Finite Group Databasebeta

← C2×#13131522 #16808188

Subgroups — C2×#13131522 #16808188

Order 3328.

2134 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131522C2116yesmaximal, direct product factor
2C2×#11719077C212yesmaximal, complemented
3C2×#11719068C212yesmaximal, complemented
4C2×#11719055C211yesmaximal, characteristic, complemented
5C2×#11719054C211yesmaximal, characteristic, complemented
6C2×#11719130C211yesmaximal, characteristic, complemented
7C2×#11719128C211yesmaximal, characteristic, complemented
8C2×#11719067C212yesmaximal, complemented
9C2×#11719146C212yesmaximal
10C2×C4×#6122C211yesmaximal, characteristic, complemented
11C22×#88110C211yesmaximal, characteristic, complemented
12C2×C4×#6129C211yesmaximal, characteristic
13C22×#6129C2211yescharacteristic
14C22×#6122C2212yes—
15C23×#1057C2212yes—
16C2×C4×D4×C13C411yescharacteristic
17C2×D4×Dic13C411yescharacteristic
18C2×C4×C22⋊D13C411yescharacteristic
19C2×C4×C22⋊D13C411yescharacteristic
20C22×C4×Dic13C2211yescharacteristic
21C22×C4×Dic13C2211yescharacteristic
22C4×#6129C418yes—
23C4×#6129C2218yes—
24C2×#88146C2211yescharacteristic
25C2×#88141C411yescharacteristic
26C2×#88135C411yescharacteristic
27C2×#88135C411yescharacteristic
28C2×#88131C2211yescharacteristic
29C2×#88131C2211yescharacteristic
30C2×C42×D13C2211yescharacteristic
31C2×C4×Dic26C411yescharacteristic
32C2×C22⋊#1057C2212yes—
33C2×C22⋊#1057C2212yes—
34#11719146C22116yes—
35#11719130C2218yescomplemented
36#11719128C2218yescomplemented
37C2×#88112C2212yes—
38C2×#88112C2212yes—
39C2×#88111C2212yes—
40C2×#88111C2212yes—
41C4×#6122C2218yescomplemented
42C2×#88110C2211yescharacteristic
43C2×#88110C2211yescharacteristic
44C2×#88110C2212yes—
45C2×#88110C2211yescharacteristic
46C2×#88110C2211yescharacteristic
47C2×#88110C2218yescomplemented
48C2×C4⋊#1057C2212yes—
49C2×C4⋊#1057C2212yes—
50C2×C4×#1057C2212yes—
51#11719077C22116yescomplemented
52#11719068C22116yescomplemented
53#11719067C22116yescomplemented
54#11719055C2218yescomplemented
55#11719054C2218yescomplemented
56C22×D4×C13C2×C411yescharacteristic
57C23×C4×C13C2×C411yescharacteristic
58C23×C4×C13C2×C411yescharacteristic
59C22×C22⋊D13C2×C411yescharacteristic
60C22×C22⋊D13C2×C411yescharacteristic
61C23×Dic13C2×C411yescharacteristic
62C23×Dic13C2×C411yescharacteristic
63C23×Dic13C2311yescharacteristic
64C23×Dic13C2311yescharacteristic
65C2×#6129—28no—
66C2×#6129C2×C414yes—
67C2×#6129C2314yes—
68C22×C4×D13C2×C411yescharacteristic
69C22×C4×D13C2×C411yescharacteristic
70C22×C4×D13C2311yescharacteristic
71C22×Dic26C2×C411yescharacteristic
72C2×#6122—28no—
73C2×#6122C2318yes—
74C22×#1057—28no—
75C22×#1057C2312yes—
76C22×#1057C2312yes—
77C22×#1057C2312yes—
78C22×#1057C2312yes—
79C22×#1057C2318yes—
80C4×D4×C13C2×C418yes—
81C2×C13×C4⋊C4C2×C411yescharacteristic
82C2×C13×C22⋊C4C2×C411yescharacteristic
83C2×C13×C22⋊C4C2×C411yescharacteristic
84C2×C42×C13C2×C411yescharacteristic
85C2×C22⋊Dic13C2×C411yescharacteristic
86C2×C22⋊Dic13C2×C411yescharacteristic
87D4×Dic13C2×C418yes—
88C4×C22⋊D13C2×C418yes—
89C4×C22⋊D13C2×C418yes—
90C2×#6103C2×C411yescharacteristic
91C2×#6103C2×C411yescharacteristic
92C2×C4⋊Dic13C2×C411yescharacteristic
93C2×#6101C2×C411yescharacteristic
94C2×#6101C2×C411yescharacteristic
95C2×#6101C2×C411yescharacteristic
96C2×#6101C2×C411yescharacteristic
97C2×C4×Dic13C2×C411yescharacteristic
98C2×C4×Dic13C2×C411yescharacteristic
99C2×C4×Dic13C2×C411yescharacteristic
100C2×C4×Dic13C2×C411yescharacteristic
101C2×C4×Dic13C2×C411yescharacteristic
102C2×C4×Dic13C2×C414yes—
103C2×C4×Dic13C2×C414yes—
104C2×C4×Dic13C2311yescharacteristic
105C2×C4×Dic13C2311yescharacteristic
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C2×C4×Dic13C2314yes—
109C2×C4×Dic13C2314yes—
110#88146C2×C414yes—
111#88146C2314yes—
112#88141C2×C418yes—
113#88135C2×C418yes—
114#88135C2×C418yes—
115#88131C2×C414yes—
116#88131C2×C414yes—
117#88131C2314yes—
118#88131C2314yes—
119C42×D13C2×C414yes—
120C42×D13C2314yes—
121C4×Dic26C2×C418yes—
122C22⋊#1057C2318yes—
123C22⋊#1057C2318yes—
124#88112C2318yes—
125#88112C2318yes—
126#88111C2318yes—
127#88111C2318yes—
128#88110C2314yes—
129#88110C2314yes—
130#88110C2314yes—
131#88110C2314yes—
132#88110C2318yes—
133C4⋊#1057C2318yes—
134C4⋊#1057C2318yes—
135C4×#1057C2318yes—
136C2×#2923—131nomaximal, Sylow
137C24×C13C22×C411yescharacteristic
138C24×C13C22×C411yescharacteristic
139C23×D13C22×C411yescharacteristic
140C2×D4×C13—24no—
141C2×D4×C13C22×C414yes—
142C22×C4×C13—42no—
143C22×C4×C13—42no—
144C22×C4×C13C22×C411yescharacteristic
145C22×C4×C13C22×C411yescharacteristic
146C22×C4×C13C22×C411yescharacteristic
147C22×C4×C13C22×C411yescharacteristic
148C22×C4×C13C22×C411yescharacteristic
149C22×C4×C13C22×C414yes—
150C22×C4×C13C22×C414yes—
151C2×C22⋊D13—24no—
152C2×C22⋊D13—24no—
153C2×C22⋊D13C22×C414yes—
154C2×C22⋊D13C22×C414yes—
155C22×Dic13—24no—
156C22×Dic13—42no—
157C22×Dic13—42no—
158C22×Dic13C22×C412yes—
159C22×Dic13C22×C411yescharacteristic
160C22×Dic13C22×C411yescharacteristic
161C22×Dic13C22×C411yescharacteristic
162C22×Dic13C22×C411yescharacteristic
163C22×Dic13C22×C411yescharacteristic
164C22×Dic13C22×C414yes—
165C22×Dic13C22×C414yes—
166C22×Dic13C22×C412yes—
167C22×Dic13C22×C411yescharacteristic
168C22×Dic13#4118yes—
169C22×Dic13C2412yes—
170C22×Dic13C2412yes—
171C22×Dic13C2411yescharacteristic
172#6129—216no—
173#6129—216no—
174C2×C4×D13—24no—
175C2×C4×D13—42no—
176C2×C4×D13—42no—
177C2×C4×D13C22×C412yes—
178C2×C4×D13C22×C414yes—
179C2×C4×D13C22×C414yes—
180C2×C4×D13C2412yes—
181C2×Dic26—24no—
182C2×Dic26C22×C414yes—
183#6122—216no—
184C2×#1057—24no—
185C2×#1057—24no—
186C2×#1057—216no—
187C2×#1057—24no—
188C2×#1057—24no—
189C2×#1057C2414yes—
190C2×#1057C2414yes—
191C2×#1057C2414yes—
192C2×#1057C2414yes—
193C13×C4⋊C4C22×C414yes—
194C13×C22⋊C4C22×C414yes—
195C13×C22⋊C4C22×C414yes—
196C42×C13C22×C414yes—
197C22⋊Dic13C22×C414yes—
198C22⋊Dic13C22×C414yes—
199#6103C22×C414yes—
200#6103C22×C414yes—
201C4⋊Dic13C22×C414yes—
202#6101C22×C414yes—
203#6101C22×C414yes—
204#6101C22×C414yes—
205#6101C22×C414yes—
206C4×Dic13C22×C414yes—
207C4×Dic13C22×C414yes—
208C4×Dic13C22×C412yes—
209C4×Dic13C22×C412yes—
210C4×Dic13C22×C414yes—
211C4×Dic13C22×C412yes—
212C4×Dic13C22×C412yes—
213C4×Dic13C22×C414yes—
214C4×Dic13C22×C414yes—
215C4×Dic13C2412yes—
216C4×Dic13C2412yes—
217C4×Dic13C2412yes—
218C4×Dic13C2412yes—
219C2×C4×#41—131no—
220#2923—1316no—
221C2×#433—132no—
222C2×#433—132no—
223C2×#432—131no—
224C2×#432—131no—
225C2×#432—131no—
226C2×#432—131no—
227C2×#424—132no—
228C2×#408—132no—
229C2×C4×M16—131no—
230C22×#97—131no—
231C23×C13—22no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13—22no—
235C23×C13—22no—
236C23×C13#11714yes—
237C23×C13C23×C412yes—
238C23×C13C23×C412yes—
239C23×C13C23×C411yescharacteristic
240C22×D13—22no—
241C22×D13—22no—
242C22×D13—22no—
243C22×D13C23×C412yes—
244D4×C13—28no—
245C2×C4×C13—22no—
246C2×C4×C13—22no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—44no—
251C2×C4×C13—44no—
252C2×C4×C13C23×C412yes—
253C2×C4×C13C23×C412yes—
254C2×C4×C13C23×C412yes—
255C2×C4×C13C23×C412yes—
256C2×C4×C13C23×C412yes—
257C22⋊D13—28no—
258C22⋊D13—28no—
259C2×Dic13—22no—
260C2×Dic13—22no—
261C2×Dic13—22no—
262C2×Dic13—22no—
263C2×Dic13—22no—
264C2×Dic13—24no—
265C2×Dic13—24no—
266C2×Dic13—44no—
267C2×Dic13—44no—
268C2×Dic13#11714yes—
269C2×Dic13#11718yes—
270C2×Dic13C23×C412yes—
271C2×Dic13C23×C412yes—
272C2×Dic13C23×C412yes—
273C2×Dic13C23×C412yes—
274C2×Dic13C23×C412yes—
275C2×Dic13C23×C411yescharacteristic
276C2×Dic13C23×C412yes—
277C2×Dic13C2×#4114yes—
278C2×Dic13C2×#4118yes—
279C2×Dic13C2511yescharacteristic
280C4×D13—24no—
281C4×D13—24no—
282C4×D13—44no—
283C4×D13—44no—
284Dic26—28no—
285#1057—24no—
286#1057—24no—
287#1057—24no—
288#1057—24no—
289C22×#41—131no—
290C22×M16—132no—
291C23×C8—132no—
292C4×#41—138no—
293C4×#41—138no—
294C2×C4×Q8—131no—
295C2×C4×Q8—131no—
296C2×C4×D4—131no—
297C2×C4×D4—131no—
298C2×C4×D4—131no—
299C2×C4×D4—131no—
300C2×C4×D4—131no—
301C2×C4×D4#26111yescharacteristic
302C2×#117—131no—
303C2×#117—131no—
304C2×#117—131no—
305C22×C42—131no—
306C22×C42—131no—
307C22×C42—131no—
308#433—1316no—
309#433—1316no—
310#432—138no—
311#432—138no—
312#432—138no—
313#432—138no—
314#424—1316no—
315C2×C4⋊C8—132no—
316C2×C4⋊C8—132no—
317C2×C4⋊C8—132no—
318C2×C4⋊C8—132no—
319#408—1316no—
320C2×C22⋊C8—132no—
321C2×C22⋊C8—132no—
322C2×C22⋊C8—132no—
323C2×C22⋊C8—132no—
324C4×M16—138no—
325C2×#97—131no—
326C2×#97—131no—
327C2×#97—132no—
328C2×#97—138no—
329C2×#97—131no—
330C2×#97—131no—
331C2×C4×C8—132no—
332C22×C13—28no—
333C22×C13—22no—
334C22×C13—22no—
335C22×C13—22no—
336C22×C13—22no—
337C22×C13—22no—
338C22×C13—44no—
339C22×C13C2×#11714yes—
340C22×C13C2×#11712yes—
341C22×C13#56812yes—
342C22×C13#56812yes—
343C22×C13C24×C411yescharacteristic
344C2×D13—22no—
345C2×D13—22no—
346C2×D13—22no—
347C2×D13—44no—
348C4×C13—22no—
349C4×C13—22no—
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—22no—
353Dic13—22no—
354Dic13—22no—
355Dic13—22no—
356Dic13—22no—
357Dic13—22no—
358Dic13C2×#11714yes—
359Dic13C2×#11712yes—
360Dic13C22×#4112yes—
361C2×#41—134no—
362C2×#41—134no—
363C2×#41—268no—
364C22×Q8—131no—
365C22×D4—131no—
366C22×D4—131no—
367C22×D4C2×#26111yescharacteristic
368C23×C4—131no—
369C23×C4—131no—
370C23×C4—131no—
371C23×C4—131no—
372C23×C4—131no—
373C23×C4—131no—
374C23×C4—131no—
375C23×C4C2×#26111yescharacteristic
376C23×C4C2×#26111yescharacteristic
377C2×M16—138no—
378C2×M16—268no—
379C22×C8—132no—
380C22×C8—132no—
381C22×C8—138no—
382C22×C8—132no—
383C22×C8—132no—
384C22×C8—268no—
385C4×Q8—138no—
386C4×Q8—138no—
387C4×D4—138no—
388C4×D4—138no—
389C4×D4—138no—
390C4×D4—138no—
391C4×D4—138no—
392C4×D4C2×#26118yes—
393#117—134no—
394#117—134no—
395#117—134no—
396#117—134no—
397#117—134no—
398#117—134no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—131no—
402C2×C4⋊C4—131no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4C2×#26111yescharacteristic
405C2×C22⋊C4—131no—
406C2×C22⋊C4—131no—
407C2×C22⋊C4—131no—
408C2×C22⋊C4—131no—
409C2×C22⋊C4C2×#26111yescharacteristic
410C2×C22⋊C4C2×#26111yescharacteristic
411C2×C42—131no—
412C2×C42—131no—
413C2×C42—131no—
414C2×C42—131no—
415C2×C42—131no—
416C2×C42—131no—
417C2×C42—131no—
418C2×C42—131no—
419C2×C42—134no—
420C2×C42—134no—
421C2×C42—134no—
422C2×C42—134no—
423C2×C42—131no—
424C2×C42—134no—
425C2×C42—134no—
426C2×C42C2×#26111yescharacteristic
427C4⋊C8—138no—
428C4⋊C8—138no—
429C4⋊C8—138no—
430C4⋊C8—138no—
431C22⋊C8—138no—
432C22⋊C8—138no—
433C22⋊C8—138no—
434C22⋊C8—138no—
435#97—134no—
436#97—134no—
437#97—138no—
438#97—134no—
439#97—134no—
440C4×C8—138no—
441C2×C13—24no—
442C2×C13—42no—
443C2×C13#292314yescomplemented
444C2×C13C22×#11711yescharacteristic
445C2×C13C2×#56811yescharacteristic
446C2×C13C2×#56811yescharacteristic
447D13—42no—
448C24—131no—
449C24C22×#26111yescharacteristic
450C24C22×#26111yescharacteristic
451#41—2616no—
452#41—2616no—
453C2×Q8—134no—
454C2×Q8—264no—
455C2×D4—24no—
456C2×D4—134no—
457C2×D4—134no—
458C2×D4—264no—
459C2×D4—264no—
460C2×D4C22×#26114yes—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—138no—
464C22×C4—132no—
465C22×C4—132no—
466C22×C4—131no—
467C22×C4—131no—
468C22×C4—132no—
469C22×C4—132no—
470C22×C4—131no—
471C22×C4—131no—
472C22×C4—131no—
473C22×C4—131no—
474C22×C4—131no—
475C22×C4—134no—
476C22×C4—134no—
477C22×C4—132no—
478C22×C4—132no—
479C22×C4—134no—
480C22×C4—134no—
481C22×C4—264no—
482C22×C4—264no—
483C22×C4—522no—
484C22×C4—522no—
485C22×C4—522no—
486C22×C4—522no—
487C22×C4C22×#26111yescharacteristic
488C22×C4C22×#26111yescharacteristic
489C22×C4C22×#26111yescharacteristic
490C22×C4C22×#26111yescharacteristic
491C22×C4C22×#26111yescharacteristic
492C22×C4C22×#26114yes—
493C22×C4C22×#26114yes—
494M16—2616no—
495C2×C8—134no—
496C2×C8—134no—
497C2×C8—134no—
498C2×C8—134no—
499C2×C8—264no—
500C2×C8—264no—
501C2×C8—2616no—
502C2×C8—264no—
503C2×C8—264no—
504C4⋊C4—134no—
505C4⋊C4—134no—
506C4⋊C4—134no—
507C4⋊C4—134no—
508C4⋊C4—134no—
509C4⋊C4C22×#26114yes—
510C22⋊C4—134no—
511C22⋊C4—134no—
512C22⋊C4—134no—
513C22⋊C4—134no—
514C22⋊C4C22×#26114yes—
515C22⋊C4C22×#26114yes—
516C42—132no—
517C42—132no—
518C42—132no—
519C42—132no—
520C42—132no—
521C42—132no—
522C42—132no—
523C42—132no—
524C42—134no—
525C42—134no—
526C42—134no—
527C42—134no—
528C42—134no—
529C42C22×#26114yes—
530C13C2×#292311yescharacteristic, complemented, Sylow
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—132no—
537C23—262no—
538C23—262no—
539C23—262no—
540C23#8823714yes—
541C23C23×#26112yes—
542C23C23×#26112yes—
543C23C23×#26111yescharacteristic
544Q8—268no—
545D4—28no—
546D4—268no—
547D4—268no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—134no—
556C2×C4—138no—
557C2×C4—134no—
558C2×C4—138no—
559C2×C4—131no—
560C2×C4—131no—
561C2×C4—132no—
562C2×C4—132no—
563C2×C4—132no—
564C2×C4—132no—
565C2×C4—132no—
566C2×C4—132no—
567C2×C4—264no—
568C2×C4—264no—
569C2×C4—264no—
570C2×C4—264no—
571C2×C4—262no—
572C2×C4—262no—
573C2×C4—262no—
574C2×C4—262no—
575C2×C4—262no—
576C2×C4—524no—
577C2×C4—524no—
578C2×C4—524no—
579C2×C4—524no—
580C2×C4C23×#26112yes—
581C2×C4C23×#26112yes—
582C2×C4C23×#26112yes—
583C2×C4C23×#26112yes—
584C2×C4C23×#26112yes—
585C8—264no—
586C8—264no—
587C8—264no—
588C8—264no—
589C22—28no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—44no—
596C22—262no—
597C22—262no—
598C22—262no—
599C22—524no—
600C22C2×#8823714yes—
601C22C2×#8823712yes—
602C22#1171962112yes—
603C22#1171962912yes—
604C22C24×#26111yescharacteristic
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—132no—
611C4—134no—
612C4—132no—
613C4—262no—
614C4—262no—
615C4—262no—
616C4—262no—
617C4—262no—
618C2—24no—
619C2—42no—
620C2—522no—
621C2#1313152214yesdirect product factor
622C2C22×#8823711yescharacteristic
623C2C2×#1171962111yescharacteristic
624C2C2×#1171962911yescharacteristic
subgroupG/H
centerC23C23×#261
commutatorC22×C13C24×C4
FrattiniC22C24×#261
FittingC2×C4×D4×C13C4
socleC23×C13C23×C4
prime radicalC22×#6129C22
cosocleC2×Dic13C25
hypercenterC2×C4×D4#261
hypocenterC13C2×#2923
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #13131522C2direct (N × H)
2. C2×#11719077C2semidirect
3. C2×#11719068C2semidirect
4. C2×#11719055C2semidirect
5. C2×#11719054C2semidirect
6. C2×#11719130C2semidirect
7. C2×#11719128C2semidirect
8. C2×#11719067C2semidirect
10. C2×C4×#6122C2semidirect
11. C22×#88110C2semidirect
35. #11719130C22semidirect
36. #11719128C22semidirect
41. C4×#6122C22semidirect
47. C2×#88110C22semidirect
51. #11719077C22semidirect
52. #11719068C22semidirect
53. #11719067C22semidirect
54. #11719055C22semidirect
55. #11719054C22semidirect
443. C2×C13#2923semidirect
530. C13C2×#2923semidirect
621. C2#13131522direct (N × H)

Maximal subgroups

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

subgroupindex
#131315222
C2×#117190772
C2×#117190682
C2×#117190552
C2×#117190542
C2×#117191302
C2×#117191282
C2×#117190672
C2×#117191462
C2×C4×#61222
C22×#881102
C2×C4×#61292
C2×#292313