358#390—6116no—
359#388—6116no—
360#388—6116no—
361#384—618no—
362#380—618no—
363#380—618no—
364C4×C4⋊C4—618no—
365C4×C4⋊C4—6116no—
366C2×#95—612no—
367C2×#95—612no—
368C2×#95—612no—
369C61C2×#235411yescharacteristic, complemented, Sylow
370C25—611no—
371C25C22×D6111yescharacteristic
372C22×D4—618no—
373C22×D4—614no—
374C22×D4—1224no—
375C22×D4—1224no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—611no—
381C23×C4—618no—
382C23×C4—614no—
383C23×C4—612no—
384C23×C4—612no—
385C23×C4—1224no—
386C23×C4—1224no—
387C23×C4C4×D6118yes—
388C23×C4C22×D6112yes—
389C23×C4C22×D6114yes—
390C23×C4C22×D6111yescharacteristic
391C23×C4C22×D6112yes—
392C23×C4C22×D6111yescharacteristic
393C4×D4—6164no—
394#117—232no—
395#117—12232no—
396C2×C4⋊C4—24no—
397C2×C4⋊C4—618no—
398C2×C4⋊C4—614no—
399C2×C4⋊C4—614no—
400C2×C4⋊C4—614no—
401C2×C4⋊C4—618no—
402C2×C4⋊C4—618no—
403C2×C4⋊C4—1228no—
404C2×C4⋊C4—1224no—
405C2×C4⋊C4—1224no—
406C2×C4⋊C4—2444no—
407C2×C4⋊C4C22⋊D6118yescomplemented
408C2×C4⋊C4C22×D6114yes—
409C2×C4⋊C4C22×D6114yes—
410C2×C22⋊C4—28no—
411C2×C22⋊C4—618no—
412C2×C22⋊C4—614no—
413C2×C22⋊C4—618no—
414C2×C22⋊C4—614no—
415C2×C22⋊C4—618no—
416C2×C22⋊C4—618no—
417C2×C22⋊C4—618no—
418C2×C22⋊C4—1228no—
419C2×C22⋊C4—2444no—
420C2×C22⋊C4—2444no—
421C2×C22⋊C4C22×D6118yes—
422C2×C42—28no—
423C2×C42—618no—
424C2×C42—614no—
425C2×C42—618no—
426C2×C42—618no—
427C2×C42—1228no—
428C2×C42—1228no—
429C2×C42C22×D6118yes—
430#95—618no—
431#95—618no—
432#95—618no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—614no—
440C24—612no—
441C24—1222no—
442C24—1222no—
443C24—1222no—
444C24—1222no—
445C24—1222no—
446C24—1222no—
447C24C2×C4×D6114yes—
448C24C23×D6112yes—
449C24C23×D6111yescharacteristic
450C2×D4—6116no—
451C2×D4—12232no—
452C2×D4—1228no—
453C2×D4—1228no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—232no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—614no—
465C22×C4—614no—
466C22×C4—614no—
467C22×C4—612no—
468C22×C4—614no—
469C22×C4—614no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—6116no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4—1228no—
480C22×C4—1228no—
481C22×C4—12232no—
482C22×C4—1224no—
483C22×C4—1224no—
484C22×C4—1224no—
485C22×C4—1224no—
486C22×C4—1224no—
487C22×C4—1224no—
488C22×C4C2×C4×D6114yes—
489C22×C4C2×C4×D61116yes—
490C22×C4C2×C4×D6114yes—
491C22×C4#1175822214yes—
492C22×C4C2×C22⋊D6118yes—
493C22×C4C2×C22⋊D6114yes—
494C22×C4C23×D6114yes—
495C22×C4C23×D6112yes—
496C22×C4C23×D6114yes—
497C22×C4C23×D6112yes—
498C4⋊C4—28no—
499C4⋊C4—6116no—
500C4⋊C4—12216no—
501C4⋊C4—1228no—
502C4⋊C4—1228no—
503C4⋊C4—2448no—
504C4⋊C4C2×C22⋊D61116yescomplemented
505C22⋊C4—216no—
506C22⋊C4—6116no—
507C22⋊C4—6116no—
508C22⋊C4—12216no—
509C22⋊C4—2448no—
510C22⋊C4—2448no—
511C42—216no—
512C42—6116no—
513C42—12216no—
514C42—12216no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—614no—
529C23—1228no—
530C23—1228no—
531C23—1228no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23—1222no—
537C23—1222no—
538C23—2444no—
539C23—2444no—
540C23—2444no—
541C23—2444no—
542C23C22×C4×D6114yes—
543C23C22×C4×D6112yes—
544C23C22×C22⋊D6112yes—
545C23#41×D6112yes—
546C23#41×D6112yes—
547C23C2×#1175822212yes—
548C23#1451982312yes—
549C23#1451982412yes—
550C23C24×D6111yescharacteristic
551D4—12232no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—216no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—6116no—
564C2×C4—614no—
565C2×C4—614no—
566C2×C4—614no—
567C2×C4—614no—
568C2×C4—614no—
569C2×C4—1228no—
570C2×C4—1228no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1222no—
575C2×C4—12232no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1224no—
580C2×C4—1224no—
581C2×C4—1224no—
582C2×C4—2448no—
583C2×C4—2448no—
584C2×C4C4×C22⋊D61116yes—
585C2×C4C22×C4×D6114yes—
586C2×C4C22×C4×D6114yes—
587C2×C4C22×C22⋊D6118yes—
588C2×C4C22×C22⋊D6114yes—
589C2×C4C2×#1175822214yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—416no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—1224no—
599C22—1224no—
600C22—1224no—
601C22—24416no—
602C22—2442no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22#117×D6114yes—
607C22C2×C4×C22⋊D6114yes—
608C22#1684024214yes—
609C22#1684041114yes—
610C22#1684043114yes—
611C22#1684047514yes—
612C22#1684048014yes—
613C22C23×C4×D6111yescharacteristic
614C22C23×C22⋊D6111yescharacteristic
615C22C2×#41×D6111yescharacteristic
616C22C2×#41×D6111yescharacteristic
617C22C22×#1175822211yescharacteristic
618C22C2×#1451982311yescharacteristic
619C22C2×#1451982411yescharacteristic
620C4—24no—
621C4—28no—
622C4—44no—
623C4—618no—
624C4—1224no—
625C4—1224no—
626C4—2444no—
627C4—2444no—
628C4C2×C4×C22⋊D6118yes—
629C2—44no—
630C2—2444no—
631C2#2144876918yesdirect product factor
632C2C2×#117×D6111yescharacteristic
633C2C22×C4×C22⋊D6111yescharacteristic
634C2C2×#1684024211yescharacteristic
635C2C2×#1684041111yescharacteristic
636C2C2×#1684043111yescharacteristic
637C2C2×#1684047511yescharacteristic
638C2C2×#1684048011yescharacteristic
subgroupG/H
centerC24C23×D61
commutatorC22×C61C24×C4
FrattiniC23C24×D61
FittingC22×#117×C61C2
socleC24×C61C24
prime radicalC23×C22⋊D61C22
cosocleC23×C61C25
hypercenterC22×#117D61
hypocenterC61C2×#2354
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C22×#14519735C2semidirect
3. #21448769C2direct (N × H)
5. C2×#16839653C2semidirect
6. C2×#16839650C2semidirect
7. C2×#16839631C2semidirect
8. C2×#16839622C2semidirect
9. C2×#16839619C2semidirect
10. C2×C4⋊C4×Dic61C2semidirect
11. C2×C4×#11758196C2semidirect
12. C22×#117×C61C2semidirect
15. C2×C4×C22⋊D61C4semidirect
18. C2×#14519735C22semidirect
33. #16839653C22semidirect
34. #16839650C22semidirect
35. #16839631C22semidirect
36. #16839622C22semidirect
37. #16839619C22semidirect
38. C4⋊C4×Dic61C22semidirect
39. C4×#11758196C22semidirect
45. C2×#117×C61C22semidirect
64. C4×C22⋊D61C2×C4semidirect
106. C2×C4⋊C4×C61D4semidirect
179. C4⋊C4×C61C2×D4semidirect
263. C2×Dic61#117semidirect
309. Dic61C2×#117semidirect
312. C22×#117D61semidirect
323. C2×C61#2354semidirect
340. C2×#117C2×D61semidirect
369. C61C2×#2354semidirect
407. C2×C4⋊C4C22⋊D61semidirect
504. C4⋊C4C2×C22⋊D61semidirect
631. C2#21448769direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D612
C22×#145197352
#214487692
C2×#168396622
C2×#168396532
C2×#168396502
C2×#168396312
C2×#168396222
C2×#168396192
C2×C4⋊C4×Dic612
C2×C4×#117581962
C22×#117×C612
C2×#235461