Finite Group Databasebeta

← C2×#21456853 #30551313

Subgroups — C2×#21456853 #30551313

Order 15616.

2198 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic, complemented
2C2×#16840291C211yesmaximal, characteristic, complemented
3#21456853C2116yesmaximal, direct product factor
4C2×#16840152C211yesmaximal, characteristic, complemented
5C2×#16840130C211yesmaximal, characteristic, complemented
6C2×#16840127C211yesmaximal, characteristic, complemented
7C2×#16839986C212yesmaximal, complemented
8C2×#16839979C212yesmaximal, complemented
9C2×#16839923C212yesmaximal, complemented
10C2×#16839906C212yesmaximal, complemented
11C2×#16839817C211yesmaximal, characteristic, complemented
12C2×C61×#497C211yesmaximal, characteristic, complemented
13C22×#11758226C2212yes—
14C22×#11758201C2212yes—
15C22×#11758199C2212yes—
16#16840313—24no—
17#16840313C2218yescomplemented
18C2×#14519774C2211yescharacteristic
19#16840291—24no—
20#16840291C2218yescomplemented
21C2×#14519770—21no—
22C2×#14519769C2211yescharacteristic
23C2×#14519766C2212yes—
24C2×#14519759—21no—
25C2×#14519753C2211yescharacteristic
26C2×#14519752—21no—
27C2×#14519752—21no—
28C2×#14519748C2211yescharacteristic
29C2×Q8×Dic61C2211yescharacteristic
30C2×D4×Dic61C2211yescharacteristic
31C2×#14519739—21no—
32C2×#14519739—21no—
33C2×#14519737—21no—
34C2×#14519733—21no—
35C2×C42×D61C2211yescharacteristic
36#16840152C2218yescomplemented
37#16840130C2218yescomplemented
38#16840127C2218yescomplemented
39#16839986C22116yescomplemented
40#16839979C22116yescomplemented
41#16839923C22116yescomplemented
42#16839906C22116yescomplemented
43#16839817C2218yescomplemented
44C2×#14519706C2212yes—
45C2×#14519701C2212yes—
46C2×#14519701C2212yes—
47C2×#14519699C2212yes—
48C2×#14519699C2212yes—
49C2×#14519689C2212yes—
50C2×#14519688C2211yescharacteristic
51C22×Q16×C61C2212yes—
52C22×SD16×C61C2212yes—
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×#124×C61C2211yescharacteristic
55C61×#497C2218yescomplemented
56C2×#97×C61C2211yescharacteristic
57C2×#11758226—24no—
58C2×#11758226—44no—
59C2×#11758226—44no—
60C2×#11758226C2318yes—
61C22×D244—22no—
62C22×D244—21no—
63C22×D244—21no—
64C22×D244C2312yes—
65C22×C22⋊D61—21no—
66C23×Dic61—21no—
67C22×C4×D61—22no—
68C22×C4×D61D411yescharacteristic
69C22×C4×D61D411yescharacteristic
70C22×C4×D61C2311yescharacteristic
71C2×#11758201—44no—
72C2×#11758201C2318yes—
73C2×#11758199—44no—
74C2×#11758199C2318yes—
75C22×#92418C2312yes—
76#14519774—22no—
77#14519774C2314yes—
78#14519770—28no—
79#14519769—22no—
80#14519769C2314yes—
81#14519766—24no—
82#14519766C2318yes—
83#14519759—28no—
84#14519753—22no—
85#14519753C2314yes—
86#14519752—28no—
87#14519752—28no—
88#14519748—22no—
89#14519748C2314yes—
90Q8×Dic61—22no—
91Q8×Dic61C2314yes—
92D4×Dic61—22no—
93D4×Dic61C2314yes—
94#14519739—28no—
95#14519739—28no—
96#14519737—28no—
97#14519733—28no—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2311yescharacteristic
100C2×C4⋊Dic61C2311yescharacteristic
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×#11758198—21no—
107C2×#11758198—22no—
108C2×#11758198—22no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61C2311yescharacteristic
113C42×D61D414yes—
114C42×D61C2314yes—
115#14519706C2318yes—
116#14519701C2318yes—
117#14519701C2318yes—
118#14519699C2318yes—
119#14519699C2318yes—
120#14519689C2318yes—
121#14519688C2314yes—
122C22×Q8×C61C2312yes—
123C22×Q8×C61C2311yescharacteristic
124C22×D4×C61C2311yescharacteristic
125C2×Q16×C61—44no—
126C2×Q16×C61C2318yes—
127C2×SD16×C61—44no—
128C2×SD16×C61C2318yes—
129C22×C8×C61C2312yes—
130C4⋊Q8×C61—22no—
131C4⋊Q8×C61C2314yes—
132#124×C61—22no—
133#124×C61C2314yes—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61—21no—
136C2×C22⋊C4×C61—21no—
137C2×C22⋊C4×C61—21no—
138C2×C42×C61C2311yescharacteristic
139#97×C61C2314yes—
140C24×C61—21no—
141C23×D61—22no—
142C23×D61C2×D411yescharacteristic
143C2×Q8×C61—22no—
144C2×Q8×C61—21no—
145C2×Q8×C61—24no—
146C2×Q8×C61—24no—
147C2×Q8×C61—24no—
148C2×Q8×C61C2414yes—
149C2×Q8×C61C2412yes—
150C2×D4×C61—21no—
151C2×D4×C61—24no—
152C2×D4×C61C2412yes—
153C22×C4×C61—22no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—24no—
159C2×C22⋊D61—44no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—24no—
163C22×Dic61—44no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167#11758226—48no—
168#11758226—48no—
169#11758226—416no—
170C2×D244—24no—
171C2×D244—24no—
172C2×D244—28no—
173C2×D244—24no—
174C2×D244—24no—
175C2×D244—22no—
176C2×D244—44no—
177C2×D244—44no—
178C2×D244—48no—
179C2×D244C2414yes—
180C2×C4×D61—24no—
181C2×C4×D61—28no—
182C2×C4×D61—42no—
183C2×C4×D61—42no—
184C2×C4×D61—48no—
185C2×C4×D61C2×D412yes—
186C2×C4×D61C2×D414yes—
187C2×C4×D61C2×D414yes—
188C2×C4×D61C2412yes—
189Q16×C61—48no—
190SD16×C61—48no—
191C2×C8×C61—24no—
192C2×C8×C61C2414yes—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—24no—
195C22⋊C4×C61—24no—
196C22⋊C4×C61—24no—
197C42×C61C2×D412yes—
198C42×C61C2412yes—
199C22⋊Dic61—24no—
200C22⋊Dic61—24no—
201#11758201—48no—
202#11758199—48no—
203#11758198—24no—
204#11758198—28no—
205#11758198—28no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—24no—
209C4⋊Dic61C2412yes—
210C4⋊Dic61C2412yes—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214C4×Dic61—24no—
215C4×Dic61C2×D412yes—
216C4×Dic61C2×D414yes—
217C4×Dic61C2×D414yes—
218C4×Dic61C2412yes—
219C2×#92418—24no—
220C2×#92418C2414yes—
221C23×C61—22no—
222C23×C61—42no—
223C23×C61—42no—
224C23×C61—42no—
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—24no—
230C22×D61—44no—
231C22×D61—44no—
232C22×D61—44no—
233C22×D61C22×D412yes—
234Q8×C61—24no—
235Q8×C61—24no—
236Q8×C61—24no—
237Q8×C61—44no—
238Q8×C61—42no—
239D4×C61—24no—
240D4×C61—42no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—24no—
245C2×C4×C61—44no—
246C2×C4×C61—42no—
247C2×C4×C61C22×D412yes—
248C2×C4×C61C22×D412yes—
249C2×C4×C61C22×D411yescharacteristic
250C2×C4×C61H(5,𝔽2)14yes—
251C2×C4×C61C2511yescharacteristic
252C22⋊D61—48no—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—48no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61C22×D412yes—
262C2×Dic61C22×D412yes—
263C2×Dic61C22×D412yes—
264D244—24no—
265D244—24no—
266D244—48no—
267D244—48no—
268D244—416no—
269D244—44no—
270C4×D61—24no—
271C4×D61—24no—
272C4×D61—44no—
273C4×D61—44no—
274C4×D61—416no—
275C8×C61—24no—
276#92418—24no—
277C2×#3219—611nomaximal, Sylow
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—84no—
282C22×C61#53512yes—
283C22×C61#57512yes—
284C22×C61#57512yes—
285C22×C61C23×D411yescharacteristic
286C2×D61—22no—
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—44no—
290C2×D61—44no—
291C2×D61—44no—
292C2×D61—44no—
293C2×D61—88no—
294C4×C61—22no—
295C4×C61—22no—
296C4×C61—44no—
297C4×C61—42no—
298C4×C61#53512yes—
299C4×C61C2×H(5,𝔽2)12yes—
300Dic61—22no—
301Dic61—22no—
302Dic61—22no—
303Dic61—42no—
304Dic61—42no—
305C2×#532—611no—
306C2×#529—611no—
307#3219—6116no—
308C2×#497D6111yescharacteristic, complemented
309C2×#496—611no—
310C2×#491—611no—
311C2×#490—611no—
312C2×#464—612no—
313C2×#463—612no—
314C2×#452—612no—
315C2×#449—612no—
316C2×#432—611no—
317C2×C61—82no—
318C2×C61#321914yescomplemented
319C2×C61C2×#53511yescharacteristic
320C2×C61C2×#57511yescharacteristic
321C2×C61C2×#57511yescharacteristic
322D61—42no—
323D61—84no—
324C22×#41—612no—
325C22×Q16C2×D6112yes—
326C22×SD16—612no—
327C22×SD16C2×D6112yes—
328C22×D8—612no—
329#532—618no—
330#532—1224no—
331C2×C4⋊Q8C2×D6111yescharacteristic
332C2×#127—611no—
333#529—618no—
334#529—1224no—
335C2×#126—1221no—
336C2×#125—611no—
337C2×#124—612no—
338C2×#124C2×D6111yescharacteristic
339C2×#123—1221no—
340C2×C22⋊Q8—611no—
341C2×#121—611no—
342C2×#121—1221no—
343C2×#121—1221no—
344C2×C4×Q8—611no—
345C2×C4×D4—611no—
346C2×C4×D4—1221no—
347C2×C4×D4—1221no—
348C2×C4×D4—1221no—
349C2×#117—1221no—
350C22×C42—611no—
351#497C2×D6118yescomplemented
352#496—618no—
353#491—618no—
354#490—618no—
355#464—6116no—
356#463—6116no—
357#452—6116no—
358#449—6116no—
359#432—618no—
360C2×C4⋊C8—612no—
361C2×#103—612no—
362C2×#103—612no—
363C2×#102—612no—
364C2×#102—612no—
365C2×C22⋊C8—612no—
366C2×#97—611no—
367C2×#97C2×D6111yescharacteristic
368C61C2×#321911yescharacteristic, complemented, Sylow
369C2×#41—618no—
370C2×#41—1224no—
371C2×#41—2444no—
372C2×#41—2444no—
373C22×Q8C22×D6111yescharacteristic
374C22×Q8C22×D6112yes—
375C22×D4—612no—
376C22×D4—1222no—
377C22×D4—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C22×D4C22×D6111yescharacteristic
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—611no—
384C23×C4—1222no—
385C23×C4—1221no—
386C2×Q16—44no—
387C2×Q16C22×D6118yes—
388C2×SD16—44no—
389C2×SD16—618no—
390C2×SD16—2444no—
391C2×SD16C22×D6118yes—
392C2×D8—618no—
393C2×D8—2444no—
394C22×C8—612no—
395C22×C8C22×D6112yes—
396C4⋊Q8—22no—
397C4⋊Q8C22×D6114yes—
398#127—614no—
399#127—1222no—
400#126—1228no—
401#125—614no—
402#125—1222no—
403#124—22no—
404#124—618no—
405#124—1224no—
406#124C22×D6114yes—
407#123—1228no—
408C22⋊Q8—614no—
409C22⋊Q8—1222no—
410#121—614no—
411#121—1228no—
412#121—1228no—
413#121—1222no—
414C4×Q8—614no—
415C4×Q8—1222no—
416C4×D4—614no—
417C4×D4—1228no—
418C4×D4—1228no—
419C4×D4—1222no—
420C4×D4—1228no—
421#117—1228no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1222no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—614no—
441C2×C42—614no—
442C2×C42—1221no—
443C2×C42C22×D6111yescharacteristic
444C4⋊C8—618no—
445#103—618no—
446#103—618no—
447#102—618no—
448#102—618no—
449C22⋊C8—618no—
450#97—614no—
451#97C22×D6114yes—
452C24—21no—
453C24—611no—
454C24—1222no—
455#41—2448no—
456#41—2448no—
457#41—24416no—
458C2×Q8—21no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8C23×D6114yes—
464C2×Q8C23×D6112yes—
465C2×D4—21no—
466C2×D4—24no—
467C2×D4—614no—
468C2×D4—1228no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—1222no—
473C2×D4—1224no—
474C2×D4—1224no—
475C2×D4—2444no—
476C2×D4—2444no—
477C2×D4—2448no—
478C2×D4—2444no—
479C2×D4C23×D6112yes—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—612no—
486C22×C4—612no—
487C22×C4—614no—
488C22×C4—614no—
489C22×C4—1228no—
490C22×C4—1224no—
491C22×C4—1224no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—2442no—
495C22×C4—2442no—
496C22×C4—2448no—
497C22×C4—2444no—
498C22×C4D4×D6111yescharacteristic
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501Q16—48no—
502SD16—48no—
503SD16—2448no—
504D8—2448no—
505C2×C8—24no—
506C2×C8—614no—
507C2×C8—1224no—
508C2×C8C23×D6114yes—
509C4⋊C4—24no—
510C4⋊C4—24no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1221no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C22⋊C4—24no—
520C22⋊C4—24no—
521C22⋊C4—1228no—
522C22⋊C4—1228no—
523C22⋊C4—1224no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C42—614no—
527C42—614no—
528C42—612no—
529C42—612no—
530C42—1224no—
531C42D4×D6112yes—
532C42C23×D6112yes—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—612no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—1224no—
542C23—2444no—
543C23—2444no—
544C23—2444no—
545C23C2×D4×D6111yescharacteristic
546Q8—24no—
547Q8—24no—
548Q8—24no—
549Q8—42no—
550Q8—44no—
551D4—24no—
552D4—42no—
553D4—1224no—
554D4—1224no—
555D4—2448no—
556D4—2448no—
557D4—24416no—
558D4—2448no—
559D4—2444no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—1222no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1224no—
573C2×C4—1224no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4—24416no—
579C2×C4—2448no—
580C2×C4—2442no—
581C2×C4—2442no—
582C2×C4C2×D4×D6112yes—
583C2×C4C2×D4×D6112yes—
584C2×C4C2×D4×D6111yescharacteristic
585C2×C4#1451982214yes—
586C2×C4C24×D6111yescharacteristic
587C8—24no—
588C8—1224no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—84no—
593C22—1222no—
594C22—1222no—
595C22—1222no—
596C22—4888no—
597C22—2444no—
598C22—2444no—
599C22—2444no—
600C22—2444no—
601C22#1684032512yes—
602C22#1684056112yes—
603C22#1684056112yes—
604C22C22×D4×D6111yescharacteristic
605C4—22no—
606C4—22no—
607C4—44no—
608C4—42no—
609C4—1222no—
610C4—1222no—
611C4—1222no—
612C4—2442no—
613C4—2442no—
614C4#1684032512yes—
615C4C2×#1451982212yes—
616C2—82no—
617C2—4884no—
618C2—2442no—
619C2#2145685314yesdirect product factor
620C2C2×#1684032511yescharacteristic
621C2C2×#1684056111yescharacteristic
622C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#497C2
socleC23×C61C22×D4
hypercenterC2×#497D61
hypocenterC61C2×#3219
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
1. C2×#16840313C2semidirect
2. C2×#16840291C2semidirect
3. #21456853C2direct (N × H)
4. C2×#16840152C2semidirect
5. C2×#16840130C2semidirect
6. C2×#16840127C2semidirect
7. C2×#16839986C2semidirect
8. C2×#16839979C2semidirect
9. C2×#16839923C2semidirect
10. C2×#16839906C2semidirect
11. C2×#16839817C2semidirect
12. C2×C61×#497C2semidirect
17. #16840313C22semidirect
20. #16840291C22semidirect
36. #16840152C22semidirect
37. #16840130C22semidirect
38. #16840127C22semidirect
39. #16839986C22semidirect
40. #16839979C22semidirect
41. #16839923C22semidirect
42. #16839906C22semidirect
43. #16839817C22semidirect
55. C61×#497C22semidirect
308. C2×#497D61semidirect
318. C2×C61#3219semidirect
351. #497C2×D61semidirect
368. C61C2×#3219semidirect
619. C2#21456853direct (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×#168402912
#214568532
C2×#168401522
C2×#168401302
C2×#168401272
C2×#168399862
C2×#168399792
C2×#168399232
C2×#168399062
C2×#168398172
C2×C61×#4972
C2×#321961