Finite Group Databasebeta

← C2×#20257784 #27124070

Subgroups — C2×#20257784 #27124070

Order 12032.

7810 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257784C2132yesmaximal, direct product factor
2C2×#15689871C211yesmaximal, characteristic, complemented
3C2×#15689867C212yesmaximal, complemented
4C2×#15689827C214yesmaximal, complemented
5C2×#15689791C214yesmaximal, complemented
6C2×#15689771C214yesmaximal, complemented
7C2×#15689725C214yesmaximal, complemented
8C2×#15689703C214yesmaximal, complemented
9C2×#15689700C214yesmaximal, complemented
10C2×#15689679C211yesmaximal, characteristic, complemented
11C2×#15689672C212yesmaximal, complemented
12C2×C47×#562C211yesmaximal, characteristic, complemented
13C22×#10625354C2212yes—
14C22×#10625350C2214yes—
15C22×#10625352C2214yes—
16C22×#10625352C2212yes—
17#15689871C22116yescomplemented
18#15689867C22132yescomplemented
19#15689827C22164yescomplemented
20#15689791C22164yescomplemented
21#15689771C22164yescomplemented
22#15689725C22164yescomplemented
23#15689703C22164yescomplemented
24#15689700C22164yescomplemented
25#15689679C22116yescomplemented
26#15689672C22132yescomplemented
27C2×#13094023C2212yes—
28C2×#13094022C2212yes—
29C2×#13094020C2212yes—
30C2×#13094020C2214yes—
31C2×#13094016C2211yescharacteristic
32C2×#13094014C2214yes—
33C2×#13094012C2212yes—
34C2×#13094009C2214yes—
35C2×#13094008C2214yes—
36C2×#13094006C2214yes—
37C2×#13094002C2214yes—
38C2×#13094002C2214yes—
39C2×#13094002C2214yes—
40C2×#13094001C2214yes—
41C2×#13094001C2214yes—
42C2×#13094000C2214yes—
43C2×#13093999C2214yes—
44C2×#13093999C2214yes—
45C2×#13093998C2214yes—
46C2×#13093997C2214yes—
47C2×#13093997C2214yes—
48C2×#13093997C2214yes—
49C2×#13093997C2214yes—
50C2×#13093997C2214yes—
51C2×#13093996C2214yes—
52C2×#13093996C2214yes—
53C2×#13093996C2214yes—
54C2×#13093989C2214yes—
55C2×C4×C22⋊D47C2214yes—
56C2×(C22⋊D47)⋊C4C2214yes—
57C2×(C22⋊D47)⋊C4C2214yes—
58C2×(C22⋊D47)⋊C4C2214yes—
59C2×#13093982C2212yes—
60C22×Dic47⋊C4C2214yes—
61C2×C4⋊C4×D47C2214yes—
62C2×C42×D47C2211yescharacteristic
63C47×#562C22116yescomplemented
64C2×#125×C47C2211yescharacteristic
65C2×#124×C47C2212yes—
66C2×#123×C47C2214yes—
67C2×C22⋊Q8×C47C2214yes—
68C2×#121×C47C2214yes—
69C2×#10625354—48no—
70C2×#10625354C23116yes—
71C2×#10625350—416no—
72C2×#10625350C23132yes—
73C2×#10625352—416no—
74C2×#10625352—48no—
75C2×#10625352C23132yes—
76C2×#10625352C23116yes—
77C22×Dic94C2314yes—
78C22×Dic94C2312yes—
79C22×D188C2314yes—
80C22×D188C2312yes—
81C22×D188C2314yes—
82C22×C22⋊D47C2314yes—
83C22×C22⋊D47C2314yes—
84C22×C22⋊D47C2314yes—
85C22×C22⋊D47C2314yes—
86C22×C22⋊D47C2314yes—
87C23×Dic47C2314yes—
88C23×Dic47C2314yes—
89C22×C4×D47C2311yescharacteristic
90C22×C4×D47C2312yes—
91C22×C4×D47C2314yes—
92C22×C4×D47C2314yes—
93C22×C4×D47C2314yes—
94#13094023C23116yes—
95#13094022C23116yes—
96#13094020C23116yes—
97#13094020C23132yes—
98#13094016C2318yes—
99#13094014C23132yes—
100#13094012C23116yes—
101#13094009C23132yes—
102#13094008C23132yes—
103#13094006C23132yes—
104#13094002C23132yes—
105#13094002C23132yes—
106#13094002C23132yes—
107#13094001C23132yes—
108#13094001C23132yes—
109#13094000C23132yes—
110#13093999C23132yes—
111#13093999C23132yes—
112#13093998C23132yes—
113#13093997C23132yes—
114#13093997C23132yes—
115#13093997C23132yes—
116#13093997C23132yes—
117#13093997C23132yes—
118#13093996C23132yes—
119#13093996C23132yes—
120#13093996C23132yes—
121#13093989C23132yes—
122C4×C22⋊D47C23132yes—
123(C22⋊D47)⋊C4C23132yes—
124(C22⋊D47)⋊C4C23132yes—
125(C22⋊D47)⋊C4C23132yes—
126#13093982C23116yes—
127C2×C4⋊Dic47C2314yes—
128C2×C4⋊Dic47C2312yes—
129C2×Dic47⋊C4C23132yes—
130C2×Dic47⋊C4C2314yes—
131C2×Dic47⋊C4C2314yes—
132C2×Dic47⋊C4C2314yes—
133C2×Dic47⋊C4C2314yes—
134C2×Dic47⋊C4C2314yes—
135C2×Dic47⋊C4C2314yes—
136C4⋊C4×D47C23132yes—
137C2×C22⋊Dic47C2314yes—
138C2×(C2×D47)⋊C4C2314yes—
139C2×(C2×D47)⋊C4C2314yes—
140C2×(C2×D47)⋊C4C2314yes—
141C2×(C2×D47)⋊C4C2314yes—
142C2×(C2×D47)⋊C4C2314yes—
143C2×C4×Dic47C2311yescharacteristic
144C2×C4×Dic47C2312yes—
145C2×C4×Dic47C2314yes—
146C42×D47C2318yes—
147C22×Q8×C47C2312yes—
148C22×D4×C47C2314yes—
149C22×D4×C47C2312yes—
150C23×C4×C47C2314yes—
151#125×C47C2318yes—
152#124×C47C23116yes—
153#123×C47C23132yes—
154C22⋊Q8×C47C23132yes—
155#121×C47C23132yes—
156C2×C4⋊C4×C47C2314yes—
157C2×C4⋊C4×C47C2314yes—
158C2×C4⋊C4×C47C2312yes—
159C2×C22⋊C4×C47C2314yes—
160C2×C22⋊C4×C47C2314yes—
161C2×C22⋊C4×C47C2314yes—
162C2×C42×C47C2311yescharacteristic
163C24×C47C2414yes—
164C23×D47C2414yes—
165C23×D47C2411yescharacteristic
166C2×Q8×C47—44no—
167C2×Q8×C47C2418yes—
168C2×D4×C47—48no—
169C2×D4×C47—44no—
170C2×D4×C47C24116yes—
171C2×D4×C47C2418yes—
172C22×C4×C47—48no—
173C22×C4×C47C2411yescharacteristic
174C22×C4×C47C2412yes—
175C22×C4×C47C2414yes—
176C22×C4×C47C2414yes—
177C22×C4×C47C24116yes—
178C2×C22⋊D47—216no—
179C2×C22⋊D47—216no—
180C2×C22⋊D47—48no—
181C2×C22⋊D47—48no—
182C2×C22⋊D47—48no—
183C2×C22⋊D47C24116yes—
184C2×C22⋊D47C24116yes—
185C2×C22⋊D47C24116yes—
186C2×C22⋊D47C24116yes—
187C2×C22⋊D47C24116yes—
188C22×Dic47—216no—
189C22×Dic47—48no—
190C22×Dic47C2414yes—
191C22×Dic47C2414yes—
192C22×Dic47C24116yes—
193C22×Dic47C24116yes—
194C22×Dic47C2412yes—
195C22×Dic47C2411yescharacteristic
196#10625354—432no—
197#10625352—464no—
198#10625352—432no—
199#10625350—464no—
200C2×D188—48no—
201C2×D188—44no—
202C2×D188—48no—
203C2×D188C24116yes—
204C2×D188C2418yes—
205C2×D188C24116yes—
206C2×C4×D47—28no—
207C2×C4×D47—216no—
208C2×C4×D47—24no—
209C2×C4×D47—48no—
210C2×C4×D47—48no—
211C2×C4×D47C2414yes—
212C2×C4×D47C2418yes—
213C2×C4×D47C24116yes—
214C2×C4×D47C24116yes—
215C2×C4×D47C24116yes—
216C2×Dic94—216no—
217C2×Dic94—44no—
218C2×Dic94C24116yes—
219C2×Dic94C2418yes—
220C4⋊C4×C47C24116yes—
221C4⋊C4×C47C24116yes—
222C4⋊C4×C47C2418yes—
223C22⋊C4×C47C24116yes—
224C22⋊C4×C47C24116yes—
225C22⋊C4×C47C24116yes—
226C42×C47C2414yes—
227C22⋊Dic47C24116yes—
228(C2×D47)⋊C4C24116yes—
229(C2×D47)⋊C4C24116yes—
230(C2×D47)⋊C4C24116yes—
231(C2×D47)⋊C4C24116yes—
232(C2×D47)⋊C4C24116yes—
233C4⋊Dic47C24116yes—
234C4⋊Dic47C2418yes—
235Dic47⋊C4C24116yes—
236Dic47⋊C4C24116yes—
237Dic47⋊C4C24116yes—
238Dic47⋊C4C24116yes—
239Dic47⋊C4C24116yes—
240Dic47⋊C4C24116yes—
241C4×Dic47C2414yes—
242C4×Dic47C2418yes—
243C4×Dic47C24116yes—
244C23×C47—28no—
245C23×C47—28no—
246C23×C47—28no—
247C23×C47C2518yes—
248C23×C47C2511yescharacteristic
249C22×D47—28no—
250C22×D47—24no—
251C22×D47—28no—
252C22×D47—22no—
253C22×D47—28no—
254C22×D47C2518yes—
255C22×D47C2512yes—
256Q8×C47—48no—
257D4×C47—416no—
258D4×C47—48no—
259C2×C4×C47—28no—
260C2×C4×C47—24no—
261C2×C4×C47—28no—
262C2×C4×C47—22no—
263C2×C4×C47—416no—
264C2×C4×C47C2514yes—
265C2×C4×C47C2518yes—
266C2×C4×C47C2512yes—
267C2×C4×C47C2518yes—
268C22⋊D47—232no—
269C22⋊D47—232no—
270C22⋊D47—416no—
271C22⋊D47—416no—
272C22⋊D47—416no—
273C2×Dic47—28no—
274C2×Dic47—28no—
275C2×Dic47—232no—
276C2×Dic47—416no—
277C2×Dic47H(5,𝔽2)18yes—
278C2×Dic472−1+414yes—
279C2×Dic47C2518yes—
280C2×Dic47C2518yes—
281C2×Dic47C2514yes—
282C2×Dic47C2512yes—
283D188—416no—
284D188—48no—
285D188—416no—
286C4×D47—232no—
287C4×D47—216no—
288C4×D47—28no—
289C4×D47—416no—
290C4×D47—416no—
291Dic94—232no—
292Dic94—48no—
293C2×#3548—471nomaximal, Sylow
294C22×C47—28no—
295C22×C47—28no—
296C22×C47—28no—
297C22×C47—416no—
298C22×C47C2×H(5,𝔽2)14yes—
299C22×C47C2×2−1+412yes—
300C22×C47C2611yescharacteristic
301C2×D47—28no—
302C2×D47—24no—
303C2×D47—28no—
304C2×D47—22no—
305C2×D47—28no—
306C2×D47—416no—
307C2×D47—44no—
308C4×C47—28no—
309C4×C47—24no—
310C4×C47—28no—
311C4×C47—22no—
312Dic47—28no—
313Dic47—28no—
314Dic47C2×H(5,𝔽2)18yes—
315Dic47C2×2−1+414yes—
316#3548—4732no—
317C2×#562—471no—
318C2×#562—472no—
319C2×#562D4711yescharacteristic, complemented
320C2×#554—474no—
321C2×#550—474no—
322C2×#547—474no—
323C2×#540—474no—
324C2×#537—474no—
325C2×#537—474no—
326C2×#533—471no—
327C2×#532—472no—
328C2×C47—48no—
329C2×C47#354814yescomplemented
330C2×C47C22×H(5,𝔽2)12yes—
331C2×C47C22×2−1+411yescharacteristic
332D47—48no—
333D47—42no—
334C22×#41—472no—
335C22×#41—472no—
336C22×#41—474no—
337C22×#41—474no—
338#562—4716no—
339#562—4732no—
340#562C2×D47116yescomplemented
341#554—4764no—
342#550—4764no—
343#547—4764no—
344#540—4764no—
345#537—4764no—
346#537—4764no—
347#533—4716no—
348#532—4732no—
349C2×C4⋊Q8—472no—
350C2×C4⋊Q8—472no—
351C2×#127—472no—
352C2×#127—474no—
353C2×#125—474no—
354C2×#125—471no—
355C2×#125C2×D4711yescharacteristic
356C2×#124—472no—
357C2×#124C2×D4712yes—
358C2×#123—474no—
359C2×#123—474no—
360C2×#123—474no—
361C2×#123C2×D4714yes—
362C2×C22⋊Q8—474no—
363C2×C22⋊Q8—474no—
364C2×C22⋊Q8—474no—
365C2×C22⋊Q8—474no—
366C2×C22⋊Q8—474no—
367C2×C22⋊Q8—474no—
368C2×C22⋊Q8C2×D4714yes—
369C2×#121—474no—
370C2×#121—474no—
371C2×#121—474no—
372C2×#121—474no—
373C2×#121—474no—
374C2×#121—474no—
375C2×#121—474no—
376C2×#121—474no—
377C2×#121—474no—
378C2×#121—474no—
379C2×#121—474no—
380C2×#121C2×D4714yes—
381C2×C4×Q8—474no—
382C2×C4×D4—474no—
383C2×C4×D4—474no—
384C2×C4×D4—474no—
385C2×C4×D4—474no—
386C2×#117—472no—
387C22×C4⋊C4—474no—
388C22×C4⋊C4—474no—
389C22×C42—471no—
390C47C2×#354811yescharacteristic, complemented, Sylow
391C2×#41—4716no—
392C2×#41—4716no—
393C2×#41—4732no—
394C2×#41—4732no—
395C2×#41—18816no—
396C2×#41—1888no—
397C2×#41—1888no—
398C2×#41—18816no—
399C22×Q8—474no—
400C22×Q8—472no—
401C22×Q8C22×D4712yes—
402C22×D4—474no—
403C22×D4—474no—
404C22×D4—474no—
405C22×D4—474no—
406C22×D4—474no—
407C22×D4—474no—
408C22×D4—474no—
409C22×D4—472no—
410C22×D4C22×D4714yes—
411C22×D4C22×D4712yes—
412C23×C4—474no—
413C23×C4—474no—
414C23×C4—474no—
415C23×C4—471no—
416C23×C4—472no—
417C23×C4—474no—
418C23×C4—474no—
419C23×C4C22×D4714yes—
420C4⋊Q8—4716no—
421C4⋊Q8—4716no—
422#127—4716no—
423#127—4732no—
424#125—4732no—
425#125—478no—
426#125C22×D4718yes—
427#124—4716no—
428#124C22×D47116yes—
429#123—4732no—
430#123—4732no—
431#123—4732no—
432#123C22×D47132yes—
433C22⋊Q8—4732no—
434C22⋊Q8—4732no—
435C22⋊Q8—4732no—
436C22⋊Q8—4732no—
437C22⋊Q8—4732no—
438C22⋊Q8—4732no—
439C22⋊Q8C22×D47132yes—
440#121—4732no—
441#121—4732no—
442#121—4732no—
443#121—4732no—
444#121—4732no—
445#121—4732no—
446#121—4732no—
447#121—4732no—
448#121—4732no—
449#121—4732no—
450#121—4732no—
451#121C22×D47132yes—
452C4×Q8—4732no—
453C4×D4—4732no—
454C4×D4—4732no—
455C4×D4—4732no—
456C4×D4—4732no—
457#117—4716no—
458C2×C4⋊C4—474no—
459C2×C4⋊C4—474no—
460C2×C4⋊C4—474no—
461C2×C4⋊C4—474no—
462C2×C4⋊C4—474no—
463C2×C4⋊C4—4732no—
464C2×C4⋊C4—472no—
465C2×C4⋊C4—474no—
466C2×C4⋊C4—474no—
467C2×C4⋊C4—4732no—
468C2×C4⋊C4C22×D4714yes—
469C2×C4⋊C4C22×D4712yes—
470C2×C4⋊C4C22×D4714yes—
471C2×C22⋊C4—474no—
472C2×C22⋊C4—474no—
473C2×C22⋊C4—474no—
474C2×C22⋊C4—474no—
475C2×C22⋊C4—474no—
476C2×C22⋊C4—474no—
477C2×C22⋊C4C22×D4714yes—
478C2×C22⋊C4C22×D4714yes—
479C2×C22⋊C4C22×D4714yes—
480C2×C42—474no—
481C2×C42—471no—
482C2×C42—472no—
483C2×C42—478no—
484C2×C42C22×D4711yescharacteristic
485C24—471no—
486C24—474no—
487C24C23×D4714yes—
488#41—18864no—
489#41—18832no—
490#41—18832no—
491#41—18864no—
492C2×Q8—44no—
493C2×Q8—4716no—
494C2×Q8—478no—
495C2×Q8—9416no—
496C2×Q8—1884no—
497C2×Q8C23×D4718yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—4716no—
501C2×D4—4716no—
502C2×D4—4716no—
503C2×D4—4716no—
504C2×D4—4716no—
505C2×D4—4716no—
506C2×D4—4716no—
507C2×D4—478no—
508C2×D4—9416no—
509C2×D4—9416no—
510C2×D4—1888no—
511C2×D4—1888no—
512C2×D4—1884no—
513C2×D4—1888no—
514C2×D4—1888no—
515C2×D4—1888no—
516C2×D4C23×D47116yes—
517C2×D4C23×D4718yes—
518C22×C4—48no—
519C22×C4—471no—
520C22×C4—472no—
521C22×C4—4716no—
522C22×C4—474no—
523C22×C4—474no—
524C22×C4—4716no—
525C22×C4—474no—
526C22×C4—478no—
527C22×C4—4716no—
528C22×C4—4716no—
529C22×C4—4716no—
530C22×C4—944no—
531C22×C4—9416no—
532C22×C4—948no—
533C22×C4—9416no—
534C22×C4—1888no—
535C22×C4—1888no—
536C22×C4—1888no—
537C22×C4C23×D4711yescharacteristic
538C22×C4C23×D4712yes—
539C22×C4C23×D4714yes—
540C22×C4C23×D4714yes—
541C22×C4C23×D47116yes—
542C4⋊C4—4716no—
543C4⋊C4—4716no—
544C4⋊C4—4716no—
545C4⋊C4—4716no—
546C4⋊C4—4716no—
547C4⋊C4—4716no—
548C4⋊C4—4716no—
549C4⋊C4—478no—
550C4⋊C4C23×D47116yes—
551C4⋊C4C23×D4718yes—
552C4⋊C4C23×D47116yes—
553C22⋊C4—4716no—
554C22⋊C4—4716no—
555C22⋊C4—4716no—
556C22⋊C4—4716no—
557C22⋊C4—4716no—
558C22⋊C4—4716no—
559C22⋊C4C23×D47116yes—
560C22⋊C4C23×D47116yes—
561C22⋊C4C23×D47116yes—
562C42—474no—
563C42—478no—
564C42—4716no—
565C42C23×D4714yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—472no—
570C23—478no—
571C23—948no—
572C23—944no—
573C23—948no—
574C23—942no—
575C23—948no—
576C23C24×D4718yes—
577C23C24×D4711yescharacteristic
578Q8—48no—
579Q8—9432no—
580Q8—1888no—
581D4—416no—
582D4—48no—
583D4—9432no—
584D4—9432no—
585D4—18816no—
586D4—18816no—
587D4—1888no—
588D4—18816no—
589D4—18816no—
590D4—18816no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—478no—
597C2×C4—474no—
598C2×C4—472no—
599C2×C4—474no—
600C2×C4—478no—
601C2×C4—478no—
602C2×C4—948no—
603C2×C4—9432no—
604C2×C4—9416no—
605C2×C4—9432no—
606C2×C4—948no—
607C2×C4—948no—
608C2×C4—18816no—
609C2×C4—18816no—
610C2×C4—18816no—
611C2×C4C24×D4712yes—
612C2×C4C24×D4714yes—
613C2×C4C24×D4718yes—
614C2×C4C24×D4718yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—948no—
620C22—944no—
621C22—948no—
622C22—942no—
623C22—948no—
624C22—1884no—
625C22—18816no—
626C22H(5,𝔽2)×D4714yes—
627C222−1+4×D4712yes—
628C22C25×D4711yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—478no—
634C4—474no—
635C4—948no—
636C4—948no—
637C2—48no—
638C2—1882no—
639C2—1888no—
640C2#2025778414yesdirect product factor
641C2C2×H(5,𝔽2)×D4712yes—
642C2C2×2−1+4×D4711yescharacteristic
subgroupG/H
centerC23C24×D47
commutator, cosocleC22×C47C26
FrattiniC22C25×D47
FittingC2×C47×#562C2
socleC23×C47C25
hypercenterC2×#562D47
hypocenterC47C2×#3548
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20257784C2direct (N × H)
2. C2×#15689871C2semidirect
3. C2×#15689867C2semidirect
4. C2×#15689827C2semidirect
5. C2×#15689791C2semidirect
6. C2×#15689771C2semidirect
7. C2×#15689725C2semidirect
8. C2×#15689703C2semidirect
9. C2×#15689700C2semidirect
10. C2×#15689679C2semidirect
11. C2×#15689672C2semidirect
12. C2×C47×#562C2semidirect
17. #15689871C22semidirect
18. #15689867C22semidirect
19. #15689827C22semidirect
20. #15689791C22semidirect
21. #15689771C22semidirect
22. #15689725C22semidirect
23. #15689703C22semidirect
24. #15689700C22semidirect
25. #15689679C22semidirect
26. #15689672C22semidirect
63. C47×#562C22semidirect
319. C2×#562D47semidirect
329. C2×C47#3548semidirect
340. #562C2×D47semidirect
390. C47C2×#3548semidirect
640. C2#20257784direct (N × H)

Maximal subgroups

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

subgroupindex
#202577842
C2×#156898712
C2×#156898672
C2×#156898272
C2×#156897912
C2×#156897712
C2×#156897252
C2×#156897032
C2×#156897002
C2×#156896792
C2×#156896722
C2×C47×#5622
C2×#354847