Finite Group Databasebeta

← C2×#21454637 #30544086

Subgroups — C2×#21454637 #30544086

Order 15616.

2810 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519810C211yesmaximal, characteristic
2C2×#16840230C211yesmaximal, characteristic
3C22×#14519742C211yesmaximal, characteristic, complemented
4#21454637C2116yesmaximal, direct product factor
5C2×#16839868C212yesmaximal, complemented
6C2×#16839861C212yesmaximal, complemented
7C2×#16839856C212yesmaximal, complemented
8C2×#16839849C212yesmaximal, complemented
9C2×#16839761C211yesmaximal, characteristic
10C2×#16839770C211yesmaximal, characteristic, complemented
11C2×M16×Dic61C211yesmaximal, characteristic, complemented
12C2×C61×#428C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C23×Dic122C2211yescharacteristic
15C2×#14519810C4116yescomplemented
16C2×#14519810C2218yes—
17C22×Dic244C2212yes—
18C22×#11758190C2212yes—
19C22×#11758194C2211yescharacteristic
20#16840230—24no—
21#16840230C2218yes—
22C2×#14519742—21no—
23C2×#14519742—21no—
24C2×#14519742—24no—
25C2×#14519742C2212yes—
26C2×#14519742C2211yescharacteristic
27C2×#14519742C2212yes—
28C2×#14519742C2211yescharacteristic
29C2×#14519742C2218yescomplemented
30C2×#14519739C2211yescharacteristic
31C2×#14519739C2211yescharacteristic
32C2×#14519737—21no—
33C2×#14519737—21no—
34C2×#14519734—21no—
35C22×#11758196—21no—
36C22×C4×Dic61—21no—
37C22×C4×Dic61C2211yescharacteristic
38C2×C42×D61—21no—
39#16839868C22116yescomplemented
40#16839861C22116yescomplemented
41#16839856C22116yescomplemented
42#16839849C22116yescomplemented
43#16839770C2218yescomplemented
44#16839761C2218yes—
45C2×#14519700C2212yes—
46C2×#14519700C2212yes—
47C2×#14519700C2212yes—
48C2×#14519697C2212yes—
49M16×Dic61C2218yescomplemented
50C2×#14519687C2212yes—
51C2×C8×Dic61C2212yes—
52C22×M16×C61C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C22×C4⋊C4×C61C2211yescharacteristic
55C61×#428C2218yescomplemented
56C2×#107×C61C2212yes—
57C2×#106×C61C2212yes—
58C2×#11758222—22no—
59C2×#11758222—24no—
60C2×#11758222C2×C418yes—
61C2×#11758222C2314yes—
62C22×Dic122—21no—
63C22×Dic122—22no—
64C22×Dic122—24no—
65C22×Dic122C2×C418yescomplemented
66C22×Dic122C2312yes—
67C22×Dic122C2311yescharacteristic
68C22×Dic122C2314yes—
69C22×D244C2311yescharacteristic
70C22×C22⋊D61—21no—
71C23×Dic61—21no—
72C23×Dic61D411yescharacteristic
73C23×Dic61D411yescharacteristic
74C22×C4×D61—21no—
75C22×C4×D61—21no—
76C22×C4×D61—21no—
77#14519810C2×C4164yescomplemented
78C2×Dic244C2×C4116yescomplemented
79C2×Dic244C2318yes—
80C2×#11758190C2×C4116yescomplemented
81C2×#11758190C2318yes—
82C2×#11758194—24no—
83C2×#11758194C2314yes—
84C22×#92418C2312yes—
85#14519742—24no—
86#14519742—24no—
87#14519742—28no—
88#14519742—28no—
89#14519742—22no—
90#14519742—22no—
91#14519742C2318yes—
92#14519742C2318yes—
93#14519742C2314yes—
94#14519742C2314yes—
95#14519739—22no—
96#14519739—22no—
97#14519739C2314yes—
98#14519739C2314yes—
99#14519737—28no—
100#14519737—28no—
101#14519734—28no—
102C2×#11758196—22no—
103C2×#11758196—22no—
104C2×#11758196—28no—
105C2×#11758196—21no—
106C2×#11758196—21no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61—22no—
112C2×C4×Dic61—22no—
113C2×C4×Dic61—28no—
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D414yes—
117C2×C4×Dic61C2311yescharacteristic
118C2×C4×Dic61C2311yescharacteristic
119C2×C4×Dic61C2314yes—
120C42×D61—28no—
121#14519700C2318yes—
122#14519700C2318yes—
123#14519700C2318yes—
124#14519697C2318yes—
125#14519687C2318yes—
126C8×Dic61C2318yes—
127C23×C4×C61—21no—
128C23×C4×C61C2311yescharacteristic
129C2×M16×C61C2×C418yescomplemented
130C2×M16×C61C2314yes—
131C22×C8×C61C2312yes—
132#117×C61—22no—
133#117×C61C2314yes—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61—22no—
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C4⋊C4×C61C2312yes—
139C2×C4⋊C4×C61C2314yes—
140C2×C22⋊C4×C61—21no—
141C2×C42×C61—21no—
142#107×C61C2318yes—
143#106×C61C2318yes—
144C24×C61C2×D411yescharacteristic
145C23×D61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—24no—
149C22×C4×C61—22no—
150C22×C4×C61—44no—
151C22×C4×C61C22×C414yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4114yes—
155C22×C4×C61C2411yescharacteristic
156C22×C4×C61C2412yes—
157C2×C22⋊D61—28no—
158C2×C22⋊D61—24no—
159C22×Dic61—28no—
160C22×Dic61—21no—
161C22×Dic61—22no—
162C22×Dic61—24no—
163C22×Dic61—24no—
164C22×Dic61—42no—
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D414yes—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D414yes—
170C22×Dic61C2×D411yescharacteristic
171#11758222—216no—
172#11758222—28no—
173#11758222C22×C4116yes—
174C2×D244—21no—
175C2×D244—22no—
176C2×D244C22×C414yes—
177C2×D244C2412yes—
178C2×C4×D61—28no—
179C2×C4×D61—24no—
180C2×C4×D61—24no—
181C2×C4×D61—24no—
182C2×C4×D61—44no—
183C2×C4×D61—44no—
184C2×Dic122—216no—
185C2×Dic122—28no—
186C2×Dic122—22no—
187C2×Dic122—24no—
188C2×Dic122—21no—
189C2×Dic122—24no—
190C2×Dic122—28no—
191C2×Dic122—22no—
192C2×Dic122C22×C418yes—
193C2×Dic122C22×C414yes—
194C2×Dic122C22×C4116yescomplemented
195C2×Dic122C2414yes—
196C2×Dic122C2412yes—
197M16×C61C22×C4116yescomplemented
198C2×C8×C61C22×C418yes—
199C2×C8×C61C2414yes—
200C4⋊C4×C61—24no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61C2412yes—
205C4⋊C4×C61C2414yes—
206C4⋊C4×C61C2412yes—
207C22⋊C4×C61—24no—
208C42×C61—24no—
209#11758198—24no—
210#11758198—24no—
211#11758196—28no—
212#11758196—28no—
213#11758196—24no—
214#11758196—24no—
215C4×Dic61—24no—
216C4×Dic61—24no—
217C4×Dic61—28no—
218C4×Dic61—28no—
219C4×Dic61C2×D414yes—
220C4×Dic61C2×D414yes—
221C4×Dic61C2×D412yes—
222C4×Dic61C2×D412yes—
223C4×Dic61C2412yes—
224C4×Dic61C2412yes—
225#11758194—28no—
226C2×#92418—24no—
227C2×#92418C2414yes—
228Dic244C22×C4132yescomplemented
229#11758190C22×C4132yescomplemented
230C23×C61—22no—
231C23×C61—22no—
232C23×C61C4×D414yes—
233C23×C61C22×D412yes—
234C23×C61C22×D411yescharacteristic
235C22×D61—24no—
236C22×D61—22no—
237C22×D61—42no—
238C22×D61—42no—
239C2×C4×C61—24no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—48no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61—44no—
246C2×C4×C61C4×D414yes—
247C2×C4×C61C4×D418yes—
248C2×C4×C61C23×C412yes—
249C2×C4×C61C23×C414yes—
250C2×C4×C61C22×D412yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C2×#4112yes—
253C2×C4×C61C2×#4114yes—
254C2×C4×C61C2511yescharacteristic
255C22⋊D61—216no—
256C2×Dic61—28no—
257C2×Dic61—216no—
258C2×Dic61—24no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—24no—
263C2×Dic61—22no—
264C2×Dic61—28no—
265C2×Dic61—44no—
266C2×Dic61#13714yescomplemented
267C2×Dic61C22×D412yes—
268C2×Dic61C22×D412yes—
269C2×Dic61C22×D412yes—
270C2×Dic61C22×D412yes—
271D244—24no—
272D244—22no—
273D244C23×C414yes—
274C4×D61—216no—
275C4×D61—48no—
276C4×D61—48no—
277Dic122—28no—
278Dic122—216no—
279Dic122—24no—
280Dic122—24no—
281Dic122—22no—
282Dic122C23×C418yes—
283Dic122C23×C414yes—
284C8×C61C23×C418yes—
285#92418—24no—
286C2×C4×#137—611nomaximal, Sylow
287C22×C61—28no—
288C22×C61—22no—
289C22×C61—22no—
290C22×C61C2×C4×D414yes—
291C22×C61C2×C4×D412yes—
292C22×C61C2×#13712yes—
293C22×C61#57512yes—
294C22×C61C23×D411yescharacteristic
295C2×D61—24no—
296C2×D61—48no—
297C2×D61—42no—
298C2×D61—42no—
299C4×C61—42no—
300C4×C61—42no—
301C4×C61—44no—
302C4×C61C2×C4×D412yes—
303C4×C61C2×C4×D414yes—
304C4×C61C24×C411yescharacteristic
305C4×C61C22×#4111yescharacteristic
306Dic61—28no—
307Dic61—24no—
308Dic61—22no—
309Dic61—22no—
310Dic61—22no—
311Dic61C2×#13714yescomplemented
312C22×#137—611no—
313C2×C4×#41—611no—
314C22×C4×Q8—611no—
315C4×#137—6116no—
316C2×#441—612no—
317C2×#440—612no—
318C2×C4×Q16—612no—
319C2×C4×SD16—612no—
320C2×#428D6111yescharacteristic, complemented
321C2×#417—611no—
322C2×#419—611no—
323C2×C4×M16—611no—
324C2×C61—24no—
325C2×C61C4×#13714yescomplemented
326C2×C61C22×C4×D411yescharacteristic
327C2×C61C22×#13711yescharacteristic
328C2×C61C2×#57511yescharacteristic
329D61—44no—
330C22×#41—611no—
331C23×Q8—611no—
332C2×#137—6116no—
333C2×#137—618no—
334C22×Q16—612no—
335C22×SD16—612no—
336C22×M16—611no—
337C22×M16C2×D6111yescharacteristic
338C4×#41—618no—
339C4×#41—1224no—
340C2×C4×Q8—612no—
341C2×C4×Q8—612no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—618no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1224no—
347C2×C4×Q8—1221no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—1221no—
351C2×C4×D4—1221no—
352C2×#117—1221no—
353C2×#117C2×D6111yescharacteristic
354C22×C4⋊C4—1221no—
355C22×C4⋊C4C2×D6111yescharacteristic
356C22×C42—611no—
357C22×C42—1221no—
358C22×C42—1221no—
359#441—6116no—
360#440—6116no—
361C4×Q16—6116no—
362C4×SD16—6116no—
363#428C2×D6118yescomplemented
364C2×#107C2×D6112yes—
365C2×#106C2×D6112yes—
366#419—618no—
367#417—618no—
368C2×#103—612no—
369C2×#103—612no—
370C2×#103—612no—
371C2×#102—612no—
372C4×M16—618no—
373C2×#97—612no—
374C2×C4×C8—612no—
375C61C2×C4×#13711yescharacteristic, complemented, Sylow
376C2×#41—618no—
377C2×#41—614no—
378C2×#41—1222no—
379C2×#41—1224no—
380C22×Q8—618no—
381C22×Q8—612no—
382C22×Q8—611no—
383C22×Q8—614no—
384C22×Q8—1222no—
385C22×Q8—1221no—
386C22×Q8—1224no—
387C22×D4—611no—
388C22×D4—1221no—
389C23×C4—21no—
390C23×C4—611no—
391C23×C4—611no—
392C23×C4—1221no—
393C23×C4—1221no—
394C23×C4—1221no—
395C23×C4—1221no—
396C23×C4C22×D6111yescharacteristic
397#137—6164no—
398C2×Q16—6116no—
399C2×Q16—618no—
400C2×SD16—6116no—
401C2×SD16—618no—
402C2×M16—614no—
403C2×M16—1224no—
404C2×M16C4×D6118yescomplemented
405C2×M16C22×D6114yes—
406C22×C8—612no—
407C22×C8C22×D6112yes—
408C4×Q8—618no—
409C4×Q8—618no—
410C4×Q8—614no—
411C4×Q8—614no—
412C4×Q8—1228no—
413C4×Q8—1228no—
414C4×Q8—1224no—
415C4×Q8—1224no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×D4—614no—
419C4×D4—614no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1228no—
423C4×D4—1228no—
424#117—22no—
425#117—1228no—
426#117C22×D6114yes—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—1228no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6112yes—
435C2×C4⋊C4C22×D6114yes—
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—611no—
443C2×C42—614no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—614no—
448C2×C42—1228no—
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1228no—
452C2×C42—1222no—
453C2×C42—1222no—
454#107C22×D6118yes—
455#106C22×D6118yes—
456#103—618no—
457#103—618no—
458#103—618no—
459#102—618no—
460#97—618no—
461C4×C8—618no—
462C24—1221no—
463C24D4×D6111yescharacteristic
464#41—6116no—
465#41—1228no—
466#41—12216no—
467C2×Q8—618no—
468C2×Q8—614no—
469C2×Q8—6116no—
470C2×Q8—614no—
471C2×Q8—612no—
472C2×Q8—1224no—
473C2×Q8—1222no—
474C2×Q8—1221no—
475C2×Q8—1224no—
476C2×Q8—1228no—
477C2×Q8—1222no—
478C2×Q8—1228no—
479C2×Q8—12216no—
480C2×D4—614no—
481C2×D4—612no—
482C2×D4—1224no—
483C2×D4—1228no—
484C2×D4—1221no—
485C2×D4—1222no—
486C22×C4—24no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—611no—
492C22×C4—614no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—614no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1224no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1222no—
503C22×C4—1228no—
504C22×C4—1224no—
505C22×C4—1228no—
506C22×C4—2444no—
507C22×C4—2444no—
508C22×C4—2442no—
509C22×C4C2×C4×D6114yes—
510C22×C4D4×D6112yes—
511C22×C4D4×D6111yescharacteristic
512C22×C4#1175822614yes—
513C22×C4C23×D6112yes—
514C22×C4C23×D6111yescharacteristic
515Q16—6132no—
516SD16—6132no—
517M16—1228no—
518M16C2×C4×D61116yescomplemented
519C2×C8—614no—
520C2×C8—1224no—
521C2×C8C2×C4×D6118yes—
522C2×C8C23×D6114yes—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—1228no—
528C4⋊C4—1228no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4C23×D6114yes—
532C4⋊C4C23×D6112yes—
533C4⋊C4C23×D6112yes—
534C22⋊C4—24no—
535C22⋊C4—1224no—
536C22⋊C4—1224no—
537C42—24no—
538C42—614no—
539C42—614no—
540C42—612no—
541C42—612no—
542C42—612no—
543C42—612no—
544C42—1224no—
545C42—1224no—
546C42—1228no—
547C42—1228no—
548C23—22no—
549C23—22no—
550C23—1224no—
551C23—1222no—
552C23—2442no—
553C23—2442no—
554C23#1451973914yes—
555C23C2×D4×D6112yes—
556C23C2×D4×D6111yescharacteristic
557Q8—618no—
558Q8—614no—
559Q8—12216no—
560Q8—1228no—
561Q8—1224no—
562Q8—1224no—
563Q8—1222no—
564D4—614no—
565D4—12216no—
566D4—1224no—
567D4—1222no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—614no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—612no—
579C2×C4—612no—
580C2×C4—1228no—
581C2×C4—1224no—
582C2×C4—12216no—
583C2×C4—1228no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1224no—
589C2×C4—12216no—
590C2×C4—2448no—
591C2×C4—2448no—
592C2×C4—2444no—
593C2×C4#1451973914yes—
594C2×C4#1451973918yes—
595C2×C4C22×C4×D6112yes—
596C2×C4C22×C4×D6114yes—
597C2×C4C2×D4×D6112yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×#1175822612yes—
600C2×C4C2×#1175822614yes—
601C2×C4C24×D6111yescharacteristic
602C8—1224no—
603C8C22×C4×D6118yes—
604C22—28no—
605C22—22no—
606C22—22no—
607C22—1224no—
608C22—2448no—
609C22—2442no—
610C22—2442no—
611C22C2×#1451973914yes—
612C22C2×#1451973912yes—
613C22#137×D6112yes—
614C22#1684055812yes—
615C22C22×D4×D6111yescharacteristic
616C4—42no—
617C4—44no—
618C4—42no—
619C4—614no—
620C4—1224no—
621C4—1228no—
622C4—1222no—
623C4—1222no—
624C4—1222no—
625C4C2×#1451973914yes—
626C4C2×#1451973912yes—
627C4C23×C4×D6111yescharacteristic
628C4C22×#1175822611yescharacteristic
629C2—24no—
630C2—2444no—
631C2#2145463714yesdirect product factor
632C2C22×#1451973911yescharacteristic
633C2C2×#137×D6111yescharacteristic
634C2C2×#1684055811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC2×C4×C61C25
hypercenterC2×#428D61
hypocenterC61C2×C4×#137
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. C22×#14519742C2semidirect
4. #21454637C2direct (N × H)
5. C2×#16839868C2semidirect
6. C2×#16839861C2semidirect
7. C2×#16839856C2semidirect
8. C2×#16839849C2semidirect
10. C2×#16839770C2semidirect
11. C2×M16×Dic61C2semidirect
12. C2×C61×#428C2semidirect
15. C2×#14519810C4semidirect
29. C2×#14519742C22semidirect
39. #16839868C22semidirect
40. #16839861C22semidirect
41. #16839856C22semidirect
42. #16839849C22semidirect
43. #16839770C22semidirect
49. M16×Dic61C22semidirect
55. C61×#428C22semidirect
65. C22×Dic122C2×C4semidirect
77. #14519810C2×C4semidirect
78. C2×Dic244C2×C4semidirect
80. C2×#11758190C2×C4semidirect
129. C2×M16×C61C2×C4semidirect
194. C2×Dic122C22×C4semidirect
197. M16×C61C22×C4semidirect
228. Dic244C22×C4semidirect
229. #11758190C22×C4semidirect
266. C2×Dic61#137semidirect
311. Dic61C2×#137semidirect
320. C2×#428D61semidirect
325. C2×C61C4×#137semidirect
363. #428C2×D61semidirect
375. C61C2×C4×#137semidirect
404. C2×M16C4×D61semidirect
518. M16C2×C4×D61semidirect
631. C2#21454637direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145198102
C2×#168402302
C22×#145197422
#214546372
C2×#168398682
C2×#168398612
C2×#168398562
C2×#168398492
C2×#168397612
C2×#168397702
C2×M16×Dic612
C2×C61×#4282
C2×C4×#13761