Finite Group Databasebeta

← C2×#21457006 #30552115

Subgroups — C2×#21457006 #30552115

Order 15616.

2050 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840307C211yesmaximal, characteristic
2C22×#14519768C211yesmaximal, characteristic, complemented
3#21457006C2116yesmaximal, direct product factor
4C2×#16840170C212yesmaximal, complemented
5C2×#16840125C212yesmaximal, complemented
6C2×#16840081C211yesmaximal, characteristic, complemented
7C2×#16840078C211yesmaximal, characteristic, complemented
8C2×#16840075C211yesmaximal, characteristic, complemented
9C2×#16840065C212yesmaximal, complemented
10C2×#16840064C212yesmaximal, complemented
11C2×#16839816C211yesmaximal, characteristic, complemented
12C2×C61×#480C211yesmaximal, characteristic, complemented
13C22×#11758224—21no—
14#16840307—24no—
15#16840307C2218yes—
16C2×#14519776C2211yescharacteristic
17C2×#14519773C2211yescharacteristic
18C2×#14519768—21no—
19C2×#14519768—21no—
20C2×#14519768—24no—
21C2×#14519768C2212yes—
22C2×#14519768C2212yes—
23C2×#14519768C2218yescomplemented
24C2×#14519763—21no—
25C2×#14519751C2211yescharacteristic
26C2×#14519749—21no—
27C2×D4×Dic61C2211yescharacteristic
28C2×#14519739C2211yescharacteristic
29C2×#14519737—21no—
30C22×C4⋊Dic61—21no—
31C22×C4⋊Dic61C2211yescharacteristic
32C22×#11758196—21no—
33C22×C4×Dic61C2211yescharacteristic
34#16840170C22116yescomplemented
35#16840125C22116yescomplemented
36#16840081C2218yescomplemented
37#16840078C2218yescomplemented
38#16840075C2218yescomplemented
39#16840065C22116yescomplemented
40#16840064C22116yescomplemented
41#16839816C2218yescomplemented
42C2×#14519710C2212yes—
43C2×#14519709C2212yes—
44C2×#14519708C2212yes—
45C2×#14519706C2212yes—
46C2×#14519699C2212yes—
47C2×#14519698C2212yes—
48C2×#14519697C2212yes—
49C2×C22⋊#92418C2211yescharacteristic
50C2×#14519687C2212yes—
51C2×#121×C61C2211yescharacteristic
52C22×C4⋊C4×C61C2211yescharacteristic
53C61×#480C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×#102×C61C2212yes—
56C2×C22⋊C8×C61C2211yescharacteristic
57C2×#11758224—28no—
58C2×#11758224—48no—
59C22×Dic122—21no—
60C22×D244C2311yescharacteristic
61C22×C22⋊D61—21no—
62C22×C22⋊D61—21no—
63C22×C22⋊D61—21no—
64C23×Dic61—21no—
65C23×Dic61—21no—
66C23×Dic61D411yescharacteristic
67C23×Dic61D411yescharacteristic
68C22×C4×D61—21no—
69C22×#92418C2312yes—
70#14519776—22no—
71#14519776C2314yes—
72#14519773—22no—
73#14519773C2314yes—
74#14519768—24no—
75#14519768—24no—
76#14519768—28no—
77#14519768—28no—
78#14519768C2318yes—
79#14519768C2318yes—
80#14519763—28no—
81#14519751—22no—
82#14519751C2314yes—
83#14519749—28no—
84D4×Dic61—22no—
85D4×Dic61C2314yes—
86#14519739—22no—
87#14519739C2314yes—
88#14519737—28no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61—22no—
92C2×C4⋊Dic61—22no—
93C2×C4⋊Dic61—28no—
94C2×C4⋊Dic61C2312yes—
95C2×C4⋊Dic61C2314yes—
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—22no—
98C2×#11758196—22no—
99C2×#11758196—28no—
100C2×#11758196—21no—
101C2×C22⋊Dic61—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D414yes—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109C2×C4×Dic61C2314yes—
110#14519710C2318yes—
111#14519709C2318yes—
112#14519708C2318yes—
113#14519706C2318yes—
114#14519699C2318yes—
115#14519698C2318yes—
116#14519697C2318yes—
117C22⋊#92418C2314yes—
118#14519687C2318yes—
119C22×D4×C61—21no—
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C23×C4×C61C2311yescharacteristic
123C22×C8×C61C2312yes—
124#121×C61—22no—
125#121×C61C2314yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2312yes—
130C2×C4⋊C4×C61C2314yes—
131C2×C22⋊C4×C61—21no—
132#107×C61C2318yes—
133#102×C61C2318yes—
134C22⋊C8×C61C2314yes—
135C24×C61—21no—
136C24×C61C2×D411yescharacteristic
137C23×D61—21no—
138C2×D4×C61—24no—
139C2×D4×C61—21no—
140C2×D4×C61—24no—
141C2×D4×C61—44no—
142C2×D4×C61C2412yes—
143C22×C4×C61—24no—
144C22×C4×C61—21no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—44no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2412yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—24no—
153C2×C22⋊D61—24no—
154C2×C22⋊D61—24no—
155C2×C22⋊D61—44no—
156C2×C22⋊D61—44no—
157C2×C22⋊D61—44no—
158C22×Dic61—24no—
159C22×Dic61—21no—
160C22×Dic61—24no—
161C22×Dic61—22no—
162C22×Dic61—24no—
163C22×Dic61—42no—
164C22×Dic61—44no—
165C22×Dic61—44no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D414yes—
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D414yes—
172#11758224—432no—
173C2×D244—24no—
174C2×D244—21no—
175C2×D244C2412yes—
176C2×C4×D61—24no—
177C2×C4×D61—44no—
178C2×Dic122—24no—
179C2×Dic122—44no—
180C2×C8×C61—24no—
181C2×C8×C61C2414yes—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61C2414yes—
186C4⋊C4×C61C2412yes—
187C22⋊C4×C61—24no—
188C22⋊Dic61—24no—
189#11758198—24no—
190#11758198—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61—22no—
193C4⋊Dic61—21no—
194C4⋊Dic61—28no—
195C4⋊Dic61—28no—
196C4⋊Dic61C2414yes—
197C4⋊Dic61C2412yes—
198#11758196—28no—
199#11758196—28no—
200#11758196—24no—
201C4×Dic61C2×D414yes—
202C4×Dic61C2×D412yes—
203C4×Dic61C2×D414yes—
204C4×Dic61C2×D412yes—
205C4×Dic61C2412yes—
206C4×Dic61C2412yes—
207C2×#92418—24no—
208C2×#92418C2414yes—
209C23×C61—22no—
210C23×C61—22no—
211C23×C61—22no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61—42no—
215C23×C61AGL(1,ℤ/8)14yes—
216C23×C61C22×D412yes—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—42no—
220C22×D61—42no—
221C22×D61—42no—
222D4×C61—24no—
223D4×C61—48no—
224D4×C61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—24no—
228C2×C4×C61—24no—
229C2×C4×C61—22no—
230C2×C4×C61—42no—
231C2×C4×C61—48no—
232C2×C4×C61—44no—
233C2×C4×C61C22×D412yes—
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C612−1+414yes—
236C2×C4×C61C2511yescharacteristic
237C22⋊D61—48no—
238C22⋊D61—48no—
239C22⋊D61—48no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—28no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—24no—
246C2×Dic61—22no—
247C2×Dic61—44no—
248C2×Dic61—48no—
249C2×Dic61—48no—
250C2×Dic61—42no—
251C2×Dic61—44no—
252C2×Dic61C22×D412yes—
253C2×Dic61C22×D412yes—
254C2×Dic61C22×D412yes—
255C2×Dic61C22×D412yes—
256D244—24no—
257D244—42no—
258C4×D61—48no—
259Dic122—48no—
260C8×C61—24no—
261#92418—24no—
262C2×#3234—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—28no—
265C22×C61—22no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—84no—
270C22×C61#53612yes—
271C22×C61C2×AGL(1,ℤ/8)14yes—
272C22×C61C2×AGL(1,ℤ/8)12yes—
273C22×C61#57512yes—
274C22×C61C23×D411yescharacteristic
275C2×D61—42no—
276C2×D61—42no—
277C2×D61—42no—
278C2×D61—84no—
279C4×C61—22no—
280C4×C61—42no—
281C4×C61—44no—
282C4×C61#53612yes—
283C4×C61C2×2−1+412yes—
284Dic61—22no—
285Dic61—22no—
286Dic61—22no—
287Dic61—22no—
288Dic61—42no—
289Dic61—44no—
290C2×#532—611no—
291C22×#125—611no—
292#3234—6116no—
293C2×#501—612no—
294C2×#490—612no—
295C2×#481—611no—
296C2×#481—611no—
297C2×#480—611no—
298C2×#480D6111yescharacteristic, complemented
299C2×#478—612no—
300C2×#478—612no—
301C2×#432—611no—
302C2×C61—24no—
303C2×C61—82no—
304C2×C61#323414yescomplemented
305C2×C61C2×#53611yescharacteristic
306C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
307C2×C61C2×#57511yescharacteristic
308D61—82no—
309C22×#41—1221no—
310#532—618no—
311#532—1224no—
312C2×C4⋊Q8—611no—
313C2×#127—611no—
314C2×#125—612no—
315C2×#125—612no—
316C2×#125—618no—
317C2×#125—1221no—
318C2×#125—1221no—
319C2×#125—1224no—
320C2×#124—1221no—
321C2×#121—611no—
322C2×#121—1221no—
323C2×#121C2×D6111yescharacteristic
324C2×C4×D4—611no—
325C2×C4×D4—611no—
326C2×C4×D4—1221no—
327C22×C4⋊C4—611no—
328C22×C4⋊C4—1221no—
329C22×C4⋊C4—1221no—
330C22×C4⋊C4C2×D6111yescharacteristic
331C22×C42—611no—
332#501—6116no—
333#490—6116no—
334#481—618no—
335#481—618no—
336#480—618no—
337#480C2×D6118yescomplemented
338#478—6116no—
339#478—6116no—
340#432—618no—
341C2×#107—612no—
342C2×#107C2×D6112yes—
343C2×#106—612no—
344C2×#106—612no—
345C2×C4⋊C8—612no—
346C2×#102—612no—
347C2×#102—612no—
348C2×#102—612no—
349C2×#102C2×D6112yes—
350C2×C22⋊C8—611no—
351C2×C22⋊C8C2×D6111yescharacteristic
352C2×#97—612no—
353C61C2×#323411yescharacteristic, complemented, Sylow
354C2×#41—1228no—
355C2×#41—2448no—
356C22×Q8—1221no—
357C22×D4—21no—
358C22×D4—611no—
359C22×D4—1221no—
360C22×D4—1221no—
361C22×D4—1221no—
362C22×D4C22×D6111yescharacteristic
363C23×C4—21no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—1221no—
367C23×C4—1221no—
368C23×C4—1221no—
369C23×C4C22×D6111yescharacteristic
370C22×C8—612no—
371C22×C8C22×D6112yes—
372C4⋊Q8—614no—
373C4⋊Q8—1222no—
374#127—614no—
375#127—1222no—
376#125—618no—
377#125—618no—
378#125—1224no—
379#125—1228no—
380#125—1228no—
381#125—1224no—
382#124—1228no—
383#121—22no—
384#121—614no—
385#121—1228no—
386#121—1222no—
387#121C22×D6114yes—
388C4×D4—614no—
389C4×D4—614no—
390C4×D4—1228no—
391C4×D4—1222no—
392C4×D4—1222no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—614no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—1228no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1228no—
402C2×C4⋊C4—1222no—
403C2×C4⋊C4—1222no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1222no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C22×D6112yes—
408C2×C4⋊C4C22×D6114yes—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C42—611no—
415C2×C42—611no—
416C2×C42—614no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—614no—
420#107—618no—
421#107C22×D6118yes—
422#106—618no—
423#106—618no—
424C4⋊C8—618no—
425#102—618no—
426#102—618no—
427#102—618no—
428#102C22×D6118yes—
429C22⋊C8—614no—
430C22⋊C8C22×D6114yes—
431#97—618no—
432C24—21no—
433C24—1221no—
434C24D4×D6111yescharacteristic
435#41—24432no—
436C2×Q8—1224no—
437C2×Q8—2444no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—24no—
441C2×D4—44no—
442C2×D4—612no—
443C2×D4—1224no—
444C2×D4—1224no—
445C2×D4—1224no—
446C2×D4—1221no—
447C2×D4—1224no—
448C2×D4—2444no—
449C2×D4—2444no—
450C2×D4—2444no—
451C2×D4C23×D6112yes—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—614no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—614no—
463C22×C4—1224no—
464C22×C4—1224no—
465C22×C4—1224no—
466C22×C4—1224no—
467C22×C4—1221no—
468C22×C4—1222no—
469C22×C4—2444no—
470C22×C4—2442no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4D4×D6112yes—
474C22×C4D4×D6111yescharacteristic
475C22×C4C23×D6112yes—
476C22×C4C23×D6111yescharacteristic
477C2×C8—24no—
478C2×C8—614no—
479C2×C8—1224no—
480C2×C8C23×D6114yes—
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—614no—
485C4⋊C4—612no—
486C4⋊C4—1228no—
487C4⋊C4—1228no—
488C4⋊C4—1228no—
489C4⋊C4—1228no—
490C4⋊C4—1224no—
491C4⋊C4—1224no—
492C4⋊C4—1222no—
493C4⋊C4—1221no—
494C4⋊C4C23×D6114yes—
495C4⋊C4C23×D6112yes—
496C22⋊C4—24no—
497C22⋊C4—1224no—
498C22⋊C4—1224no—
499C22⋊C4—1224no—
500C42—612no—
501C42—614no—
502C42—614no—
503C42—612no—
504C42—612no—
505C42—612no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—1222no—
513C23—2442no—
514C23—2442no—
515C23—2442no—
516C23#1451980114yes—
517C23C2×D4×D6112yes—
518C23C2×D4×D6111yescharacteristic
519Q8—2448no—
520D4—24no—
521D4—48no—
522D4—42no—
523D4—1224no—
524D4—2448no—
525D4—2448no—
526D4—2448no—
527D4—2442no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—48no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1228no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1224no—
547C2×C4—2448no—
548C2×C4—2444no—
549C2×C4—2448no—
550C2×C4—2448no—
551C2×C4—2442no—
552C2×C4—2444no—
553C2×C4C2×D4×D6112yes—
554C2×C4C2×D4×D6111yescharacteristic
555C2×C4#1451982414yes—
556C2×C4C24×D6111yescharacteristic
557C8—24no—
558C8—1224no—
559C22—22no—
560C22—28no—
561C22—22no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—84no—
566C22—4884no—
567C22—2442no—
568C22—2442no—
569C22—2442no—
570C22#1684033412yes—
571C22C2×#1451980114yes—
572C22C2×#1451980112yes—
573C22#1684056112yes—
574C22C22×D4×D6111yescharacteristic
575C4—22no—
576C4—44no—
577C4—42no—
578C4—1222no—
579C4—1222no—
580C4—1222no—
581C4—1222no—
582C4—2442no—
583C4—2444no—
584C4#1684033412yes—
585C4C2×#1451982412yes—
586C2—24no—
587C2—82no—
588C2—4882no—
589C2#2145700614yesdirect product factor
590C2C2×#1684033411yescharacteristic
591C2C22×#1451980111yescharacteristic
592C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#480C2
socleC23×C61C22×D4
prime radicalC2×#16840307C2
hypercenterC2×#480D61
hypocenterC61C2×#3234
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. C22×#14519768C2semidirect
3. #21457006C2direct (N × H)
4. C2×#16840170C2semidirect
5. C2×#16840125C2semidirect
6. C2×#16840081C2semidirect
7. C2×#16840078C2semidirect
8. C2×#16840075C2semidirect
9. C2×#16840065C2semidirect
10. C2×#16840064C2semidirect
11. C2×#16839816C2semidirect
12. C2×C61×#480C2semidirect
23. C2×#14519768C22semidirect
34. #16840170C22semidirect
35. #16840125C22semidirect
36. #16840081C22semidirect
37. #16840078C22semidirect
38. #16840075C22semidirect
39. #16840065C22semidirect
40. #16840064C22semidirect
41. #16839816C22semidirect
53. C61×#480C22semidirect
298. C2×#480D61semidirect
304. C2×C61#3234semidirect
337. #480C2×D61semidirect
353. C61C2×#3234semidirect
589. C2#21457006direct (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×#145197682
#214570062
C2×#168401702
C2×#168401252
C2×#168400812
C2×#168400782
C2×#168400752
C2×#168400652
C2×#168400642
C2×#168398162
C2×C61×#4802
C2×#323461