Finite Group Databasebeta

← C2×#21449226 #29848715

Subgroups — C2×#21449226 #29848715

Order 15616.

3570 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
1C22×#14519760C211yesmaximal, characteristic
2C22×#14519760C211yesmaximal, characteristic
3C22×#14519736C211yesmaximal, characteristic, complemented
4#21449226C2116yesmaximal, direct product factor
5C2×#16839695C212yesmaximal, complemented
6C2×#16839684C212yesmaximal, complemented
7C2×#16839657C212yesmaximal, complemented
8C2×#16839652C212yesmaximal, complemented
9C2×#16839641C211yesmaximal, characteristic
10C2×#16839635C211yesmaximal, characteristic, complemented
11C2×#16839625C211yesmaximal, characteristic, complemented
12C23×C4⋊C4×C61C211yesmaximal, characteristic, complemented
13C23×C22⋊D61C2211yescharacteristic
14C2×#14519760—28no—
15C2×#14519760—28no—
16C2×#14519760C2218yes—
17C2×#14519760C2218yes—
18C2×#14519736—28no—
19C2×#14519736C2218yescomplemented
20C22×C4⋊Dic61C2212yes—
21C22×#11758196C2212yes—
22C22×#11758196C2211yescharacteristic
23C22×#11758196C2212yes—
24C22×#11758196C2211yescharacteristic
25C22×C22⋊Dic61C2211yescharacteristic
26C22×C22⋊Dic61C2211yescharacteristic
27C22×C22⋊Dic61C2211yescharacteristic
28C22×#11758198C2212yes—
29C22×#11758198C2211yescharacteristic
30C22×#11758198C2212yes—
31C22×#11758198C2211yescharacteristic
32C22×C4×Dic61C2212yes—
33#16839695C22116yescomplemented
34#16839684C22116yescomplemented
35#16839657C22116yescomplemented
36#16839652C22116yescomplemented
37#16839641C2218yes—
38#16839635C2218yescomplemented
39#16839625C2218yescomplemented
40C2×#14519684C2212yes—
41C2×#14519684C2212yes—
42C2×#14519684C2212yes—
43C2×#14519684C2212yes—
44C24×C4×C61C2211yescharacteristic
45C24×C4×C61C2211yescharacteristic
46C24×C4×C61C2211yescharacteristic
47C22×C4⋊C4×C61—28no—
48C22×C4⋊C4×C61C2212yes—
49C22×C4⋊C4×C61C2212yes—
50C22×C4⋊C4×C61C2218yescomplemented
51C24×D61C2311yescharacteristic
52C22×C22⋊D61—24no—
53C22×C22⋊D61—24no—
54C22×C22⋊D61—24no—
55C22×C22⋊D61C2314yes—
56C23×Dic61C2312yes—
57C23×Dic61C2311yescharacteristic
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C23×Dic61C2312yes—
61#14519760—232no—
62#14519760—232no—
63#14519736—232no—
64C2×C4⋊Dic61—28no—
65C2×C4⋊Dic61C2318yes—
66C2×#11758196—24no—
67C2×#11758196—24no—
68C2×#11758196—28no—
69C2×#11758196—28no—
70C2×#11758196C2318yes—
71C2×#11758196C2314yes—
72C2×#11758196C2318yes—
73C2×#11758196C2314yes—
74C2×C22⋊Dic61—24no—
75C2×C22⋊Dic61—24no—
76C2×C22⋊Dic61D418yes—
77C2×C22⋊Dic61C2314yes—
78C2×C22⋊Dic61C2314yes—
79C2×C22⋊Dic61C2314yes—
80C2×#11758198—24no—
81C2×#11758198—24no—
82C2×#11758198—28no—
83C2×#11758198—28no—
84C2×#11758198C2314yes—
85C2×#11758198C2318yes—
86C2×#11758198C2318yes—
87C2×#11758198C2314yes—
88C2×C4×Dic61—28no—
89C2×C4×Dic61C2318yes—
90#14519684C2318yes—
91#14519684C2318yes—
92#14519684C2318yes—
93#14519684C2318yes—
94C25×C61C2311yescharacteristic
95C23×C4×C61—24no—
96C23×C4×C61—24no—
97C23×C4×C61—24no—
98C23×C4×C61—28no—
99C23×C4×C61—24no—
100C23×C4×C61—24no—
101C23×C4×C61—24no—
102C23×C4×C61D418yes—
103C23×C4×C61C2312yes—
104C23×C4×C61C2314yes—
105C23×C4×C61C2314yes—
106C23×C4×C61C2311yescharacteristic
107C23×C4×C61C2311yescharacteristic
108C23×C4×C61C2314yes—
109C23×C4×C61C2312yes—
110C2×C4⋊C4×C61—28no—
111C2×C4⋊C4×C61—28no—
112C2×C4⋊C4×C61—232no—
113C2×C4⋊C4×C61C2318yes—
114C2×C4⋊C4×C61C2318yes—
115C24×C61—24no—
116C24×C61—24no—
117C24×C61C2×D414yes—
118C24×C61C2×D414yes—
119C24×C61#4114yes—
120C24×C61C2412yes—
121C24×C61C2411yescharacteristic
122C23×D61—22no—
123C23×D61—22no—
124C23×D61—22no—
125C23×D61—24no—
126C23×D61—24no—
127C23×D61C2412yes—
128C22×C4×C61—232no—
129C22×C4×C61—232no—
130C22×C4×C61—28no—
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—28no—
135C22×C4×C61—28no—
136C22×C4×C61—22no—
137C22×C4×C61—22no—
138C22×C4×C61—24no—
139C22×C4×C61—216no—
140C22×C4×C61—28no—
141C22×C4×C61—24no—
142C22×C4×C61—24no—
143C22×C4×C61—28no—
144C22×C4×C61—28no—
145C22×C4×C61—24no—
146C22×C4×C61—416no—
147C22×C4×C61C2×D414yes—
148C22×C4×C61C2×D414yes—
149C22×C4×C61C2×D4116yes—
150C22×C4×C61#4118yes—
151C22×C4×C61C2414yes—
152C22×C4×C61C2412yes—
153C22×C4×C61C2412yes—
154C22×C4×C61C2414yes—
155C2×C22⋊D61—28no—
156C2×C22⋊D61—28no—
157C2×C22⋊D61—28no—
158C2×C22⋊D61—416no—
159C22×Dic61—24no—
160C22×Dic61—24no—
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—24no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—22no—
167C22×Dic61—24no—
168C22×Dic61—24no—
169C22×Dic61—24no—
170C22×Dic61—24no—
171C22×Dic61C2×D418yes—
172C22×Dic61C2412yes—
173C22×Dic61C2412yes—
174C22×Dic61C2414yes—
175C22×Dic61C2412yes—
176C22×Dic61C2414yes—
177C4⋊C4×C61—216no—
178C4⋊C4×C61—216no—
179C22⋊Dic61—28no—
180C22⋊Dic61—28no—
181C22⋊Dic61C2×D4116yes—
182#11758198—28no—
183#11758198—28no—
184#11758198—216no—
185#11758198—216no—
186C4⋊Dic61—216no—
187#11758196—28no—
188#11758196—28no—
189#11758196—216no—
190#11758196—216no—
191C4×Dic61—216no—
192C23×C61—28no—
193C23×C61—28no—
194C23×C61—216no—
195C23×C61—216no—
196C23×C61—24no—
197C23×C61—24no—
198C23×C61#121116yes—
199C23×C61C22×D414yes—
200C23×C61C22×D414yes—
201C23×C61C22×D412yes—
202C23×C61C22×D412yes—
203C23×C61C2×#4114yes—
204C23×C61C2×#4112yes—
205C23×C61C2×#4114yes—
206C23×C612−1+414yes—
207C23×C61C2511yescharacteristic
208C22×D61—22no—
209C22×D61—22no—
210C22×D61—24no—
211C22×D61—22no—
212C22×D61—24no—
213C22×D61—44no—
214C22×D61—44no—
215C22×D61—44no—
216C22×D61—48no—
217C22×D61—48no—
218C2×C4×C61—232no—
219C2×C4×C61—232no—
220C2×C4×C61—28no—
221C2×C4×C61—28no—
222C2×C4×C61—216no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—24no—
229C2×C4×C61—24no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—416no—
233C2×C4×C61—48no—
234C2×C4×C61C22×D414yes—
235C2×C4×C61C22×D414yes—
236C2×C4×C61C2×#4118yes—
237C22⋊D61—416no—
238C2×Dic61—216no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—22no—
249C2×Dic61—24no—
250C2×Dic61—24no—
251C2×Dic61—44no—
252C2×Dic61—44no—
253C2×Dic61—44no—
254C2×Dic61—48no—
255C2×Dic61C22×D418yes—
256C2×#2413—611nomaximal, Sylow
257C22×C61—24no—
258C22×C61—232no—
259C22×C61—24no—
260C22×C61—28no—
261C22×C61—28no—
262C22×C61C2×#12118yes—
263C22×C61C2×#12114yes—
264C22×C61#52514yes—
265C22×C61#53614yes—
266C22×C61#54718yes—
267C22×C61#55218yes—
268C22×C61C23×D411yescharacteristic
269C22×C61C23×D411yescharacteristic
270C22×C61C22×#4111yescharacteristic
271C22×C61C22×#4112yes—
272C22×C61C2×2−1+412yes—
273C2×D61—42no—
274C2×D61—42no—
275C2×D61—42no—
276C2×D61—44no—
277C2×D61—44no—
278C2×D61—88no—
279C4×C61—24no—
280C4×C61—24no—
281C4×C61—28no—
282C4×C61—44no—
283Dic61—28no—
284Dic61—42no—
285Dic61—42no—
286Dic61—42no—
287Dic61—44no—
288C22×#123—611no—
289C22×#123—611no—
290C22×#117—611no—
291C23×C4⋊C4D6111yescharacteristic, complemented
292#2413—6116no—
293C2×#398—612no—
294C2×#396—612no—
295C2×#389—612no—
296C2×#388—612no—
297C2×#386—611no—
298C2×#385—611no—
299C2×#381—611no—
300C2×C61—28no—
301C2×C61#241318yescomplemented
302C2×C61C22×#12111yescharacteristic
303C2×C61C2×#52511yescharacteristic
304C2×C61C2×#53611yescharacteristic
305C2×C61C2×#54712yes—
306C2×C61C2×#55212yes—
307D61—82no—
308C23×D4—611no—
309C24×C4C2×D6111yescharacteristic
310C24×C4C2×D6111yescharacteristic
311C24×C4C2×D6111yescharacteristic
312C2×#123—618no—
313C2×#123—618no—
314C2×#123—1228no—
315C2×#123—1228no—
316C2×#117—618no—
317C2×#117—1228no—
318C22×C4⋊C4—28no—
319C22×C4⋊C4—612no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4—612no—
322C22×C4⋊C4—611no—
323C22×C4⋊C4—612no—
324C22×C4⋊C4C2×D6112yes—
325C22×C4⋊C4C2×D6112yes—
326C22×C4⋊C4C2×D6118yescomplemented
327C22×C22⋊C4—611no—
328C22×C22⋊C4—611no—
329C22×C22⋊C4—611no—
330C22×C22⋊C4—611no—
331C22×C22⋊C4—612no—
332C22×C22⋊C4—612no—
333C22×C22⋊C4—611no—
334C22×C42—612no—
335#398—6116no—
336#396—6116no—
337#389—6116no—
338#388—6116no—
339#386—618no—
340#385—618no—
341#381—618no—
342C2×#95—612no—
343C2×#95—612no—
344C2×#95—612no—
345C2×#95—612no—
346C61C2×#241311yescharacteristic, complemented, Sylow
347C25—611no—
348C25C22×D6111yescharacteristic
349C22×D4—614no—
350C22×D4—1224no—
351C22×D4—1224no—
352C22×D4—1224no—
353C23×C4—24no—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—28no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—24no—
360C23×C4—612no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—612no—
364C23×C4—611no—
365C23×C4C22⋊D6118yes—
366C23×C4C22×D6114yes—
367C23×C4C22×D6114yes—
368C23×C4C22×D6112yes—
369C23×C4C22×D6112yes—
370C23×C4C22×D6114yes—
371C23×C4C22×D6111yescharacteristic
372C23×C4C22×D6111yescharacteristic
373#123—12232no—
374#123—12232no—
375#117—12232no—
376C2×C4⋊C4—28no—
377C2×C4⋊C4—28no—
378C2×C4⋊C4—232no—
379C2×C4⋊C4—618no—
380C2×C4⋊C4—614no—
381C2×C4⋊C4—618no—
382C2×C4⋊C4—614no—
383C2×C4⋊C4—618no—
384C2×C4⋊C4—1228no—
385C2×C4⋊C4—1228no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1224no—
389C2×C4⋊C4C22×D6118yes—
390C2×C4⋊C4C22×D6118yes—
391C2×C22⋊C4—618no—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—614no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—614no—
396C2×C22⋊C4—618no—
397C2×C22⋊C4—614no—
398C2×C22⋊C4—618no—
399C2×C22⋊C4—1228no—
400C2×C22⋊C4—1224no—
401C2×C22⋊C4—1224no—
402C2×C22⋊C4—1228no—
403C2×C22⋊C4—1224no—
404C2×C22⋊C4—1224no—
405C2×C42—618no—
406C2×C42—1228no—
407#95—618no—
408#95—618no—
409#95—618no—
410#95—618no—
411C24—24no—
412C24—24no—
413C24—612no—
414C24—1222no—
415C24—1222no—
416C24—1222no—
417C24—1224no—
418C24—1224no—
419C24D4×D6114yes—
420C24#1175822414yes—
421C24C2×C22⋊D6114yes—
422C24C23×D6112yes—
423C24C23×D6111yescharacteristic
424C2×D4—1228no—
425C2×D4—1228no—
426C2×D4—1228no—
427C2×D4—24416no—
428C22×C4—232no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—232no—
439C22×C4—216no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—24no—
446C22×C4—416no—
447C22×C4—618no—
448C22×C4—614no—
449C22×C4—612no—
450C22×C4—614no—
451C22×C4—612no—
452C22×C4—612no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—1224no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1224no—
462C22×C4—1224no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4#1175822218yes—
466C22×C4C2×C22⋊D6114yes—
467C22×C4C2×C22⋊D6114yes—
468C22×C4C2×C22⋊D61116yes—
469C22×C4C23×D6114yes—
470C22×C4C23×D6112yes—
471C22×C4C23×D6112yes—
472C22×C4C23×D6114yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—12216no—
476C4⋊C4—12216no—
477C4⋊C4—1228no—
478C4⋊C4—1228no—
479C4⋊C4—12216no—
480C22⋊C4—6116no—
481C22⋊C4—12216no—
482C22⋊C4—1228no—
483C22⋊C4—1228no—
484C22⋊C4—1228no—
485C22⋊C4—1228no—
486C22⋊C4—12216no—
487C42—12216no—
488C23—28no—
489C23—24no—
490C23—28no—
491C23—24no—
492C23—216no—
493C23—216no—
494C23—1222no—
495C23—1222no—
496C23—1222no—
497C23—1224no—
498C23—1224no—
499C23—2444no—
500C23—2444no—
501C23—2444no—
502C23—2448no—
503C23—2448no—
504C23#14519749116yes—
505C23C2×D4×D6114yes—
506C23C2×D4×D6112yes—
507C23C22×C22⋊D6114yes—
508C23C22×C22⋊D6112yes—
509C23C2×#1175822414yes—
510C23C2×#1175822412yes—
511C23C2×#1175822214yes—
512C23#1451982514yes—
513C23C24×D6111yescharacteristic
514D4—24416no—
515C2×C4—232no—
516C2×C4—232no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—216no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—416no—
530C2×C4—48no—
531C2×C4—618no—
532C2×C4—1224no—
533C2×C4—1224no—
534C2×C4—12216no—
535C2×C4—1224no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2444no—
548C2×C4—2448no—
549C2×C4C22×C22⋊D6114yes—
550C2×C4C22×C22⋊D6114yes—
551C2×C4C2×#1175822218yes—
552C22—232no—
553C22—28no—
554C22—24no—
555C22—28no—
556C22—24no—
557C22—4888no—
558C22—2444no—
559C22—2442no—
560C22—2442no—
561C22—2444no—
562C22—2442no—
563C22C2×#1451974918yes—
564C22C2×#1451974914yes—
565C22#1684027314yes—
566C22#1684033614yes—
567C22#1684041318yes—
568C22#1684044718yes—
569C22C22×D4×D6111yescharacteristic
570C22C23×C22⋊D6111yescharacteristic
571C22C22×#1175822411yescharacteristic
572C22C22×#1175822212yes—
573C22C2×#1451982512yes—
574C4—24no—
575C4—24no—
576C4—28no—
577C4—44no—
578C4—1228no—
579C4—2444no—
580C4—2442no—
581C4—2442no—
582C4—2442no—
583C2—28no—
584C2—4882no—
585C2#2144922618yesdirect product factor
586C2C22×#1451974911yescharacteristic
587C2C2×#1684027311yescharacteristic
588C2C2×#1684033611yescharacteristic
589C2C2×#1684041312yes—
590C2C2×#1684044712yes—
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC23×C4⋊C4×C61C2
socleC24×C61C24
prime radicalC23×C22⋊D61C22
hypercenterC23×C4⋊C4D61
hypocenterC61C2×#2413
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×#14519736C2semidirect
4. #21449226C2direct (N × H)
5. C2×#16839695C2semidirect
6. C2×#16839684C2semidirect
7. C2×#16839657C2semidirect
8. C2×#16839652C2semidirect
10. C2×#16839635C2semidirect
11. C2×#16839625C2semidirect
12. C23×C4⋊C4×C61C2semidirect
19. C2×#14519736C22semidirect
33. #16839695C22semidirect
34. #16839684C22semidirect
35. #16839657C22semidirect
36. #16839652C22semidirect
38. #16839635C22semidirect
39. #16839625C22semidirect
50. C22×C4⋊C4×C61C22semidirect
291. C23×C4⋊C4D61semidirect
301. C2×C61#2413semidirect
326. C22×C4⋊C4C2×D61semidirect
346. C61C2×#2413semidirect
585. C2#21449226direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197602
C22×#145197362
#214492262
C2×#168396952
C2×#168396842
C2×#168396572
C2×#168396522
C2×#168396412
C2×#168396352
C2×#168396252
C23×C4⋊C4×C612
C2×#241361