Finite Group Databasebeta

← C2×#21456210 #30549300

Subgroups — C2×#21456210 #30549300

Order 15616.

2398 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840308C211yesmaximal, characteristic
2C22×#14519776C211yesmaximal, characteristic, complemented
3#21456210C2116yesmaximal, direct product factor
4C2×#16840123C212yesmaximal, complemented
5C2×#16840120C212yesmaximal, complemented
6C2×#16840111C212yesmaximal, complemented
7C2×#16840107C212yesmaximal, complemented
8C2×#16839764C211yesmaximal, characteristic, complemented
9C2×#16839763C211yesmaximal, characteristic, complemented
10C2×#16839761C211yesmaximal, characteristic, complemented
11C2×M16×Dic61C211yesmaximal, characteristic, complemented
12C2×C61×#417C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C23×Dic122—21no—
15C22×#11758194C2211yescharacteristic
16#16840308—24no—
17#16840308C2218yes—
18C2×#14519776—21no—
19C2×#14519776—21no—
20C2×#14519776—24no—
21C2×#14519776C2212yes—
22C2×#14519776C2212yes—
23C2×#14519776C2218yescomplemented
24C2×#14519775C2211yescharacteristic
25C2×#14519773C2211yescharacteristic
26C2×#14519766C2211yescharacteristic
27C2×#14519764C2211yescharacteristic
28C2×#14519749—21no—
29C2×#14519749—21no—
30C2×C4×C22⋊D61—21no—
31C2×C4×C22⋊D61—21no—
32C22×C4⋊Dic61C2211yescharacteristic
33C22×#11758196—21no—
34C22×C4×Dic61C2211yescharacteristic
35#16840123C22116yescomplemented
36#16840120C22116yescomplemented
37#16840111C22116yescomplemented
38#16840107C22116yescomplemented
39#16839764C2218yescomplemented
40#16839763C2218yescomplemented
41#16839761C2218yescomplemented
42C2×#14519702C2212yes—
43C2×#14519701C2212yes—
44C2×#14519700C2212yes—
45C2×#14519699C2212yes—
46C2×#14519698C2212yes—
47C2×#14519697C2212yes—
48M16×Dic61C2218yescomplemented
49C2×#14519687C2212yes—
50C2×C8×Dic61C2212yes—
51C22×#41×C61C2211yescharacteristic
52C22×M16×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#417C2218yescomplemented
55C2×#103×C61C2212yes—
56C2×#102×C61C2212yes—
57C2×#11758222—22no—
58C2×#11758222—24no—
59C2×#11758222—24no—
60C2×#11758222C2314yes—
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—28no—
64C22×Dic122—48no—
65C22×Dic122C2311yescharacteristic
66C22×D244C2311yescharacteristic
67C22×C22⋊D61—21no—
68C22×C22⋊D61—21no—
69C22×C22⋊D61—21no—
70C23×Dic61—21no—
71C23×Dic61D411yescharacteristic
72C23×Dic61D411yescharacteristic
73C22×C4×D61—21no—
74C2×#11758194—24no—
75C2×#11758194C2314yes—
76C22×#92418C2312yes—
77#14519776—24no—
78#14519776—24no—
79#14519776—28no—
80#14519776—28no—
81#14519776C2318yes—
82#14519776C2318yes—
83#14519775—22no—
84#14519775C2314yes—
85#14519773—22no—
86#14519773C2314yes—
87#14519766—22no—
88#14519766C2314yes—
89#14519764—22no—
90#14519764C2314yes—
91#14519749—28no—
92#14519749—28no—
93C4×C22⋊D61—28no—
94C4×C22⋊D61—28no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61—22no—
97C2×C4⋊Dic61C2312yes—
98C2×C4⋊Dic61C2314yes—
99C2×#11758196—22no—
100C2×#11758196—22no—
101C2×#11758196—28no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D414yes—
111C2×C4×Dic61C2311yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C2×C4×Dic61C2314yes—
114#14519702C2318yes—
115#14519701C2318yes—
116#14519700C2318yes—
117#14519699C2318yes—
118#14519698C2318yes—
119#14519697C2318yes—
120#14519687C2318yes—
121C8×Dic61C2318yes—
122C2×#41×C61—22no—
123C2×#41×C61—24no—
124C2×#41×C61—24no—
125C2×#41×C61C2314yes—
126C22×Q8×C61C2311yescharacteristic
127C22×D4×C61—21no—
128C22×D4×C61C2311yescharacteristic
129C23×C4×C61—21no—
130C23×C4×C61—21no—
131C23×C4×C61C2311yescharacteristic
132C2×M16×C61—24no—
133C2×M16×C61C2314yes—
134C22×C8×C61C2312yes—
135C2×C4⋊C4×C61—21no—
136C2×C4⋊C4×C61—22no—
137C2×C4⋊C4×C61C2312yes—
138C2×C4⋊C4×C61C2314yes—
139#103×C61C2318yes—
140#102×C61C2318yes—
141C24×C61—21no—
142C24×C61C2×D411yescharacteristic
143C23×D61—21no—
144#41×C61—28no—
145#41×C61—28no—
146#41×C61—48no—
147C2×Q8×C61—21no—
148C2×Q8×C61—22no—
149C2×Q8×C61—22no—
150C2×Q8×C61C2412yes—
151C2×D4×C61—24no—
152C2×D4×C61—21no—
153C2×D4×C61—22no—
154C2×D4×C61—22no—
155C2×D4×C61—44no—
156C2×D4×C61C2412yes—
157C22×C4×C61—24no—
158C22×C4×C61—22no—
159C22×C4×C61—21no—
160C22×C4×C61—24no—
161C22×C4×C61—44no—
162C22×C4×C61—44no—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D412yes—
165C22×C4×C61#4114yes—
166C22×C4×C61#4114yes—
167C22×C4×C61C2411yescharacteristic
168C22×C4×C61C2412yes—
169C2×C22⋊D61—24no—
170C2×C22⋊D61—24no—
171C2×C22⋊D61—24no—
172C2×C22⋊D61—44no—
173C2×C22⋊D61—44no—
174C2×C22⋊D61—44no—
175C22×Dic61—24no—
176C22×Dic61—24no—
177C22×Dic61—21no—
178C22×Dic61—24no—
179C22×Dic61—22no—
180C22×Dic61—44no—
181C22×Dic61C2×D411yescharacteristic
182C22×Dic61C2×D414yes—
183C22×Dic61C2×D411yescharacteristic
184C22×Dic61C2×D414yes—
185C22×Dic61C2×D411yescharacteristic
186C22×Dic61C2×D411yescharacteristic
187#11758222—28no—
188#11758222—28no—
189#11758222—48no—
190C2×D244—22no—
191C2×D244—22no—
192C2×D244—21no—
193C2×D244C2412yes—
194C2×C4×D61—24no—
195C2×C4×D61—44no—
196C2×Dic122—22no—
197C2×Dic122—22no—
198C2×Dic122—21no—
199C2×Dic122—28no—
200C2×Dic122—28no—
201C2×Dic122—48no—
202C2×Dic122—432no—
203C2×Dic122—48no—
204C2×Dic122C2412yes—
205M16×C61—28no—
206C2×C8×C61—24no—
207C2×C8×C61C2414yes—
208C4⋊C4×C61—22no—
209C4⋊C4×C61—24no—
210C4⋊C4×C61C2414yes—
211C22⋊Dic61—24no—
212C22⋊Dic61—24no—
213#11758198—24no—
214#11758198—24no—
215C4⋊Dic61—22no—
216C4⋊Dic61—24no—
217C4⋊Dic61C2414yes—
218#11758196—28no—
219#11758196—28no—
220#11758196—24no—
221#11758196—24no—
222C4×Dic61C2×D414yes—
223C4×Dic61C2×D414yes—
224C4×Dic61C2×D412yes—
225C4×Dic61C2×D412yes—
226C4×Dic61C2412yes—
227C4×Dic61C2412yes—
228#11758194—28no—
229C2×#92418—24no—
230C2×#92418C2414yes—
231C23×C61—22no—
232C23×C61—22no—
233C23×C61—22no—
234C23×C61—42no—
235C23×C61—42no—
236C23×C61—42no—
237C23×C61#12414yes—
238C23×C61C22×D412yes—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241C22×D61—42no—
242C22×D61—42no—
243C22×D61—42no—
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61—42no—
247D4×C61—22no—
248D4×C61—22no—
249D4×C61—48no—
250D4×C61—42no—
251C2×C4×C61—22no—
252C2×C4×C61—24no—
253C2×C4×C61—48no—
254C2×C4×C61—42no—
255C2×C4×C61—44no—
256C2×C4×C61—48no—
257C2×C4×C61#12414yes—
258C2×C4×C61#12418yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D412yes—
261C2×C4×C61C2×#4112yes—
262C2×C4×C61C2×#4114yes—
263C2×C4×C61C2×#4112yes—
264C2×C4×C61C2×#4114yes—
265C2×C4×C61C2511yescharacteristic
266C22⋊D61—48no—
267C22⋊D61—48no—
268C22⋊D61—48no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—28no—
272C2×Dic61—28no—
273C2×Dic61—24no—
274C2×Dic61—22no—
275C2×Dic61—48no—
276C2×Dic61—42no—
277C2×Dic61—44no—
278C2×Dic61AGL(1,ℤ/8)14yes—
279C2×Dic61#13714yes—
280C2×Dic61C22×D412yes—
281C2×Dic61C22×D412yes—
282C2×Dic61C22×D412yes—
283C2×Dic61C22×D412yes—
284D244—22no—
285D244—22no—
286D244—42no—
287C4×D61—48no—
288Dic122—22no—
289Dic122—22no—
290Dic122—416no—
291Dic122—42no—
292Dic122—416no—
293C8×C61—24no—
294#92418—24no—
295C2×#3141—611nomaximal, Sylow
296C22×C61—28no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—84no—
303C22×C61C2×#12414yes—
304C22×C61C2×#12412yes—
305C22×C61C2×AGL(1,ℤ/8)12yes—
306C22×C61C2×#13712yes—
307C22×C61C23×D411yescharacteristic
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—84no—
312C4×C61—42no—
313C4×C61—44no—
314C4×C61C2×#12412yes—
315C4×C61C2×#12414yes—
316C4×C61C22×#4111yescharacteristic
317C4×C61C22×#4111yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—44no—
322Dic61C2×AGL(1,ℤ/8)14yes—
323Dic61C2×#13714yes—
324C2×#532—611no—
325C22×C4⋊Q8—611no—
326#3141—6116no—
327C2×#489—612no—
328C2×#489—612no—
329C2×#487—612no—
330C2×#486—612no—
331C2×#417—611no—
332C2×#417—611no—
333C2×#417—611no—
334C2×#417D6111yescharacteristic, complemented
335C2×C4×M16—611no—
336C2×C61—24no—
337C2×C61—82no—
338C2×C61#314114yescomplemented
339C2×C61C22×#12411yescharacteristic
340C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
341C2×C61C22×#13711yescharacteristic
342D61—82no—
343C22×#41—611no—
344C22×#41C2×D6111yescharacteristic
345C23×Q8—1221no—
346C22×M16—611no—
347C22×M16C2×D6111yescharacteristic
348#532—618no—
349#532—1224no—
350C2×C4⋊Q8—612no—
351C2×C4⋊Q8—612no—
352C2×C4⋊Q8—618no—
353C2×C4⋊Q8—611no—
354C2×C4⋊Q8—1221no—
355C2×C4⋊Q8—1221no—
356C2×C4⋊Q8—1224no—
357C2×#127—611no—
358C2×#124—611no—
359C2×#124—611no—
360C2×#121—1221no—
361C2×#121—1221no—
362C2×C4×D4—1221no—
363C2×C4×D4—1221no—
364C22×C4⋊C4—611no—
365C22×C4⋊C4—1221no—
366C22×C4⋊C4C2×D6111yescharacteristic
367C22×C42—611no—
368#489—6116no—
369#489—6116no—
370#487—6116no—
371#486—6116no—
372#417—618no—
373#417—618no—
374#417—618no—
375#417C2×D6118yescomplemented
376C2×#103—612no—
377C2×#103—612no—
378C2×#103—612no—
379C2×#103C2×D6112yes—
380C2×#102—612no—
381C2×#102—612no—
382C2×#102—612no—
383C2×#102C2×D6112yes—
384C4×M16—618no—
385C2×#97—612no—
386C2×C4×C8—612no—
387C61C2×#314111yescharacteristic, complemented, Sylow
388C2×#41—22no—
389C2×#41—24no—
390C2×#41—24no—
391C2×#41—614no—
392C2×#41—1222no—
393C2×#41—1224no—
394C2×#41—1224no—
395C2×#41C22×D6114yes—
396C22×Q8—611no—
397C22×Q8—1228no—
398C22×Q8—1222no—
399C22×Q8—1222no—
400C22×Q8—2448no—
401C22×Q8C22×D6111yescharacteristic
402C22×D4—21no—
403C22×D4—611no—
404C22×D4—1221no—
405C22×D4—1221no—
406C22×D4—1221no—
407C22×D4C22×D6111yescharacteristic
408C23×C4—21no—
409C23×C4—21no—
410C23×C4—611no—
411C23×C4—611no—
412C23×C4—1221no—
413C23×C4—1221no—
414C23×C4C22×D6111yescharacteristic
415C2×M16—24no—
416C2×M16—614no—
417C2×M16—1224no—
418C2×M16C22×D6114yes—
419C22×C8—612no—
420C22×C8C22×D6112yes—
421C4⋊Q8—618no—
422C4⋊Q8—618no—
423C4⋊Q8—614no—
424C4⋊Q8—1228no—
425C4⋊Q8—1228no—
426C4⋊Q8—1224no—
427C4⋊Q8—1224no—
428C4⋊Q8—1222no—
429#127—614no—
430#127—1222no—
431#124—614no—
432#124—614no—
433#124—1222no—
434#124—1222no—
435#121—1228no—
436#121—1228no—
437C4×D4—1228no—
438C4×D4—1228no—
439C2×C4⋊C4—22no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—612no—
442C2×C4⋊C4—614no—
443C2×C4⋊C4—1228no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1222no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22×D6112yes—
451C2×C4⋊C4C22×D6114yes—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—614no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—614no—
462#103—618no—
463#103—618no—
464#103—618no—
465#103C22×D6118yes—
466#102—618no—
467#102—618no—
468#102—618no—
469#102C22×D6118yes—
470#97—618no—
471C4×C8—618no—
472C24—21no—
473C24—1221no—
474C24D4×D6111yescharacteristic
475#41—28no—
476#41—28no—
477#41—48no—
478#41—1228no—
479#41—1228no—
480#41—2448no—
481C2×Q8—21no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—612no—
485C2×Q8—1228no—
486C2×Q8—1228no—
487C2×Q8—1221no—
488C2×Q8—1222no—
489C2×Q8—1222no—
490C2×Q8—2448no—
491C2×Q8—24432no—
492C2×Q8—2448no—
493C2×Q8C23×D6112yes—
494C2×D4—24no—
495C2×D4—21no—
496C2×D4—22no—
497C2×D4—22no—
498C2×D4—44no—
499C2×D4—612no—
500C2×D4—1224no—
501C2×D4—1224no—
502C2×D4—1224no—
503C2×D4—1221no—
504C2×D4—1222no—
505C2×D4—1222no—
506C2×D4—2444no—
507C2×D4—2444no—
508C2×D4—2444no—
509C2×D4C23×D6112yes—
510C22×C4—24no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—24no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—614no—
519C22×C4—614no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—1224no—
523C22×C4—1224no—
524C22×C4—1224no—
525C22×C4—1221no—
526C22×C4—1222no—
527C22×C4—1224no—
528C22×C4—2444no—
529C22×C4—2444no—
530C22×C4#1175822214yes—
531C22×C4D4×D6112yes—
532C22×C4D4×D6111yescharacteristic
533C22×C4#1175822414yes—
534C22×C4C23×D6112yes—
535C22×C4C23×D6111yescharacteristic
536M16—28no—
537M16—1228no—
538C2×C8—24no—
539C2×C8—614no—
540C2×C8—1224no—
541C2×C8C23×D6114yes—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—614no—
545C4⋊C4—1228no—
546C4⋊C4—1228no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1222no—
551C4⋊C4C23×D6114yes—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C22⋊C4—1224no—
555C22⋊C4—1224no—
556C42—614no—
557C42—614no—
558C42—612no—
559C42—612no—
560C42—612no—
561C42—612no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—1222no—
569C23—2442no—
570C23—2442no—
571C23—2442no—
572C23#1451976314yes—
573C23C2×D4×D6112yes—
574C23C2×D4×D6111yescharacteristic
575Q8—22no—
576Q8—22no—
577Q8—42no—
578Q8—1222no—
579Q8—1222no—
580Q8—24416no—
581Q8—24416no—
582Q8—2442no—
583D4—22no—
584D4—22no—
585D4—48no—
586D4—42no—
587D4—1222no—
588D4—1222no—
589D4—2448no—
590D4—2448no—
591D4—2448no—
592D4—2442no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—48no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—614no—
600C2×C4—614no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1228no—
606C2×C4—1228no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1224no—
611C2×C4—2448no—
612C2×C4—2442no—
613C2×C4—2444no—
614C2×C4—2448no—
615C2×C4#1451976314yes—
616C2×C4#1451976318yes—
617C2×C4C2×D4×D6112yes—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×#1175822412yes—
620C2×C4C2×#1175822414yes—
621C2×C4C2×#1175822212yes—
622C2×C4C2×#1175822214yes—
623C2×C4C24×D6111yescharacteristic
624C8—24no—
625C8—1224no—
626C22—22no—
627C22—22no—
628C22—28no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—84no—
633C22—4884no—
634C22—2442no—
635C22—2442no—
636C22—2442no—
637C22C2×#1451976314yes—
638C22C2×#1451976312yes—
639C22AGL(1,ℤ/8)×D6112yes—
640C22#137×D6112yes—
641C22C22×D4×D6111yescharacteristic
642C4—44no—
643C4—42no—
644C4—614no—
645C4—614no—
646C4—1222no—
647C4—1222no—
648C4—2444no—
649C4—2442no—
650C4C2×#1451976314yes—
651C4C2×#1451976312yes—
652C4C22×#1175822411yescharacteristic
653C4C22×#1175822211yescharacteristic
654C2—24no—
655C2—82no—
656C2—4882no—
657C2#2145621014yesdirect product factor
658C2C22×#1451976311yescharacteristic
659C2C2×AGL(1,ℤ/8)×D6111yescharacteristic
660C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#417C2
socleC23×C61C22×D4
prime radicalC2×#16840308C2
hypercenterC2×#417D61
hypocenterC61C2×#3141
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#14519776C2semidirect
3. #21456210C2direct (N × H)
4. C2×#16840123C2semidirect
5. C2×#16840120C2semidirect
6. C2×#16840111C2semidirect
7. C2×#16840107C2semidirect
8. C2×#16839764C2semidirect
9. C2×#16839763C2semidirect
10. C2×#16839761C2semidirect
11. C2×M16×Dic61C2semidirect
12. C2×C61×#417C2semidirect
23. C2×#14519776C22semidirect
35. #16840123C22semidirect
36. #16840120C22semidirect
37. #16840111C22semidirect
38. #16840107C22semidirect
39. #16839764C22semidirect
40. #16839763C22semidirect
41. #16839761C22semidirect
48. M16×Dic61C22semidirect
54. C61×#417C22semidirect
334. C2×#417D61semidirect
338. C2×C61#3141semidirect
375. #417C2×D61semidirect
387. C61C2×#3141semidirect
657. C2#21456210direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403082
C22×#145197762
#214562102
C2×#168401232
C2×#168401202
C2×#168401112
C2×#168401072
C2×#168397642
C2×#168397632
C2×#168397612
C2×M16×Dic612
C2×C61×#4172
C2×#314161