Finite Group Databasebeta

← C2×#21459529 #30587955

Subgroups — C2×#21459529 #30587955

Order 15616.

7910 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459529C2132yesmaximal, direct product factor
2C2×#16840473C214yesmaximal, complemented
3C2×#16840465C214yesmaximal, complemented
4C2×#16840450C214yesmaximal, complemented
5C2×#16840415C214yesmaximal, complemented
6C2×#16840361C214yesmaximal, complemented
7C2×#16840343C211yesmaximal, characteristic, complemented
8C2×#16840275C214yesmaximal, complemented
9C2×#16840274C212yesmaximal, complemented
10C2×#16840242C212yesmaximal, complemented
11C22×C4⋊C4×D61C211yesmaximal, characteristic, complemented
12C2×C61×#544C211yesmaximal, characteristic, complemented
13C22×#11758226C2214yes—
14C22×#11758222C2212yes—
15C23×C4×D61C2211yescharacteristic
16C23×C4×D61C2212yes—
17#16840473C22164yescomplemented
18#16840465C22164yescomplemented
19#16840450C22164yescomplemented
20#16840415C22164yescomplemented
21#16840361C22164yescomplemented
22#16840343C22116yescomplemented
23C2×#14519772C2218yescomplemented
24C2×#14519768C2218yescomplemented
25#16840275C22164yescomplemented
26#16840274C22132yescomplemented
27C2×#14519762C2218yes—
28C2×#14519762C2218yescomplemented
29C2×#14519761C2214yes—
30C2×#14519757C2212yes—
31C2×#14519757C2212yes—
32C2×#14519757C2214yescomplemented
33C2×#14519753C2214yes—
34C2×#14519753C2218yescomplemented
35C2×#14519752C2212yes—
36C2×#14519752C2212yes—
37C2×#14519752C2214yes—
38C2×#14519752C2214yescomplemented
39C2×#14519751C2214yes—
40C2×#14519747C2214yes—
41#16840242C22132yescomplemented
42C2×C4×Dic122C2214yescomplemented
43C2×C4×D244C2214yes—
44C2×C4×D244C2214yes—
45C2×C4×D244C2214yescomplemented
46C2×#14519739C2218yes—
47C2×#14519737C2218yes—
48C2×#14519735C2214yes—
49C2×#14519735C2214yescomplemented
50C22×C4⋊Dic61C2211yescharacteristic
51C22×#11758196C2212yes—
52C2×C4⋊C4×D61C2214yes—
53C2×C4⋊C4×D61C2214yes—
54C2×C4⋊C4×D61C22116yescomplemented
55C2×C4⋊C4×D61C2214yescomplemented
56C61×#544C22116yescomplemented
57C2×C4⋊Q8×C61C2214yescomplemented
58C2×#125×C61C2214yescomplemented
59C2×C22⋊Q8×C61C2214yes—
60C2×#117×C61C2212yes—
61C22×C4⋊C4×C61C2211yescharacteristic
62C24×D61C2311yescharacteristic
63C2×#11758226—416no—
64C2×#11758226C23132yes—
65C2×#11758222—216no—
66C2×#11758222C23116yes—
67C22×Dic122C2312yes—
68C22×D244C2314yes—
69C22×D244C2314yes—
70C22×D244C2312yes—
71C22×D244C2314yes—
72C22×C22⋊D61C2314yes—
73C23×Dic61C2311yescharacteristic
74C23×Dic61C2312yes—
75C22×C4×D61—28no—
76C22×C4×D61—48no—
77C22×C4×D61C2314yes—
78C22×C4×D61C2312yes—
79C22×C4×D61C2312yes—
80C22×C4×D61C2318yes—
81C22×C4×D61C2314yes—
82C22×C4×D61C2314yes—
83C22×C4×D61C23116yes—
84C22×C4×D61C2318yes—
85#14519772C23164yescomplemented
86#14519768C23164yescomplemented
87#14519762C23164yes—
88#14519762C23164yescomplemented
89#14519761C23132yes—
90#14519757C23116yes—
91#14519757C23116yes—
92#14519757C23132yescomplemented
93#14519753C23132yes—
94#14519753C23164yescomplemented
95#14519752C23116yes—
96#14519752C23132yes—
97#14519752C23116yes—
98#14519752C23132yescomplemented
99#14519751C23132yes—
100#14519747C23132yes—
101C4×Dic122C23132yescomplemented
102C4×D244C23132yes—
103C4×D244C23132yes—
104C4×D244C23132yescomplemented
105#14519739C23164yes—
106#14519737C23164yes—
107#14519735C23132yes—
108#14519735C23132yescomplemented
109C2×C4⋊Dic61C2314yes—
110C2×C4⋊Dic61C2318yes—
111C2×C4⋊Dic61C2314yes—
112C2×C4⋊Dic61C2314yes—
113C2×C4⋊Dic61C2314yes—
114C2×#11758196C2318yes—
115C2×#11758196C23116yes—
116C2×#11758196C2318yes—
117C4⋊C4×D61C23132yes—
118C4⋊C4×D61C23132yes—
119C4⋊C4×D61C23132yescomplemented
120C2×#11758198C2314yes—
121C2×#11758198C2318yes—
122C2×#11758198C2314yes—
123C2×#11758198C2318yes—
124C2×#11758198C2314yes—
125C2×#11758198C2314yes—
126C2×C4×Dic61C2318yes—
127C22×Q8×C61C2314yes—
128C23×C4×C61C2312yes—
129C23×C4×C61C2311yescharacteristic
130C4⋊Q8×C61C23132yescomplemented
131#125×C61C23132yescomplemented
132C22⋊Q8×C61C23132yes—
133#117×C61C23116yes—
134C2×C4⋊C4×C61C2312yes—
135C2×C4⋊C4×C61C2314yes—
136C2×C4⋊C4×C61C2318yes—
137C2×C4⋊C4×C61C2318yes—
138C2×C4⋊C4×C61C2314yes—
139C2×C4⋊C4×C61C2312yes—
140C2×C22⋊C4×C61C2314yes—
141C2×C42×C61C2314yes—
142C24×C61C2411yescharacteristic
143C23×D61—24no—
144C23×D61—24no—
145C23×D61—24no—
146C23×D61C2414yes—
147C23×D61C2414yes—
148C23×D61C2411yescharacteristic
149C23×D61C2411yescharacteristic
150C2×Q8×C61—48no—
151C2×Q8×C61C24116yes—
152C22×C4×C61—44no—
153C22×C4×C61#4118yescomplemented
154C22×C4×C61C2414yes—
155C22×C4×C61C2414yes—
156C22×C4×C61C2414yes—
157C22×C4×C61C2418yes—
158C22×C4×C61C2412yes—
159C22×C4×C61C2414yes—
160C2×C22⋊D61—216no—
161C2×C22⋊D61C24116yes—
162C22×Dic61—24no—
163C22×Dic61—44no—
164C22×Dic61C2414yes—
165C22×Dic61C2412yes—
166C22×Dic61C2414yes—
167C22×Dic61C2414yes—
168C22×Dic61C2418yes—
169#11758226—464no—
170#11758222—264no—
171C2×D244—216no—
172C2×D244—216no—
173C2×D244—28no—
174C2×D244—48no—
175C2×D244C2418yes—
176C2×D244C24116yes—
177C2×D244C24116yes—
178C2×D244C24116yes—
179C2×C4×D61—216no—
180C2×C4×D61—28no—
181C2×C4×D61—28no—
182C2×C4×D61—216no—
183C2×C4×D61—232no—
184C2×C4×D61—432no—
185C2×C4×D61—48no—
186C2×C4×D61—416no—
187C2×C4×D61C24116yes—
188C2×C4×D61C2418yes—
189C2×C4×D61C2418yes—
190C2×C4×D61C24116yes—
191C2×C4×D61C24116yes—
192C2×C4×D61C24132yes—
193C2×Dic122—28no—
194C2×Dic122C2418yes—
195C4⋊C4×C61C24116yes—
196C4⋊C4×C61C2418yes—
197C4⋊C4×C61C2418yes—
198C4⋊C4×C61C24116yes—
199C4⋊C4×C61C24132yes—
200C22⋊C4×C61C24116yes—
201C42×C61C24116yes—
202#11758198C24116yes—
203#11758198C24132yes—
204#11758198C24116yes—
205#11758198C24132yes—
206#11758198C24116yes—
207#11758198C24116yes—
208C4⋊Dic61C24116yes—
209C4⋊Dic61C24116yes—
210C4⋊Dic61C24116yes—
211C4⋊Dic61C24116yes—
212#11758196C24132yes—
213#11758196C24132yes—
214C4×Dic61C24132yes—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61C2×#4114yes—
218C23×C61C2512yes—
219C23×C61C2511yescharacteristic
220C22×D61—232no—
221C22×D61—28no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—28no—
227C22×D61—28no—
228C22×D61—28no—
229C22×D61—28no—
230C22×D61—28no—
231C22×D61H(5,𝔽2)14yes—
232C22×D612−1+414yes—
233C22×D61C2518yes—
234C22×D61C2512yes—
235C22×D61C2512yes—
236Q8×C61—416no—
237C2×C4×C61—28no—
238C2×C4×C61—28no—
239C2×C4×C61—28no—
240C2×C4×C61—48no—
241C2×C4×C61C2×#4118yes—
242C2×C4×C61C2×#41116yescomplemented
243C2×C4×C61C2518yes—
244C2×C4×C61C2518yes—
245C2×C4×C61C2518yes—
246C2×C4×C61C2514yes—
247C22⋊D61—232no—
248C2×Dic61—28no—
249C2×Dic61—28no—
250C2×Dic61—24no—
251C2×Dic61—28no—
252C2×Dic61—48no—
253C2×Dic61C2518yes—
254C2×Dic61C2518yes—
255C2×Dic61C2514yes—
256D244—232no—
257D244—232no—
258D244—216no—
259D244—416no—
260C4×D61—232no—
261C4×D61—216no—
262C4×D61—216no—
263C4×D61—232no—
264C4×D61—416no—
265C4×D61—432no—
266Dic122—216no—
267C2×#3492—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—28no—
271C22×C61C22×#4114yes—
272C22×C61C22×#4112yes—
273C22×C61C2×H(5,𝔽2)12yes—
274C22×C61C2×2−1+412yes—
275C22×C61C2611yescharacteristic
276C2×D61—232no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—28no—
282C2×D61—28no—
283C2×D61—28no—
284C2×D61—28no—
285C2×D61—28no—
286C2×D61—416no—
287C2×D61C2×H(5,𝔽2)14yes—
288C2×D61C2×2−1+414yes—
289C4×C61—28no—
290C4×C61—28no—
291C4×C61—28no—
292C4×C61C22×#4118yes—
293Dic61—28no—
294Dic61—24no—
295Dic61—28no—
296#3492—6132no—
297C2×#554—614no—
298C2×#553—614no—
299C2×#552—614no—
300C2×#547—614no—
301C2×#544D6111yescharacteristic, complemented
302C2×#539—614no—
303C2×#537—611no—
304C2×#525—614no—
305C2×#525—612no—
306C2×#520—612no—
307C23×C4⋊C4—611no—
308C2×C61—24no—
309C2×C61#349214yescomplemented
310C2×C61C23×#4111yescharacteristic
311C2×C61C22×H(5,𝔽2)11yescharacteristic
312C2×C61C22×2−1+411yescharacteristic
313D61—24no—
314D61—24no—
315D61—48no—
316C22×#41—614no—
317C22×#41—612no—
318C24×C4—611no—
319C24×C4—612no—
320#554—6164no—
321#553—6164no—
322#552—6164no—
323#547—6164no—
324#544C2×D61116yescomplemented
325#539—6164no—
326#537—6116no—
327C2×C4⋊Q8C2×D6114yescomplemented
328C2×#126—618no—
329C2×#125—618no—
330C2×#125C2×D6114yescomplemented
331#525—6164no—
332#525—6132no—
333C2×#123—618no—
334C2×#123—618no—
335C2×#123—614no—
336C2×C22⋊Q8—614no—
337C2×C22⋊Q8—618no—
338C2×C22⋊Q8—612no—
339C2×C22⋊Q8—612no—
340C2×C22⋊Q8—614no—
341C2×C22⋊Q8C2×D6114yes—
342C2×#121—614no—
343C2×#121—612no—
344C2×#121—614no—
345C2×#121—612no—
346C2×#121—614no—
347C2×#120—614no—
348#520—6132no—
349C2×C4×Q8—614no—
350C2×C4×D4—618no—
351C2×C4×D4—618no—
352C2×C4×D4—614no—
353C2×C4×D4—614no—
354C2×C4×D4—614no—
355C2×#117—614no—
356C2×#117—614no—
357C2×#117C2×D6112yes—
358C22×C4⋊C4—612no—
359C22×C4⋊C4—611no—
360C22×C4⋊C4—614no—
361C22×C4⋊C4—614no—
362C22×C4⋊C4—6116no—
363C22×C4⋊C4—614no—
364C22×C4⋊C4C2×D6111yescharacteristic
365C61C2×#349211yescharacteristic, complemented, Sylow
366C25—611no—
367C2×#41—6132no—
368C2×#41—6116no—
369C2×#41—12216no—
370C2×#41—24416no—
371C22×Q8—612no—
372C22×Q8C22×D6114yes—
373C22×D4—614no—
374C22×D4—614no—
375C22×D4—612no—
376C22×D4—614no—
377C22×D4—614no—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4—614no—
381C23×C4—612no—
382C23×C4—612no—
383C23×C4—618no—
384C23×C4—614no—
385C23×C4—614no—
386C23×C4—6116no—
387C23×C4—618no—
388C23×C4—1228no—
389C23×C4—2448no—
390C23×C4C22×D6112yes—
391C23×C4C22×D6111yescharacteristic
392C4⋊Q8C22×D61132yescomplemented
393#126—6164no—
394#125—6164no—
395#125C22×D61132yescomplemented
396#123—6164no—
397#123—6164no—
398#123—6132no—
399C22⋊Q8—6132no—
400C22⋊Q8—6164no—
401C22⋊Q8—6116no—
402C22⋊Q8—6116no—
403C22⋊Q8—6132no—
404C22⋊Q8C22×D61132yes—
405#121—6132no—
406#121—6116no—
407#121—6132no—
408#121—6116no—
409#121—6132no—
410#120—6132no—
411C4×Q8—6132no—
412C4×D4—6164no—
413C4×D4—6164no—
414C4×D4—6132no—
415C4×D4—6132no—
416C4×D4—6132no—
417#117—6132no—
418#117—6132no—
419#117C22×D61116yes—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—618no—
422C2×C4⋊C4—618no—
423C2×C4⋊C4—6116no—
424C2×C4⋊C4—618no—
425C2×C4⋊C4—614no—
426C2×C4⋊C4—614no—
427C2×C4⋊C4—614no—
428C2×C4⋊C4—6132no—
429C2×C4⋊C4—6132no—
430C2×C4⋊C4—6132no—
431C2×C4⋊C4C22×D6114yes—
432C2×C4⋊C4C22×D6118yes—
433C2×C4⋊C4C22×D6112yes—
434C2×C4⋊C4C22×D6112yes—
435C2×C4⋊C4C22×D6114yes—
436C2×C4⋊C4C22×D6118yes—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—618no—
439C2×C22⋊C4—614no—
440C2×C22⋊C4—618no—
441C2×C22⋊C4—614no—
442C2×C22⋊C4—614no—
443C2×C22⋊C4C22×D6114yes—
444C2×C42—618no—
445C2×C42C22×D6114yes—
446C24—611no—
447C24—611no—
448C24—614no—
449C24—614no—
450C24—1224no—
451C24—1224no—
452C24—1224no—
453C24C23×D6111yescharacteristic
454#41—12264no—
455#41—24464no—
456C2×Q8—48no—
457C2×Q8—618no—
458C2×Q8—1228no—
459C2×Q8C23×D61116yes—
460C2×D4—6116no—
461C2×D4—6116no—
462C2×D4—618no—
463C2×D4—6116no—
464C2×D4—6116no—
465C2×D4—12216no—
466C2×D4—12216no—
467C2×D4—12216no—
468C2×D4—1228no—
469C2×D4—2448no—
470C22×C4—44no—
471C22×C4—614no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—612no—
475C22×C4—618no—
476C22×C4—6116no—
477C22×C4—618no—
478C22×C4—618no—
479C22×C4—6116no—
480C22×C4—6116no—
481C22×C4—6132no—
482C22×C4—12216no—
483C22×C4—1224no—
484C22×C4—1228no—
485C22×C4—1228no—
486C22×C4—12216no—
487C22×C4—12232no—
488C22×C4—2448no—
489C22×C4—24416no—
490C22×C4—24432no—
491C22×C4—2444no—
492C22×C4#1175822618yescomplemented
493C22×C4C23×D6114yes—
494C22×C4C23×D6114yes—
495C22×C4C23×D6114yes—
496C22×C4C23×D6118yes—
497C22×C4C23×D6112yes—
498C22×C4C23×D6114yes—
499C4⋊C4—6116no—
500C4⋊C4—6132no—
501C4⋊C4—6132no—
502C4⋊C4—6116no—
503C4⋊C4—6116no—
504C4⋊C4—6116no—
505C4⋊C4C23×D61116yes—
506C4⋊C4C23×D6118yes—
507C4⋊C4C23×D6118yes—
508C4⋊C4C23×D61116yes—
509C4⋊C4C23×D61132yes—
510C22⋊C4—6116no—
511C22⋊C4—6132no—
512C22⋊C4—6116no—
513C22⋊C4—6132no—
514C22⋊C4—6116no—
515C22⋊C4—6116no—
516C22⋊C4C23×D61116yes—
517C42—6132no—
518C42C23×D61116yes—
519C23—22no—
520C23—22no—
521C23—614no—
522C23—614no—
523C23—612no—
524C23—612no—
525C23—618no—
526C23—1228no—
527C23—1228no—
528C23—12232no—
529C23—1228no—
530C23—1222no—
531C23—1222no—
532C23—1222no—
533C23—1222no—
534C23—1228no—
535C23—1228no—
536C23—1228no—
537C23C2×#1175822614yes—
538C23C24×D6112yes—
539C23C24×D6111yescharacteristic
540Q8—416no—
541Q8—12216no—
542D4—12232no—
543D4—12232no—
544D4—12232no—
545D4—12216no—
546D4—24416no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—48no—
551C2×C4—618no—
552C2×C4—618no—
553C2×C4—614no—
554C2×C4—12232no—
555C2×C4—1228no—
556C2×C4—1224no—
557C2×C4—1228no—
558C2×C4—1228no—
559C2×C4—12216no—
560C2×C4—12216no—
561C2×C4—12232no—
562C2×C4—24416no—
563C2×C4—24432no—
564C2×C4—2448no—
565C2×C4C2×#1175822618yes—
566C2×C4C2×#11758226116yescomplemented
567C2×C4C24×D6118yes—
568C2×C4C24×D6118yes—
569C2×C4C24×D6118yes—
570C2×C4C24×D6114yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—614no—
575C22—614no—
576C22—12232no—
577C22—1228no—
578C22—1228no—
579C22—1222no—
580C22—1222no—
581C22—1222no—
582C22—1222no—
583C22—1228no—
584C22—1228no—
585C22—1228no—
586C22—24416no—
587C22C22×#1175822614yes—
588C22C22×#1175822612yes—
589C22H(5,𝔽2)×D6112yes—
590C222−1+4×D6112yes—
591C22C25×D6111yescharacteristic
592C4—28no—
593C4—28no—
594C4—28no—
595C4—1224no—
596C4—1228no—
597C4—1228no—
598C4C22×#1175822618yes—
599C2—24no—
600C2—1224no—
601C2—1224no—
602C2—2448no—
603C2#2145952914yesdirect product factor
604C2C23×#1175822611yescharacteristic
605C2C2×H(5,𝔽2)×D6111yescharacteristic
606C2C2×2−1+4×D6111yescharacteristic
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#544C2
socleC23×C61C25
hypercenterC2×#544D61
hypocenterC61C2×#3492
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. #21459529C2direct (N × H)
2. C2×#16840473C2semidirect
3. C2×#16840465C2semidirect
4. C2×#16840450C2semidirect
5. C2×#16840415C2semidirect
6. C2×#16840361C2semidirect
7. C2×#16840343C2semidirect
8. C2×#16840275C2semidirect
9. C2×#16840274C2semidirect
10. C2×#16840242C2semidirect
11. C22×C4⋊C4×D61C2semidirect
12. C2×C61×#544C2semidirect
17. #16840473C22semidirect
18. #16840465C22semidirect
19. #16840450C22semidirect
20. #16840415C22semidirect
21. #16840361C22semidirect
22. #16840343C22semidirect
23. C2×#14519772C22semidirect
24. C2×#14519768C22semidirect
25. #16840275C22semidirect
26. #16840274C22semidirect
28. C2×#14519762C22semidirect
32. C2×#14519757C22semidirect
34. C2×#14519753C22semidirect
38. C2×#14519752C22semidirect
41. #16840242C22semidirect
42. C2×C4×Dic122C22semidirect
45. C2×C4×D244C22semidirect
49. C2×#14519735C22semidirect
54. C2×C4⋊C4×D61C22semidirect
55. C2×C4⋊C4×D61C22semidirect
56. C61×#544C22semidirect
57. C2×C4⋊Q8×C61C22semidirect
58. C2×#125×C61C22semidirect
85. #14519772C23semidirect
86. #14519768C23semidirect
88. #14519762C23semidirect
92. #14519757C23semidirect
94. #14519753C23semidirect
98. #14519752C23semidirect
101. C4×Dic122C23semidirect
104. C4×D244C23semidirect
108. #14519735C23semidirect
119. C4⋊C4×D61C23semidirect
130. C4⋊Q8×C61C23semidirect
131. #125×C61C23semidirect
153. C22×C4×C61#41semidirect
242. C2×C4×C61C2×#41semidirect
301. C2×#544D61semidirect
309. C2×C61#3492semidirect
324. #544C2×D61semidirect
327. C2×C4⋊Q8C2×D61semidirect
330. C2×#125C2×D61semidirect
365. C61C2×#3492semidirect
392. C4⋊Q8C22×D61semidirect
395. #125C22×D61semidirect
492. C22×C4#11758226semidirect
566. C2×C4C2×#11758226semidirect
603. C2#21459529direct (N × H)

Maximal subgroups

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

subgroupindex
#214595292
C2×#168404732
C2×#168404652
C2×#168404502
C2×#168404152
C2×#168403612
C2×#168403432
C2×#168402752
C2×#168402742
C2×#168402422
C22×C4⋊C4×D612
C2×C61×#5442
C2×#349261