Finite Group Databasebeta

← C2×#21457208 #30552887

Subgroups — C2×#21457208 #30552887

Order 15616.

2146 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic, complemented
2C2×#16840309C211yesmaximal, characteristic
3#21457208C2116yesmaximal, direct product factor
4C2×#16840056C212yesmaximal, complemented
5C2×#16840039C211yesmaximal, characteristic, complemented
6C2×#16840009C212yesmaximal, complemented
7C2×#16840007C212yesmaximal, complemented
8C2×#16839961C212yesmaximal, complemented
9C2×#16839968C211yesmaximal, characteristic, complemented
10C2×#16839967C211yesmaximal, characteristic, complemented
11C2×#16839784C211yesmaximal, characteristic, complemented
12C2×C61×#474C211yesmaximal, characteristic, complemented
13C22×#11758222—21no—
14C22×#11758190C2212yes—
15C22×#11758200C2212yes—
16C22×#11758189C2212yes—
17#16840317—24no—
18#16840317C2218yescomplemented
19#16840309—24no—
20#16840309C2218yes—
21C2×#14519777C2211yescharacteristic
22C2×#14519776C2211yescharacteristic
23C2×#14519773C2211yescharacteristic
24C2×#14519768—21no—
25C2×#14519763—21no—
26C2×#14519757C2212yes—
27C2×#14519753—21no—
28C2×#14519752C2211yescharacteristic
29C2×#14519750—21no—
30C2×#14519748C2211yescharacteristic
31C2×Q8×Dic61—21no—
32C2×#14519742C2212yes—
33C2×C4×D244C2211yescharacteristic
34C2×C4×C22⋊D61—21no—
35C2×#14519735—21no—
36C2×C42×D61C2211yescharacteristic
37#16840056C22116yescomplemented
38#16840039C2218yescomplemented
39#16840009C22116yescomplemented
40#16840007C22116yescomplemented
41#16839968C2218yescomplemented
42#16839967C2218yescomplemented
43#16839961C22116yescomplemented
44C2×#14519710C2212yes—
45#16839784C2218yescomplemented
46C2×C4⋊#92418C2211yescharacteristic
47C2×#14519706C2212yes—
48C2×#14519702C2212yes—
49C2×#14519700C2212yes—
50C2×#14519697C2212yes—
51C2×C4⋊Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C61×#474C2218yescomplemented
54C2×#107×C61C2212yes—
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2212yes—
57C2×#11758222—28no—
58C2×#11758222—48no—
59C22×Dic122—21no—
60C22×Dic122C2312yes—
61C22×D244—21no—
62C22×D244C2311yescharacteristic
63C22×C22⋊D61—21no—
64C22×C22⋊D61—21no—
65C22×C22⋊D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61D411yescharacteristic
68C22×C4×D61D411yescharacteristic
69C22×C4×D61C2311yescharacteristic
70C2×#11758190—44no—
71C2×#11758190C2318yes—
72C2×#11758200—44no—
73C2×#11758200C2318yes—
74C2×#11758189—44no—
75C2×#11758189C2318yes—
76C22×#92418C2312yes—
77#14519777—22no—
78#14519777C2314yes—
79#14519776—22no—
80#14519776C2314yes—
81#14519773—22no—
82#14519773C2314yes—
83#14519768—28no—
84#14519763—28no—
85#14519757—24no—
86#14519757C2318yes—
87#14519753—28no—
88#14519752—22no—
89#14519752C2314yes—
90#14519750—28no—
91#14519748—22no—
92#14519748C2314yes—
93Q8×Dic61—28no—
94#14519742—24no—
95#14519742C2318yes—
96C4×D244—22no—
97C4×D244C2314yes—
98C4×C22⋊D61—28no—
99#14519735—28no—
100C2×C4⋊Dic61—21no—
101C2×C4⋊Dic61—21no—
102C2×C4⋊Dic61—22no—
103C2×C4⋊Dic61C2311yescharacteristic
104C2×#11758196—21no—
105C2×#11758196—22no—
106C2×C22⋊Dic61—21no—
107C2×#11758198—22no—
108C2×#11758198—21no—
109C2×#11758198—21no—
110C2×C4×Dic61—22no—
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61C2311yescharacteristic
114C42×D61D414yes—
115C42×D61C2314yes—
116#14519710C2318yes—
117C4⋊#92418C2314yes—
118#14519706C2318yes—
119#14519702C2318yes—
120#14519700C2318yes—
121#14519697C2318yes—
122C22×Q8×C61—21no—
123C22×D4×C61C2311yescharacteristic
124C23×C4×C61—21no—
125C22×C8×C61C2312yes—
126C4⋊Q8×C61—22no—
127C4⋊Q8×C61C2314yes—
128C4×D4×C61—22no—
129C4×D4×C61C2314yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2312yes—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C22⋊C4×C61—21no—
134C2×C42×C61C2311yescharacteristic
135#107×C61C2318yes—
136C4⋊C8×C61C2314yes—
137#102×C61C2318yes—
138C24×C61—21no—
139C23×D61—21no—
140C23×D61C2×D411yescharacteristic
141C2×Q8×C61—24no—
142C2×Q8×C61—44no—
143C2×D4×C61—21no—
144C2×D4×C61—22no—
145C2×D4×C61—22no—
146C2×D4×C61C2412yes—
147C22×C4×C61—21no—
148C22×C4×C61—22no—
149C22×C4×C61—24no—
150C22×C4×C61—44no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—24no—
155C2×C22⋊D61—24no—
156C2×C22⋊D61—24no—
157C2×C22⋊D61—44no—
158C2×C22⋊D61—44no—
159C2×C22⋊D61—44no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165#11758222—432no—
166C2×D244—22no—
167C2×D244—22no—
168C2×D244—24no—
169C2×D244—21no—
170C2×D244—44no—
171C2×D244C2412yes—
172C2×C4×D61—24no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—24no—
176C2×C4×D61—24no—
177C2×C4×D61—44no—
178C2×C4×D61C2×D414yes—
179C2×C4×D61C2×D414yes—
180C2×C4×D61C2×D412yes—
181C2×C4×D61C2412yes—
182C2×Dic122—24no—
183C2×Dic122—24no—
184C2×Dic122—22no—
185C2×Dic122—24no—
186C2×Dic122—44no—
187C2×Dic122C2414yes—
188C2×C8×C61—24no—
189C2×C8×C61C2414yes—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61C2414yes—
194C4⋊C4×C61C2412yes—
195C22⋊C4×C61—24no—
196C42×C61C2×D412yes—
197C42×C61C2412yes—
198C22⋊Dic61—24no—
199#11758200—48no—
200#11758198—28no—
201#11758198—24no—
202#11758198—24no—
203C4⋊Dic61—24no—
204C4⋊Dic61—24no—
205C4⋊Dic61—21no—
206C4⋊Dic61—28no—
207C4⋊Dic61C2412yes—
208#11758196—24no—
209#11758196—28no—
210C4×Dic61—28no—
211C4×Dic61C2×D412yes—
212C4×Dic61C2×D414yes—
213C4×Dic61C2×D414yes—
214C4×Dic61C2412yes—
215C2×#92418—24no—
216C2×#92418C2414yes—
217#11758190—48no—
218#11758189—48no—
219C23×C61—22no—
220C23×C61—42no—
221C23×C61—42no—
222C23×C61—42no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—42no—
229C22×D61—42no—
230C22×D61—42no—
231C22×D61C22×D412yes—
232Q8×C61—48no—
233D4×C61—22no—
234D4×C61—22no—
235D4×C61—42no—
236C2×C4×C61—22no—
237C2×C4×C61—24no—
238C2×C4×C61—22no—
239C2×C4×C61—42no—
240C2×C4×C61—44no—
241C2×C4×C61—48no—
242C2×C4×C61AGL(1,ℤ/8)14yes—
243C2×C4×C61C22×D412yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D412yes—
246C2×C4×C61H(5,𝔽2)12yes—
247C2×C4×C612−1+412yes—
248C2×C4×C61C2511yescharacteristic
249C22⋊D61—48no—
250C22⋊D61—48no—
251C22⋊D61—48no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—24no—
255C2×Dic61—22no—
256C2×Dic61—44no—
257C2×Dic61—42no—
258C2×Dic61#13714yes—
259C2×Dic61C22×D412yes—
260C2×Dic61C22×D412yes—
261C2×Dic61C22×D412yes—
262D244—22no—
263D244—22no—
264D244—48no—
265D244—42no—
266C4×D61—24no—
267C4×D61—28no—
268C4×D61—22no—
269C4×D61—22no—
270C4×D61—28no—
271C4×D61—48no—
272Dic122—24no—
273Dic122—24no—
274Dic122—48no—
275Dic122—44no—
276C8×C61—24no—
277#92418—24no—
278C2×#3254—611nomaximal, Sylow
279C22×C61—42no—
280C22×C61—42no—
281C22×C61—42no—
282C22×C61—84no—
283C22×C61#53712yes—
284C22×C61C2×AGL(1,ℤ/8)12yes—
285C22×C61C2×#13712yes—
286C22×C61C23×D411yescharacteristic
287C2×D61—22no—
288C2×D61—22no—
289C2×D61—22no—
290C2×D61—44no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—84no—
295C4×C61—22no—
296C4×C61—42no—
297C4×C61—44no—
298C4×C61#53712yes—
299C4×C61C2×AGL(1,ℤ/8)14yes—
300C4×C61C2×H(5,𝔽2)11yescharacteristic
301C4×C61C2×2−1+411yescharacteristic
302Dic61—22no—
303Dic61—22no—
304Dic61—44no—
305Dic61—42no—
306Dic61C2×#13714yes—
307C2×#533—611no—
308C2×#532—611no—
309#3254—6116no—
310C2×#477—612no—
311C2×#474—611no—
312C2×#474D6111yescharacteristic, complemented
313C2×#468—612no—
314C2×#468—612no—
315C2×#460—612no—
316C2×#461—611no—
317C2×#461—611no—
318C2×#423—611no—
319C2×C61—82no—
320C2×C61#325414yescomplemented
321C2×C61C2×#53711yescharacteristic
322C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
323C2×C61C22×#13711yescharacteristic
324D61—42no—
325D61—82no—
326C22×#41—1221no—
327C22×SD16—612no—
328C22×SD16—612no—
329C22×M16—612no—
330#533—618no—
331#533—1224no—
332#532—618no—
333#532—1224no—
334C2×C4⋊Q8—611no—
335C2×C4⋊Q8—611no—
336C2×C4⋊Q8C2×D6111yescharacteristic
337C2×#127—611no—
338C2×#125—1221no—
339C2×#124—1221no—
340C2×C22⋊Q8—612no—
341C2×C22⋊Q8—1221no—
342C2×#121—611no—
343C2×#121—611no—
344C2×#121—1221no—
345C2×C4×Q8—612no—
346C2×C4×Q8—1221no—
347C2×C4×D4—611no—
348C2×C4×D4—1221no—
349C2×C4×D4C2×D6111yescharacteristic
350C2×#117—1221no—
351C22×C42—611no—
352#477—6116no—
353#474—618no—
354#474C2×D6118yescomplemented
355#468—6116no—
356#468—6116no—
357#461—618no—
358#461—618no—
359#460—6116no—
360C2×#107—612no—
361C2×#107C2×D6112yes—
362#423—618no—
363C2×C4⋊C8—612no—
364C2×C4⋊C8—611no—
365C2×C4⋊C8C2×D6111yescharacteristic
366C2×#103—612no—
367C2×#103—612no—
368C2×#102—612no—
369C2×#102C2×D6112yes—
370C61C2×#325411yescharacteristic, complemented, Sylow
371C2×#41—1228no—
372C2×#41—2448no—
373C22×Q8—21no—
374C22×Q8—612no—
375C22×Q8—1221no—
376C22×D4—611no—
377C22×D4—1221no—
378C22×D4—1221no—
379C22×D4—1221no—
380C22×D4—1221no—
381C22×D4C22×D6111yescharacteristic
382C23×C4—21no—
383C23×C4—611no—
384C23×C4—611no—
385C23×C4—611no—
386C23×C4—1221no—
387C2×SD16—618no—
388C2×SD16—618no—
389C2×SD16—2444no—
390C2×SD16—2444no—
391C2×M16—618no—
392C2×M16—2444no—
393C22×C8—612no—
394C22×C8C22×D6112yes—
395C4⋊Q8—22no—
396C4⋊Q8—614no—
397C4⋊Q8—614no—
398C4⋊Q8—1222no—
399C4⋊Q8—1222no—
400C4⋊Q8C22×D6114yes—
401#127—614no—
402#127—1222no—
403#125—1228no—
404#124—1228no—
405C22⋊Q8—618no—
406C22⋊Q8—1224no—
407C22⋊Q8—1228no—
408#121—614no—
409#121—614no—
410#121—1228no—
411#121—1222no—
412#121—1222no—
413C4×Q8—618no—
414C4×Q8—1224no—
415C4×Q8—1228no—
416C4×D4—22no—
417C4×D4—614no—
418C4×D4—1228no—
419C4×D4—1222no—
420C4×D4C22×D6114yes—
421#117—1228no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1222no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6112yes—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1222no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—614no—
440C2×C42—614no—
441C2×C42—1222no—
442C2×C42C22×D6111yescharacteristic
443#107—618no—
444#107C22×D6118yes—
445C4⋊C8—618no—
446C4⋊C8—614no—
447C4⋊C8C22×D6114yes—
448#103—618no—
449#103—618no—
450#102—618no—
451#102C22×D6118yes—
452C24—21no—
453C24—611no—
454C24—1221no—
455#41—24432no—
456C2×Q8—24no—
457C2×Q8—44no—
458C2×Q8—614no—
459C2×Q8—1224no—
460C2×Q8—1222no—
461C2×Q8—1224no—
462C2×Q8—1224no—
463C2×Q8—2444no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—612no—
468C2×D4—1224no—
469C2×D4—1224no—
470C2×D4—1224no—
471C2×D4—1224no—
472C2×D4—1221no—
473C2×D4—1222no—
474C2×D4—1222no—
475C2×D4—2444no—
476C2×D4—2444no—
477C2×D4—2444no—
478C2×D4—2444no—
479C2×D4C23×D6112yes—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—44no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—614no—
488C22×C4—612no—
489C22×C4—614no—
490C22×C4—612no—
491C22×C4—1224no—
492C22×C4—1222no—
493C22×C4—1222no—
494C22×C4—1224no—
495C22×C4—1222no—
496C22×C4—1221no—
497C22×C4—1224no—
498C22×C4—2444no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—2448no—
503SD16—2448no—
504M16—2448no—
505C2×C8—24no—
506C2×C8—614no—
507C2×C8—1224no—
508C2×C8C23×D6114yes—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1228no—
515C4⋊C4—1228no—
516C4⋊C4—1221no—
517C4⋊C4—1224no—
518C4⋊C4—1224no—
519C4⋊C4C23×D6114yes—
520C4⋊C4C23×D6112yes—
521C22⋊C4—24no—
522C22⋊C4—1224no—
523C22⋊C4—1228no—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C42—614no—
527C42—612no—
528C42—614no—
529C42—612no—
530C42—1228no—
531C42D4×D6112yes—
532C42C23×D6112yes—
533C23—22no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—612no—
538C23—1222no—
539C23—1222no—
540C23—1222no—
541C23—1222no—
542C23—2442no—
543C23—2442no—
544C23—2442no—
545C23C2×D4×D6111yescharacteristic
546Q8—48no—
547Q8—1224no—
548Q8—1224no—
549Q8—2448no—
550Q8—2444no—
551D4—22no—
552D4—22no—
553D4—42no—
554D4—1222no—
555D4—1222no—
556D4—2448no—
557D4—2448no—
558D4—2448no—
559D4—2448no—
560D4—2442no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—614no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1228no—
572C2×C4—1222no—
573C2×C4—1222no—
574C2×C4—1228no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1224no—
578C2×C4—1224no—
579C2×C4—1222no—
580C2×C4—2442no—
581C2×C4—2444no—
582C2×C4—2448no—
583C2×C4#1451980114yes—
584C2×C4C2×D4×D6112yes—
585C2×C4C2×D4×D6112yes—
586C2×C4C2×D4×D6111yescharacteristic
587C2×C4#1451982312yes—
588C2×C4#1451982512yes—
589C2×C4C24×D6111yescharacteristic
590C8—24no—
591C8—1224no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—84no—
596C22—1222no—
597C22—1222no—
598C22—1222no—
599C22—4884no—
600C22—2442no—
601C22—2442no—
602C22—2442no—
603C22—2444no—
604C22#1684034212yes—
605C22C2×#1451980112yes—
606C22#137×D6112yes—
607C22C22×D4×D6111yescharacteristic
608C4—22no—
609C4—42no—
610C4—44no—
611C4—614no—
612C4—1222no—
613C4—1222no—
614C4—2442no—
615C4—2444no—
616C4#1684034212yes—
617C4C2×#1451980114yes—
618C4C2×#1451982311yescharacteristic
619C4C2×#1451982511yescharacteristic
620C2—82no—
621C2—4882no—
622C2—2442no—
623C2#2145720814yesdirect product factor
624C2C2×#1684034211yescharacteristic
625C2C22×#1451980111yescharacteristic
626C2C2×#137×D6111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC2×#16840309C2
hypercenterC2×#474D61
hypocenterC61C2×#3254
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#16840317C2semidirect
3. #21457208C2direct (N × H)
4. C2×#16840056C2semidirect
5. C2×#16840039C2semidirect
6. C2×#16840009C2semidirect
7. C2×#16840007C2semidirect
8. C2×#16839961C2semidirect
9. C2×#16839968C2semidirect
10. C2×#16839967C2semidirect
11. C2×#16839784C2semidirect
12. C2×C61×#474C2semidirect
18. #16840317C22semidirect
37. #16840056C22semidirect
38. #16840039C22semidirect
39. #16840009C22semidirect
40. #16840007C22semidirect
41. #16839968C22semidirect
42. #16839967C22semidirect
43. #16839961C22semidirect
45. #16839784C22semidirect
53. C61×#474C22semidirect
312. C2×#474D61semidirect
320. C2×C61#3254semidirect
354. #474C2×D61semidirect
370. C61C2×#3254semidirect
623. C2#21457208direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168403172
C2×#168403092
#214572082
C2×#168400562
C2×#168400392
C2×#168400092
C2×#168400072
C2×#168399612
C2×#168399682
C2×#168399672
C2×#168397842
C2×C61×#4742
C2×#325461