Finite Group Databasebeta

← C2×#21455742 #30547791

Subgroups — C2×#21455742 #30547791

Order 15616.

2650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2#21455742C2116yesmaximal, direct product factor
3C2×#16840097C212yesmaximal, complemented
4C2×#16840080C212yesmaximal, complemented
5C2×#16840092C212yesmaximal, complemented
6C2×#16840075C212yesmaximal, complemented
7C2×#16839802C212yesmaximal, complemented
8C2×#16839761C212yesmaximal, complemented
9C2×#16839726C211yesmaximal, characteristic, complemented
10C23×C4⋊C4×C61C211yesmaximal, characteristic, complemented
11C22×#11758222C2211yescharacteristic
12C22×#11758194C2212yes—
13#16840273—24no—
14#16840273C2218yes—
15C2×#14519760—21no—
16C2×#14519756C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×#14519751C2211yescharacteristic
19C2×#14519751C2211yescharacteristic
20C2×#14519746—21no—
21C2×C4×C22⋊D61—21no—
22C2×C4×C22⋊D61—21no—
23C2×#14519736C2211yescharacteristic
24#16840097C22116yescomplemented
25#16840092C22116yescomplemented
26#16840080C22116yescomplemented
27#16840075C22116yescomplemented
28#16839802C22116yescomplemented
29C2×#14519710C2214yescomplemented
30C2×#14519708C2214yescomplemented
31#16839761C22116yescomplemented
32C2×#14519700C2214yescomplemented
33C2×#14519697C2214yescomplemented
34#16839726C2218yescomplemented
35C2×C22⋊#92418C2212yes—
36C2×C22⋊#92418C2212yes—
37C24×C4×C61—21no—
38C24×C4×C61C2211yescharacteristic
39C22×C4⋊C4×C61—24no—
40C22×C4⋊C4×C61—21no—
41C22×C4⋊C4×C61—21no—
42C22×C4⋊C4×C61—21no—
43C22×C4⋊C4×C61C2212yes—
44C22×C4⋊C4×C61C2212yes—
45C22×C4⋊C4×C61C2212yes—
46C22×C4⋊C4×C61C2218yescomplemented
47C2×#11758222—22no—
48C2×#11758222—28no—
49C2×#11758222C2314yes—
50C22×Dic122C2311yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61—21no—
53C22×C22⋊D61—21no—
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C2×#11758194—28no—
57C2×#11758194C2318yes—
58C22×#92418C2314yes—
59#14519760—28no—
60#14519756—22no—
61#14519756—22no—
62#14519756C2314yes—
63#14519756C2314yes—
64#14519751—22no—
65#14519751—22no—
66#14519751C2314yes—
67#14519751C2314yes—
68#14519746—28no—
69C4×C22⋊D61—28no—
70C4×C22⋊D61—28no—
71#14519736—22no—
72#14519736C2314yes—
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×#11758198—21no—
81C2×#11758198—21no—
82C2×C4×Dic61—21no—
83#14519710C23116yescomplemented
84#14519708C23116yescomplemented
85#14519700C23116yescomplemented
86#14519697C23116yescomplemented
87C22⋊#92418C2318yes—
88C22⋊#92418C2318yes—
89C25×C61D411yescharacteristic
90C23×C4×C61—28no—
91C23×C4×C61—22no—
92C23×C4×C61—22no—
93C23×C4×C61—22no—
94C23×C4×C61—28no—
95C23×C4×C61—48no—
96C23×C4×C61D411yescharacteristic
97C23×C4×C61D411yescharacteristic
98C23×C4×C61D411yescharacteristic
99C23×C4×C61D414yes—
100C23×C4×C61C2311yescharacteristic
101C23×C4×C61C2311yescharacteristic
102C23×C4×C61C2314yes—
103C23×C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61—22no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—24no—
107C2×C4⋊C4×C61—24no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—24no—
110C2×C4⋊C4×C61—28no—
111C2×C4⋊C4×C61—28no—
112C2×C4⋊C4×C61—28no—
113C2×C4⋊C4×C61C2314yes—
114C2×C4⋊C4×C61C2318yes—
115C2×C4⋊C4×C61C2318yes—
116C2×C4⋊C4×C61C2318yes—
117C24×C61—24no—
118C24×C61—24no—
119C24×C61—24no—
120C24×C61C2×D414yes—
121C24×C61C2×D411yescharacteristic
122C24×C61C2×D411yescharacteristic
123C24×C61C2×D411yescharacteristic
124C23×D61—21no—
125C22×C4×C61—24no—
126C22×C4×C61—24no—
127C22×C4×C61—24no—
128C22×C4×C61—216no—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—216no—
132C22×C4×C61—28no—
133C22×C4×C61—28no—
134C22×C4×C61—28no—
135C22×C4×C61—24no—
136C22×C4×C61—48no—
137C22×C4×C61—48no—
138C22×C4×C61—432no—
139C22×C4×C61—48no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2×D412yes—
146C22×C4×C61C2×D414yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D412yes—
149C22×C4×C61#4118yes—
150C22×C4×C61C2412yes—
151C22×C4×C61C2412yes—
152C22×C4×C61C2411yescharacteristic
153C22×C4×C61C2412yes—
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—21no—
162C22×Dic61—21no—
163#11758222—216no—
164#11758222—48no—
165C2×D244—24no—
166C2×D244—21no—
167C2×D244C2412yes—
168C2×C4×D61—24no—
169C2×C4×D61—44no—
170C2×Dic122—24no—
171C2×Dic122—21no—
172C2×Dic122C2412yes—
173C4⋊C4×C61—28no—
174C4⋊C4×C61—28no—
175C4⋊C4×C61—28no—
176C4⋊C4×C61—24no—
177C4⋊C4×C61C2418yes—
178C22⋊Dic61—24no—
179C22⋊Dic61—24no—
180C22⋊Dic61—24no—
181#11758198—24no—
182#11758198—24no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61C2412yes—
186C4⋊Dic61C2412yes—
187#11758196—24no—
188#11758196—24no—
189C4×Dic61—24no—
190#11758194—216no—
191C2×#92418—28no—
192C2×#92418C2418yes—
193C23×C61—232no—
194C23×C61—28no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—28no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—28no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61#12314yes—
204C23×C61AGL(1,ℤ/8)14yes—
205C23×C61#13714yes—
206C23×C61C22×D412yes—
207C23×C61C22×D412yes—
208C23×C61C22×D412yes—
209C23×C61C22×D411yescharacteristic
210C22×D61—22no—
211C22×D61—42no—
212C22×D61—42no—
213C22×D61—42no—
214C2×C4×C61—28no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—28no—
220C2×C4×C61—28no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—28no—
224C2×C4×C61—416no—
225C2×C4×C61—416no—
226C2×C4×C61—416no—
227C2×C4×C61—48no—
228C2×C4×C61#12314yes—
229C2×C4×C61#12318yes—
230C2×C4×C61C22×D412yes—
231C2×C4×C61C22×D412yes—
232C2×C4×C61C22×D412yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4114yes—
235C2×C4×C61C2×#4118yes—
236C2×C4×C61C2511yescharacteristic
237C22⋊D61—48no—
238C22⋊D61—48no—
239C22⋊D61—48no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246D244—24no—
247D244—42no—
248C4×D61—48no—
249Dic122—24no—
250Dic122—42no—
251#92418—28no—
252C2×#3092—611nomaximal, Sylow
253C22×C61—232no—
254C22×C61—28no—
255C22×C61—28no—
256C22×C61—28no—
257C22×C61—22no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—22no—
261C22×C61—22no—
262C22×C61—22no—
263C22×C61C2×#12314yes—
264C22×C61C2×#12312yes—
265C22×C61C2×AGL(1,ℤ/8)14yes—
266C22×C61C2×AGL(1,ℤ/8)12yes—
267C22×C61C2×#13714yes—
268C22×C61C2×#13712yes—
269C22×C61C23×D411yescharacteristic
270C2×D61—42no—
271C2×D61—42no—
272C2×D61—42no—
273C2×D61—84no—
274C4×C61—22no—
275C4×C61—22no—
276C4×C61—48no—
277C4×C61C2×#12312yes—
278C4×C61C2×#12314yes—
279C4×C61C22×#4112yes—
280Dic61—42no—
281Dic61—42no—
282Dic61—42no—
283C2×#525—611no—
284C23×C4⋊C4D6111yescharacteristic, complemented
285#3092—6116no—
286C2×#484—612no—
287C2×#481—612no—
288C2×#483—612no—
289C2×#480—612no—
290C2×#428—612no—
291C2×#417—612no—
292C2×#407—611no—
293C2×C61—24no—
294C2×C61—24no—
295C2×C61—24no—
296C2×C61#309214yescomplemented
297C2×C61C22×#12311yescharacteristic
298C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
299C2×C61C22×#13711yescharacteristic
300D61—82no—
301C22×#41—611no—
302C24×C4—21no—
303C24×C4C2×D6111yescharacteristic
304C22×M16—612no—
305#525—618no—
306#525—1224no—
307C2×#123—1221no—
308C2×C22⋊Q8—611no—
309C2×C22⋊Q8—611no—
310C2×#121—611no—
311C2×#121—611no—
312C2×#120—1221no—
313C2×C4×D4—1221no—
314C2×C4×D4—1221no—
315C2×#117—611no—
316C22×C4⋊C4—24no—
317C22×C4⋊C4—21no—
318C22×C4⋊C4—21no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4C2×D6112yes—
321C22×C4⋊C4C2×D6112yes—
322C22×C4⋊C4C2×D6112yes—
323C22×C4⋊C4C2×D6118yescomplemented
324#484—6116no—
325#483—6116no—
326#481—6116no—
327#480—6116no—
328#428—6116no—
329C2×#107—614no—
330C2×#106—614no—
331#417—6116no—
332C2×#103—614no—
333C2×#102—614no—
334#407—618no—
335C2×C22⋊C8—612no—
336C2×C22⋊C8—612no—
337C61C2×#309211yescharacteristic, complemented, Sylow
338C25C22⋊D6111yescharacteristic
339C2×#41—614no—
340C2×#41—1222no—
341C2×#41—1228no—
342C22×Q8—611no—
343C22×D4—611no—
344C22×D4—1221no—
345C22×D4—1221no—
346C22×D4—1221no—
347C23×C4—28no—
348C23×C4—22no—
349C23×C4—22no—
350C23×C4—22no—
351C23×C4—28no—
352C23×C4—48no—
353C23×C4—1221no—
354C23×C4C22⋊D6111yescharacteristic
355C23×C4C22⋊D6111yescharacteristic
356C23×C4C22⋊D6111yescharacteristic
357C23×C4C22⋊D6114yes—
358C23×C4C22×D6111yescharacteristic
359C23×C4C22×D6111yescharacteristic
360C23×C4C22×D6114yes—
361C23×C4C22×D6111yescharacteristic
362C2×M16—618no—
363C2×M16—1228no—
364C22×C8—614no—
365#123—1228no—
366C22⋊Q8—614no—
367C22⋊Q8—614no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370#121—614no—
371#121—614no—
372#121—1222no—
373#121—1222no—
374#120—1228no—
375C4×D4—1228no—
376C4×D4—1228no—
377#117—614no—
378#117—1222no—
379C2×C4⋊C4—24no—
380C2×C4⋊C4—24no—
381C2×C4⋊C4—24no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—28no—
386C2×C4⋊C4—28no—
387C2×C4⋊C4—28no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4C22×D6114yes—
393C2×C4⋊C4C22×D6118yes—
394C2×C4⋊C4C22×D6118yes—
395C2×C4⋊C4C22×D6118yes—
396C2×C22⋊C4—1221no—
397C2×C22⋊C4—1221no—
398C2×C22⋊C4—1221no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—1221no—
402#107—6116no—
403#106—6116no—
404#103—6116no—
405#102—6116no—
406C22⋊C8—618no—
407C22⋊C8—618no—
408C24—24no—
409C24—24no—
410C24—24no—
411C24—1221no—
412C24C2×C22⋊D6114yes—
413C24C2×C22⋊D6111yescharacteristic
414C24C2×C22⋊D6111yescharacteristic
415C24C2×C22⋊D6111yescharacteristic
416#41—12216no—
417#41—2448no—
418C2×Q8—612no—
419C2×Q8—1221no—
420C2×Q8—1224no—
421C2×D4—612no—
422C2×D4—1224no—
423C2×D4—1224no—
424C2×D4—1224no—
425C2×D4—1221no—
426C2×D4—1224no—
427C2×D4—2444no—
428C2×D4—2444no—
429C2×D4—2444no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—216no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—216no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—28no—
440C22×C4—24no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1224no—
449C22×C4—2444no—
450C22×C4#1175822218yes—
451C22×C4C2×C22⋊D6114yes—
452C22×C4C2×C22⋊D6114yes—
453C22×C4C2×C22⋊D6112yes—
454C22×C4C2×C22⋊D6112yes—
455C22×C4C2×C22⋊D6114yes—
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6112yes—
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C23×D6112yes—
461C22×C4C23×D6112yes—
462C22×C4C23×D6112yes—
463C22×C4C23×D6111yescharacteristic
464M16—12216no—
465C2×C8—618no—
466C2×C8—1228no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—612no—
472C4⋊C4—612no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1224no—
476C4⋊C4—1224no—
477C4⋊C4C23×D6118yes—
478C22⋊C4—1224no—
479C22⋊C4—1224no—
480C22⋊C4—1224no—
481C22⋊C4—1224no—
482C22⋊C4—1224no—
483C42—1224no—
484C23—232no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—28no—
492C23—22no—
493C23—22no—
494C23—1222no—
495C23—2442no—
496C23—2442no—
497C23—2442no—
498C23#1451976014yes—
499C23#1451980314yes—
500C23#1451980814yes—
501C23C22×C22⋊D6112yes—
502C23C22×C22⋊D6112yes—
503C23C22×C22⋊D6112yes—
504C23C22×C22⋊D6111yescharacteristic
505Q8—1224no—
506Q8—2442no—
507D4—1224no—
508D4—2448no—
509D4—2448no—
510D4—2448no—
511D4—2442no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—48no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—2442no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2448no—
533C2×C4#1451976014yes—
534C2×C4#1451976018yes—
535C2×C4C22×C22⋊D6112yes—
536C2×C4C22×C22⋊D6112yes—
537C2×C4C22×C22⋊D6112yes—
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C2×#1175822214yes—
540C2×C4C2×#1175822218yes—
541C2×C4C24×D6111yescharacteristic
542C8—1228no—
543C22—232no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—28no—
551C22—28no—
552C22—28no—
553C22—4884no—
554C22—2442no—
555C22—2442no—
556C22—2442no—
557C22C2×#1451976014yes—
558C22C2×#1451976012yes—
559C22C2×#1451980314yes—
560C22C2×#1451980312yes—
561C22C2×#1451980814yes—
562C22C2×#1451980812yes—
563C22C23×C22⋊D6111yescharacteristic
564C4—22no—
565C4—22no—
566C4—48no—
567C4—2442no—
568C4—2442no—
569C4—2442no—
570C4C2×#1451976014yes—
571C4C2×#1451976012yes—
572C4C22×#1175822212yes—
573C2—24no—
574C2—24no—
575C2—24no—
576C2—4882no—
577C2#2145574214yesdirect product factor
578C2C22×#1451976011yescharacteristic
579C2C22×#1451980311yescharacteristic
580C2C22×#1451980811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC23×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
hypercenterC23×C4⋊C4D61
hypocenterC61C2×#3092
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21455742C2direct (N × H)
3. C2×#16840097C2semidirect
4. C2×#16840080C2semidirect
5. C2×#16840092C2semidirect
6. C2×#16840075C2semidirect
7. C2×#16839802C2semidirect
8. C2×#16839761C2semidirect
9. C2×#16839726C2semidirect
10. C23×C4⋊C4×C61C2semidirect
24. #16840097C22semidirect
25. #16840092C22semidirect
26. #16840080C22semidirect
27. #16840075C22semidirect
28. #16839802C22semidirect
29. C2×#14519710C22semidirect
30. C2×#14519708C22semidirect
31. #16839761C22semidirect
32. C2×#14519700C22semidirect
33. C2×#14519697C22semidirect
34. #16839726C22semidirect
46. C22×C4⋊C4×C61C22semidirect
83. #14519710C23semidirect
84. #14519708C23semidirect
85. #14519700C23semidirect
86. #14519697C23semidirect
284. C23×C4⋊C4D61semidirect
296. C2×C61#3092semidirect
323. C22×C4⋊C4C2×D61semidirect
337. C61C2×#3092semidirect
577. C2#21455742direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402732
#214557422
C2×#168400972
C2×#168400802
C2×#168400922
C2×#168400752
C2×#168398022
C2×#168397612
C2×#168397262
C23×C4⋊C4×C612
C2×#309261