Finite Group Databasebeta

← C2×#21459526 #30587953

Subgroups — C2×#21459526 #30587953

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#21459526C2132yesmaximal, direct product factor
2C2×#16840470C214yesmaximal, complemented
3C2×#16840459C214yesmaximal, complemented
4C2×#16840446C214yesmaximal, complemented
5C2×#16840412C214yesmaximal, complemented
6C2×#16840393C211yesmaximal, characteristic, complemented
7C2×#16840357C214yesmaximal, complemented
8C2×#16840272C214yesmaximal, complemented
9C2×#16840269C212yesmaximal, complemented
10C2×#16840245C212yesmaximal, complemented
11C22×C4⋊C4×D61C211yesmaximal, characteristic, complemented
12C2×C61×#537C211yesmaximal, characteristic, complemented
13C22×#11758224C2214yes—
14C23×C4×D61C2211yescharacteristic
15C23×C4×D61C2212yes—
16#16840470C22164yescomplemented
17#16840459C22164yescomplemented
18#16840446C22164yescomplemented
19#16840412C22164yescomplemented
20#16840393C22116yescomplemented
21#16840357C22164yescomplemented
22C2×#14519776C2214yescomplemented
23C2×#14519770C2218yescomplemented
24C2×#14519768C2214yescomplemented
25C2×#14519767C2218yescomplemented
26#16840272C22164yescomplemented
27#16840269C22132yescomplemented
28C2×#14519759C2218yes—
29C2×#14519758C2214yes—
30C2×#14519758C2218yescomplemented
31C2×#14519757C2214yes—
32C2×#14519756C2214yes—
33C2×#14519754C2218yescomplemented
34C2×#14519753C2212yes—
35C2×#14519753C2212yes—
36C2×#14519748C2212yes—
37C2×#14519748C2212yes—
38C2×#14519748C2214yes—
39C2×#14519745C2214yes—
40#16840245C22132yescomplemented
41C2×Q8×Dic61C2214yescomplemented
42C2×D4×Dic61C2218yes—
43C2×D4×Dic61C2214yescomplemented
44C2×C4×C22⋊D61C2214yes—
45C2×C4×C22⋊D61C2214yes—
46C2×#14519737C2218yes—
47C2×#14519736C2212yes—
48C2×#14519736C2214yescomplemented
49C2×#14519735C2214yes—
50C22×C4⋊Dic61C2211yescharacteristic
51C22×C4⋊Dic61C2214yescomplemented
52C22×#11758196C2212yes—
53C2×C4⋊C4×D61C2214yes—
54C2×C4⋊C4×D61C2214yes—
55C2×C4⋊C4×D61C22116yescomplemented
56C22×#41×C61C2212yes—
57C61×#537C22116yescomplemented
58C2×C22⋊Q8×C61C2214yescomplemented
59C2×#121×C61C2214yes—
60C2×#121×C61C2214yescomplemented
61C22×C4⋊C4×C61C2211yescharacteristic
62C24×D61C2311yescharacteristic
63C2×#11758224—416no—
64C2×#11758224C23132yes—
65C22×Dic122C2314yes—
66C22×C22⋊D61C2314yes—
67C22×C22⋊D61C2314yes—
68C23×Dic61C2311yescharacteristic
69C23×Dic61C2312yes—
70C23×Dic61C2318yes—
71C22×C4×D61—28no—
72C22×C4×D61—48no—
73C22×C4×D61C2314yes—
74C22×C4×D61C2312yes—
75C22×C4×D61C2312yes—
76C22×C4×D61C2318yes—
77C22×C4×D61C2314yes—
78C22×C4×D61C23116yes—
79#14519776C23132yescomplemented
80#14519770C23164yescomplemented
81#14519768C23132yescomplemented
82#14519767C23164yescomplemented
83#14519759C23164yes—
84#14519758C23132yes—
85#14519758C23164yescomplemented
86#14519757C23132yes—
87#14519756C23132yes—
88#14519754C23164yescomplemented
89#14519753C23116yes—
90#14519753C23116yes—
91#14519748C23116yes—
92#14519748C23132yes—
93#14519748C23116yes—
94#14519745C23132yes—
95Q8×Dic61C23132yescomplemented
96D4×Dic61C23164yes—
97D4×Dic61C23132yescomplemented
98C4×C22⋊D61C23132yes—
99C4×C22⋊D61C23132yes—
100#14519737C23164yes—
101#14519736C23116yes—
102#14519736C23132yescomplemented
103#14519735C23132yes—
104C2×C4⋊Dic61C2314yes—
105C2×C4⋊Dic61C2318yes—
106C2×C4⋊Dic61C2312yes—
107C2×C4⋊Dic61C2318yes—
108C2×C4⋊Dic61C2314yes—
109C2×C4⋊Dic61C2312yes—
110C2×C4⋊Dic61C23132yescomplemented
111C2×#11758196C2318yes—
112C2×#11758196C23116yes—
113C2×#11758196C2318yes—
114C2×#11758196C2314yes—
115C2×#11758196C2314yes—
116C4⋊C4×D61C23132yes—
117C4⋊C4×D61C23132yes—
118C2×C22⋊Dic61C2314yes—
119C2×C22⋊Dic61C2318yes—
120C2×C22⋊Dic61C2314yes—
121C2×C22⋊Dic61C2314yes—
122C2×#11758198C2314yes—
123C2×#11758198C2314yes—
124C2×C4×Dic61C2318yes—
125C2×C4×Dic61C2314yes—
126C2×#41×C61—216no—
127C2×#41×C61C23116yes—
128C22×Q8×C61C2312yes—
129C22×D4×C61C2314yes—
130C22×D4×C61C2312yes—
131C22×D4×C61C2314yes—
132C23×C4×C61C2314yes—
133C23×C4×C61C2312yes—
134C23×C4×C61C2311yescharacteristic
135C22⋊Q8×C61C23132yescomplemented
136#121×C61C23132yes—
137#121×C61C23132yescomplemented
138C2×C4⋊C4×C61C2318yes—
139C2×C4⋊C4×C61C2314yes—
140C2×C4⋊C4×C61C2314yes—
141C2×C22⋊C4×C61C2318yes—
142C24×C61C2414yes—
143C24×C61C2411yescharacteristic
144C23×D61—24no—
145C23×D61—24no—
146C23×D61—24no—
147C23×D61C2414yes—
148C23×D61C2411yescharacteristic
149C23×D61C2411yescharacteristic
150#41×C61—264no—
151C2×Q8×C61—28no—
152C2×Q8×C61C2418yes—
153C2×D4×C61—216no—
154C2×D4×C61—28no—
155C2×D4×C61—48no—
156C2×D4×C61C24116yes—
157C2×D4×C61C2418yes—
158C2×D4×C61C24116yes—
159C22×C4×C61—216no—
160C22×C4×C61—44no—
161C22×C4×C61#4118yescomplemented
162C22×C4×C61C2414yes—
163C22×C4×C61C2414yes—
164C22×C4×C61C24116yes—
165C22×C4×C61C2418yes—
166C22×C4×C61C2412yes—
167C22×C4×C61C2414yes—
168C2×C22⋊D61—216no—
169C2×C22⋊D61—216no—
170C2×C22⋊D61C24116yes—
171C2×C22⋊D61C24116yes—
172C22×Dic61—24no—
173C22×Dic61—44no—
174C22×Dic61—416no—
175C22×Dic61C2414yes—
176C22×Dic61C2412yes—
177C22×Dic61C2414yes—
178C22×Dic61C2414yes—
179C22×Dic61C2414yes—
180C22×Dic61C2418yes—
181C22×Dic61C24132yes—
182#11758224—464no—
183C2×C4×D61—216no—
184C2×C4×D61—28no—
185C2×C4×D61—28no—
186C2×C4×D61—232no—
187C2×C4×D61—432no—
188C2×C4×D61—48no—
189C2×C4×D61C24116yes—
190C2×C4×D61C2418yes—
191C2×C4×D61C2418yes—
192C2×C4×D61C24116yes—
193C2×Dic122—48no—
194C2×Dic122C24116yes—
195C4⋊C4×C61C24116yes—
196C4⋊C4×C61C24116yes—
197C22⋊C4×C61C24132yes—
198C22⋊Dic61C24116yes—
199C22⋊Dic61C24132yes—
200C22⋊Dic61C24116yes—
201C22⋊Dic61C24116yes—
202#11758198C24116yes—
203#11758198C24116yes—
204C4⋊Dic61C24116yes—
205C4⋊Dic61C2418yes—
206C4⋊Dic61C24132yes—
207C4⋊Dic61C24116yes—
208C4⋊Dic61C2418yes—
209#11758196C24132yes—
210#11758196C24132yes—
211#11758196C24116yes—
212#11758196C24116yes—
213C4×Dic61C24132yes—
214C4×Dic61C24116yes—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—28no—
218C23×C61—28no—
219C23×C61—28no—
220C23×C61C2×#4114yes—
221C23×C61C2518yes—
222C23×C61C2512yes—
223C23×C61C2511yescharacteristic
224C22×D61—232no—
225C22×D61—28no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—28no—
231C22×D61—28no—
232C22×D61H(5,𝔽2)14yes—
233C22×D612−1+414yes—
234C22×D61C2512yes—
235C22×D61C2512yes—
236Q8×C61—216no—
237D4×C61—232no—
238D4×C61—216no—
239D4×C61—416no—
240C2×C4×C61—28no—
241C2×C4×C61—28no—
242C2×C4×C61—232no—
243C2×C4×C61—48no—
244C2×C4×C61C2×#4118yes—
245C2×C4×C61C2×#41116yescomplemented
246C2×C4×C61C2518yes—
247C2×C4×C61C2518yes—
248C2×C4×C61C2514yes—
249C22⋊D61—232no—
250C22⋊D61—232no—
251C2×Dic61—28no—
252C2×Dic61—24no—
253C2×Dic61—28no—
254C2×Dic61—28no—
255C2×Dic61—28no—
256C2×Dic61—48no—
257C2×Dic61—432no—
258C2×Dic61C2518yes—
259C2×Dic61C2518yes—
260C2×Dic61C2518yes—
261C2×Dic61C2514yes—
262C4×D61—232no—
263C4×D61—216no—
264C4×D61—216no—
265C4×D61—416no—
266Dic122—416no—
267C2×#3492—611nomaximal, Sylow
268C22×C61—22no—
269C22×C61—22no—
270C22×C61—28no—
271C22×C61—28no—
272C22×C61—28no—
273C22×C61—28no—
274C22×C61—416no—
275C22×C61C22×#4114yes—
276C22×C61C22×#4112yes—
277C22×C61C2×H(5,𝔽2)12yes—
278C22×C61C2×2−1+412yes—
279C22×C61C2611yescharacteristic
280C2×D61—232no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—22no—
285C2×D61—28no—
286C2×D61—28no—
287C2×D61C2×H(5,𝔽2)14yes—
288C2×D61C2×2−1+414yes—
289C4×C61—28no—
290C4×C61—28no—
291C4×C61C22×#4118yes—
292Dic61—24no—
293Dic61—28no—
294Dic61—28no—
295Dic61—28no—
296#3492—6132no—
297C2×#554—614no—
298C2×#553—614no—
299C2×#552—614no—
300C2×#547—614no—
301C2×#544—611no—
302C2×#539—614no—
303C2×#537D6111yescharacteristic, complemented
304C2×#525—612no—
305C2×#525—614no—
306C2×#520—612no—
307C23×C4⋊C4—611no—
308C2×C61—24no—
309C2×C61—48no—
310C2×C61#349214yescomplemented
311C2×C61C23×#4111yescharacteristic
312C2×C61C22×H(5,𝔽2)11yescharacteristic
313C2×C61C22×2−1+411yescharacteristic
314D61—24no—
315D61—24no—
316C22×#41—614no—
317C22×#41C2×D6112yes—
318C24×C4—611no—
319C24×C4—612no—
320#554—6164no—
321#553—6164no—
322#552—6164no—
323#547—6164no—
324#544—6116no—
325#539—6164no—
326#537C2×D61116yescomplemented
327C2×C4⋊Q8—614no—
328C2×#126—618no—
329C2×#125—618no—
330C2×#125—614no—
331#525—6132no—
332#525—6164no—
333C2×#123—614no—
334C2×#123—618no—
335C2×#123—618no—
336C2×C22⋊Q8—612no—
337C2×C22⋊Q8—612no—
338C2×C22⋊Q8—614no—
339C2×C22⋊Q8—614no—
340C2×C22⋊Q8—618no—
341C2×C22⋊Q8C2×D6114yescomplemented
342C2×#121—612no—
343C2×#121—614no—
344C2×#121—612no—
345C2×#121C2×D6114yes—
346C2×#121C2×D6114yescomplemented
347C2×#120—614no—
348#520—6132no—
349C2×C4×Q8—614no—
350C2×C4×D4—618no—
351C2×C4×D4—614no—
352C2×C4×D4—614no—
353C2×C4×D4—614no—
354C2×C4×D4—618no—
355C2×#117—612no—
356C2×#117—614no—
357C2×#117—614no—
358C22×C4⋊C4—612no—
359C22×C4⋊C4—611no—
360C22×C4⋊C4—614no—
361C22×C4⋊C4—614no—
362C22×C4⋊C4—614no—
363C22×C4⋊C4—6116no—
364C22×C4⋊C4C2×D6111yescharacteristic
365C61C2×#349211yescharacteristic, complemented, Sylow
366C25—611no—
367C2×#41—216no—
368C2×#41—6132no—
369C2×#41—24416no—
370C2×#41C22×D61116yes—
371C22×Q8—614no—
372C22×Q8C22×D6112yes—
373C22×D4—614no—
374C22×D4—614no—
375C22×D4C22×D6114yes—
376C22×D4C22×D6112yes—
377C22×D4C22×D6114yes—
378C23×C4—611no—
379C23×C4—612no—
380C23×C4—618no—
381C23×C4—614no—
382C23×C4—612no—
383C23×C4—612no—
384C23×C4—618no—
385C23×C4—614no—
386C23×C4—6116no—
387C23×C4—1228no—
388C23×C4—2448no—
389C23×C4C22×D6114yes—
390C23×C4C22×D6112yes—
391C23×C4C22×D6111yescharacteristic
392C4⋊Q8—6132no—
393#126—6164no—
394#125—6164no—
395#125—6132no—
396#123—6132no—
397#123—6164no—
398#123—6164no—
399C22⋊Q8—6116no—
400C22⋊Q8—6116no—
401C22⋊Q8—6132no—
402C22⋊Q8—6132no—
403C22⋊Q8—6164no—
404C22⋊Q8C22×D61132yescomplemented
405#121—6116no—
406#121—6132no—
407#121—6116no—
408#121C22×D61132yes—
409#121C22×D61132yescomplemented
410#120—6132no—
411C4×Q8—6132no—
412C4×D4—6164no—
413C4×D4—6132no—
414C4×D4—6132no—
415C4×D4—6132no—
416C4×D4—6164no—
417#117—6116no—
418#117—6132no—
419#117—6132no—
420C2×C4⋊C4—614no—
421C2×C4⋊C4—618no—
422C2×C4⋊C4—618no—
423C2×C4⋊C4—6116no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—618no—
426C2×C4⋊C4—6132no—
427C2×C4⋊C4—612no—
428C2×C4⋊C4—614no—
429C2×C4⋊C4—614no—
430C2×C4⋊C4—618no—
431C2×C4⋊C4—614no—
432C2×C4⋊C4—6132no—
433C2×C4⋊C4—6132no—
434C2×C4⋊C4C22×D6114yes—
435C2×C4⋊C4C22×D6118yes—
436C2×C4⋊C4C22×D6114yes—
437C2×C22⋊C4—614no—
438C2×C22⋊C4—618no—
439C2×C22⋊C4—614no—
440C2×C22⋊C4—614no—
441C2×C22⋊C4—614no—
442C2×C22⋊C4—614no—
443C2×C22⋊C4C22×D6118yes—
444C2×C42—618no—
445C2×C42—614no—
446C24—611no—
447C24—611no—
448C24—614no—
449C24—1224no—
450C24—1224no—
451C24—1224no—
452C24C23×D6114yes—
453C24C23×D6111yescharacteristic
454#41—264no—
455#41—24464no—
456C2×Q8—28no—
457C2×Q8—6116no—
458C2×Q8—2448no—
459C2×Q8C23×D6118yes—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—48no—
463C2×D4—6116no—
464C2×D4—6116no—
465C2×D4—12216no—
466C2×D4—12216no—
467C2×D4C23×D61116yes—
468C2×D4C23×D6118yes—
469C2×D4C23×D61116yes—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—614no—
473C22×C4—614no—
474C22×C4—614no—
475C22×C4—614no—
476C22×C4—612no—
477C22×C4—618no—
478C22×C4—6132no—
479C22×C4—6116no—
480C22×C4—618no—
481C22×C4—618no—
482C22×C4—6116no—
483C22×C4—12216no—
484C22×C4—1224no—
485C22×C4—1228no—
486C22×C4—1228no—
487C22×C4—12232no—
488C22×C4—2448no—
489C22×C4—24432no—
490C22×C4—2444no—
491C22×C4—24416no—
492C22×C4#1175822418yescomplemented
493C22×C4C23×D61116yes—
494C22×C4C23×D6114yes—
495C22×C4C23×D6114yes—
496C22×C4C23×D6118yes—
497C22×C4C23×D6112yes—
498C22×C4C23×D6114yes—
499C4⋊C4—6116no—
500C4⋊C4—6132no—
501C4⋊C4—618no—
502C4⋊C4—6132no—
503C4⋊C4—6116no—
504C4⋊C4—6132no—
505C4⋊C4—6116no—
506C4⋊C4—618no—
507C4⋊C4—6116no—
508C4⋊C4C23×D61116yes—
509C4⋊C4C23×D61116yes—
510C22⋊C4—6116no—
511C22⋊C4—6132no—
512C22⋊C4—6116no—
513C22⋊C4—6116no—
514C22⋊C4—6116no—
515C22⋊C4—6116no—
516C22⋊C4C23×D61132yes—
517C42—6132no—
518C42—6116no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—614no—
525C23—614no—
526C23—612no—
527C23—612no—
528C23—1228no—
529C23—1228no—
530C23—12232no—
531C23—1228no—
532C23—1222no—
533C23—1222no—
534C23—1222no—
535C23—1222no—
536C23C2×#1175822414yes—
537C23C24×D6118yes—
538C23C24×D6112yes—
539C23C24×D6111yescharacteristic
540Q8—216no—
541Q8—24416no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—12232no—
546D4—12232no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—618no—
552C2×C4—618no—
553C2×C4—618no—
554C2×C4—614no—
555C2×C4—12232no—
556C2×C4—1228no—
557C2×C4—1224no—
558C2×C4—1228no—
559C2×C4—1228no—
560C2×C4—1228no—
561C2×C4—12216no—
562C2×C4—12216no—
563C2×C4—24416no—
564C2×C4—2448no—
565C2×C4—24432no—
566C2×C4C2×#1175822418yes—
567C2×C4C2×#11758224116yescomplemented
568C2×C4C24×D6118yes—
569C2×C4C24×D6118yes—
570C2×C4C24×D6114yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—614no—
579C22—614no—
580C22—12232no—
581C22—1228no—
582C22—1228no—
583C22—1222no—
584C22—1222no—
585C22—1222no—
586C22—1222no—
587C22C22×#1175822414yes—
588C22C22×#1175822412yes—
589C22H(5,𝔽2)×D6112yes—
590C222−1+4×D6112yes—
591C22C25×D6111yescharacteristic
592C4—28no—
593C4—28no—
594C4—1224no—
595C4—1228no—
596C4—1228no—
597C4—1228no—
598C4C22×#1175822418yes—
599C2—24no—
600C2—48no—
601C2—1224no—
602C2—1224no—
603C2#2145952614yesdirect product factor
604C2C23×#1175822411yescharacteristic
605C2C2×H(5,𝔽2)×D6111yescharacteristic
606C2C2×2−1+4×D6111yescharacteristic
subgroupG/H
centerC23C24×D61
commutator, cosocleC22×C61C26
FrattiniC22C25×D61
FittingC2×C61×#537C2
socleC23×C61C25
hypercenterC2×#537D61
hypocenterC61C2×#3492
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21459526C2direct (N × H)
2. C2×#16840470C2semidirect
3. C2×#16840459C2semidirect
4. C2×#16840446C2semidirect
5. C2×#16840412C2semidirect
6. C2×#16840393C2semidirect
7. C2×#16840357C2semidirect
8. C2×#16840272C2semidirect
9. C2×#16840269C2semidirect
10. C2×#16840245C2semidirect
11. C22×C4⋊C4×D61C2semidirect
12. C2×C61×#537C2semidirect
16. #16840470C22semidirect
17. #16840459C22semidirect
18. #16840446C22semidirect
19. #16840412C22semidirect
20. #16840393C22semidirect
21. #16840357C22semidirect
22. C2×#14519776C22semidirect
23. C2×#14519770C22semidirect
24. C2×#14519768C22semidirect
25. C2×#14519767C22semidirect
26. #16840272C22semidirect
27. #16840269C22semidirect
30. C2×#14519758C22semidirect
33. C2×#14519754C22semidirect
40. #16840245C22semidirect
41. C2×Q8×Dic61C22semidirect
43. C2×D4×Dic61C22semidirect
48. C2×#14519736C22semidirect
51. C22×C4⋊Dic61C22semidirect
55. C2×C4⋊C4×D61C22semidirect
57. C61×#537C22semidirect
58. C2×C22⋊Q8×C61C22semidirect
60. C2×#121×C61C22semidirect
79. #14519776C23semidirect
80. #14519770C23semidirect
81. #14519768C23semidirect
82. #14519767C23semidirect
85. #14519758C23semidirect
88. #14519754C23semidirect
95. Q8×Dic61C23semidirect
97. D4×Dic61C23semidirect
102. #14519736C23semidirect
110. C2×C4⋊Dic61C23semidirect
135. C22⋊Q8×C61C23semidirect
137. #121×C61C23semidirect
161. C22×C4×C61#41semidirect
245. C2×C4×C61C2×#41semidirect
303. C2×#537D61semidirect
310. C2×C61#3492semidirect
326. #537C2×D61semidirect
341. C2×C22⋊Q8C2×D61semidirect
346. C2×#121C2×D61semidirect
365. C61C2×#3492semidirect
404. C22⋊Q8C22×D61semidirect
409. #121C22×D61semidirect
492. C22×C4#11758224semidirect
567. C2×C4C2×#11758224semidirect
603. C2#21459526direct (N × H)

Maximal subgroups

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

subgroupindex
#214595262
C2×#168404702
C2×#168404592
C2×#168404462
C2×#168404122
C2×#168403932
C2×#168403572
C2×#168402722
C2×#168402692
C2×#168402452
C22×C4⋊C4×D612
C2×C61×#5372
C2×#349261