Finite Group Databasebeta

← C2×#13117767 #16022248

Subgroups — C2×#13117767 #16022248

Order 3328.

3894 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
1C22×C4×C22⋊D13C211yesmaximal, characteristic
2C22×#88146C211yesmaximal, characteristic, complemented
3C23×C4⋊Dic13C211yesmaximal, characteristic, complemented
4#13117767C2116yesmaximal, direct product factor
5C2×#11718331C212yesmaximal, complemented
6C2×#11718316C212yesmaximal, complemented
7C2×#11718619C212yesmaximal
8C2×#11718638C211yesmaximal, characteristic, complemented
9C2×#11718648C211yesmaximal, characteristic, complemented
10C2×C4⋊C4×Dic13C211yesmaximal, characteristic, complemented
11C2×C22⋊C4×Dic13C212yesmaximal, complemented
12C2×C13×#380C211yesmaximal, characteristic, complemented
13C24×C4×C13C2211yescharacteristic
14C23×C22⋊D13C2211yescharacteristic
15C24×Dic13C2212yes—
16C23×C4×D13C2211yescharacteristic
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C13×C4⋊C4C2211yescharacteristic
19C22×C13×C22⋊C4C2212yes—
20C22×C22⋊Dic13C2211yescharacteristic
21C2×C4×C22⋊D13C4116yescomplemented
22C2×C4×C22⋊D13C2218yes—
23C22×#6103C2212yes—
24C22×#6103C2212yes—
25C22×#6103C2211yescharacteristic
26C22×C4⋊Dic13—28no—
27C22×C4⋊Dic13C2211yescharacteristic
28C22×C4⋊Dic13C2212yes—
29C22×C4⋊Dic13C2211yescharacteristic
30C22×C4⋊Dic13C2218yescomplemented
31C22×#6101C2211yescharacteristic
32C22×C4×Dic13C2212yes—
33C22×C4×Dic13C2212yes—
34C22×C4×Dic13C2211yescharacteristic
35C2×#88146—28no—
36C2×#88146C2218yescomplemented
37C13×#380C2218yescomplemented
38C2×C13×#95C2212yes—
39C2×#88071C2212yes—
40C2×#88071C2212yes—
41C2×#88071C2212yes—
42C2×#88071C2212yes—
43#11718648C2218yescomplemented
44#11718638C2218yescomplemented
45C4⋊C4×Dic13C2218yescomplemented
46#11718619C22116yes—
47C22⋊C4×Dic13C22116yescomplemented
48#11718331C22116yescomplemented
49#11718316C22116yescomplemented
50C25×C13C2311yescharacteristic
51C24×D13C2311yescharacteristic
52C23×C4×C13—24no—
53C23×C4×C13—24no—
54C23×C4×C13C2×C418yes—
55C23×C4×C13C2312yes—
56C23×C4×C13C2312yes—
57C23×C4×C13C2314yes—
58C23×C4×C13C2311yescharacteristic
59C23×C4×C13C2311yescharacteristic
60C22×C22⋊D13—24no—
61C22×C22⋊D13—24no—
62C22×C22⋊D13C2×C418yes—
63C22×C22⋊D13C2314yes—
64C23×Dic13—28no—
65C23×Dic13—28no—
66C23×Dic13—44no—
67C23×Dic13C2312yes—
68C23×Dic13C2318yes—
69C23×Dic13C2311yescharacteristic
70C23×Dic13C2311yescharacteristic
71C23×Dic13C2311yescharacteristic
72C23×Dic13C2312yes—
73C22×C4×D13—24no—
74C22×C4×D13—24no—
75C22×C4×D13C2×C418yescomplemented
76C22×C4×D13C2314yes—
77C2×C13×C4⋊C4—24no—
78C2×C13×C4⋊C4—24no—
79C2×C13×C4⋊C4C2314yes—
80C2×C13×C4⋊C4C2314yes—
81C2×C13×C22⋊C4—28no—
82C2×C13×C22⋊C4C2318yes—
83C2×C22⋊Dic13C2×C418yes—
84C2×C22⋊Dic13C2314yes—
85C4×C22⋊D13C2×C4164yescomplemented
86C2×#6103—28no—
87C2×#6103—44no—
88C2×#6103C2×C418yescomplemented
89C2×#6103C2318yes—
90C2×#6103C2318yes—
91C2×#6103C2314yes—
92C2×C4⋊Dic13—24no—
93C2×C4⋊Dic13—28no—
94C2×C4⋊Dic13—232no—
95C2×C4⋊Dic13D418yescomplemented
96C2×C4⋊Dic13C2318yes—
97C2×C4⋊Dic13C2314yes—
98C2×C4⋊Dic13C2314yes—
99C2×#6101C2×C418yescomplemented
100C2×#6101C2314yes—
101C2×C4×Dic13—28no—
102C2×C4×Dic13—44no—
103C2×C4×Dic13C2×C418yescomplemented
104C2×C4×Dic13C2318yes—
105C2×C4×Dic13C2318yes—
106C2×C4×Dic13C2314yes—
107#88146—232no—
108C13×#95C2318yes—
109#88071C2318yes—
110#88071C2318yes—
111#88071C2318yes—
112#88071C2318yes—
113C2×#2352—131nomaximal, Sylow
114C24×C13—22no—
115C24×C13—22no—
116C24×C13—22no—
117C24×C13—22no—
118C24×C13C22×C414yes—
119C24×C13#4114yes—
120C24×C13#4114yes—
121C24×C13C2412yes—
122C24×C13C2411yescharacteristic
123C23×D13—22no—
124C23×D13—22no—
125C23×D13—22no—
126C23×D13—22no—
127C23×D13—22no—
128C23×D13—22no—
129C23×D13C22×C414yes—
130C23×D13C2412yes—
131C22×C4×C13—22no—
132C22×C4×C13—22no—
133C22×C4×C13—232no—
134C22×C4×C13—28no—
135C22×C4×C13—28no—
136C22×C4×C13—24no—
137C22×C4×C13—24no—
138C22×C4×C13—24no—
139C22×C4×C13—24no—
140C22×C4×C13—24no—
141C22×C4×C13—24no—
142C22×C4×C13C22×C414yes—
143C22×C4×C13C22×C4116yes—
144C22×C4×C13C22×C414yes—
145C22×C4×C13C2×D414yes—
146C22×C4×C13#4114yes—
147C22×C4×C13C2414yes—
148C22×C4×C13C2414yes—
149C22×C4×C13C2412yes—
150C22×C4×C13C2412yes—
151C2×C22⋊D13—232no—
152C2×C22⋊D13—28no—
153C2×C22⋊D13—28no—
154C2×C22⋊D13C22×C4116yes—
155C22×Dic13—22no—
156C22×Dic13—22no—
157C22×Dic13—24no—
158C22×Dic13—22no—
159C22×Dic13—22no—
160C22×Dic13—22no—
161C22×Dic13—24no—
162C22×Dic13—22no—
163C22×Dic13—24no—
164C22×Dic13—216no—
165C22×Dic13—24no—
166C22×Dic13—24no—
167C22×Dic13—216no—
168C22×Dic13—48no—
169C22×Dic13—432no—
170C22×Dic13C22×C414yes—
171C22×Dic13C22×C414yes—
172C22×Dic13C22×C414yes—
173C22×Dic13C2×D418yes—
174C22×Dic13C2414yes—
175C22×Dic13C2412yes—
176C22×Dic13C2412yes—
177C22×Dic13C2412yes—
178C22×Dic13C2414yes—
179C2×C4×D13—232no—
180C2×C4×D13—28no—
181C2×C4×D13—28no—
182C2×C4×D13C22×C4116yescomplemented
183C13×C4⋊C4—28no—
184C13×C4⋊C4—28no—
185C13×C22⋊C4—216no—
186C22⋊Dic13C22×C4116yes—
187#6103—216no—
188#6103—48no—
189#6103C22×C4116yescomplemented
190C4⋊Dic13—28no—
191C4⋊Dic13—216no—
192C4⋊Dic13C2×D4116yescomplemented
193#6101C22×C4116yescomplemented
194C4×Dic13—216no—
195C4×Dic13—48no—
196C4×Dic13C22×C4116yescomplemented
197C22×C4×D4—131no—
198C22×#117—131no—
199C23×C4⋊C4—131no—
200#2352—1316no—
201C2×#388—132no—
202C2×#387—132no—
203C2×#386—132no—
204C2×#382—131no—
205C2×#380—131no—
206C2×#380D1311yescharacteristic, complemented
207C2×C4×C4⋊C4—131no—
208C2×C4×C22⋊C4—132no—
209C23×C13—28no—
210C23×C13—28no—
211C23×C13—28no—
212C23×C13—28no—
213C23×C13—28no—
214C23×C13—28no—
215C23×C13—22no—
216C23×C13—22no—
217C23×C13—22no—
218C23×C13—22no—
219C23×C13#117116yescomplemented
220C23×C13C23×C414yes—
221C23×C13C23×C412yes—
222C23×C13C22×D412yes—
223C23×C13C2×#4114yes—
224C23×C13C2×#4114yes—
225C23×C13C2×#4112yes—
226C23×C13C2×#4112yes—
227C23×C13C2×#4112yes—
228C23×C13H(5,𝔽2)12yes—
229C23×C132−1+412yes—
230C23×C13C2511yescharacteristic
231C22×D13—28no—
232C22×D13—28no—
233C22×D13—28no—
234C22×D13—22no—
235C22×D13—22no—
236C22×D13—22no—
237C22×D13—22no—
238C22×D13—22no—
239C22×D13—22no—
240C22×D13—44no—
241C22×D13—44no—
242C22×D13—44no—
243C22×D13—44no—
244C22×D13C23×C414yes—
245C2×C4×C13—28no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—232no—
249C2×C4×C13—24no—
250C2×C4×C13—24no—
251C2×C4×C13—24no—
252C2×C4×C13—24no—
253C2×C4×C13—24no—
254C2×C4×C13—24no—
255C2×C4×C13—48no—
256C2×C4×C13—48no—
257C2×C4×C13C4×D4116yes—
258C2×C4×C13C23×C414yes—
259C2×C4×C13C23×C414yes—
260C2×C4×C13C22×D414yes—
261C2×C4×C13C2×#4114yes—
262C22⋊D13—232no—
263C2×Dic13—28no—
264C2×Dic13—28no—
265C2×Dic13—28no—
266C2×Dic13—216no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—24no—
270C2×Dic13—22no—
271C2×Dic13—22no—
272C2×Dic13—22no—
273C2×Dic13—24no—
274C2×Dic13—22no—
275C2×Dic13—24no—
276C2×Dic13—24no—
277C2×Dic13—24no—
278C2×Dic13—432no—
279C2×Dic13—48no—
280C2×Dic13C23×C414yes—
281C2×Dic13C23×C414yes—
282C2×Dic13C23×C414yes—
283C2×Dic13C22×D418yes—
284C4×D13—232no—
285C23×D4—131no—
286C24×C4—131no—
287C24×C4—132no—
288C24×C4C2×D1311yescharacteristic
289C2×C4×D4—1316no—
290C2×C4×D4—138no—
291C2×#117—138no—
292C2×#117—268no—
293C22×C4⋊C4—131no—
294C22×C4⋊C4—131no—
295C22×C4⋊C4—138no—
296C22×C4⋊C4—131no—
297C22×C4⋊C4—132no—
298C22×C4⋊C4—268no—
299C22×C4⋊C4C2×D1311yescharacteristic
300C22×C4⋊C4C2×D1311yescharacteristic
301C22×C22⋊C4—131no—
302C22×C22⋊C4—131no—
303C22×C22⋊C4—132no—
304C22×C22⋊C4—132no—
305C22×C22⋊C4C2×D1312yes—
306C22×C42—131no—
307C22×C42—132no—
308C22×C42—132no—
309#388—1316no—
310#387—1316no—
311#386—1316no—
312#382—138no—
313#380—138no—
314#380C2×D1318yescomplemented
315C4×C4⋊C4—138no—
316C4×C22⋊C4—1316no—
317C2×#95—132no—
318C2×#95—132no—
319C2×#95—132no—
320C2×#95—132no—
321C2×#95C2×D1312yes—
322C22×C13—232no—
323C22×C13—24no—
324C22×C13—24no—
325C22×C13—24no—
326C22×C13—24no—
327C22×C13—24no—
328C22×C13—24no—
329C22×C13C2×#11714yes—
330C22×C13C2×#11718yescomplemented
331C22×C13C2×C4×D414yes—
332C22×C13#52014yes—
333C22×C13#54614yes—
334C22×C13#54814yes—
335C22×C13#55214yes—
336C22×C13#55314yes—
337C22×C13C24×C411yescharacteristic
338C22×C13C23×D411yescharacteristic
339C22×C13C22×#4111yescharacteristic
340C22×C13C22×#4111yescharacteristic
341C22×C13C22×#4111yescharacteristic
342C22×C13C2×H(5,𝔽2)11yescharacteristic
343C22×C13C2×2−1+411yescharacteristic
344C2×D13—24no—
345C2×D13—24no—
346C2×D13—24no—
347C2×D13—416no—
348C2×D13—42no—
349C2×D13—42no—
350C2×D13—42no—
351C2×D13—42no—
352C4×C13—24no—
353C4×C13—44no—
354C4×C13—44no—
355C4×C13C2×C4×D418yes—
356Dic13—24no—
357Dic13—24no—
358Dic13—24no—
359Dic13—28no—
360Dic13—44no—
361C25—131no—
362C25C22×D1311yescharacteristic
363C22×D4—138no—
364C22×D4—134no—
365C22×D4—264no—
366C22×D4—264no—
367C23×C4—24no—
368C23×C4—24no—
369C23×C4—131no—
370C23×C4—132no—
371C23×C4—131no—
372C23×C4—131no—
373C23×C4—138no—
374C23×C4—134no—
375C23×C4—132no—
376C23×C4—138no—
377C23×C4—268no—
378C23×C4—264no—
379C23×C4—264no—
380C23×C4—268no—
381C23×C4—524no—
382C23×C4C4×D1318yes—
383C23×C4C22×D1312yes—
384C23×C4C22×D1312yes—
385C23×C4C22×D1314yes—
386C23×C4C22×D1311yescharacteristic
387C23×C4C22×D1311yescharacteristic
388C4×D4—1364no—
389#117—2632no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—138no—
393C2×C4⋊C4—138no—
394C2×C4⋊C4—134no—
395C2×C4⋊C4—134no—
396C2×C4⋊C4—134no—
397C2×C4⋊C4—138no—
398C2×C4⋊C4—2632no—
399C2×C4⋊C4—268no—
400C2×C4⋊C4—264no—
401C2×C4⋊C4C22×D1314yes—
402C2×C4⋊C4C22×D1314yes—
403C2×C22⋊C4—28no—
404C2×C22⋊C4—138no—
405C2×C22⋊C4—134no—
406C2×C22⋊C4—138no—
407C2×C22⋊C4—134no—
408C2×C22⋊C4—138no—
409C2×C22⋊C4—138no—
410C2×C22⋊C4—268no—
411C2×C22⋊C4—524no—
412C2×C22⋊C4C22×D1318yes—
413C2×C42—138no—
414C2×C42—134no—
415C2×C42—138no—
416C2×C42—138no—
417C2×C42—268no—
418C2×C42—524no—
419#95—138no—
420#95—138no—
421#95—138no—
422#95—138no—
423#95C22×D1318yes—
424C2×C13—28no—
425C2×C13#235218yescomplemented
426C2×C13C22×#11711yescharacteristic
427C2×C13C22×C4×D411yescharacteristic
428C2×C13C2×#52011yescharacteristic
429C2×C13C2×#54611yescharacteristic
430C2×C13C2×#54811yescharacteristic
431C2×C13C2×#55211yescharacteristic
432C2×C13C2×#55311yescharacteristic
433D13—44no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—22no—
438C24—134no—
439C24—132no—
440C24—262no—
441C24—262no—
442C24—262no—
443C24—262no—
444C24—262no—
445C24—262no—
446C24C2×C4×D1314yes—
447C24#612914yes—
448C24#613114yes—
449C24C23×D1312yes—
450C24C23×D1311yescharacteristic
451C2×D4—1316no—
452C2×D4—2632no—
453C2×D4—268no—
454C2×D4—268no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—232no—
458C22×C4—28no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—134no—
467C22×C4—134no—
468C22×C4—134no—
469C22×C4—138no—
470C22×C4—132no—
471C22×C4—132no—
472C22×C4—132no—
473C22×C4—134no—
474C22×C4—1316no—
475C22×C4—134no—
476C22×C4—2616no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4—264no—
481C22×C4—262no—
482C22×C4—262no—
483C22×C4—264no—
484C22×C4—262no—
485C22×C4—2632no—
486C22×C4—268no—
487C22×C4—268no—
488C22×C4—264no—
489C22×C4—2616no—
490C22×C4—264no—
491C22×C4—264no—
492C22×C4—528no—
493C22×C4—5232no—
494C22×C4C2×C4×D1314yes—
495C22×C4C2×C4×D13116yes—
496C22×C4C2×C4×D1314yes—
497C22×C4D4×D1314yes—
498C22×C4#612914yes—
499C22×C4C23×D1314yes—
500C22×C4C23×D1314yes—
501C22×C4C23×D1312yes—
502C22×C4C23×D1312yes—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—1316no—
506C4⋊C4—1316no—
507C4⋊C4—2616no—
508C4⋊C4—268no—
509C22⋊C4—216no—
510C22⋊C4—1316no—
511C22⋊C4—1316no—
512C22⋊C4—2616no—
513C22⋊C4—528no—
514C42—1316no—
515C42—2616no—
516C42—528no—
517C13C2×#235211yescharacteristic, complemented, Sylow
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—134no—
529C23—268no—
530C23—268no—
531C23—268no—
532C23—262no—
533C23—262no—
534C23—262no—
535C23—262no—
536C23—262no—
537C23—262no—
538C23—524no—
539C23—524no—
540C23—524no—
541C23—524no—
542C23#88146116yescomplemented
543C23C22×C4×D1314yes—
544C23C22×C4×D1312yes—
545C23C2×D4×D1312yes—
546C23C2×#612914yes—
547C23C2×#612912yes—
548C23C2×#612912yes—
549C23C2×#613114yes—
550C23C2×#613112yes—
551C23#8825612yes—
552C23#8825712yes—
553C23C24×D1311yescharacteristic
554D4—2632no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—232no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—134no—
568C2×C4—134no—
569C2×C4—134no—
570C2×C4—138no—
571C2×C4—268no—
572C2×C4—268no—
573C2×C4—268no—
574C2×C4—2616no—
575C2×C4—262no—
576C2×C4—262no—
577C2×C4—262no—
578C2×C4—264no—
579C2×C4—262no—
580C2×C4—262no—
581C2×C4—264no—
582C2×C4—262no—
583C2×C4—2632no—
584C2×C4—264no—
585C2×C4—264no—
586C2×C4—264no—
587C2×C4—5232no—
588C2×C4—528no—
589C2×C4#88135116yes—
590C2×C4C22×C4×D1314yes—
591C2×C4C22×C4×D1314yes—
592C2×C4C2×D4×D1314yes—
593C2×C4C2×#612914yes—
594C22—24no—
595C22—232no—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—264no—
602C22—264no—
603C22—264no—
604C22—5216no—
605C22—522no—
606C22—522no—
607C22—522no—
608C22—522no—
609C22C2×#8813514yes—
610C22C2×#8814614yes—
611C22C2×#8814618yescomplemented
612C22#1171922814yes—
613C22#1171930314yes—
614C22#1171932314yes—
615C22#1171933514yes—
616C22#1171934714yes—
617C22C23×C4×D1311yescharacteristic
618C22C22×D4×D1311yescharacteristic
619C22C22×#612911yescharacteristic
620C22C22×#612911yescharacteristic
621C22C22×#613111yescharacteristic
622C22C2×#8825611yescharacteristic
623C22C2×#8825711yescharacteristic
624C4—24no—
625C4—44no—
626C4—44no—
627C4—264no—
628C4—264no—
629C4—268no—
630C4—264no—
631C4—524no—
632C4C2×#8813518yes—
633C2—28no—
634C2—524no—
635C2#1311776718yesdirect product factor
636C2C22×#8814611yescharacteristic
637C2C22×#8813511yescharacteristic
638C2C2×#1171922811yescharacteristic
639C2C2×#1171933511yescharacteristic
640C2C2×#1171930311yescharacteristic
641C2C2×#1171934711yescharacteristic
642C2C2×#1171932311yescharacteristic
subgroupG/H
centerC24C23×D13
commutatorC22×C13C24×C4
FrattiniC23C24×D13
FittingC2×C13×#380C2
socleC24×C13C24
prime radicalC23×C22⋊D13C22
cosocleC23×C13C25
hypercenterC2×#380D13
hypocenterC13C2×#2352
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#88146C2semidirect
3. C23×C4⋊Dic13C2semidirect
4. #13117767C2direct (N × H)
5. C2×#11718331C2semidirect
6. C2×#11718316C2semidirect
8. C2×#11718638C2semidirect
9. C2×#11718648C2semidirect
10. C2×C4⋊C4×Dic13C2semidirect
11. C2×C22⋊C4×Dic13C2semidirect
12. C2×C13×#380C2semidirect
21. C2×C4×C22⋊D13C4semidirect
30. C22×C4⋊Dic13C22semidirect
36. C2×#88146C22semidirect
37. C13×#380C22semidirect
43. #11718648C22semidirect
44. #11718638C22semidirect
45. C4⋊C4×Dic13C22semidirect
47. C22⋊C4×Dic13C22semidirect
48. #11718331C22semidirect
49. #11718316C22semidirect
75. C22×C4×D13C2×C4semidirect
85. C4×C22⋊D13C2×C4semidirect
88. C2×#6103C2×C4semidirect
95. C2×C4⋊Dic13D4semidirect
99. C2×#6101C2×C4semidirect
103. C2×C4×Dic13C2×C4semidirect
182. C2×C4×D13C22×C4semidirect
189. #6103C22×C4semidirect
192. C4⋊Dic13C2×D4semidirect
193. #6101C22×C4semidirect
196. C4×Dic13C22×C4semidirect
206. C2×#380D13semidirect
219. C23×C13#117semidirect
314. #380C2×D13semidirect
330. C22×C13C2×#117semidirect
425. C2×C13#2352semidirect
517. C13C2×#2352semidirect
542. C23#88146semidirect
611. C22C2×#88146semidirect
635. C2#13117767direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D132
C22×#881462
C23×C4⋊Dic132
#131177672
C2×#117183312
C2×#117183162
C2×#117186192
C2×#117186382
C2×#117186482
C2×C4⋊C4×Dic132
C2×C22⋊C4×Dic132
C2×C13×#3802
C2×#235213