Finite Group Databasebeta

← C2×#19148991 #26004389

Subgroups — C2×#19148991 #26004389

Order 11008.

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#19148991C2132yesmaximal, direct product factor
2C2×#14583647C211yesmaximal, characteristic, complemented
3C2×#14583643C212yesmaximal, complemented
4C2×#14583603C214yesmaximal, complemented
5C2×#14583567C214yesmaximal, complemented
6C2×#14583547C214yesmaximal, complemented
7C2×#14583501C214yesmaximal, complemented
8C2×#14583479C214yesmaximal, complemented
9C2×#14583476C214yesmaximal, complemented
10C2×#14583455C211yesmaximal, characteristic, complemented
11C2×#14583448C212yesmaximal, complemented
12C2×C43×#562C211yesmaximal, characteristic, complemented
13C22×#10621902C2212yes—
14C22×#10621898C2214yes—
15C22×#10621900C2214yes—
16C22×#10621900C2212yes—
17#14583647C22116yescomplemented
18#14583643C22132yescomplemented
19#14583603C22164yescomplemented
20#14583567C22164yescomplemented
21#14583547C22164yescomplemented
22#14583501C22164yescomplemented
23#14583479C22164yescomplemented
24#14583476C22164yescomplemented
25#14583455C22116yescomplemented
26#14583448C22132yescomplemented
27C2×#13063324C2212yes—
28C2×#13063323C2212yes—
29C2×#13063321C2212yes—
30C2×#13063321C2214yes—
31C2×#13063317C2211yescharacteristic
32C2×#13063315C2214yes—
33C2×#13063313C2212yes—
34C2×#13063310C2214yes—
35C2×#13063309C2214yes—
36C2×#13063307C2214yes—
37C2×#13063303C2214yes—
38C2×#13063303C2214yes—
39C2×#13063303C2214yes—
40C2×#13063302C2214yes—
41C2×#13063302C2214yes—
42C2×#13063301C2214yes—
43C2×#13063300C2214yes—
44C2×#13063300C2214yes—
45C2×#13063299C2214yes—
46C2×#13063298C2214yes—
47C2×#13063298C2214yes—
48C2×#13063298C2214yes—
49C2×#13063298C2214yes—
50C2×#13063298C2214yes—
51C2×#13063297C2214yes—
52C2×#13063297C2214yes—
53C2×#13063297C2214yes—
54C2×#13063290C2214yes—
55C2×C4×C22⋊D43C2214yes—
56C2×(C22⋊D43)⋊C4C2214yes—
57C2×(C22⋊D43)⋊C4C2214yes—
58C2×(C22⋊D43)⋊C4C2214yes—
59C2×#13063283C2212yes—
60C22×Dic43⋊C4C2214yes—
61C2×C4⋊C4×D43C2214yes—
62C2×C42×D43C2211yescharacteristic
63C43×#562C22116yescomplemented
64C2×#125×C43C2211yescharacteristic
65C2×#124×C43C2212yes—
66C2×#123×C43C2214yes—
67C2×C22⋊Q8×C43C2214yes—
68C2×#121×C43C2214yes—
69C2×#10621902—48no—
70C2×#10621902C23116yes—
71C2×#10621898—416no—
72C2×#10621898C23132yes—
73C2×#10621900—416no—
74C2×#10621900—48no—
75C2×#10621900C23132yes—
76C2×#10621900C23116yes—
77C22×Dic86C2314yes—
78C22×Dic86C2312yes—
79C22×D172C2314yes—
80C22×D172C2312yes—
81C22×D172C2314yes—
82C22×C22⋊D43C2314yes—
83C22×C22⋊D43C2314yes—
84C22×C22⋊D43C2314yes—
85C22×C22⋊D43C2314yes—
86C22×C22⋊D43C2314yes—
87C23×Dic43C2314yes—
88C23×Dic43C2314yes—
89C22×C4×D43C2311yescharacteristic
90C22×C4×D43C2312yes—
91C22×C4×D43C2314yes—
92C22×C4×D43C2314yes—
93C22×C4×D43C2314yes—
94#13063324C23116yes—
95#13063323C23116yes—
96#13063321C23116yes—
97#13063321C23132yes—
98#13063317C2318yes—
99#13063315C23132yes—
100#13063313C23116yes—
101#13063310C23132yes—
102#13063309C23132yes—
103#13063307C23132yes—
104#13063303C23132yes—
105#13063303C23132yes—
106#13063303C23132yes—
107#13063302C23132yes—
108#13063302C23132yes—
109#13063301C23132yes—
110#13063300C23132yes—
111#13063300C23132yes—
112#13063299C23132yes—
113#13063298C23132yes—
114#13063298C23132yes—
115#13063298C23132yes—
116#13063298C23132yes—
117#13063298C23132yes—
118#13063297C23132yes—
119#13063297C23132yes—
120#13063297C23132yes—
121#13063290C23132yes—
122C4×C22⋊D43C23132yes—
123(C22⋊D43)⋊C4C23132yes—
124(C22⋊D43)⋊C4C23132yes—
125(C22⋊D43)⋊C4C23132yes—
126#13063283C23116yes—
127C2×C4⋊Dic43C2314yes—
128C2×C4⋊Dic43C2312yes—
129C2×Dic43⋊C4C23132yes—
130C2×Dic43⋊C4C2314yes—
131C2×Dic43⋊C4C2314yes—
132C2×Dic43⋊C4C2314yes—
133C2×Dic43⋊C4C2314yes—
134C2×Dic43⋊C4C2314yes—
135C2×Dic43⋊C4C2314yes—
136C4⋊C4×D43C23132yes—
137C2×C22⋊Dic43C2314yes—
138C2×(C2×D43)⋊C4C2314yes—
139C2×(C2×D43)⋊C4C2314yes—
140C2×(C2×D43)⋊C4C2314yes—
141C2×(C2×D43)⋊C4C2314yes—
142C2×(C2×D43)⋊C4C2314yes—
143C2×C4×Dic43C2311yescharacteristic
144C2×C4×Dic43C2312yes—
145C2×C4×Dic43C2314yes—
146C42×D43C2318yes—
147C22×Q8×C43C2312yes—
148C22×D4×C43C2314yes—
149C22×D4×C43C2312yes—
150C23×C4×C43C2314yes—
151#125×C43C2318yes—
152#124×C43C23116yes—
153#123×C43C23132yes—
154C22⋊Q8×C43C23132yes—
155#121×C43C23132yes—
156C2×C4⋊C4×C43C2314yes—
157C2×C4⋊C4×C43C2314yes—
158C2×C4⋊C4×C43C2312yes—
159C2×C22⋊C4×C43C2314yes—
160C2×C22⋊C4×C43C2314yes—
161C2×C22⋊C4×C43C2314yes—
162C2×C42×C43C2311yescharacteristic
163C24×C43C2414yes—
164C23×D43C2414yes—
165C23×D43C2411yescharacteristic
166C2×Q8×C43—44no—
167C2×Q8×C43C2418yes—
168C2×D4×C43—44no—
169C2×D4×C43—48no—
170C2×D4×C43C24116yes—
171C2×D4×C43C2418yes—
172C22×C4×C43—48no—
173C22×C4×C43C2411yescharacteristic
174C22×C4×C43C2412yes—
175C22×C4×C43C2414yes—
176C22×C4×C43C2414yes—
177C22×C4×C43C24116yes—
178C2×C22⋊D43—216no—
179C2×C22⋊D43—216no—
180C2×C22⋊D43—48no—
181C2×C22⋊D43—48no—
182C2×C22⋊D43—48no—
183C2×C22⋊D43C24116yes—
184C2×C22⋊D43C24116yes—
185C2×C22⋊D43C24116yes—
186C2×C22⋊D43C24116yes—
187C2×C22⋊D43C24116yes—
188C22×Dic43—216no—
189C22×Dic43—48no—
190C22×Dic43C2414yes—
191C22×Dic43C2414yes—
192C22×Dic43C24116yes—
193C22×Dic43C24116yes—
194C22×Dic43C2412yes—
195C22×Dic43C2411yescharacteristic
196#10621902—432no—
197#10621900—464no—
198#10621900—432no—
199#10621898—464no—
200C2×D172—48no—
201C2×D172—44no—
202C2×D172—48no—
203C2×D172C24116yes—
204C2×D172C2418yes—
205C2×D172C24116yes—
206C2×C4×D43—28no—
207C2×C4×D43—216no—
208C2×C4×D43—24no—
209C2×C4×D43—48no—
210C2×C4×D43—48no—
211C2×C4×D43C2414yes—
212C2×C4×D43C2418yes—
213C2×C4×D43C24116yes—
214C2×C4×D43C24116yes—
215C2×C4×D43C24116yes—
216C2×Dic86—216no—
217C2×Dic86—44no—
218C2×Dic86C24116yes—
219C2×Dic86C2418yes—
220C4⋊C4×C43C24116yes—
221C4⋊C4×C43C24116yes—
222C4⋊C4×C43C2418yes—
223C22⋊C4×C43C24116yes—
224C22⋊C4×C43C24116yes—
225C22⋊C4×C43C24116yes—
226C42×C43C2414yes—
227C22⋊Dic43C24116yes—
228(C2×D43)⋊C4C24116yes—
229(C2×D43)⋊C4C24116yes—
230(C2×D43)⋊C4C24116yes—
231(C2×D43)⋊C4C24116yes—
232(C2×D43)⋊C4C24116yes—
233C4⋊Dic43C24116yes—
234C4⋊Dic43C2418yes—
235Dic43⋊C4C24116yes—
236Dic43⋊C4C24116yes—
237Dic43⋊C4C24116yes—
238Dic43⋊C4C24116yes—
239Dic43⋊C4C24116yes—
240Dic43⋊C4C24116yes—
241C4×Dic43C2414yes—
242C4×Dic43C2418yes—
243C4×Dic43C24116yes—
244C23×C43—28no—
245C23×C43—28no—
246C23×C43—28no—
247C23×C43C2518yes—
248C23×C43C2511yescharacteristic
249C22×D43—28no—
250C22×D43—24no—
251C22×D43—28no—
252C22×D43—22no—
253C22×D43—28no—
254C22×D43C2518yes—
255C22×D43C2512yes—
256Q8×C43—48no—
257D4×C43—416no—
258D4×C43—48no—
259C2×C4×C43—28no—
260C2×C4×C43—24no—
261C2×C4×C43—28no—
262C2×C4×C43—22no—
263C2×C4×C43—416no—
264C2×C4×C43C2514yes—
265C2×C4×C43C2518yes—
266C2×C4×C43C2512yes—
267C2×C4×C43C2518yes—
268C22⋊D43—232no—
269C22⋊D43—232no—
270C22⋊D43—416no—
271C22⋊D43—416no—
272C22⋊D43—416no—
273C2×Dic43—28no—
274C2×Dic43—28no—
275C2×Dic43—232no—
276C2×Dic43—416no—
277C2×Dic43H(5,𝔽2)18yes—
278C2×Dic432−1+414yes—
279C2×Dic43C2518yes—
280C2×Dic43C2518yes—
281C2×Dic43C2514yes—
282C2×Dic43C2512yes—
283D172—416no—
284D172—48no—
285D172—416no—
286C4×D43—232no—
287C4×D43—216no—
288C4×D43—28no—
289C4×D43—416no—
290C4×D43—416no—
291Dic86—232no—
292Dic86—48no—
293C2×#3548—431nomaximal, Sylow
294C22×C43—28no—
295C22×C43—28no—
296C22×C43—28no—
297C22×C43—416no—
298C22×C43C2×H(5,𝔽2)14yes—
299C22×C43C2×2−1+412yes—
300C22×C43C2611yescharacteristic
301C2×D43—28no—
302C2×D43—24no—
303C2×D43—28no—
304C2×D43—22no—
305C2×D43—28no—
306C2×D43—416no—
307C2×D43—44no—
308C4×C43—28no—
309C4×C43—24no—
310C4×C43—28no—
311C4×C43—22no—
312Dic43—28no—
313Dic43—28no—
314Dic43C2×H(5,𝔽2)18yes—
315Dic43C2×2−1+414yes—
316#3548—4332no—
317C2×#562—431no—
318C2×#562—432no—
319C2×#562D4311yescharacteristic, complemented
320C2×#554—434no—
321C2×#550—434no—
322C2×#547—434no—
323C2×#540—434no—
324C2×#537—434no—
325C2×#537—434no—
326C2×#533—431no—
327C2×#532—432no—
328C2×C43—48no—
329C2×C43#354814yescomplemented
330C2×C43C22×H(5,𝔽2)12yes—
331C2×C43C22×2−1+411yescharacteristic
332D43—48no—
333D43—42no—
334C22×#41—432no—
335C22×#41—432no—
336C22×#41—434no—
337C22×#41—434no—
338#562—4316no—
339#562—4332no—
340#562C2×D43116yescomplemented
341#554—4364no—
342#550—4364no—
343#547—4364no—
344#540—4364no—
345#537—4364no—
346#537—4364no—
347#533—4316no—
348#532—4332no—
349C2×C4⋊Q8—432no—
350C2×C4⋊Q8—432no—
351C2×#127—432no—
352C2×#127—434no—
353C2×#125—434no—
354C2×#125—431no—
355C2×#125C2×D4311yescharacteristic
356C2×#124—432no—
357C2×#124C2×D4312yes—
358C2×#123—434no—
359C2×#123—434no—
360C2×#123—434no—
361C2×#123C2×D4314yes—
362C2×C22⋊Q8—434no—
363C2×C22⋊Q8—434no—
364C2×C22⋊Q8—434no—
365C2×C22⋊Q8—434no—
366C2×C22⋊Q8—434no—
367C2×C22⋊Q8—434no—
368C2×C22⋊Q8C2×D4314yes—
369C2×#121—434no—
370C2×#121—434no—
371C2×#121—434no—
372C2×#121—434no—
373C2×#121—434no—
374C2×#121—434no—
375C2×#121—434no—
376C2×#121—434no—
377C2×#121—434no—
378C2×#121—434no—
379C2×#121—434no—
380C2×#121C2×D4314yes—
381C2×C4×Q8—434no—
382C2×C4×D4—434no—
383C2×C4×D4—434no—
384C2×C4×D4—434no—
385C2×C4×D4—434no—
386C2×#117—432no—
387C22×C4⋊C4—434no—
388C22×C4⋊C4—434no—
389C22×C42—431no—
390C43C2×#354811yescharacteristic, complemented, Sylow
391C2×#41—4316no—
392C2×#41—4316no—
393C2×#41—4332no—
394C2×#41—4332no—
395C2×#41—17216no—
396C2×#41—1728no—
397C2×#41—1728no—
398C2×#41—17216no—
399C22×Q8—434no—
400C22×Q8—432no—
401C22×Q8C22×D4312yes—
402C22×D4—434no—
403C22×D4—434no—
404C22×D4—434no—
405C22×D4—434no—
406C22×D4—434no—
407C22×D4—434no—
408C22×D4—434no—
409C22×D4—432no—
410C22×D4C22×D4314yes—
411C22×D4C22×D4312yes—
412C23×C4—434no—
413C23×C4—434no—
414C23×C4—434no—
415C23×C4—431no—
416C23×C4—432no—
417C23×C4—434no—
418C23×C4—434no—
419C23×C4C22×D4314yes—
420C4⋊Q8—4316no—
421C4⋊Q8—4316no—
422#127—4316no—
423#127—4332no—
424#125—4332no—
425#125—438no—
426#125C22×D4318yes—
427#124—4316no—
428#124C22×D43116yes—
429#123—4332no—
430#123—4332no—
431#123—4332no—
432#123C22×D43132yes—
433C22⋊Q8—4332no—
434C22⋊Q8—4332no—
435C22⋊Q8—4332no—
436C22⋊Q8—4332no—
437C22⋊Q8—4332no—
438C22⋊Q8—4332no—
439C22⋊Q8C22×D43132yes—
440#121—4332no—
441#121—4332no—
442#121—4332no—
443#121—4332no—
444#121—4332no—
445#121—4332no—
446#121—4332no—
447#121—4332no—
448#121—4332no—
449#121—4332no—
450#121—4332no—
451#121C22×D43132yes—
452C4×Q8—4332no—
453C4×D4—4332no—
454C4×D4—4332no—
455C4×D4—4332no—
456C4×D4—4332no—
457#117—4316no—
458C2×C4⋊C4—434no—
459C2×C4⋊C4—434no—
460C2×C4⋊C4—434no—
461C2×C4⋊C4—434no—
462C2×C4⋊C4—434no—
463C2×C4⋊C4—4332no—
464C2×C4⋊C4—432no—
465C2×C4⋊C4—434no—
466C2×C4⋊C4—434no—
467C2×C4⋊C4—4332no—
468C2×C4⋊C4C22×D4314yes—
469C2×C4⋊C4C22×D4312yes—
470C2×C4⋊C4C22×D4314yes—
471C2×C22⋊C4—434no—
472C2×C22⋊C4—434no—
473C2×C22⋊C4—434no—
474C2×C22⋊C4—434no—
475C2×C22⋊C4—434no—
476C2×C22⋊C4—434no—
477C2×C22⋊C4C22×D4314yes—
478C2×C22⋊C4C22×D4314yes—
479C2×C22⋊C4C22×D4314yes—
480C2×C42—434no—
481C2×C42—431no—
482C2×C42—432no—
483C2×C42—438no—
484C2×C42C22×D4311yescharacteristic
485C24—431no—
486C24—434no—
487C24C23×D4314yes—
488#41—17264no—
489#41—17232no—
490#41—17232no—
491#41—17264no—
492C2×Q8—44no—
493C2×Q8—4316no—
494C2×Q8—438no—
495C2×Q8—8616no—
496C2×Q8—1724no—
497C2×Q8C23×D4318yes—
498C2×D4—48no—
499C2×D4—44no—
500C2×D4—4316no—
501C2×D4—4316no—
502C2×D4—4316no—
503C2×D4—4316no—
504C2×D4—4316no—
505C2×D4—4316no—
506C2×D4—4316no—
507C2×D4—438no—
508C2×D4—8616no—
509C2×D4—8616no—
510C2×D4—1728no—
511C2×D4—1728no—
512C2×D4—1724no—
513C2×D4—1728no—
514C2×D4—1728no—
515C2×D4—1728no—
516C2×D4C23×D43116yes—
517C2×D4C23×D4318yes—
518C22×C4—48no—
519C22×C4—431no—
520C22×C4—432no—
521C22×C4—4316no—
522C22×C4—434no—
523C22×C4—434no—
524C22×C4—4316no—
525C22×C4—434no—
526C22×C4—438no—
527C22×C4—4316no—
528C22×C4—4316no—
529C22×C4—4316no—
530C22×C4—8616no—
531C22×C4—868no—
532C22×C4—864no—
533C22×C4—8616no—
534C22×C4—1728no—
535C22×C4—1728no—
536C22×C4—1728no—
537C22×C4C23×D4311yescharacteristic
538C22×C4C23×D4312yes—
539C22×C4C23×D4314yes—
540C22×C4C23×D4314yes—
541C22×C4C23×D43116yes—
542C4⋊C4—4316no—
543C4⋊C4—4316no—
544C4⋊C4—4316no—
545C4⋊C4—4316no—
546C4⋊C4—4316no—
547C4⋊C4—438no—
548C4⋊C4—4316no—
549C4⋊C4—4316no—
550C4⋊C4C23×D43116yes—
551C4⋊C4C23×D4318yes—
552C4⋊C4C23×D43116yes—
553C22⋊C4—4316no—
554C22⋊C4—4316no—
555C22⋊C4—4316no—
556C22⋊C4—4316no—
557C22⋊C4—4316no—
558C22⋊C4—4316no—
559C22⋊C4C23×D43116yes—
560C22⋊C4C23×D43116yes—
561C22⋊C4C23×D43116yes—
562C42—434no—
563C42—438no—
564C42—4316no—
565C42C23×D4314yes—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—432no—
570C23—438no—
571C23—868no—
572C23—864no—
573C23—868no—
574C23—862no—
575C23—868no—
576C23C24×D4318yes—
577C23C24×D4311yescharacteristic
578Q8—48no—
579Q8—8632no—
580Q8—1728no—
581D4—416no—
582D4—48no—
583D4—8632no—
584D4—8632no—
585D4—17216no—
586D4—17216no—
587D4—1728no—
588D4—17216no—
589D4—17216no—
590D4—17216no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—416no—
596C2×C4—438no—
597C2×C4—434no—
598C2×C4—432no—
599C2×C4—434no—
600C2×C4—438no—
601C2×C4—438no—
602C2×C4—8632no—
603C2×C4—8616no—
604C2×C4—868no—
605C2×C4—8632no—
606C2×C4—868no—
607C2×C4—868no—
608C2×C4—17216no—
609C2×C4—17216no—
610C2×C4—17216no—
611C2×C4C24×D4312yes—
612C2×C4C24×D4314yes—
613C2×C4C24×D4318yes—
614C2×C4C24×D4318yes—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—416no—
619C22—868no—
620C22—864no—
621C22—868no—
622C22—862no—
623C22—868no—
624C22—1724no—
625C22—17216no—
626C22H(5,𝔽2)×D4314yes—
627C222−1+4×D4312yes—
628C22C25×D4311yescharacteristic
629C4—28no—
630C4—28no—
631C4—24no—
632C4—22no—
633C4—438no—
634C4—434no—
635C4—868no—
636C4—868no—
637C2—48no—
638C2—1722no—
639C2—1728no—
640C2#1914899114yesdirect product factor
641C2C2×H(5,𝔽2)×D4312yes—
642C2C2×2−1+4×D4311yescharacteristic
subgroupG/H
centerC23C24×D43
commutator, cosocleC22×C43C26
FrattiniC22C25×D43
FittingC2×C43×#562C2
socleC23×C43C25
hypercenterC2×#562D43
hypocenterC43C2×#3548
LCSG ▷ C22×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #19148991C2direct (N × H)
2. C2×#14583647C2semidirect
3. C2×#14583643C2semidirect
4. C2×#14583603C2semidirect
5. C2×#14583567C2semidirect
6. C2×#14583547C2semidirect
7. C2×#14583501C2semidirect
8. C2×#14583479C2semidirect
9. C2×#14583476C2semidirect
10. C2×#14583455C2semidirect
11. C2×#14583448C2semidirect
12. C2×C43×#562C2semidirect
17. #14583647C22semidirect
18. #14583643C22semidirect
19. #14583603C22semidirect
20. #14583567C22semidirect
21. #14583547C22semidirect
22. #14583501C22semidirect
23. #14583479C22semidirect
24. #14583476C22semidirect
25. #14583455C22semidirect
26. #14583448C22semidirect
63. C43×#562C22semidirect
319. C2×#562D43semidirect
329. C2×C43#3548semidirect
340. #562C2×D43semidirect
390. C43C2×#3548semidirect
640. C2#19148991direct (N × H)

Maximal subgroups

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

subgroupindex
#191489912
C2×#145836472
C2×#145836432
C2×#145836032
C2×#145835672
C2×#145835472
C2×#145835012
C2×#145834792
C2×#145834762
C2×#145834552
C2×#145834482
C2×C43×#5622
C2×#354843