Finite Group Databasebeta

← C2×#21449420 #29848034

Subgroups — C2×#21449420 #29848034

Order 15616.

3246 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×#14519775C211yesmaximal, characteristic, complemented
2C22×#14519756C211yesmaximal, characteristic
3C22×Q8×Dic61C211yesmaximal, characteristic, complemented
4#21449420C2116yesmaximal, direct product factor
5C2×#16839695C212yesmaximal, complemented
6C2×#16839685C212yesmaximal
7C2×#16839667C211yesmaximal, characteristic, complemented
8C2×#16839658C211yesmaximal, characteristic, complemented
9C2×#16839657C212yesmaximal, complemented
10C2×#16839656C212yesmaximal
11C2×C4×C22⋊Dic61C211yesmaximal, characteristic
12C22×C22⋊Q8×C61C211yesmaximal, characteristic
13C23×Dic122C2211yescharacteristic
14C2×#14519775—28no—
15C2×#14519775C2218yescomplemented
16C2×#14519756—28no—
17C2×#14519756C2218yes—
18C2×Q8×Dic61—28no—
19C2×Q8×Dic61C2218yescomplemented
20C22×C4⋊Dic61C2212yes—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×#11758196C2212yes—
23C22×#11758196C2212yes—
24C22×#11758196C2212yes—
25C22×#11758196C2212yes—
26C22×C22⋊Dic61C2212yes—
27C22×C22⋊Dic61C2211yescharacteristic
28C22×C22⋊Dic61C2211yescharacteristic
29C22×C4×Dic61C2212yes—
30C22×C4×Dic61C2211yescharacteristic
31C22×C4×Dic61C2211yescharacteristic
32#16839695C22116yescomplemented
33#16839685C22116yes—
34#16839667C2218yescomplemented
35#16839658C2218yescomplemented
36#16839657C22116yescomplemented
37#16839656C22116yes—
38C4×C22⋊Dic61C2218yes—
39C2×#14519684C2212yes—
40C2×#14519684C2212yes—
41C2×#14519684C2211yescharacteristic
42C2×#14519684C2212yes—
43C2×#14519684C2211yescharacteristic
44C23×Q8×C61C2211yescharacteristic
45C24×C4×C61C2211yescharacteristic
46C2×C22⋊Q8×C61—28no—
47C2×C22⋊Q8×C61C2218yes—
48C22×C4⋊C4×C61C2212yes—
49C22×C4⋊C4×C61C2211yescharacteristic
50C22×C22⋊C4×C61C2212yes—
51C22×Dic122—24no—
52C22×Dic122—24no—
53C22×Dic122—24no—
54C22×Dic122C2314yes—
55C23×Dic61C2311yescharacteristic
56C23×Dic61C2312yes—
57C23×Dic61C2312yes—
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C23×Dic61C2311yescharacteristic
61#14519775—232no—
62#14519756—232no—
63Q8×Dic61—232no—
64C2×C4⋊Dic61—28no—
65C2×C4⋊Dic61D418yes—
66C2×C4⋊Dic61C2318yes—
67C2×C4⋊Dic61C2314yes—
68C2×#11758196—28no—
69C2×#11758196—28no—
70C2×#11758196—28no—
71C2×#11758196—28no—
72C2×#11758196C2318yes—
73C2×#11758196C2318yes—
74C2×#11758196C2318yes—
75C2×#11758196C2318yes—
76C2×C22⋊Dic61—28no—
77C2×C22⋊Dic61—24no—
78C2×C22⋊Dic61Q818yescomplemented
79C2×C22⋊Dic61C2314yes—
80C2×C22⋊Dic61C2318yes—
81C2×C22⋊Dic61C2314yes—
82C2×C4×Dic61—24no—
83C2×C4×Dic61—24no—
84C2×C4×Dic61—28no—
85C2×C4×Dic61C2314yes—
86C2×C4×Dic61C2318yes—
87C2×C4×Dic61C2314yes—
88#14519684C2318yes—
89#14519684C2314yes—
90#14519684C2318yes—
91#14519684C2314yes—
92#14519684C2318yes—
93C25×C61C2311yescharacteristic
94C22×Q8×C61—24no—
95C22×Q8×C61—24no—
96C22×Q8×C61D418yes—
97C22×Q8×C61C2314yes—
98C23×C4×C61—24no—
99C23×C4×C61—24no—
100C23×C4×C61—24no—
101C23×C4×C61C2311yescharacteristic
102C23×C4×C61C2312yes—
103C23×C4×C61C2314yes—
104C23×C4×C61C2312yes—
105C23×C4×C61C2311yescharacteristic
106C22⋊Q8×C61—232no—
107C2×C4⋊C4×C61—24no—
108C2×C4⋊C4×C61—28no—
109C2×C4⋊C4×C61C2318yes—
110C2×C4⋊C4×C61C2314yes—
111C2×C22⋊C4×C61—28no—
112C2×C22⋊C4×C61C2318yes—
113C24×C61—22no—
114C24×C61—22no—
115C24×C61—22no—
116C24×C61—22no—
117C24×C61—22no—
118C24×C61—22no—
119C24×C61C2×Q814yes—
120C24×C61C2412yes—
121C24×C61C2411yescharacteristic
122C2×Q8×C61—232no—
123C2×Q8×C61—28no—
124C2×Q8×C61—28no—
125C2×Q8×C61C2×D4116yes—
126C22×C4×C61—24no—
127C22×C4×C61—24no—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—24no—
133C22×C4×C61—24no—
134C22×C4×C61—28no—
135C22×C4×C61—28no—
136C22×C4×C61—28no—
137C22×C4×C61—416no—
138C22×C4×C61C2×D418yes—
139C22×C4×C61C2×D414yes—
140C22×C4×C61C2×D414yes—
141C22×C4×C61#4114yes—
142C22×C4×C61C2414yes—
143C22×C4×C61C2412yes—
144C22×C4×C61C2414yes—
145C22×C4×C61C2412yes—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—24no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—24no—
155C22×Dic61—24no—
156C22×Dic61—22no—
157C22×Dic61—24no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61C2×D418yes—
161C22×Dic61C2×Q814yes—
162C22×Dic61C2×Q814yes—
163C22×Dic61#4114yes—
164C22×Dic61C2414yes—
165C22×Dic61C2412yes—
166C22×Dic61C2414yes—
167C22×Dic61C2412yes—
168C22×Dic61C2412yes—
169C22×Dic61C2412yes—
170C2×Dic122—28no—
171C2×Dic122—28no—
172C2×Dic122—28no—
173C2×Dic122—416no—
174C4⋊C4×C61—28no—
175C4⋊C4×C61—216no—
176C22⋊C4×C61—216no—
177C22⋊Dic61—216no—
178C22⋊Dic61—28no—
179C22⋊Dic61C2×Q8116yescomplemented
180C4⋊Dic61—216no—
181C4⋊Dic61C2×D4116yes—
182#11758196—216no—
183#11758196—216no—
184#11758196—216no—
185#11758196—216no—
186C4×Dic61—28no—
187C4×Dic61—28no—
188C4×Dic61—216no—
189C23×C61—28no—
190C23×C61—28no—
191C23×C61—28no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—44no—
199C23×C61—44no—
200C23×C61—44no—
201C23×C61—44no—
202C23×C61C22×D412yes—
203C23×C61C22×D412yes—
204C23×C61C22×Q814yes—
205C23×C61C22×Q812yes—
206C23×C61C2×#4112yes—
207C23×C61C2×#4112yes—
208C23×C612−1+412yes—
209C23×C612−1+412yes—
210C23×C61C2511yescharacteristic
211Q8×C61—232no—
212C2×C4×C61—216no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—24no—
217C2×C4×C61—22no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—24no—
221C2×C4×C61—44no—
222C2×C4×C61—48no—
223C2×C4×C61—416no—
224C2×C4×C61#121116yes—
225C2×C4×C61C22×D418yes—
226C2×C4×C61C22×D414yes—
227C2×C4×C61C22×D414yes—
228C2×C4×C61C2×#4114yes—
229C2×Dic61—216no—
230C2×Dic61—28no—
231C2×Dic61—28no—
232C2×Dic61—28no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—24no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—48no—
248C2×Dic61—44no—
249C2×Dic61C22×D418yes—
250C2×Dic61C22×Q814yes—
251C2×Dic61C22×Q814yes—
252C2×Dic61C2×#4114yes—
253Dic122—416no—
254C2×#2429—611nomaximal, Sylow
255C22×C61—24no—
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—42no—
259C22×C61—416no—
260C22×C61—42no—
261C22×C61—42no—
262C22×C61—42no—
263C22×C61C2×#12114yes—
264C22×C61#53314yes—
265C22×C61#54114yes—
266C22×C61#54714yes—
267C22×C61#54814yes—
268C22×C61D4×Q814yes—
269C22×C61D4×Q814yes—
270C22×C61C23×D411yescharacteristic
271C22×C61C23×D411yescharacteristic
272C22×C61C23×Q811yescharacteristic
273C22×C61C22×#4111yescharacteristic
274C22×C61C22×#4111yescharacteristic
275C22×C61C2×2−1+411yescharacteristic
276C22×C61C2×2−1+411yescharacteristic
277C4×C61—28no—
278C4×C61—42no—
279C4×C61—44no—
280C4×C61C2×#12118yes—
281Dic61—28no—
282Dic61—24no—
283Dic61—24no—
284Dic61—24no—
285Dic61—44no—
286Dic61—42no—
287C22×C4⋊Q8—611no—
288C22×C22⋊Q8—611no—
289C22×C22⋊Q8D6111yescharacteristic
290C22×C4×Q8—611no—
291#2429—6116no—
292C2×#398—612no—
293C2×#396—612no—
294C2×#391—611no—
295C2×#389—612no—
296C2×#389—611no—
297C2×#388—612no—
298C2×C4×C22⋊C4—611no—
299C2×C61—44no—
300C2×C61#242918yescomplemented
301C2×C61C22×#12111yescharacteristic
302C2×C61C2×#53311yescharacteristic
303C2×C61C2×#54111yescharacteristic
304C2×C61C2×#54711yescharacteristic
305C2×C61C2×#54811yescharacteristic
306C2×C61C2×D4×Q811yescharacteristic
307C2×C61C2×D4×Q811yescharacteristic
308C23×Q8—611no—
309C23×Q8C2×D6111yescharacteristic
310C24×C4C2×D6111yescharacteristic
311C2×C4⋊Q8—618no—
312C2×C4⋊Q8—1228no—
313C2×C22⋊Q8—28no—
314C2×C22⋊Q8—618no—
315C2×C22⋊Q8—1228no—
316C2×C22⋊Q8C2×D6118yes—
317C2×C4×Q8—618no—
318C2×C4×Q8—1228no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—612no—
321C22×C4⋊C4—612no—
322C22×C4⋊C4—612no—
323C22×C4⋊C4—612no—
324C22×C4⋊C4—612no—
325C22×C4⋊C4C2×D6112yes—
326C22×C4⋊C4C2×D6111yescharacteristic
327C22×C22⋊C4—611no—
328C22×C22⋊C4—611no—
329C22×C22⋊C4—612no—
330C22×C22⋊C4C2×D6112yes—
331C22×C42—611no—
332C22×C42—612no—
333C22×C42—611no—
334#398—6116no—
335#396—6116no—
336#391—618no—
337#389—6116no—
338#389—618no—
339#388—6116no—
340C4×C22⋊C4—618no—
341C2×#95—612no—
342C2×#95—611no—
343C2×#95—612no—
344C2×#95—611no—
345C2×#95—612no—
346C61C2×#242911yescharacteristic, complemented, Sylow
347C25C22×D6111yescharacteristic
348C22×Q8—24no—
349C22×Q8—24no—
350C22×Q8—614no—
351C22×Q8—1224no—
352C22×Q8—1224no—
353C22×Q8—1224no—
354C22×Q8C22⋊D6118yes—
355C22×Q8C22×D6114yes—
356C23×C4—24no—
357C23×C4—24no—
358C23×C4—24no—
359C23×C4—612no—
360C23×C4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—612no—
364C23×C4—611no—
365C23×C4C22×D6114yes—
366C23×C4C22×D6112yes—
367C23×C4C22×D6111yescharacteristic
368C23×C4C22×D6112yes—
369C23×C4C22×D6111yescharacteristic
370C4⋊Q8—12232no—
371C22⋊Q8—232no—
372C22⋊Q8—12232no—
373C4×Q8—12232no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—28no—
376C2×C4⋊C4—618no—
377C2×C4⋊C4—614no—
378C2×C4⋊C4—618no—
379C2×C4⋊C4—618no—
380C2×C4⋊C4—618no—
381C2×C4⋊C4—618no—
382C2×C4⋊C4—618no—
383C2×C4⋊C4—1228no—
384C2×C4⋊C4—1228no—
385C2×C4⋊C4—1228no—
386C2×C4⋊C4—1228no—
387C2×C4⋊C4—1228no—
388C2×C4⋊C4C22×D6114yes—
389C2×C4⋊C4C22×D6118yes—
390C2×C22⋊C4—28no—
391C2×C22⋊C4—618no—
392C2×C22⋊C4—614no—
393C2×C22⋊C4—618no—
394C2×C22⋊C4—614no—
395C2×C22⋊C4—1228no—
396C2×C22⋊C4—1224no—
397C2×C22⋊C4C22×D6118yes—
398C2×C42—614no—
399C2×C42—618no—
400C2×C42—614no—
401C2×C42—1228no—
402C2×C42—1224no—
403C2×C42—1224no—
404#95—614no—
405#95—614no—
406#95—618no—
407#95—618no—
408#95—618no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24Q8×D6114yes—
416C24C23×D6112yes—
417C24C23×D6111yescharacteristic
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—232no—
421C2×Q8—1228no—
422C2×Q8—1228no—
423C2×Q8—1228no—
424C2×Q8—24416no—
425C2×Q8C2×C22⋊D61116yes—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—416no—
438C22×C4—618no—
439C22×C4—614no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—614no—
445C22×C4—612no—
446C22×C4—612no—
447C22×C4—614no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4D4×D6114yes—
463C22×C4#1175822414yes—
464C22×C4C2×C22⋊D6118yes—
465C22×C4C2×C22⋊D6114yes—
466C22×C4C23×D6114yes—
467C22×C4C23×D6112yes—
468C22×C4C23×D6114yes—
469C22×C4C23×D6112yes—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—6116no—
473C4⋊C4—12216no—
474C4⋊C4—12216no—
475C4⋊C4—12216no—
476C4⋊C4—12216no—
477C4⋊C4—12216no—
478C22⋊C4—216no—
479C22⋊C4—6116no—
480C22⋊C4—12216no—
481C22⋊C4—1228no—
482C42—12216no—
483C42—1228no—
484C42—1228no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—28no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23C2×D4×D6112yes—
499C23C22×C22⋊D6112yes—
500C23C2×Q8×D6114yes—
501C23C2×Q8×D6112yes—
502C23#41×D6112yes—
503C23C2×#1175822412yes—
504C23#1451982412yes—
505C23#1451982512yes—
506C23C24×D6111yescharacteristic
507Q8—232no—
508Q8—24416no—
509C2×C4—216no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—44no—
519C2×C4—48no—
520C2×C4—416no—
521C2×C4—618no—
522C2×C4—614no—
523C2×C4—614no—
524C2×C4—614no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1224no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1224no—
532C2×C4—1222no—
533C2×C4—12216no—
534C2×C4—1228no—
535C2×C4—1228no—
536C2×C4—1228no—
537C2×C4—1222no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1222no—
541C2×C4—1224no—
542C2×C4—1222no—
543C2×C4—2448no—
544C2×C4—2444no—
545C2×C4#14519749116yes—
546C2×C4C2×D4×D6114yes—
547C2×C4C22×C22⋊D6118yes—
548C2×C4C22×C22⋊D6114yes—
549C2×C4C2×#1175822414yes—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—416no—
557C22—42no—
558C22C2×#1451974914yes—
559C22#1684031414yes—
560C22#1684037214yes—
561C22#1684041414yes—
562C22#1684041914yes—
563C22#1684042514yes—
564C22Q8×C22⋊D6114yes—
565C22C22×D4×D6111yescharacteristic
566C22C23×C22⋊D6111yescharacteristic
567C22C22×Q8×D6111yescharacteristic
568C22C2×#41×D6111yescharacteristic
569C22C22×#1175822411yescharacteristic
570C22C2×#1451982411yescharacteristic
571C22C2×#1451982511yescharacteristic
572C4—28no—
573C4—44no—
574C4—42no—
575C4—1228no—
576C4—1224no—
577C4—1224no—
578C4—1224no—
579C4—2442no—
580C4—2444no—
581C4C2×#1451974918yes—
582C2—44no—
583C2#2144942018yesdirect product factor
584C2C22×#1451974911yescharacteristic
585C2C2×#1684031411yescharacteristic
586C2C2×#1684037211yescharacteristic
587C2C2×#1684041411yescharacteristic
588C2C2×#1684041911yescharacteristic
589C2C2×#1684042511yescharacteristic
590C2C2×Q8×C22⋊D6111yescharacteristic
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×C22⋊Q8×C61C2
socleC24×C61C24
prime radicalC25×C61C23
hypercenterC22×C22⋊Q8D61
hypocenterC61C2×#2429
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14519775C2semidirect
3. C22×Q8×Dic61C2semidirect
4. #21449420C2direct (N × H)
5. C2×#16839695C2semidirect
7. C2×#16839667C2semidirect
8. C2×#16839658C2semidirect
9. C2×#16839657C2semidirect
15. C2×#14519775C22semidirect
19. C2×Q8×Dic61C22semidirect
32. #16839695C22semidirect
34. #16839667C22semidirect
35. #16839658C22semidirect
36. #16839657C22semidirect
78. C2×C22⋊Dic61Q8semidirect
179. C22⋊Dic61C2×Q8semidirect
300. C2×C61#2429semidirect
346. C61C2×#2429semidirect
583. C2#21449420direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197752
C22×#145197562
C22×Q8×Dic612
#214494202
C2×#168396952
C2×#168396852
C2×#168396672
C2×#168396582
C2×#168396572
C2×#168396562
C2×C4×C22⋊Dic612
C22×C22⋊Q8×C612
C2×#242961