Finite Group Databasebeta

← C2×#20257095 #27124218

Subgroups — C2×#20257095 #27124218

Order 12032.

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#20257095C2132yesmaximal, direct product factor
2C2×#15689831C214yesmaximal, complemented
3C2×#15689820C214yesmaximal, complemented
4C2×#15689807C214yesmaximal, complemented
5C2×#15689773C214yesmaximal, complemented
6C2×#15689754C211yesmaximal, characteristic, complemented
7C2×#15689718C214yesmaximal, complemented
8C2×#15689633C214yesmaximal, complemented
9C2×#15689630C212yesmaximal, complemented
10C2×#15689606C212yesmaximal, complemented
11C22×C4⋊C4×D47C211yesmaximal, characteristic, complemented
12C2×C47×#537C211yesmaximal, characteristic, complemented
13C22×#10625352C2214yes—
14C23×C4×D47C2211yescharacteristic
15C23×C4×D47C2212yes—
16#15689831C22164yescomplemented
17#15689820C22164yescomplemented
18#15689807C22164yescomplemented
19#15689773C22164yescomplemented
20#15689754C22116yescomplemented
21#15689718C22164yescomplemented
22C2×#13094023C2214yescomplemented
23C2×#13094017C2218yescomplemented
24C2×#13094015C2214yescomplemented
25C2×#13094014C2218yescomplemented
26#15689633C22164yescomplemented
27#15689630C22132yescomplemented
28C2×#13094006C2218yes—
29C2×#13094005C2214yes—
30C2×#13094005C2218yescomplemented
31C2×#13094004C2214yes—
32C2×#13094003C2214yes—
33C2×#13094001C2218yescomplemented
34C2×#13094000C2212yes—
35C2×#13094000C2212yes—
36C2×#13093995C2212yes—
37C2×#13093995C2212yes—
38C2×#13093995C2214yes—
39C2×#13093992C2214yes—
40#15689606C22132yescomplemented
41C2×Q8×Dic47C2214yescomplemented
42C2×D4×Dic47C2218yes—
43C2×D4×Dic47C2214yescomplemented
44C2×C4×C22⋊D47C2214yes—
45C2×C4×C22⋊D47C2214yes—
46C2×(C22⋊D47)⋊C4C2218yes—
47C2×#13093983C2212yes—
48C2×#13093983C2214yescomplemented
49C2×#13093982C2214yes—
50C22×C4⋊Dic47C2211yescharacteristic
51C22×C4⋊Dic47C2214yescomplemented
52C22×Dic47⋊C4C2212yes—
53C2×C4⋊C4×D47C2214yes—
54C2×C4⋊C4×D47C2214yes—
55C2×C4⋊C4×D47C22116yescomplemented
56C22×#41×C47C2212yes—
57C47×#537C22116yescomplemented
58C2×C22⋊Q8×C47C2214yescomplemented
59C2×#121×C47C2214yes—
60C2×#121×C47C2214yescomplemented
61C22×C4⋊C4×C47C2211yescharacteristic
62C24×D47C2311yescharacteristic
63C2×#10625352—416no—
64C2×#10625352C23132yes—
65C22×Dic94C2314yes—
66C22×C22⋊D47C2314yes—
67C22×C22⋊D47C2314yes—
68C23×Dic47C2311yescharacteristic
69C23×Dic47C2312yes—
70C23×Dic47C2318yes—
71C22×C4×D47—28no—
72C22×C4×D47—48no—
73C22×C4×D47C2314yes—
74C22×C4×D47C2312yes—
75C22×C4×D47C2312yes—
76C22×C4×D47C2318yes—
77C22×C4×D47C2314yes—
78C22×C4×D47C23116yes—
79#13094023C23132yescomplemented
80#13094017C23164yescomplemented
81#13094015C23132yescomplemented
82#13094014C23164yescomplemented
83#13094006C23164yes—
84#13094005C23132yes—
85#13094005C23164yescomplemented
86#13094004C23132yes—
87#13094003C23132yes—
88#13094001C23164yescomplemented
89#13094000C23116yes—
90#13094000C23116yes—
91#13093995C23116yes—
92#13093995C23116yes—
93#13093995C23132yes—
94#13093992C23132yes—
95Q8×Dic47C23132yescomplemented
96D4×Dic47C23164yes—
97D4×Dic47C23132yescomplemented
98C4×C22⋊D47C23132yes—
99C4×C22⋊D47C23132yes—
100(C22⋊D47)⋊C4C23164yes—
101#13093983C23116yes—
102#13093983C23132yescomplemented
103#13093982C23132yes—
104C2×C4⋊Dic47C2314yes—
105C2×C4⋊Dic47C2318yes—
106C2×C4⋊Dic47C2312yes—
107C2×C4⋊Dic47C2318yes—
108C2×C4⋊Dic47C2314yes—
109C2×C4⋊Dic47C2312yes—
110C2×C4⋊Dic47C23132yescomplemented
111C2×Dic47⋊C4C2318yes—
112C2×Dic47⋊C4C23116yes—
113C2×Dic47⋊C4C2318yes—
114C2×Dic47⋊C4C2314yes—
115C2×Dic47⋊C4C2314yes—
116C4⋊C4×D47C23132yes—
117C4⋊C4×D47C23132yes—
118C2×C22⋊Dic47C2314yes—
119C2×C22⋊Dic47C2318yes—
120C2×C22⋊Dic47C2314yes—
121C2×C22⋊Dic47C2314yes—
122C2×(C2×D47)⋊C4C2314yes—
123C2×(C2×D47)⋊C4C2314yes—
124C2×C4×Dic47C2318yes—
125C2×C4×Dic47C2314yes—
126C2×#41×C47—216no—
127C2×#41×C47C23116yes—
128C22×Q8×C47C2312yes—
129C22×D4×C47C2314yes—
130C22×D4×C47C2312yes—
131C22×D4×C47C2314yes—
132C23×C4×C47C2314yes—
133C23×C4×C47C2312yes—
134C23×C4×C47C2311yescharacteristic
135C22⋊Q8×C47C23132yescomplemented
136#121×C47C23132yes—
137#121×C47C23132yescomplemented
138C2×C4⋊C4×C47C2318yes—
139C2×C4⋊C4×C47C2314yes—
140C2×C4⋊C4×C47C2314yes—
141C2×C22⋊C4×C47C2318yes—
142C24×C47C2414yes—
143C24×C47C2411yescharacteristic
144C23×D47—24no—
145C23×D47—24no—
146C23×D47—24no—
147C23×D47C2414yes—
148C23×D47C2411yescharacteristic
149C23×D47C2411yescharacteristic
150#41×C47—264no—
151C2×Q8×C47—28no—
152C2×Q8×C47C2418yes—
153C2×D4×C47—216no—
154C2×D4×C47—28no—
155C2×D4×C47—48no—
156C2×D4×C47C24116yes—
157C2×D4×C47C2418yes—
158C2×D4×C47C24116yes—
159C22×C4×C47—216no—
160C22×C4×C47—44no—
161C22×C4×C47#4118yescomplemented
162C22×C4×C47C2414yes—
163C22×C4×C47C2414yes—
164C22×C4×C47C24116yes—
165C22×C4×C47C2418yes—
166C22×C4×C47C2412yes—
167C22×C4×C47C2414yes—
168C2×C22⋊D47—216no—
169C2×C22⋊D47—216no—
170C2×C22⋊D47C24116yes—
171C2×C22⋊D47C24116yes—
172C22×Dic47—24no—
173C22×Dic47—44no—
174C22×Dic47—416no—
175C22×Dic47C2414yes—
176C22×Dic47C2414yes—
177C22×Dic47C2414yes—
178C22×Dic47C2412yes—
179C22×Dic47C2414yes—
180C22×Dic47C2418yes—
181C22×Dic47C24132yes—
182#10625352—464no—
183C2×C4×D47—216no—
184C2×C4×D47—28no—
185C2×C4×D47—28no—
186C2×C4×D47—232no—
187C2×C4×D47—432no—
188C2×C4×D47—48no—
189C2×C4×D47C24116yes—
190C2×C4×D47C2418yes—
191C2×C4×D47C2418yes—
192C2×C4×D47C24116yes—
193C2×Dic94—48no—
194C2×Dic94C24116yes—
195C4⋊C4×C47C24116yes—
196C4⋊C4×C47C24116yes—
197C22⋊C4×C47C24132yes—
198C22⋊Dic47C24116yes—
199C22⋊Dic47C24132yes—
200C22⋊Dic47C24116yes—
201C22⋊Dic47C24116yes—
202(C2×D47)⋊C4C24116yes—
203(C2×D47)⋊C4C24116yes—
204C4⋊Dic47C24116yes—
205C4⋊Dic47C2418yes—
206C4⋊Dic47C24132yes—
207C4⋊Dic47C24116yes—
208C4⋊Dic47C2418yes—
209Dic47⋊C4C24132yes—
210Dic47⋊C4C24132yes—
211Dic47⋊C4C24116yes—
212Dic47⋊C4C24116yes—
213C4×Dic47C24132yes—
214C4×Dic47C24116yes—
215C23×C47—22no—
216C23×C47—22no—
217C23×C47—28no—
218C23×C47—28no—
219C23×C47—28no—
220C23×C47C2×#4114yes—
221C23×C47C2518yes—
222C23×C47C2512yes—
223C23×C47C2511yescharacteristic
224C22×D47—232no—
225C22×D47—28no—
226C22×D47—22no—
227C22×D47—22no—
228C22×D47—22no—
229C22×D47—22no—
230C22×D47—28no—
231C22×D47—28no—
232C22×D47H(5,𝔽2)14yes—
233C22×D472−1+414yes—
234C22×D47C2512yes—
235C22×D47C2512yes—
236Q8×C47—216no—
237D4×C47—232no—
238D4×C47—216no—
239D4×C47—416no—
240C2×C4×C47—28no—
241C2×C4×C47—28no—
242C2×C4×C47—232no—
243C2×C4×C47—48no—
244C2×C4×C47C2×#4118yes—
245C2×C4×C47C2×#41116yescomplemented
246C2×C4×C47C2518yes—
247C2×C4×C47C2518yes—
248C2×C4×C47C2514yes—
249C22⋊D47—232no—
250C22⋊D47—232no—
251C2×Dic47—28no—
252C2×Dic47—24no—
253C2×Dic47—28no—
254C2×Dic47—28no—
255C2×Dic47—28no—
256C2×Dic47—48no—
257C2×Dic47—432no—
258C2×Dic47C2518yes—
259C2×Dic47C2518yes—
260C2×Dic47C2518yes—
261C2×Dic47C2514yes—
262C4×D47—232no—
263C4×D47—216no—
264C4×D47—216no—
265C4×D47—416no—
266Dic94—416no—
267C2×#3492—471nomaximal, Sylow
268C22×C47—22no—
269C22×C47—22no—
270C22×C47—28no—
271C22×C47—28no—
272C22×C47—28no—
273C22×C47—28no—
274C22×C47—416no—
275C22×C47C22×#4114yes—
276C22×C47C22×#4112yes—
277C22×C47C2×H(5,𝔽2)12yes—
278C22×C47C2×2−1+412yes—
279C22×C47C2611yescharacteristic
280C2×D47—232no—
281C2×D47—22no—
282C2×D47—22no—
283C2×D47—22no—
284C2×D47—22no—
285C2×D47—28no—
286C2×D47—28no—
287C2×D47C2×H(5,𝔽2)14yes—
288C2×D47C2×2−1+414yes—
289C4×C47—28no—
290C4×C47—28no—
291C4×C47C22×#4118yes—
292Dic47—24no—
293Dic47—28no—
294Dic47—28no—
295Dic47—28no—
296#3492—4732no—
297C2×#554—474no—
298C2×#553—474no—
299C2×#552—474no—
300C2×#547—474no—
301C2×#544—471no—
302C2×#539—474no—
303C2×#537D4711yescharacteristic, complemented
304C2×#525—472no—
305C2×#525—474no—
306C2×#520—472no—
307C23×C4⋊C4—471no—
308C2×C47—24no—
309C2×C47—48no—
310C2×C47#349214yescomplemented
311C2×C47C23×#4111yescharacteristic
312C2×C47C22×H(5,𝔽2)11yescharacteristic
313C2×C47C22×2−1+411yescharacteristic
314D47—24no—
315D47—24no—
316C22×#41—474no—
317C22×#41C2×D4712yes—
318C24×C4—471no—
319C24×C4—472no—
320#554—4764no—
321#553—4764no—
322#552—4764no—
323#547—4764no—
324#544—4716no—
325#539—4764no—
326#537C2×D47116yescomplemented
327C2×C4⋊Q8—474no—
328C2×#126—478no—
329C2×#125—478no—
330C2×#125—474no—
331#525—4732no—
332#525—4764no—
333C2×#123—474no—
334C2×#123—478no—
335C2×#123—478no—
336C2×C22⋊Q8—472no—
337C2×C22⋊Q8—472no—
338C2×C22⋊Q8—474no—
339C2×C22⋊Q8—474no—
340C2×C22⋊Q8—478no—
341C2×C22⋊Q8C2×D4714yescomplemented
342C2×#121—472no—
343C2×#121—474no—
344C2×#121—472no—
345C2×#121C2×D4714yes—
346C2×#121C2×D4714yescomplemented
347C2×#120—474no—
348#520—4732no—
349C2×C4×Q8—474no—
350C2×C4×D4—478no—
351C2×C4×D4—474no—
352C2×C4×D4—474no—
353C2×C4×D4—474no—
354C2×C4×D4—478no—
355C2×#117—472no—
356C2×#117—474no—
357C2×#117—474no—
358C22×C4⋊C4—471no—
359C22×C4⋊C4—472no—
360C22×C4⋊C4—474no—
361C22×C4⋊C4—474no—
362C22×C4⋊C4—474no—
363C22×C4⋊C4—4716no—
364C22×C4⋊C4C2×D4711yescharacteristic
365C47C2×#349211yescharacteristic, complemented, Sylow
366C25—471no—
367C2×#41—216no—
368C2×#41—4732no—
369C2×#41—18816no—
370C2×#41C22×D47116yes—
371C22×Q8—474no—
372C22×Q8C22×D4712yes—
373C22×D4—474no—
374C22×D4—474no—
375C22×D4C22×D4714yes—
376C22×D4C22×D4712yes—
377C22×D4C22×D4714yes—
378C23×C4—471no—
379C23×C4—472no—
380C23×C4—478no—
381C23×C4—474no—
382C23×C4—472no—
383C23×C4—472no—
384C23×C4—478no—
385C23×C4—474no—
386C23×C4—4716no—
387C23×C4—948no—
388C23×C4—1888no—
389C23×C4C22×D4714yes—
390C23×C4C22×D4712yes—
391C23×C4C22×D4711yescharacteristic
392C4⋊Q8—4732no—
393#126—4764no—
394#125—4764no—
395#125—4732no—
396#123—4732no—
397#123—4764no—
398#123—4764no—
399C22⋊Q8—4716no—
400C22⋊Q8—4716no—
401C22⋊Q8—4732no—
402C22⋊Q8—4732no—
403C22⋊Q8—4764no—
404C22⋊Q8C22×D47132yescomplemented
405#121—4716no—
406#121—4732no—
407#121—4716no—
408#121C22×D47132yes—
409#121C22×D47132yescomplemented
410#120—4732no—
411C4×Q8—4732no—
412C4×D4—4764no—
413C4×D4—4732no—
414C4×D4—4732no—
415C4×D4—4732no—
416C4×D4—4764no—
417#117—4716no—
418#117—4732no—
419#117—4732no—
420C2×C4⋊C4—474no—
421C2×C4⋊C4—478no—
422C2×C4⋊C4—478no—
423C2×C4⋊C4—4716no—
424C2×C4⋊C4—472no—
425C2×C4⋊C4—478no—
426C2×C4⋊C4—4732no—
427C2×C4⋊C4—472no—
428C2×C4⋊C4—474no—
429C2×C4⋊C4—474no—
430C2×C4⋊C4—478no—
431C2×C4⋊C4—474no—
432C2×C4⋊C4—4732no—
433C2×C4⋊C4—4732no—
434C2×C4⋊C4C22×D4714yes—
435C2×C4⋊C4C22×D4718yes—
436C2×C4⋊C4C22×D4714yes—
437C2×C22⋊C4—474no—
438C2×C22⋊C4—478no—
439C2×C22⋊C4—474no—
440C2×C22⋊C4—474no—
441C2×C22⋊C4—474no—
442C2×C22⋊C4—474no—
443C2×C22⋊C4C22×D4718yes—
444C2×C42—478no—
445C2×C42—474no—
446C24—471no—
447C24—471no—
448C24—474no—
449C24—944no—
450C24—944no—
451C24—944no—
452C24C23×D4714yes—
453C24C23×D4711yescharacteristic
454#41—264no—
455#41—18864no—
456C2×Q8—28no—
457C2×Q8—4716no—
458C2×Q8—1888no—
459C2×Q8C23×D4718yes—
460C2×D4—216no—
461C2×D4—28no—
462C2×D4—48no—
463C2×D4—4716no—
464C2×D4—4716no—
465C2×D4—9416no—
466C2×D4—9416no—
467C2×D4C23×D47116yes—
468C2×D4C23×D4718yes—
469C2×D4C23×D47116yes—
470C22×C4—216no—
471C22×C4—44no—
472C22×C4—474no—
473C22×C4—474no—
474C22×C4—474no—
475C22×C4—474no—
476C22×C4—472no—
477C22×C4—478no—
478C22×C4—4732no—
479C22×C4—4716no—
480C22×C4—478no—
481C22×C4—478no—
482C22×C4—4716no—
483C22×C4—9416no—
484C22×C4—944no—
485C22×C4—948no—
486C22×C4—948no—
487C22×C4—9432no—
488C22×C4—1888no—
489C22×C4—18832no—
490C22×C4—1884no—
491C22×C4—18816no—
492C22×C4#1062535218yescomplemented
493C22×C4C23×D47116yes—
494C22×C4C23×D4714yes—
495C22×C4C23×D4714yes—
496C22×C4C23×D4718yes—
497C22×C4C23×D4712yes—
498C22×C4C23×D4714yes—
499C4⋊C4—4732no—
500C4⋊C4—4716no—
501C4⋊C4—4732no—
502C4⋊C4—4716no—
503C4⋊C4—4716no—
504C4⋊C4—478no—
505C4⋊C4—4732no—
506C4⋊C4—478no—
507C4⋊C4—4716no—
508C4⋊C4C23×D47116yes—
509C4⋊C4C23×D47116yes—
510C22⋊C4—4716no—
511C22⋊C4—4732no—
512C22⋊C4—4716no—
513C22⋊C4—4716no—
514C22⋊C4—4716no—
515C22⋊C4—4716no—
516C22⋊C4C23×D47132yes—
517C42—4732no—
518C42—4716no—
519C23—22no—
520C23—22no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—474no—
525C23—474no—
526C23—472no—
527C23—472no—
528C23—948no—
529C23—948no—
530C23—9432no—
531C23—948no—
532C23—942no—
533C23—942no—
534C23—942no—
535C23—942no—
536C23C2×#1062535214yes—
537C23C24×D4718yes—
538C23C24×D4712yes—
539C23C24×D4711yescharacteristic
540Q8—216no—
541Q8—18816no—
542D4—232no—
543D4—216no—
544D4—416no—
545D4—9432no—
546D4—9432no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—232no—
550C2×C4—48no—
551C2×C4—478no—
552C2×C4—478no—
553C2×C4—478no—
554C2×C4—474no—
555C2×C4—9432no—
556C2×C4—948no—
557C2×C4—944no—
558C2×C4—948no—
559C2×C4—948no—
560C2×C4—948no—
561C2×C4—9416no—
562C2×C4—9416no—
563C2×C4—18816no—
564C2×C4—1888no—
565C2×C4—18832no—
566C2×C4C2×#1062535218yes—
567C2×C4C2×#10625352116yescomplemented
568C2×C4C24×D4718yes—
569C2×C4C24×D4718yes—
570C2×C4C24×D4714yes—
571C22—28no—
572C22—22no—
573C22—22no—
574C22—28no—
575C22—28no—
576C22—28no—
577C22—416no—
578C22—474no—
579C22—474no—
580C22—9432no—
581C22—948no—
582C22—948no—
583C22—942no—
584C22—942no—
585C22—942no—
586C22—942no—
587C22C22×#1062535214yes—
588C22C22×#1062535212yes—
589C22H(5,𝔽2)×D4712yes—
590C222−1+4×D4712yes—
591C22C25×D4711yescharacteristic
592C4—28no—
593C4—28no—
594C4—944no—
595C4—948no—
596C4—948no—
597C4—948no—
598C4C22×#1062535218yes—
599C2—24no—
600C2—48no—
601C2—944no—
602C2—944no—
603C2#2025709514yesdirect product factor
604C2C23×#1062535211yescharacteristic
605C2C2×H(5,𝔽2)×D4711yescharacteristic
606C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#537C2
socleC23×C47C25
hypercenterC2×#537D47
hypocenterC47C2×#3492
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20257095C2direct (N × H)
2. C2×#15689831C2semidirect
3. C2×#15689820C2semidirect
4. C2×#15689807C2semidirect
5. C2×#15689773C2semidirect
6. C2×#15689754C2semidirect
7. C2×#15689718C2semidirect
8. C2×#15689633C2semidirect
9. C2×#15689630C2semidirect
10. C2×#15689606C2semidirect
11. C22×C4⋊C4×D47C2semidirect
12. C2×C47×#537C2semidirect
16. #15689831C22semidirect
17. #15689820C22semidirect
18. #15689807C22semidirect
19. #15689773C22semidirect
20. #15689754C22semidirect
21. #15689718C22semidirect
22. C2×#13094023C22semidirect
23. C2×#13094017C22semidirect
24. C2×#13094015C22semidirect
25. C2×#13094014C22semidirect
26. #15689633C22semidirect
27. #15689630C22semidirect
30. C2×#13094005C22semidirect
33. C2×#13094001C22semidirect
40. #15689606C22semidirect
41. C2×Q8×Dic47C22semidirect
43. C2×D4×Dic47C22semidirect
48. C2×#13093983C22semidirect
51. C22×C4⋊Dic47C22semidirect
55. C2×C4⋊C4×D47C22semidirect
57. C47×#537C22semidirect
58. C2×C22⋊Q8×C47C22semidirect
60. C2×#121×C47C22semidirect
79. #13094023C23semidirect
80. #13094017C23semidirect
81. #13094015C23semidirect
82. #13094014C23semidirect
85. #13094005C23semidirect
88. #13094001C23semidirect
95. Q8×Dic47C23semidirect
97. D4×Dic47C23semidirect
102. #13093983C23semidirect
110. C2×C4⋊Dic47C23semidirect
135. C22⋊Q8×C47C23semidirect
137. #121×C47C23semidirect
161. C22×C4×C47#41semidirect
245. C2×C4×C47C2×#41semidirect
303. C2×#537D47semidirect
310. C2×C47#3492semidirect
326. #537C2×D47semidirect
341. C2×C22⋊Q8C2×D47semidirect
346. C2×#121C2×D47semidirect
365. C47C2×#3492semidirect
404. C22⋊Q8C22×D47semidirect
409. #121C22×D47semidirect
492. C22×C4#10625352semidirect
567. C2×C4C2×#10625352semidirect
603. C2#20257095direct (N × H)

Maximal subgroups

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

subgroupindex
#202570952
C2×#156898312
C2×#156898202
C2×#156898072
C2×#156897732
C2×#156897542
C2×#156897182
C2×#156896332
C2×#156896302
C2×#156896062
C22×C4⋊C4×D472
C2×C47×#5372
C2×#349247