Finite Group Databasebeta

← C2×#20257097 #27124219

Subgroups — C2×#20257097 #27124219

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#20257097C2132yesmaximal, direct product factor
2C2×#15689828C214yesmaximal, complemented
3C2×#15689821C214yesmaximal, complemented
4C2×#15689808C214yesmaximal, complemented
5C2×#15689774C214yesmaximal, complemented
6C2×#15689754C211yesmaximal, characteristic, complemented
7C2×#15689720C214yesmaximal, complemented
8C2×#15689704C211yesmaximal, characteristic, complemented
9C2×#15689634C212yesmaximal, complemented
10C2×#15689634C214yesmaximal, complemented
11C2×#15689604C212yesmaximal, complemented
12C23×C4⋊C4×C47C211yesmaximal, characteristic, complemented
13C22×#10625350C2212yes—
14C22×#10625350C2214yes—
15#15689828C22164yescomplemented
16#15689821C22164yescomplemented
17#15689808C22164yescomplemented
18#15689774C22164yescomplemented
19#15689754C22116yescomplemented
20#15689720C22164yescomplemented
21#15689704C22116yescomplemented
22C2×#13094023C2214yescomplemented
23C2×#13094019C2218yescomplemented
24C2×#13094015C2214yescomplemented
25C2×#13094014C2218yescomplemented
26#15689634C22132yescomplemented
27#15689634C22164yescomplemented
28C2×#13094009C2218yescomplemented
29C2×#13094007C2218yes—
30C2×#13094007C2214yes—
31C2×#13094004C2214yescomplemented
32C2×#13094003C2214yes—
33C2×#13094003C2212yes—
34C2×#13094003C2212yes—
35C2×#13094003C2214yes—
36C2×#13094002C2218yescomplemented
37C2×#13093999C2214yescomplemented
38C2×#13093998C2212yes—
39C2×#13093998C2212yes—
40C2×#13093998C2214yes—
41C2×#13093998C2214yes—
42C2×#13093993C2214yes—
43#15689604C22132yescomplemented
44C2×#13093989C2214yescomplemented
45C2×#13093986C2214yescomplemented
46C2×C4×C22⋊D47C2218yes—
47C2×C4×C22⋊D47C2214yes—
48C2×C4×C22⋊D47C2214yes—
49C2×C4×C22⋊D47C2218yes—
50C2×#13093983C2212yes—
51C2×#13093983C2214yes—
52C2×#13093982C2214yescomplemented
53C2×C4⋊C4×D47C2214yescomplemented
54C24×C4×C47C2211yescharacteristic
55C24×C4×C47C2212yes—
56C22×C4⋊C4×C47C2211yescharacteristic
57C22×C4⋊C4×C47C2211yescharacteristic
58C22×C4⋊C4×C47C2214yes—
59C22×C4⋊C4×C47C2214yes—
60C22×C4⋊C4×C47C2212yes—
61C22×C4⋊C4×C47C22116yescomplemented
62C2×#10625350—216no—
63C2×#10625350—416no—
64C2×#10625350C23116yes—
65C2×#10625350C23132yes—
66C22×Dic94C2312yes—
67C22×Dic94C2314yes—
68C22×D188C2312yes—
69C22×D188C2314yes—
70C22×C22⋊D47C2314yes—
71C22×C22⋊D47C2314yes—
72C22×C22⋊D47C2314yes—
73C22×C4×D47C2314yes—
74C22×C4×D47C2318yes—
75#13094023C23132yescomplemented
76#13094019C23164yescomplemented
77#13094015C23132yescomplemented
78#13094014C23164yescomplemented
79#13094009C23164yescomplemented
80#13094007C23164yes—
81#13094007C23132yes—
82#13094004C23132yescomplemented
83#13094003C23132yes—
84#13094003C23116yes—
85#13094003C23132yes—
86#13094003C23116yes—
87#13094002C23164yescomplemented
88#13093999C23132yescomplemented
89#13093998C23116yes—
90#13093998C23116yes—
91#13093998C23132yes—
92#13093998C23132yes—
93#13093993C23132yes—
94#13093989C23132yescomplemented
95#13093986C23132yescomplemented
96C4×C22⋊D47C23132yes—
97C4×C22⋊D47C23164yes—
98C4×C22⋊D47C23132yes—
99C4×C22⋊D47C23164yes—
100#13093983C23116yes—
101#13093983C23132yes—
102#13093982C23132yescomplemented
103C2×C4⋊Dic47C2312yes—
104C2×C4⋊Dic47C2312yes—
105C2×C4⋊Dic47C2314yes—
106C2×C4⋊Dic47C2314yes—
107C2×Dic47⋊C4C2318yes—
108C2×Dic47⋊C4C2314yes—
109C2×Dic47⋊C4C2318yes—
110C2×Dic47⋊C4C2314yes—
111C4⋊C4×D47C23132yescomplemented
112C2×C22⋊Dic47C2314yes—
113C2×C22⋊Dic47C2314yes—
114C2×C22⋊Dic47C2314yes—
115C2×(C2×D47)⋊C4C2318yes—
116C2×(C2×D47)⋊C4C2318yes—
117C2×(C2×D47)⋊C4C2314yes—
118C2×(C2×D47)⋊C4C2314yes—
119C2×C4×Dic47C2318yes—
120C2×C4×Dic47C2314yes—
121C25×C47C2311yescharacteristic
122C23×C4×C47—28no—
123C23×C4×C47—48no—
124C23×C4×C47C2314yes—
125C23×C4×C47C2312yes—
126C23×C4×C47C2312yes—
127C23×C4×C47C2311yescharacteristic
128C23×C4×C47C2318yes—
129C23×C4×C47C2312yes—
130C23×C4×C47C2314yes—
131C23×C4×C47C2312yes—
132C23×C4×C47C23116yes—
133C23×C4×C47C2311yescharacteristic
134C2×C4⋊C4×C47C2318yes—
135C2×C4⋊C4×C47C23132yes—
136C2×C4⋊C4×C47C2314yes—
137C2×C4⋊C4×C47C2314yes—
138C2×C4⋊C4×C47C2318yes—
139C2×C4⋊C4×C47C23132yes—
140C2×C4⋊C4×C47C2318yes—
141C2×C4⋊C4×C47C23116yes—
142C24×C47—24no—
143C24×C47—24no—
144C24×C47—24no—
145C24×C47C2414yes—
146C24×C47C2411yescharacteristic
147C24×C47C2411yescharacteristic
148C24×C47C2411yescharacteristic
149C23×D47C2414yes—
150C22×C4×C47—24no—
151C22×C4×C47—216no—
152C22×C4×C47—28no—
153C22×C4×C47—28no—
154C22×C4×C47—232no—
155C22×C4×C47—44no—
156C22×C4×C47—48no—
157C22×C4×C47—44no—
158C22×C4×C47—432no—
159C22×C4×C47#4118yescomplemented
160C22×C4×C47C2414yes—
161C22×C4×C47C2412yes—
162C22×C4×C47C2414yes—
163C22×C4×C47C24116yes—
164C22×C4×C47C2418yes—
165C22×C4×C47C2418yes—
166C22×C4×C47C2414yes—
167C22×C4×C47C2418yes—
168C22×C4×C47C24116yes—
169C22×C4×C47C2418yes—
170C22×C4×C47C2412yes—
171C22×C4×C47C2414yes—
172C2×C22⋊D47—216no—
173C2×C22⋊D47—216no—
174C2×C22⋊D47—216no—
175C2×C22⋊D47C24116yes—
176C2×C22⋊D47C24116yes—
177C2×C22⋊D47C24116yes—
178C22×Dic47C2414yes—
179C22×Dic47C2414yes—
180C22×Dic47C2414yes—
181#10625350—264no—
182#10625350—464no—
183C2×D188—28no—
184C2×D188—48no—
185C2×D188C2418yes—
186C2×D188C24116yes—
187C2×C4×D47—216no—
188C2×C4×D47—416no—
189C2×C4×D47C24116yes—
190C2×C4×D47C24132yes—
191C2×Dic94—28no—
192C2×Dic94—48no—
193C2×Dic94C2418yes—
194C2×Dic94C24116yes—
195C4⋊C4×C47C24116yes—
196C4⋊C4×C47C24116yes—
197C4⋊C4×C47C24132yes—
198C22⋊Dic47C24116yes—
199C22⋊Dic47C24116yes—
200C22⋊Dic47C24116yes—
201(C2×D47)⋊C4C24132yes—
202(C2×D47)⋊C4C24132yes—
203(C2×D47)⋊C4C24116yes—
204(C2×D47)⋊C4C24116yes—
205C4⋊Dic47C2418yes—
206C4⋊Dic47C2418yes—
207C4⋊Dic47C24116yes—
208C4⋊Dic47C24116yes—
209Dic47⋊C4C24132yes—
210Dic47⋊C4C24116yes—
211Dic47⋊C4C24132yes—
212Dic47⋊C4C24116yes—
213C4×Dic47C24132yes—
214C4×Dic47C24116yes—
215C23×C47—232no—
216C23×C47—28no—
217C23×C47—22no—
218C23×C47—22no—
219C23×C47—28no—
220C23×C47—22no—
221C23×C47—22no—
222C23×C47—28no—
223C23×C47—22no—
224C23×C47—22no—
225C23×C47C2×#4114yes—
226C23×C47H(5,𝔽2)14yes—
227C23×C472−1+414yes—
228C23×C47C2512yes—
229C23×C47C2512yes—
230C23×C47C2512yes—
231C23×C47C2511yescharacteristic
232C22×D47—28no—
233C22×D47—28no—
234C22×D47—28no—
235C22×D47C2518yes—
236C2×C4×C47—28no—
237C2×C4×C47—24no—
238C2×C4×C47—28no—
239C2×C4×C47—28no—
240C2×C4×C47—232no—
241C2×C4×C47—216no—
242C2×C4×C47—216no—
243C2×C4×C47—48no—
244C2×C4×C47—416no—
245C2×C4×C47—48no—
246C2×C4×C47C2×#4118yes—
247C2×C4×C47C2×#41116yescomplemented
248C2×C4×C47C2518yes—
249C2×C4×C47C2514yes—
250C2×C4×C47C2518yes—
251C2×C4×C47C2514yes—
252C22⋊D47—232no—
253C22⋊D47—232no—
254C22⋊D47—232no—
255C2×Dic47—28no—
256C2×Dic47—28no—
257C2×Dic47—28no—
258C2×Dic47C2518yes—
259C2×Dic47C2518yes—
260C2×Dic47C2518yes—
261D188—216no—
262D188—416no—
263C4×D47—232no—
264C4×D47—432no—
265Dic94—216no—
266Dic94—416no—
267C2×#3492—471nomaximal, Sylow
268C22×C47—232no—
269C22×C47—22no—
270C22×C47—22no—
271C22×C47—22no—
272C22×C47—22no—
273C22×C47—22no—
274C22×C47—22no—
275C22×C47—28no—
276C22×C47—28no—
277C22×C47—28no—
278C22×C47C22×#4114yes—
279C22×C47C22×#4112yes—
280C22×C47C2×H(5,𝔽2)14yes—
281C22×C47C2×H(5,𝔽2)12yes—
282C22×C47C2×2−1+414yes—
283C22×C47C2×2−1+412yes—
284C22×C47C2611yescharacteristic
285C2×D47—28no—
286C2×D47—28no—
287C2×D47—28no—
288C2×D47—416no—
289C4×C47—24no—
290C4×C47—28no—
291C4×C47—28no—
292C4×C47C22×#4118yes—
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×#537—471no—
304C2×#525—472no—
305C2×#525—474no—
306C2×#520—472no—
307C23×C4⋊C4D4711yescharacteristic, complemented
308C2×C47—24no—
309C2×C47—24no—
310C2×C47—24no—
311C2×C47#349214yescomplemented
312C2×C47C23×#4111yescharacteristic
313C2×C47C22×H(5,𝔽2)11yescharacteristic
314C2×C47C22×2−1+411yescharacteristic
315D47—48no—
316C22×#41—472no—
317C22×#41—474no—
318C24×C4C2×D4711yescharacteristic
319C24×C4C2×D4712yes—
320#554—4764no—
321#553—4764no—
322#552—4764no—
323#547—4764no—
324#544—4716no—
325#539—4764no—
326#537—4716no—
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—474no—
337C2×C22⋊Q8—472no—
338C2×C22⋊Q8—472no—
339C2×C22⋊Q8—474no—
340C2×C22⋊Q8—474no—
341C2×C22⋊Q8—478no—
342C2×#121—474no—
343C2×#121—472no—
344C2×#121—474no—
345C2×#121—472no—
346C2×#121—474no—
347C2×#120—474no—
348#520—4732no—
349C2×C4×Q8—474no—
350C2×C4×D4—478no—
351C2×C4×D4—474no—
352C2×C4×D4—478no—
353C2×C4×D4—474no—
354C2×C4×D4—474no—
355C2×#117—472no—
356C2×#117—474no—
357C2×#117—474no—
358C22×C4⋊C4—474no—
359C22×C4⋊C4C2×D4711yescharacteristic
360C22×C4⋊C4C2×D4711yescharacteristic
361C22×C4⋊C4C2×D4712yes—
362C22×C4⋊C4C2×D4714yes—
363C22×C4⋊C4C2×D4714yes—
364C22×C4⋊C4C2×D47116yescomplemented
365C47C2×#349211yescharacteristic, complemented, Sylow
366C25C22×D4711yescharacteristic
367C2×#41—4716no—
368C2×#41—4732no—
369C2×#41—9416no—
370C2×#41—18816no—
371C22×Q8—472no—
372C22×Q8—474no—
373C22×D4—474no—
374C22×D4—474no—
375C22×D4—474no—
376C22×D4—472no—
377C22×D4—474no—
378C23×C4—28no—
379C23×C4—48no—
380C23×C4—474no—
381C23×C4—478no—
382C23×C4C22×D4714yes—
383C23×C4C22×D4712yes—
384C23×C4C22×D4712yes—
385C23×C4C22×D4711yescharacteristic
386C23×C4C22×D4718yes—
387C23×C4C22×D4712yes—
388C23×C4C22×D4714yes—
389C23×C4C22×D4712yes—
390C23×C4C22×D47116yes—
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—4732no—
400C22⋊Q8—4716no—
401C22⋊Q8—4732no—
402C22⋊Q8—4716no—
403C22⋊Q8—4732no—
404C22⋊Q8—4764no—
405#121—4732no—
406#121—4716no—
407#121—4732no—
408#121—4716no—
409#121—4732no—
410#120—4732no—
411C4×Q8—4732no—
412C4×D4—4764no—
413C4×D4—4732no—
414C4×D4—4764no—
415C4×D4—4732no—
416C4×D4—4732no—
417#117—4716no—
418#117—4732no—
419#117—4732no—
420C2×C4⋊C4—472no—
421C2×C4⋊C4—478no—
422C2×C4⋊C4—472no—
423C2×C4⋊C4—474no—
424C2×C4⋊C4—474no—
425C2×C4⋊C4—474no—
426C2×C4⋊C4—478no—
427C2×C4⋊C4—474no—
428C2×C4⋊C4—4732no—
429C2×C4⋊C4C22×D4714yes—
430C2×C4⋊C4C22×D4714yes—
431C2×C4⋊C4C22×D4718yes—
432C2×C4⋊C4C22×D4718yes—
433C2×C4⋊C4C22×D4718yes—
434C2×C4⋊C4C22×D47116yes—
435C2×C4⋊C4C22×D47132yes—
436C2×C4⋊C4C22×D47132yes—
437C2×C22⋊C4—474no—
438C2×C22⋊C4—474no—
439C2×C22⋊C4—474no—
440C2×C22⋊C4—478no—
441C2×C22⋊C4—478no—
442C2×C22⋊C4—474no—
443C2×C22⋊C4—474no—
444C2×C42—478no—
445C2×C42—474no—
446C24—24no—
447C24—24no—
448C24—24no—
449C24—474no—
450C24C23×D4714yes—
451C24C23×D4711yescharacteristic
452C24C23×D4711yescharacteristic
453C24C23×D4711yescharacteristic
454#41—9464no—
455#41—18864no—
456C2×Q8—478no—
457C2×Q8—4716no—
458C2×Q8—948no—
459C2×Q8—1888no—
460C2×D4—4716no—
461C2×D4—4716no—
462C2×D4—4716no—
463C2×D4—478no—
464C2×D4—4716no—
465C2×D4—9416no—
466C2×D4—9416no—
467C2×D4—9416no—
468C2×D4—948no—
469C2×D4—1888no—
470C22×C4—24no—
471C22×C4—216no—
472C22×C4—28no—
473C22×C4—28no—
474C22×C4—232no—
475C22×C4—44no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—432no—
479C22×C4—474no—
480C22×C4—474no—
481C22×C4—474no—
482C22×C4—4716no—
483C22×C4—4732no—
484C22×C4—9416no—
485C22×C4—18816no—
486C22×C4#1062535018yescomplemented
487C22×C4C23×D47116yes—
488C22×C4C23×D4718yes—
489C22×C4C23×D4718yes—
490C22×C4C23×D4714yes—
491C22×C4C23×D4714yes—
492C22×C4C23×D4712yes—
493C22×C4C23×D4714yes—
494C22×C4C23×D4718yes—
495C22×C4C23×D47116yes—
496C22×C4C23×D4718yes—
497C22×C4C23×D4712yes—
498C22×C4C23×D4714yes—
499C4⋊C4—4732no—
500C4⋊C4—4716no—
501C4⋊C4—4732no—
502C4⋊C4—4716no—
503C4⋊C4—478no—
504C4⋊C4—478no—
505C4⋊C4—4716no—
506C4⋊C4—4716no—
507C4⋊C4C23×D47116yes—
508C4⋊C4C23×D47116yes—
509C4⋊C4C23×D47132yes—
510C22⋊C4—4716no—
511C22⋊C4—4716no—
512C22⋊C4—4716no—
513C22⋊C4—4732no—
514C22⋊C4—4732no—
515C22⋊C4—4716no—
516C22⋊C4—4716no—
517C42—4732no—
518C42—4716no—
519C23—232no—
520C23—28no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—28no—
527C23—22no—
528C23—22no—
529C23—478no—
530C23—948no—
531C23—948no—
532C23—948no—
533C23C2×#1062535014yes—
534C23#1309406914yes—
535C23#1309407214yes—
536C23C24×D4712yes—
537C23C24×D4712yes—
538C23C24×D4712yes—
539C23C24×D4711yescharacteristic
540Q8—9416no—
541Q8—18816no—
542D4—9432no—
543D4—9432no—
544D4—9432no—
545D4—9416no—
546D4—18816no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—232no—
552C2×C4—216no—
553C2×C4—216no—
554C2×C4—48no—
555C2×C4—416no—
556C2×C4—48no—
557C2×C4—478no—
558C2×C4—478no—
559C2×C4—478no—
560C2×C4—948no—
561C2×C4—948no—
562C2×C4—948no—
563C2×C4—9432no—
564C2×C4—18832no—
565C2×C4C2×#1062535018yes—
566C2×C4C2×#10625350116yescomplemented
567C2×C4C24×D4718yes—
568C2×C4C24×D4714yes—
569C2×C4C24×D4718yes—
570C2×C4C24×D4714yes—
571C22—232no—
572C22—28no—
573C22—28no—
574C22—28no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—948no—
582C22—948no—
583C22—948no—
584C22—18816no—
585C22C22×#1062535014yes—
586C22C22×#1062535012yes—
587C22C2×#1309406914yes—
588C22C2×#1309406912yes—
589C22C2×#1309407214yes—
590C22C2×#1309407212yes—
591C22C25×D4711yescharacteristic
592C4—24no—
593C4—28no—
594C4—28no—
595C4—948no—
596C4—948no—
597C4—948no—
598C4C22×#1062535018yes—
599C2—24no—
600C2—24no—
601C2—24no—
602C2—1888no—
603C2#2025709714yesdirect product factor
604C2C23×#1062535011yescharacteristic
605C2C22×#1309406911yescharacteristic
606C2C22×#1309407211yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC23×C4⋊C4×C47C2
socleC23×C47C25
hypercenterC23×C4⋊C4D47
hypocenterC47C2×#3492
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #20257097C2direct (N × H)
2. C2×#15689828C2semidirect
3. C2×#15689821C2semidirect
4. C2×#15689808C2semidirect
5. C2×#15689774C2semidirect
6. C2×#15689754C2semidirect
7. C2×#15689720C2semidirect
8. C2×#15689704C2semidirect
9. C2×#15689634C2semidirect
10. C2×#15689634C2semidirect
11. C2×#15689604C2semidirect
12. C23×C4⋊C4×C47C2semidirect
15. #15689828C22semidirect
16. #15689821C22semidirect
17. #15689808C22semidirect
18. #15689774C22semidirect
19. #15689754C22semidirect
20. #15689720C22semidirect
21. #15689704C22semidirect
22. C2×#13094023C22semidirect
23. C2×#13094019C22semidirect
24. C2×#13094015C22semidirect
25. C2×#13094014C22semidirect
26. #15689634C22semidirect
27. #15689634C22semidirect
28. C2×#13094009C22semidirect
31. C2×#13094004C22semidirect
36. C2×#13094002C22semidirect
37. C2×#13093999C22semidirect
43. #15689604C22semidirect
44. C2×#13093989C22semidirect
45. C2×#13093986C22semidirect
52. C2×#13093982C22semidirect
53. C2×C4⋊C4×D47C22semidirect
61. C22×C4⋊C4×C47C22semidirect
75. #13094023C23semidirect
76. #13094019C23semidirect
77. #13094015C23semidirect
78. #13094014C23semidirect
79. #13094009C23semidirect
82. #13094004C23semidirect
87. #13094002C23semidirect
88. #13093999C23semidirect
94. #13093989C23semidirect
95. #13093986C23semidirect
102. #13093982C23semidirect
111. C4⋊C4×D47C23semidirect
159. C22×C4×C47#41semidirect
247. C2×C4×C47C2×#41semidirect
307. C23×C4⋊C4D47semidirect
311. C2×C47#3492semidirect
364. C22×C4⋊C4C2×D47semidirect
365. C47C2×#3492semidirect
486. C22×C4#10625350semidirect
566. C2×C4C2×#10625350semidirect
603. C2#20257097direct (N × H)

Maximal subgroups

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

subgroupindex
#202570972
C2×#156898282
C2×#156898212
C2×#156898082
C2×#156897742
C2×#156897542
C2×#156897202
C2×#156897042
C2×#156896342
C2×#156896042
C23×C4⋊C4×C472
C2×#349247