Finite Group Databasebeta

← C2×#21450116 #29849530

Subgroups — C2×#21450116 #29849530

Order 15616.

3070 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519736C211yesmaximal, characteristic, complemented
2#21450116C2116yesmaximal, direct product factor
3C2×#16839690C212yesmaximal, complemented
4C2×#16839676C212yesmaximal
5C2×#16839660C211yesmaximal, characteristic, complemented
6C2×#16839660C211yesmaximal, characteristic, complemented
7C2×#16839650C212yesmaximal, complemented
8C2×#16839643C212yesmaximal, complemented
9C2×#16839622C211yesmaximal, characteristic, complemented
10C2×#16839622C211yesmaximal, characteristic, complemented
11C2×#16839622C211yesmaximal, characteristic, complemented
12C22×C4×D4×C61C211yesmaximal, characteristic
13C2×#14519736—28no—
14C2×#14519736C2218yescomplemented
15C22×C4⋊Dic61C2211yescharacteristic
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758196C2212yes—
20C22×#11758196C2212yes—
21C22×C22⋊Dic61C2212yes—
22C22×C22⋊Dic61C2212yes—
23C22×C22⋊Dic61C2212yes—
24C22×C22⋊Dic61C2212yes—
25C22×C4×Dic61C2212yes—
26#16839690C22116yescomplemented
27#16839676C22116yes—
28#16839660C2218yescomplemented
29#16839660C2218yescomplemented
30#16839650C22116yescomplemented
31#16839643C22116yescomplemented
32#16839622C2218yescomplemented
33#16839622C2218yescomplemented
34#16839622C2218yescomplemented
35C2×#14519684C2212yes—
36C2×#14519684C2212yes—
37C2×#14519684C2212yes—
38C2×#14519684C2212yes—
39C2×#14519684C2212yes—
40C23×D4×C61C2211yescharacteristic
41C24×C4×C61C2211yescharacteristic
42C24×C4×C61C2211yescharacteristic
43C2×C4×D4×C61—28no—
44C2×C4×D4×C61C2218yes—
45C22×C4⋊C4×C61C2211yescharacteristic
46C22×C22⋊C4×C61C2211yescharacteristic
47C22×C22⋊C4×C61C2211yescharacteristic
48C22×C42×C61C2211yescharacteristic
49C23×Dic61C2312yes—
50C23×Dic61C2312yes—
51C23×Dic61C2312yes—
52C23×Dic61C2312yes—
53#14519736—232no—
54C2×C4⋊Dic61—24no—
55C2×C4⋊Dic61—24no—
56C2×C4⋊Dic61—24no—
57C2×C4⋊Dic61—24no—
58C2×C4⋊Dic61C2314yes—
59C2×C4⋊Dic61C2314yes—
60C2×C4⋊Dic61C2314yes—
61C2×C4⋊Dic61C2314yes—
62C2×#11758196—28no—
63C2×#11758196—44no—
64C2×#11758196C2318yes—
65C2×#11758196C2318yes—
66C2×C22⋊Dic61—28no—
67C2×C22⋊Dic61—28no—
68C2×C22⋊Dic61—44no—
69C2×C22⋊Dic61—44no—
70C2×C22⋊Dic61C2318yes—
71C2×C22⋊Dic61C2318yes—
72C2×C22⋊Dic61C2318yes—
73C2×C22⋊Dic61C2318yes—
74C2×C4×Dic61—28no—
75C2×C4×Dic61C2318yes—
76#14519684C2318yes—
77#14519684C2318yes—
78#14519684C2318yes—
79#14519684C2318yes—
80#14519684C2318yes—
81C25×C61C2311yescharacteristic
82C25×C61C2311yescharacteristic
83C22×D4×C61—24no—
84C22×D4×C61—24no—
85C22×D4×C61—24no—
86C22×D4×C61C2314yes—
87C23×C4×C61—24no—
88C23×C4×C61—24no—
89C23×C4×C61—24no—
90C23×C4×C61—24no—
91C23×C4×C61—24no—
92C23×C4×C61D418yes—
93C23×C4×C61C2311yescharacteristic
94C23×C4×C61C2311yescharacteristic
95C23×C4×C61C2314yes—
96C23×C4×C61C2311yescharacteristic
97C23×C4×C61C2314yes—
98C23×C4×C61C2311yescharacteristic
99C23×C4×C61C2311yescharacteristic
100C4×D4×C61—232no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61C2314yes—
103C2×C22⋊C4×C61—24no—
104C2×C22⋊C4×C61—24no—
105C2×C22⋊C4×C61C2314yes—
106C2×C22⋊C4×C61C2314yes—
107C2×C42×C61—24no—
108C2×C42×C61C2314yes—
109C24×C61—22no—
110C24×C61—22no—
111C24×C61—22no—
112C24×C61—22no—
113C24×C61—22no—
114C24×C61—22no—
115C24×C61—22no—
116C24×C61—22no—
117C24×C61—22no—
118C24×C61—22no—
119C24×C61—22no—
120C24×C61—22no—
121C24×C61C2×D414yes—
122C24×C61#4114yes—
123C24×C61C2412yes—
124C24×C61C2412yes—
125C24×C61C2411yescharacteristic
126C2×D4×C61—232no—
127C2×D4×C61—28no—
128C2×D4×C61—28no—
129C2×D4×C61—28no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—232no—
135C22×C4×C61—28no—
136C22×C4×C61—28no—
137C22×C4×C61—22no—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—28no—
144C22×C4×C61—28no—
145C22×C4×C61—28no—
146C22×C4×C61—416no—
147C22×C4×C61C2×D414yes—
148C22×C4×C61C2×D4116yes—
149C22×C4×C61C2×D414yes—
150C22×C4×C61#4114yes—
151C22×C4×C61#4114yes—
152C22×C4×C61#4114yes—
153C22×C4×C61C2412yes—
154C22×C4×C61C2412yes—
155C22×C4×C61C2412yes—
156C22×C4×C61C2412yes—
157C22×C4×C61C2412yes—
158C22×Dic61—24no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—24no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—24no—
165C22×Dic61—24no—
166C22×Dic61—24no—
167C22×Dic61—24no—
168C22×Dic61—24no—
169C22×Dic61—24no—
170C22×Dic61C2414yes—
171C22×Dic61C2414yes—
172C22×Dic61C2414yes—
173C22×Dic61C2414yes—
174C4⋊C4×C61—28no—
175C22⋊C4×C61—28no—
176C22⋊C4×C61—28no—
177C42×C61—28no—
178C22⋊Dic61—216no—
179C22⋊Dic61—216no—
180C22⋊Dic61—48no—
181C22⋊Dic61—48no—
182C4⋊Dic61—28no—
183C4⋊Dic61—28no—
184C4⋊Dic61—28no—
185C4⋊Dic61—28no—
186#11758196—216no—
187#11758196—48no—
188C4×Dic61—216no—
189C23×C61—28no—
190C23×C61—28no—
191C23×C61—28no—
192C23×C61—28no—
193C23×C61—28no—
194C23×C61—28no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61—22no—
204C23×C61—22no—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—44no—
208C23×C61—44no—
209C23×C61—44no—
210C23×C61—44no—
211C23×C61—44no—
212C23×C61—44no—
213C23×C61—44no—
214C23×C61—44no—
215C23×C61C22×D414yes—
216C23×C61C22×D412yes—
217C23×C61C2×#4114yes—
218C23×C61C2×#4112yes—
219C23×C61C2×#4112yes—
220C23×C61C2×#4112yes—
221C23×C61C2×#4112yes—
222C23×C61H(5,𝔽2)12yes—
223C23×C612−1+412yes—
224C23×C61C2511yescharacteristic
225D4×C61—232no—
226C2×C4×C61—28no—
227C2×C4×C61—28no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—232no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—44no—
240C2×C4×C61—44no—
241C2×C4×C61—416no—
242C2×C4×C61#123116yes—
243C2×C4×C61C22×D414yes—
244C2×C4×C61C22×D414yes—
245C2×C4×C61C2×#4114yes—
246C2×C4×C61C2×#4114yes—
247C2×C4×C61C2×#4114yes—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—24no—
252C2×Dic61—24no—
253C2×Dic61—24no—
254C2×Dic61—24no—
255C2×Dic61—24no—
256C2×Dic61—24no—
257C2×Dic61—24no—
258C2×Dic61—24no—
259C2×Dic61—24no—
260C2×Dic61—48no—
261C2×Dic61—48no—
262C2×Dic61—48no—
263C2×Dic61—48no—
264C2×#2485—611nomaximal, Sylow
265C22×C61—24no—
266C22×C61—24no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—24no—
270C22×C61—24no—
271C22×C61—42no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61—42no—
279C22×C61—416no—
280C22×C61—416no—
281C22×C61C2×#12314yes—
282C22×C61#52514yes—
283C22×C61#53714yes—
284C22×C61#55214yes—
285C22×C61#55314yes—
286C22×C61#55514yes—
287C22×C61#55614yes—
288C22×C61C23×D411yescharacteristic
289C22×C61C22×#4111yescharacteristic
290C22×C61C22×#4111yescharacteristic
291C22×C61C22×#4111yescharacteristic
292C22×C61C22×#4111yescharacteristic
293C22×C61C2×H(5,𝔽2)11yescharacteristic
294C22×C61C2×2−1+411yescharacteristic
295C4×C61—24no—
296C4×C61—24no—
297C4×C61—42no—
298C4×C61—42no—
299C4×C61C2×#12318yes—
300Dic61—44no—
301Dic61—44no—
302Dic61—44no—
303Dic61—44no—
304C22×C4×D4D6111yescharacteristic
305C22×#117—611no—
306#2485—6116no—
307C2×#397—612no—
308C2×#394—612no—
309C2×#389—611no—
310C2×#389—611no—
311C2×#388—612no—
312C2×#387—612no—
313C2×#380—611no—
314C2×#380—611no—
315C2×#380—611no—
316C2×C61—44no—
317C2×C61—44no—
318C2×C61#248518yescomplemented
319C2×C61C22×#12311yescharacteristic
320C2×C61C2×#52511yescharacteristic
321C2×C61C2×#53711yescharacteristic
322C2×C61C2×#55211yescharacteristic
323C2×C61C2×#55311yescharacteristic
324C2×C61C2×#55511yescharacteristic
325C2×C61C2×#55611yescharacteristic
326C23×D4C2×D6111yescharacteristic
327C24×C4C2×D6111yescharacteristic
328C24×C4C2×D6111yescharacteristic
329C2×C4×D4—28no—
330C2×C4×D4C2×D6118yes—
331C2×#117—618no—
332C2×#117—1228no—
333C22×C4⋊C4—612no—
334C22×C4⋊C4—611no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—611no—
337C22×C4⋊C4—611no—
338C22×C4⋊C4—612no—
339C22×C4⋊C4C2×D6111yescharacteristic
340C22×C22⋊C4—612no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—612no—
343C22×C22⋊C4—612no—
344C22×C22⋊C4C2×D6111yescharacteristic
345C22×C22⋊C4C2×D6111yescharacteristic
346C22×C42—612no—
347C22×C42C2×D6111yescharacteristic
348#397—6116no—
349#394—6116no—
350#389—618no—
351#389—618no—
352#388—6116no—
353#387—6116no—
354#380—618no—
355#380—618no—
356#380—618no—
357C2×#95—612no—
358C2×#95—612no—
359C2×#95—612no—
360C2×#95—612no—
361C2×#95—612no—
362C61C2×#248511yescharacteristic, complemented, Sylow
363C25C22×D6111yescharacteristic
364C25C22×D6111yescharacteristic
365C22×D4—24no—
366C22×D4—24no—
367C22×D4—24no—
368C22×D4C22×D6114yes—
369C23×C4—24no—
370C23×C4—24no—
371C23×C4—24no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—612no—
375C23×C4—612no—
376C23×C4—612no—
377C23×C4—612no—
378C23×C4C22⋊D6118yes—
379C23×C4C22×D6114yes—
380C23×C4C22×D6111yescharacteristic
381C23×C4C22×D6111yescharacteristic
382C23×C4C22×D6111yescharacteristic
383C23×C4C22×D6114yes—
384C23×C4C22×D6111yescharacteristic
385C23×C4C22×D6111yescharacteristic
386C4×D4—232no—
387#117—12232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—614no—
390C2×C4⋊C4—614no—
391C2×C4⋊C4—614no—
392C2×C4⋊C4—614no—
393C2×C4⋊C4—618no—
394C2×C4⋊C4—618no—
395C2×C4⋊C4—1228no—
396C2×C4⋊C4—1224no—
397C2×C4⋊C4—1224no—
398C2×C4⋊C4—1224no—
399C2×C4⋊C4—1224no—
400C2×C4⋊C4—2444no—
401C2×C4⋊C4C22×D6114yes—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—618no—
405C2×C22⋊C4—618no—
406C2×C22⋊C4—618no—
407C2×C22⋊C4—618no—
408C2×C22⋊C4—1228no—
409C2×C22⋊C4—1228no—
410C2×C22⋊C4—2444no—
411C2×C22⋊C4—2444no—
412C2×C22⋊C4C22×D6114yes—
413C2×C22⋊C4C22×D6114yes—
414C2×C42—24no—
415C2×C42—618no—
416C2×C42—1228no—
417C2×C42C22×D6114yes—
418#95—618no—
419#95—618no—
420#95—618no—
421#95—618no—
422#95—618no—
423C24—22no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—22no—
429C24—22no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24#1175822214yes—
436C24C2×C22⋊D6114yes—
437C24C23×D6112yes—
438C24C23×D6112yes—
439C24C23×D6111yescharacteristic
440C2×D4—232no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—232no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—416no—
461C22×C4—614no—
462C22×C4—614no—
463C22×C4—614no—
464C22×C4—614no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1224no—
468C22×C4—1224no—
469C22×C4—1224no—
470C22×C4—1224no—
471C22×C4—1224no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—1224no—
476C22×C4—1224no—
477C22×C4#1175822214yes—
478C22×C4#1175822414yes—
479C22×C4#1175822414yes—
480C22×C4C2×C22⋊D6114yes—
481C22×C4C2×C22⋊D61116yes—
482C22×C4C2×C22⋊D6114yes—
483C22×C4C23×D6112yes—
484C22×C4C23×D6112yes—
485C22×C4C23×D6112yes—
486C22×C4C23×D6112yes—
487C22×C4C23×D6112yes—
488C4⋊C4—28no—
489C4⋊C4—1228no—
490C4⋊C4—1228no—
491C4⋊C4—1228no—
492C4⋊C4—1228no—
493C4⋊C4—12216no—
494C4⋊C4—2448no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—12216no—
498C22⋊C4—12216no—
499C22⋊C4—2448no—
500C22⋊C4—2448no—
501C42—28no—
502C42—12216no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C22×C22⋊D6114yes—
530C23C22×C22⋊D6112yes—
531C23C2×#1175822412yes—
532C23C2×#1175822412yes—
533C23C2×#1175822214yes—
534C23C2×#1175822212yes—
535C23C2×#1175822212yes—
536C23#1451982312yes—
537C23#1451982412yes—
538C23C24×D6111yescharacteristic
539D4—232no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—232no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—416no—
556C2×C4—1224no—
557C2×C4—1224no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1224no—
565C2×C4—1224no—
566C2×C4—1224no—
567C2×C4—1224no—
568C2×C4—2448no—
569C2×C4—2448no—
570C2×C4—2448no—
571C2×C4—2448no—
572C2×C4#14519758116yes—
573C2×C4C22×C22⋊D6114yes—
574C2×C4C22×C22⋊D6114yes—
575C2×C4C2×#1175822414yes—
576C2×C4C2×#1175822414yes—
577C2×C4C2×#1175822214yes—
578C22—24no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—42no—
585C22—42no—
586C22—416no—
587C22—416no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22C2×#1451975814yes—
595C22#1684027314yes—
596C22#1684034014yes—
597C22#1684045214yes—
598C22#1684045814yes—
599C22#1684047714yes—
600C22#1684048214yes—
601C22C23×C22⋊D6111yescharacteristic
602C22C22×#1175822411yescharacteristic
603C22C22×#1175822411yescharacteristic
604C22C22×#1175822211yescharacteristic
605C22C22×#1175822211yescharacteristic
606C22C2×#1451982311yescharacteristic
607C22C2×#1451982411yescharacteristic
608C4—24no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—2444no—
613C4—2444no—
614C4—2444no—
615C4—2444no—
616C4C2×#1451975818yes—
617C2—44no—
618C2—44no—
619C2#2145011618yesdirect product factor
620C2C22×#1451975811yescharacteristic
621C2C2×#1684027311yescharacteristic
622C2C2×#1684034011yescharacteristic
623C2C2×#1684045211yescharacteristic
624C2C2×#1684045811yescharacteristic
625C2C2×#1684047711yescharacteristic
626C2C2×#1684048211yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×C4×D4×C61C2
socleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC22×C4×D4D61
hypocenterC61C2×#2485
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14519736C2semidirect
2. #21450116C2direct (N × H)
3. C2×#16839690C2semidirect
5. C2×#16839660C2semidirect
6. C2×#16839660C2semidirect
7. C2×#16839650C2semidirect
8. C2×#16839643C2semidirect
9. C2×#16839622C2semidirect
10. C2×#16839622C2semidirect
11. C2×#16839622C2semidirect
14. C2×#14519736C22semidirect
26. #16839690C22semidirect
28. #16839660C22semidirect
29. #16839660C22semidirect
30. #16839650C22semidirect
31. #16839643C22semidirect
32. #16839622C22semidirect
33. #16839622C22semidirect
34. #16839622C22semidirect
318. C2×C61#2485semidirect
362. C61C2×#2485semidirect
619. C2#21450116direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197362
#214501162
C2×#168396902
C2×#168396762
C2×#168396602
C2×#168396502
C2×#168396432
C2×#168396222
C22×C4×D4×C612
C2×#248561