Finite Group Databasebeta

← C2×#21456945 #30552052

Subgroups — C2×#21456945 #30552052

Order 15616.

2286 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C22×#14519776C211yesmaximal, characteristic, complemented
3#21456945C2116yesmaximal, direct product factor
4C2×#16840159C212yesmaximal, complemented
5C2×#16840107C212yesmaximal, complemented
6C2×#16840081C211yesmaximal, characteristic, complemented
7C2×#16840080C211yesmaximal, characteristic, complemented
8C2×#16840078C211yesmaximal, characteristic, complemented
9C2×#16840051C212yesmaximal, complemented
10C2×#16840050C212yesmaximal, complemented
11C2×#16839812C211yesmaximal, characteristic, complemented
12C2×C61×#481C211yesmaximal, characteristic, complemented
13C22×#11758224—21no—
14C23×Dic122—21no—
15#16840307—24no—
16#16840307C2218yes—
17C2×#14519776—21no—
18C2×#14519776—21no—
19C2×#14519776—24no—
20C2×#14519776C2212yes—
21C2×#14519776C2212yes—
22C2×#14519776C2211yescharacteristic
23C2×#14519776C2218yescomplemented
24C2×#14519773C2211yescharacteristic
25C2×#14519763—21no—
26C2×#14519751C2211yescharacteristic
27C2×#14519749—21no—
28C2×D4×Dic61C2211yescharacteristic
29C2×#14519739C2211yescharacteristic
30C2×#14519737—21no—
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196—21no—
33C22×C4×Dic61C2211yescharacteristic
34#16840159C22116yescomplemented
35#16840107C22116yescomplemented
36#16840081C2218yescomplemented
37#16840080C2218yescomplemented
38#16840078C2218yescomplemented
39#16840051C22116yescomplemented
40#16840050C22116yescomplemented
41#16839812C2218yescomplemented
42C2×#14519709C2212yes—
43C2×#14519708C2212yes—
44C2×#14519708C2212yes—
45C2×#14519706C2212yes—
46C2×#14519699C2212yes—
47C2×#14519698C2212yes—
48C2×#14519697C2212yes—
49C2×C22⋊#92418C2211yescharacteristic
50C2×C8×Dic61C2212yes—
51C2×#121×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#481C2218yescomplemented
54C2×#106×C61C2212yes—
55C2×#102×C61C2212yes—
56C2×C22⋊C8×C61C2211yescharacteristic
57C2×#11758224—28no—
58C2×#11758224—48no—
59C22×Dic122—21no—
60C22×Dic122—22no—
61C22×Dic122—22no—
62C22×Dic122—28no—
63C22×Dic122—48no—
64C22×D244C2311yescharacteristic
65C22×C22⋊D61—21no—
66C22×C22⋊D61—21no—
67C22×C22⋊D61—21no—
68C23×Dic61—21no—
69C23×Dic61—21no—
70C23×Dic61D411yescharacteristic
71C23×Dic61D411yescharacteristic
72C22×C4×D61—21no—
73C22×#92418C2312yes—
74#14519776—24no—
75#14519776—24no—
76#14519776—28no—
77#14519776—28no—
78#14519776—22no—
79#14519776C2318yes—
80#14519776C2318yes—
81#14519776C2314yes—
82#14519773—22no—
83#14519773C2314yes—
84#14519763—28no—
85#14519751—22no—
86#14519751C2314yes—
87#14519749—28no—
88D4×Dic61—22no—
89D4×Dic61C2314yes—
90#14519739—22no—
91#14519739C2314yes—
92#14519737—28no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61—22no—
95C2×C4⋊Dic61C2312yes—
96C2×C4⋊Dic61C2314yes—
97C2×C4⋊Dic61C2311yescharacteristic
98C2×#11758196—22no—
99C2×#11758196—22no—
100C2×#11758196—28no—
101C2×#11758196—21no—
102C2×C22⋊Dic61—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61D414yes—
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2314yes—
111#14519709C2318yes—
112#14519708C2318yes—
113#14519708C2318yes—
114#14519706C2318yes—
115#14519699C2318yes—
116#14519698C2318yes—
117#14519697C2318yes—
118C22⋊#92418C2314yes—
119C8×Dic61C2318yes—
120C22×D4×C61—21no—
121C22×D4×C61C2311yescharacteristic
122C23×C4×C61—21no—
123C23×C4×C61C2311yescharacteristic
124C22×C8×C61C2312yes—
125#121×C61—22no—
126#121×C61C2314yes—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—22no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2312yes—
131C2×C4⋊C4×C61C2314yes—
132C2×C22⋊C4×C61—21no—
133#106×C61C2318yes—
134#102×C61C2318yes—
135C22⋊C8×C61C2314yes—
136C24×C61—21no—
137C24×C61C2×D411yescharacteristic
138C23×D61—21no—
139C2×D4×C61—24no—
140C2×D4×C61—21no—
141C2×D4×C61—24no—
142C2×D4×C61—44no—
143C2×D4×C61C2412yes—
144C22×C4×C61—24no—
145C22×C4×C61—21no—
146C22×C4×C61—24no—
147C22×C4×C61—22no—
148C22×C4×C61—44no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2412yes—
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—24no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—44no—
159C22×Dic61—24no—
160C22×Dic61—21no—
161C22×Dic61—24no—
162C22×Dic61—22no—
163C22×Dic61—24no—
164C22×Dic61—44no—
165C22×Dic61—44no—
166C22×Dic61SD1618yes—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D414yes—
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D414yes—
172C22×Dic61C2×D411yescharacteristic
173#11758224—432no—
174C2×D244—24no—
175C2×D244—21no—
176C2×D244C2412yes—
177C2×C4×D61—24no—
178C2×C4×D61—44no—
179C2×Dic122—28no—
180C2×Dic122—28no—
181C2×Dic122—24no—
182C2×Dic122—48no—
183C2×Dic122—432no—
184C2×Dic122—44no—
185C2×Dic122—48no—
186C2×C8×C61—24no—
187C2×C8×C61C2414yes—
188C4⋊C4×C61—24no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61C2414yes—
192C4⋊C4×C61C2412yes—
193C22⋊C4×C61—24no—
194C22⋊Dic61—24no—
195#11758198—24no—
196#11758198—24no—
197C4⋊Dic61—24no—
198C4⋊Dic61—22no—
199C4⋊Dic61—21no—
200C4⋊Dic61C2414yes—
201C4⋊Dic61C2412yes—
202#11758196—28no—
203#11758196—28no—
204#11758196—24no—
205C4×Dic61C2×D414yes—
206C4×Dic61C2×D412yes—
207C4×Dic61C2×D414yes—
208C4×Dic61C2×D412yes—
209C4×Dic61C2412yes—
210C4×Dic61C2412yes—
211C2×#92418—24no—
212C2×#92418C2414yes—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61—22no—
216C23×C61—42no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61C2×SD1614yes—
220C23×C61C22×D412yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—42no—
226D4×C61—24no—
227D4×C61—48no—
228D4×C61—42no—
229C2×C4×C61—22no—
230C2×C4×C61—24no—
231C2×C4×C61—24no—
232C2×C4×C61—24no—
233C2×C4×C61—22no—
234C2×C4×C61—48no—
235C2×C4×C61—42no—
236C2×C4×C61—44no—
237C2×C4×C61C22×D412yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C612−1+414yes—
240C2×C4×C61C2511yescharacteristic
241C22⋊D61—48no—
242C22⋊D61—48no—
243C22⋊D61—48no—
244C2×Dic61—22no—
245C2×Dic61—28no—
246C2×Dic61—24no—
247C2×Dic61—22no—
248C2×Dic61—48no—
249C2×Dic61—42no—
250C2×Dic61—44no—
251C2×Dic61—48no—
252C2×Dic61C2×SD1614yes—
253C2×Dic61C2×SD1614yes—
254C2×Dic61C2×SD16116yes—
255C2×Dic61AGL(1,ℤ/8)14yes—
256C2×Dic61C22×D412yes—
257C2×Dic61C22×D412yes—
258C2×Dic61C22×D412yes—
259C2×Dic61C22×D412yes—
260D244—24no—
261D244—42no—
262C4×D61—48no—
263Dic122—416no—
264Dic122—48no—
265Dic122—416no—
266C8×C61—24no—
267#92418—24no—
268C2×#3227—611nomaximal, Sylow
269C22×C61—22no—
270C22×C61—28no—
271C22×C61—22no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—84no—
276C22×C61#53612yes—
277C22×C61C22×SD1614yes—
278C22×C61C22×SD1612yes—
279C22×C61C2×AGL(1,ℤ/8)12yes—
280C22×C61C23×D411yescharacteristic
281C2×D61—42no—
282C2×D61—42no—
283C2×D61—42no—
284C2×D61—84no—
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—44no—
288C4×C61#53612yes—
289C4×C61C2×2−1+412yes—
290Dic61—22no—
291Dic61—42no—
292Dic61—44no—
293Dic61C22×SD1614yes—
294Dic61C22×SD1614yes—
295Dic61C2×AGL(1,ℤ/8)14yes—
296C2×#532—611no—
297C22×C4⋊Q8—611no—
298#3227—6116no—
299C2×#498—612no—
300C2×#486—612no—
301C2×#481—611no—
302C2×#481—611no—
303C2×#481—611no—
304C2×#481D6111yescharacteristic, complemented
305C2×#476—612no—
306C2×#476—612no—
307C2×#431—611no—
308C2×C61—24no—
309C2×C61—82no—
310C2×C61#322714yescomplemented
311C2×C61C2×#53611yescharacteristic
312C2×C61C23×SD1611yescharacteristic
313C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
314D61—82no—
315C22×#41—1221no—
316C23×Q8—1221no—
317#532—618no—
318#532—1224no—
319C2×C4⋊Q8—612no—
320C2×C4⋊Q8—611no—
321C2×C4⋊Q8—612no—
322C2×C4⋊Q8—618no—
323C2×C4⋊Q8—1221no—
324C2×C4⋊Q8—1221no—
325C2×C4⋊Q8—1224no—
326C2×#127—611no—
327C2×#124—1221no—
328C2×#121—611no—
329C2×#121—1221no—
330C2×#121C2×D6111yescharacteristic
331C2×C4×D4—611no—
332C2×C4×D4—611no—
333C2×C4×D4—1221no—
334C22×C4⋊C4—611no—
335C22×C4⋊C4—1221no—
336C22×C4⋊C4C2×D6111yescharacteristic
337C22×C42—611no—
338#498—6116no—
339#486—6116no—
340#481—618no—
341#481—618no—
342#481—618no—
343#481C2×D6118yescomplemented
344#476—6116no—
345#476—6116no—
346#431—618no—
347C2×#106—612no—
348C2×#106—612no—
349C2×#106—612no—
350C2×#106C2×D6112yes—
351C2×C4⋊C8—612no—
352C2×#102—612no—
353C2×#102—612no—
354C2×#102—612no—
355C2×#102C2×D6112yes—
356C2×C22⋊C8—611no—
357C2×C22⋊C8C2×D6111yescharacteristic
358C2×C4×C8—612no—
359C61C2×#322711yescharacteristic, complemented, Sylow
360C2×#41—1228no—
361C2×#41—2448no—
362C22×Q8—1221no—
363C22×Q8—1228no—
364C22×Q8—1222no—
365C22×Q8—1222no—
366C22×Q8—2448no—
367C22×D4—21no—
368C22×D4—611no—
369C22×D4—1221no—
370C22×D4—1221no—
371C22×D4—1221no—
372C22×D4C22×D6111yescharacteristic
373C23×C4—21no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—1221no—
377C23×C4—1221no—
378C23×C4—1221no—
379C23×C4C22×D6111yescharacteristic
380C22×C8—612no—
381C22×C8C22×D6112yes—
382C4⋊Q8—618no—
383C4⋊Q8—618no—
384C4⋊Q8—614no—
385C4⋊Q8—1224no—
386C4⋊Q8—1228no—
387C4⋊Q8—1228no—
388C4⋊Q8—1224no—
389C4⋊Q8—1222no—
390#127—614no—
391#127—1222no—
392#124—1228no—
393#121—22no—
394#121—614no—
395#121—1228no—
396#121—1222no—
397#121C22×D6114yes—
398C4×D4—614no—
399C4×D4—614no—
400C4×D4—1228no—
401C4×D4—1222no—
402C4×D4—1222no—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—612no—
406C2×C4⋊C4—614no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1228no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1222no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1222no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4C22×D6112yes—
415C2×C4⋊C4C22×D6114yes—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—614no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—614no—
427#106—618no—
428#106—618no—
429#106—618no—
430#106C22×D6118yes—
431C4⋊C8—618no—
432#102—618no—
433#102—618no—
434#102—618no—
435#102C22×D6118yes—
436C22⋊C8—614no—
437C22⋊C8C22×D6114yes—
438C4×C8—618no—
439C24—21no—
440C24—1221no—
441C24D4×D6111yescharacteristic
442#41—24432no—
443C2×Q8—1224no—
444C2×Q8—1228no—
445C2×Q8—1228no—
446C2×Q8—2448no—
447C2×Q8—2444no—
448C2×Q8—24432no—
449C2×Q8—2448no—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—24no—
453C2×D4—44no—
454C2×D4—612no—
455C2×D4—1224no—
456C2×D4—1224no—
457C2×D4—1224no—
458C2×D4—1221no—
459C2×D4—1224no—
460C2×D4—2444no—
461C2×D4—2444no—
462C2×D4—2444no—
463C2×D4C23×D6112yes—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—44no—
469C22×C4—614no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—618no—
474C22×C4—614no—
475C22×C4—611no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1221no—
481C22×C4—1222no—
482C22×C4—2444no—
483C22×C4—2444no—
484C22×C4—2444no—
485C22×C4D4×D6112yes—
486C22×C4D4×D6111yescharacteristic
487C22×C4C23×D6112yes—
488C22×C4C23×D6111yescharacteristic
489C2×C8—24no—
490C2×C8—614no—
491C2×C8—1224no—
492C2×C8C23×D6114yes—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—614no—
497C4⋊C4—612no—
498C4⋊C4—1228no—
499C4⋊C4—1228no—
500C4⋊C4—1224no—
501C4⋊C4—1224no—
502C4⋊C4—1222no—
503C4⋊C4—1221no—
504C4⋊C4C23×D6114yes—
505C4⋊C4C23×D6112yes—
506C22⋊C4—24no—
507C22⋊C4—1224no—
508C22⋊C4—1224no—
509C22⋊C4—1224no—
510C42—614no—
511C42—612no—
512C42—614no—
513C42—612no—
514C42—612no—
515C42—612no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—1222no—
523C23—2442no—
524C23—2442no—
525C23—2442no—
526C23SD16×D6114yes—
527C23C2×D4×D6112yes—
528C23C2×D4×D6111yescharacteristic
529Q8—24416no—
530Q8—2448no—
531Q8—24416no—
532D4—24no—
533D4—48no—
534D4—42no—
535D4—1224no—
536D4—2448no—
537D4—2448no—
538D4—2448no—
539D4—2442no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—48no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—614no—
549C2×C4—614no—
550C2×C4—6116no—
551C2×C4—614no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—612no—
556C2×C4—1228no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1224no—
560C2×C4—2448no—
561C2×C4—2448no—
562C2×C4—2448no—
563C2×C4—2442no—
564C2×C4—2444no—
565C2×C4C2×D4×D6112yes—
566C2×C4C2×D4×D6111yescharacteristic
567C2×C4#1451982414yes—
568C2×C4C24×D6111yescharacteristic
569C8—24no—
570C8—1224no—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—84no—
578C22—4884no—
579C22—2442no—
580C22—2442no—
581C22—2442no—
582C22#1684033412yes—
583C22C2×SD16×D6114yes—
584C22C2×SD16×D6112yes—
585C22AGL(1,ℤ/8)×D6112yes—
586C22C22×D4×D6111yescharacteristic
587C4—22no—
588C4—42no—
589C4—44no—
590C4—614no—
591C4—614no—
592C4—614no—
593C4—1222no—
594C4—2442no—
595C4—2444no—
596C4#1684033412yes—
597C4C2×#1451982412yes—
598C2—24no—
599C2—82no—
600C2—4882no—
601C2#2145694514yesdirect product factor
602C2C2×#1684033411yescharacteristic
603C2C22×SD16×D6111yescharacteristic
604C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#481C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
hypercenterC2×#481D61
hypocenterC61C2×#3227
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
2. C22×#14519776C2semidirect
3. #21456945C2direct (N × H)
4. C2×#16840159C2semidirect
5. C2×#16840107C2semidirect
6. C2×#16840081C2semidirect
7. C2×#16840080C2semidirect
8. C2×#16840078C2semidirect
9. C2×#16840051C2semidirect
10. C2×#16840050C2semidirect
11. C2×#16839812C2semidirect
12. C2×C61×#481C2semidirect
23. C2×#14519776C22semidirect
34. #16840159C22semidirect
35. #16840107C22semidirect
36. #16840081C22semidirect
37. #16840080C22semidirect
38. #16840078C22semidirect
39. #16840051C22semidirect
40. #16840050C22semidirect
41. #16839812C22semidirect
53. C61×#481C22semidirect
304. C2×#481D61semidirect
310. C2×C61#3227semidirect
343. #481C2×D61semidirect
359. C61C2×#3227semidirect
601. C2#21456945direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403072
C22×#145197762
#214569452
C2×#168401592
C2×#168401072
C2×#168400812
C2×#168400802
C2×#168400782
C2×#168400512
C2×#168400502
C2×#168398122
C2×C61×#4812
C2×#322761