Finite Group Databasebeta

← C2×#21456519 #30549926

Subgroups — C2×#21456519 #30549926

Order 15616.

2278 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic, complemented
2C2×#16840297C211yesmaximal, characteristic, complemented
3#21456519C2116yesmaximal, direct product factor
4C2×#16840151C211yesmaximal, characteristic, complemented
5C2×#16840128C211yesmaximal, characteristic, complemented
6C2×#16840124C211yesmaximal, characteristic, complemented
7C2×#16839999C212yesmaximal, complemented
8C2×#16839992C212yesmaximal, complemented
9C2×#16839873C212yesmaximal, complemented
10C2×#16839861C212yesmaximal, complemented
11C2×#97×D61C211yesmaximal, characteristic, complemented
12C2×C61×#501C211yesmaximal, characteristic, complemented
13C22×#11758226—21no—
14C22×#11758190C2212yes—
15C22×D488C2212yes—
16C22×C8×D61C2212yes—
17#16840313—24no—
18#16840313C2218yescomplemented
19C2×#14519774C2211yescharacteristic
20#16840297—24no—
21#16840297C2218yescomplemented
22C2×#14519772—21no—
23C2×#14519766—21no—
24C2×#14519765C2211yescharacteristic
25C2×#14519762—21no—
26C2×#14519757C2211yescharacteristic
27C2×#14519752—21no—
28C2×#14519752C2212yes—
29C2×#14519752C2211yescharacteristic
30C2×#14519742C2211yescharacteristic
31C2×#14519739—21no—
32C2×#14519739—21no—
33C2×#14519739C2212yes—
34C2×#14519739C2211yescharacteristic
35C2×#14519735—21no—
36C2×C42×D61C2211yescharacteristic
37#16840151C2218yescomplemented
38#16840128C2218yescomplemented
39#16840124C2218yescomplemented
40#16839999C22116yescomplemented
41#16839992C22116yescomplemented
42#16839873C22116yescomplemented
43#16839861C22116yescomplemented
44C2×#14519700C2212yes—
45C2×#14519697C2212yes—
46C2×#14519697C2212yes—
47C2×#14519697C2212yes—
48C2×#14519688C2211yescharacteristic
49C2×#14519687C2212yes—
50#97×D61C2218yescomplemented
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C61×#501C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#106×C61C2212yes—
56C2×#97×C61C2211yescharacteristic
57C2×#11758226—28no—
58C2×#11758226—48no—
59C22×Dic122C2311yescharacteristic
60C22×D244—21no—
61C22×D244—22no—
62C22×D244—21no—
63C22×D244—21no—
64C22×D244C2311yescharacteristic
65C22×D244C2312yes—
66C22×C4×D61—21no—
67C22×C4×D61—22no—
68C22×C4×D61D411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61C2311yescharacteristic
71C2×#11758190—28no—
72C2×#11758190C2318yes—
73C2×D488—28no—
74C2×D488C2318yes—
75C22×#92418C2312yes—
76C2×C8×D61—28no—
77C2×C8×D61C2318yes—
78#14519774—22no—
79#14519774C2314yes—
80#14519772—28no—
81#14519766—28no—
82#14519765—22no—
83#14519765C2314yes—
84#14519762—28no—
85#14519757—22no—
86#14519757C2314yes—
87#14519752—28no—
88#14519752—22no—
89#14519752—24no—
90#14519752C2318yes—
91#14519752C2314yes—
92#14519742—22no—
93#14519742C2314yes—
94#14519739—28no—
95#14519739—28no—
96#14519739—22no—
97#14519739—24no—
98#14519739C2314yes—
99#14519739C2318yes—
100#14519735—28no—
101C2×C4⋊Dic61—21no—
102C2×#11758196—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—22no—
108C2×#11758198—22no—
109C2×#11758198—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C42×D61D414yes—
115C42×D61C2314yes—
116#14519700C2318yes—
117#14519697C2318yes—
118#14519697C2318yes—
119#14519697C2318yes—
120#14519688C2314yes—
121#14519687C2318yes—
122C22×Q8×C61—21no—
123C22×C8×C61C2312yes—
124C4⋊Q8×C61—22no—
125C4⋊Q8×C61C2314yes—
126#125×C61—22no—
127#125×C61C2314yes—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#107×C61C2318yes—
136#106×C61C2318yes—
137#97×C61C2314yes—
138C23×D61—22no—
139C23×D61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—24no—
142C2×Q8×C61—44no—
143C22×C4×C61—22no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—432no—
154C2×D244—24no—
155C2×D244—24no—
156C2×D244—24no—
157C2×D244—28no—
158C2×D244—24no—
159C2×D244—24no—
160C2×D244—24no—
161C2×D244—21no—
162C2×D244—22no—
163C2×D244—44no—
164C2×D244—44no—
165C2×D244—44no—
166C2×D244—48no—
167C2×D244C2412yes—
168C2×D244C2414yes—
169C2×C4×D61—24no—
170C2×C4×D61—24no—
171C2×C4×D61—28no—
172C2×C4×D61—24no—
173C2×C4×D61—42no—
174C2×C4×D61—44no—
175C2×C4×D61—48no—
176C2×C4×D61C2×D414yes—
177C2×C4×D61C2×D412yes—
178C2×C4×D61C2×D414yes—
179C2×C4×D61C2412yes—
180C2×Dic122—24no—
181C2×Dic122—21no—
182C2×Dic122C2412yes—
183C2×C8×C61#4118yescomplemented
184C2×C8×C61C2414yes—
185C4⋊C4×C61—24no—
186C4⋊C4×C61—24no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61C2412yes—
192C4⋊C4×C61C2414yes—
193C4⋊C4×C61C2412yes—
194C42×C61C2×D412yes—
195C42×C61C2412yes—
196#11758198—24no—
197#11758198—24no—
198#11758198—24no—
199#11758198—24no—
200#11758198—28no—
201#11758198—28no—
202#11758198—24no—
203C4⋊Dic61—24no—
204#11758196—24no—
205C4×Dic61—24no—
206C4×Dic61C2×D412yes—
207C4×Dic61C2×D414yes—
208C4×Dic61C2×D414yes—
209C4×Dic61C2412yes—
210C2×#92418—24no—
211C2×#92418C2414yes—
212D488—216no—
213#11758190—216no—
214C8×D61—216no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—24no—
220C22×D61—42no—
221C22×D61—44no—
222C22×D61—44no—
223C22×D61—42no—
224C22×D61—42no—
225C22×D61—44no—
226C22×D61AGL(1,ℤ/8)14yes—
227C22×D61C22×D412yes—
228Q8×C61—48no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—24no—
234C2×C4×C61—44no—
235C2×C4×C61—42no—
236C2×C4×C61—42no—
237C2×C4×C61C22×D412yes—
238C2×C4×C61C22×D412yes—
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4114yes—
241C2×C4×C61C2511yescharacteristic
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—42no—
247C2×Dic61C22×D412yes—
248C2×Dic61C22×D412yes—
249C2×Dic61C22×D412yes—
250D244—24no—
251D244—24no—
252D244—24no—
253D244—48no—
254D244—48no—
255D244—48no—
256D244—416no—
257D244—42no—
258D244—44no—
259C4×D61—28no—
260C4×D61—24no—
261C4×D61—24no—
262C4×D61—44no—
263C4×D61—48no—
264C4×D61—416no—
265Dic122—24no—
266Dic122—42no—
267C8×C61C2×#4118yescomplemented
268#92418—24no—
269C2×#3180—611nomaximal, Sylow
270C22×C61#53212yes—
271C22×C61C2×AGL(1,ℤ/8)12yes—
272C22×C61#57512yes—
273C22×C61C23×D411yescharacteristic
274C2×D61—28no—
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—44no—
278C2×D61—42no—
279C2×D61—44no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—44no—
283C2×D61—88no—
284C2×D61—84no—
285C2×D61C2×AGL(1,ℤ/8)14yes—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—44no—
289C4×C61—42no—
290C4×C61—42no—
291C4×C61#53212yes—
292C4×C61C22×#4112yes—
293Dic61—22no—
294Dic61—22no—
295Dic61—22no—
296Dic61—42no—
297C2×#532—611no—
298C2×#529—611no—
299#3180—6116no—
300C2×#501D6111yescharacteristic, complemented
301C2×#496—611no—
302C2×#490—611no—
303C2×#489—611no—
304C2×#466—612no—
305C2×#465—612no—
306C2×#442—612no—
307C2×#440—612no—
308C22×#97—611no—
309C2×C61#318014yescomplemented
310C2×C61C2×#53211yescharacteristic
311C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
312C2×C61C2×#57511yescharacteristic
313D61—24no—
314D61—84no—
315D61—82no—
316C22×#41—1221no—
317C22×SD16—612no—
318C22×D8—612no—
319C23×C8—612no—
320#532—618no—
321#532—1224no—
322C2×C4⋊Q8C2×D6111yescharacteristic
323C2×#127—611no—
324#529—618no—
325#529—1224no—
326C2×#126—1221no—
327C2×#125C2×D6111yescharacteristic
328C2×#124—611no—
329C2×#124—1221no—
330C2×#123—1221no—
331C2×C22⋊Q8—611no—
332C2×#121—611no—
333C2×#121—612no—
334C2×#121—1221no—
335C2×C4×Q8—611no—
336C2×C4×D4—611no—
337C2×C4×D4—612no—
338C2×C4×D4—1221no—
339C2×C4×D4—1221no—
340C2×#117—1221no—
341C22×C42—611no—
342#501C2×D6118yescomplemented
343#496—618no—
344#490—618no—
345#489—618no—
346#466—6116no—
347#465—6116no—
348#442—6116no—
349#440—6116no—
350C2×#107C2×D6112yes—
351C2×#106C2×D6112yes—
352C2×#103—612no—
353C2×#102—612no—
354C2×#102—612no—
355C2×#102—612no—
356C2×#97—612no—
357C2×#97—611no—
358C2×#97—618no—
359C2×#97C2×D6111yescharacteristic
360C61C2×#318011yescharacteristic, complemented, Sylow
361C2×#41—1228no—
362C2×#41—2448no—
363C22×Q8—21no—
364C22×Q8—611no—
365C22×D4—611no—
366C22×D4—612no—
367C22×D4—1221no—
368C22×D4—1222no—
369C22×D4—1221no—
370C22×D4—1221no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—611no—
374C23×C4—1221no—
375C23×C4—1222no—
376C2×SD16—618no—
377C2×SD16—1228no—
378C2×D8—618no—
379C2×D8—1228no—
380C22×C8—612no—
381C22×C8—618no—
382C22×C8—1228no—
383C22×C8C22×D6112yes—
384C4⋊Q8—22no—
385C4⋊Q8C22×D6114yes—
386#127—614no—
387#127—1222no—
388#126—1228no—
389#125—22no—
390#125C22×D6114yes—
391#124—614no—
392#124—1228no—
393#124—1222no—
394#123—1228no—
395C22⋊Q8—614no—
396C22⋊Q8—1222no—
397#121—614no—
398#121—618no—
399#121—1222no—
400#121—1224no—
401#121—1228no—
402C4×Q8—614no—
403C4×Q8—1222no—
404C4×D4—614no—
405C4×D4—618no—
406C4×D4—1222no—
407C4×D4—1224no—
408C4×D4—1228no—
409C4×D4—1228no—
410#117—1228no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6112yes—
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1222no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1222no—
423C2×C22⋊C4—1221no—
424C2×C22⋊C4—1221no—
425C2×C22⋊C4—1221no—
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—614no—
430C2×C42—614no—
431C2×C42—1221no—
432C2×C42C22×D6111yescharacteristic
433#107C22×D6118yes—
434#106C22×D6118yes—
435#103—618no—
436#102—618no—
437#102—618no—
438#102—618no—
439#97—618no—
440#97—614no—
441#97C22×D6114yes—
442C24—611no—
443C24—1221no—
444C24—1222no—
445#41—24432no—
446C2×Q8—24no—
447C2×Q8—44no—
448C2×Q8—612no—
449C2×Q8—1221no—
450C2×Q8—1224no—
451C2×D4—612no—
452C2×D4—614no—
453C2×D4—1224no—
454C2×D4—1228no—
455C2×D4—1221no—
456C2×D4—1224no—
457C2×D4—1222no—
458C2×D4—1224no—
459C2×D4—1224no—
460C2×D4—1224no—
461C2×D4—1224no—
462C2×D4—2444no—
463C2×D4—2444no—
464C2×D4—2448no—
465C2×D4—2444no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—612no—
473C22×C4—614no—
474C22×C4—612no—
475C22×C4—614no—
476C22×C4—1224no—
477C22×C4—1228no—
478C22×C4—1224no—
479C22×C4—1224no—
480C22×C4—1221no—
481C22×C4—2442no—
482C22×C4—2444no—
483C22×C4—2448no—
484C22×C4D4×D6111yescharacteristic
485C22×C4D4×D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487SD16—12216no—
488D8—12216no—
489C2×C8—614no—
490C2×C8—1224no—
491C2×C8—12216no—
492C2×C8#1175822618yescomplemented
493C2×C8C23×D6114yes—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—1224no—
501C4⋊C4—1224no—
502C4⋊C4C23×D6112yes—
503C4⋊C4C23×D6112yes—
504C4⋊C4C23×D6114yes—
505C22⋊C4—1224no—
506C22⋊C4—1228no—
507C22⋊C4—1224no—
508C22⋊C4—1228no—
509C22⋊C4—1224no—
510C22⋊C4—1224no—
511C22⋊C4—1224no—
512C42—614no—
513C42—612no—
514C42—614no—
515C42—612no—
516C42—1224no—
517C42D4×D6112yes—
518C42C23×D6112yes—
519C23—614no—
520C23—612no—
521C23—1222no—
522C23—1222no—
523C23—1222no—
524C23—1224no—
525C23—2444no—
526C23—2442no—
527C23—2442no—
528C23—2444no—
529C23—2442no—
530C23—2444no—
531C23C2×D4×D6111yescharacteristic
532Q8—48no—
533Q8—1224no—
534Q8—2442no—
535D4—1224no—
536D4—1224no—
537D4—1224no—
538D4—2448no—
539D4—2448no—
540D4—2448no—
541D4—24416no—
542D4—2442no—
543D4—2444no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—612no—
553C2×C4—612no—
554C2×C4—612no—
555C2×C4—1228no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1222no—
562C2×C4—2444no—
563C2×C4—2448no—
564C2×C4—24416no—
565C2×C4—2442no—
566C2×C4C2×D4×D6112yes—
567C2×C4C2×D4×D6112yes—
568C2×C4C2×D4×D6111yescharacteristic
569C2×C4C2×#1175822614yes—
570C2×C4C24×D6111yescharacteristic
571C8—1224no—
572C8C2×#1175822618yescomplemented
573C22—614no—
574C22—1222no—
575C22—1228no—
576C22—1222no—
577C22—4888no—
578C22—4884no—
579C22—2444no—
580C22—2442no—
581C22—2442no—
582C22—2444no—
583C22—2442no—
584C22—2444no—
585C22#1684031312yes—
586C22AGL(1,ℤ/8)×D6112yes—
587C22#1684056112yes—
588C22C22×D4×D6111yescharacteristic
589C4—22no—
590C4—22no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—1222no—
595C4—1222no—
596C4—1222no—
597C4—2442no—
598C4#1684031312yes—
599C4C22×#1175822612yes—
600C2—1224no—
601C2—4884no—
602C2—4882no—
603C2#2145651914yesdirect product factor
604C2C2×#1684031311yescharacteristic
605C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
606C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#501C2
socleC23×C61C22×D4
hypercenterC2×#501D61
hypocenterC61C2×#3180
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×#16840313C2semidirect
2. C2×#16840297C2semidirect
3. #21456519C2direct (N × H)
4. C2×#16840151C2semidirect
5. C2×#16840128C2semidirect
6. C2×#16840124C2semidirect
7. C2×#16839999C2semidirect
8. C2×#16839992C2semidirect
9. C2×#16839873C2semidirect
10. C2×#16839861C2semidirect
11. C2×#97×D61C2semidirect
12. C2×C61×#501C2semidirect
18. #16840313C22semidirect
21. #16840297C22semidirect
37. #16840151C22semidirect
38. #16840128C22semidirect
39. #16840124C22semidirect
40. #16839999C22semidirect
41. #16839992C22semidirect
42. #16839873C22semidirect
43. #16839861C22semidirect
50. #97×D61C22semidirect
53. C61×#501C22semidirect
183. C2×C8×C61#41semidirect
267. C8×C61C2×#41semidirect
300. C2×#501D61semidirect
309. C2×C61#3180semidirect
342. #501C2×D61semidirect
360. C61C2×#3180semidirect
492. C2×C8#11758226semidirect
572. C8C2×#11758226semidirect
603. C2#21456519direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#168402972
#214565192
C2×#168401512
C2×#168401282
C2×#168401242
C2×#168399992
C2×#168399922
C2×#168398732
C2×#168398612
C2×#97×D612
C2×C61×#5012
C2×#318061