Finite Group Databasebeta

← #15885222

Subgroups — #15885222

Order 3328.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×Dic13C211yesmaximal, characteristic, complemented
2C2×SD16×Dic13C211yesmaximal, characteristic
3#13117911C211yesmaximal, characteristic
4#13117757C211yesmaximal, characteristic, complemented
5#13115648C211yesmaximal, characteristic
6#13115606C211yesmaximal, characteristic, complemented
7#13115521C211yesmaximal, characteristic
8#13115514C211yesmaximal, characteristic, complemented
9#13115193C211yesmaximal, characteristic
10#13115179C211yesmaximal, characteristic, complemented
11#13115106C211yesmaximal, characteristic, complemented
12#13115085C211yesmaximal, characteristic
13#13114701C211yesmaximal, characteristic, complemented
14C4×#88075C211yesmaximal, characteristic, complemented
15C2×C13×#464C211yesmaximal, characteristic
16C22×C13×Q16C2211yescharacteristic
17C22×C13×SD16C2211yescharacteristic
18C2×C13×#124C2211yescharacteristic
19C2×C4×Q8×C13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×Q8×Dic13C2211yescharacteristic
22C2×Q8×Dic13C2211yescharacteristic
23C2×D4×Dic13C2211yescharacteristic
24C13×#464C418yescomplemented
25C2×C13×C4⋊C8C2211yescharacteristic
26C2×C13×#103C2211yescharacteristic
27C2×C13×#102C2211yescharacteristic
28#11718891C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33#11718882C2211yescharacteristic
34C2×#88072C2211yescharacteristic
35C2×#88072C2211yescharacteristic
36Q16×Dic13—24no—
37SD16×Dic13—24no—
38C2×#88058C2211yescharacteristic
39C2×#88057C2211yescharacteristic
40C2×C8×Dic13C2211yescharacteristic
41C2×C8×Dic13C2211yescharacteristic
42#11718646—21no—
43#11718646—21no—
44#11718646C2211yescharacteristic
45#11718646C2211yescharacteristic
46#11718612—21no—
47#11718612—21no—
48#11718611—21no—
49C22⋊C4×Dic13—21no—
50#11718597—21no—
51#11718597C2211yescharacteristic
52C4×C4⋊Dic13—21no—
53C4×C4⋊Dic13C2211yescharacteristic
54C4×C4⋊Dic13C2211yescharacteristic
55C2×C4×#1055C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718142C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C22×Q8×C13C2311yescharacteristic
63C22×Q8×C13C2311yescharacteristic
64C22×D4×C13C2311yescharacteristic
65C23×Dic13—21no—
66C2×C13×Q16—22no—
67C2×C13×Q16—22no—
68C2×C13×Q16C2×C414yescomplemented
69C2×C13×SD16—22no—
70C2×C13×SD16—22no—
71C2×C13×SD16C2×C414yes—
72C22×C8×C13C2311yescharacteristic
73C22×C8×C13C2311yescharacteristic
74C13×#124—22no—
75C13×#124C2×C414yes—
76C4×Q8×C13—22no—
77C4×Q8×C13C2×C414yescomplemented
78C2×C13×C4⋊C4—21no—
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C42×C13—21no—
83C2×C42×C13C2311yescharacteristic
84C2×C22⋊Dic13—21no—
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87Q8×Dic13—22no—
88Q8×Dic13—22no—
89Q8×Dic13—22no—
90Q8×Dic13—41no—
91Q8×Dic13—41no—
92Q8×Dic13—41no—
93D4×Dic13—22no—
94D4×Dic13—41no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13—21no—
98C2×C4⋊Dic13—21no—
99C2×C4⋊Dic13—21no—
100C2×C4⋊Dic13D411yescharacteristic
101C2×C4⋊Dic13D411yescharacteristic
102C2×C4⋊Dic13C2311yescharacteristic
103C2×C4⋊Dic13C2311yescharacteristic
104C2×C4⋊Dic13C2311yescharacteristic
105C2×C4⋊Dic13C2311yescharacteristic
106C2×C4×Dic13—21no—
107C2×C4×Dic13—21no—
108C2×C4×Dic13—21no—
109C2×C4×Dic13—21no—
110C2×C4×Dic13C2311yescharacteristic
111C2×C4×Dic13C2311yescharacteristic
112C22×#1055C2311yescharacteristic
113C22×#1055C2311yescharacteristic
114C13×C4⋊C8C2×C414yescomplemented
115C13×#103C2×C414yescomplemented
116C13×#102C2×C414yes—
117#88075—22no—
118#88075—22no—
119#88075—41no—
120#88075D414yes—
121#88072—22no—
122#88072—41no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071—21no—
127#88071—21no—
128#88071—21no—
129#88071—21no—
130#88071D411yescharacteristic
131#88071D411yescharacteristic
132#88058—22no—
133#88057—22no—
134C8×Dic13—22no—
135C8×Dic13—22no—
136C4×#1055—22no—
137#23000—131nomaximal, Sylow
138C24×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13—21no—
142C2×Q8×C13—21no—
143C2×Q8×C13—21no—
144C2×Q8×C13—21no—
145C2×Q8×C13—21no—
146C2×Q8×C13—21no—
147C2×Q8×C13C22×C412yes—
148C2×Q8×C13C22×C412yes—
149C2×Q8×C13C2×D412yes—
150C2×Q8×C13#4112yes—
151C2×D4×C13—21no—
152C2×D4×C13—21no—
153C2×D4×C13—21no—
154C2×D4×C13—21no—
155C2×D4×C13—21no—
156C2×D4×C13C22×C412yes—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13C2×D411yescharacteristic
161C22×C4×C13C2×D411yescharacteristic
162C22×C4×C13C2411yescharacteristic
163C22×Dic13—21no—
164C22×Dic13—21no—
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—42no—
168C22×Dic13—42no—
169C22×Dic13—81no—
170C22×Dic13C2×D411yescharacteristic
171C22×Dic13C2×D411yescharacteristic
172C22×Dic13C2×D411yescharacteristic
173C22×Dic13C2×D411yescharacteristic
174C13×Q16—28no—
175C13×SD16—28no—
176C2×C8×C13—21no—
177C2×C8×C13—21no—
178C2×C8×C13—21no—
179C2×C8×C13—21no—
180C2×C8×C13C22×C412yes—
181C2×C8×C13C22×C412yes—
182C13×C4⋊C4—24no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4C22×C412yes—
185C13×C22⋊C4—24no—
186C13×C22⋊C4—24no—
187C42×C13—24no—
188C42×C13C22×C412yes—
189C42×C13#4112yes—
190C22⋊Dic13—42no—
191C22⋊Dic13—42no—
192C22⋊Dic13—81no—
193C4⋊Dic13—21no—
194C4⋊Dic13—22no—
195C4⋊Dic13—22no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13—21no—
201C4⋊Dic13—21no—
202C4⋊Dic13—42no—
203C4⋊Dic13—42no—
204C4⋊Dic13—42no—
205C4⋊Dic13—42no—
206C4⋊Dic13—42no—
207C4⋊Dic13C2×D412yes—
208C4×Dic13—21no—
209C4×Dic13—21no—
210C4×Dic13—21no—
211C4×Dic13—21no—
212C4×Dic13—42no—
213C4×Dic13—42no—
214C4×Dic13—42no—
215C4×Dic13—81no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055C2×D412yes—
222C2×#464D1311yescharacteristic
223C2×C4×Q16—131no—
224C2×C4×SD16—131no—
225#2367—131no—
226#2351—131no—
227#1958—131no—
228#1949—131no—
229#1934—131no—
230#1933—131no—
231#1880—131no—
232#1878—131no—
233#1865—131no—
234#1862—131no—
235C8×C4⋊C4—131no—
236C4×#103—131no—
237C23×C13—22no—
238C23×C13—41no—
239C23×C13—41no—
240C23×C13—41no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—41no—
244C23×C13C22×D411yescharacteristic
245Q8×C13—22no—
246Q8×C13—22no—
247Q8×C13—22no—
248Q8×C13—22no—
249Q8×C13—22no—
250Q8×C13—41no—
251Q8×C13—41no—
252Q8×C13C4×D414yes—
253D4×C13—22no—
254D4×C13—22no—
255D4×C13—41no—
256D4×C13—41no—
257C2×C4×C13—22no—
258C2×C4×C13—22no—
259C2×C4×C13—22no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—22no—
265C2×C4×C13—22no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13C4×D412yes—
271C2×C4×C13C4×D412yes—
272C2×C4×C13C23×C411yescharacteristic
273C2×C4×C13C22×D411yescharacteristic
274C2×C4×C13C2×#4111yescharacteristic
275C2×C4×C13C2×#4111yescharacteristic
276C2×C4×C13H(5,𝔽2)11yescharacteristic
277C2×C4×C132−1+411yescharacteristic
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—22no—
285C2×Dic13—21no—
286C2×Dic13—21no—
287C2×Dic13—21no—
288C2×Dic13—21no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—84no—
301C2×Dic13#13512yes—
302C2×Dic13#13512yes—
303C8×C13—22no—
304C8×C13—22no—
305#1055—22no—
306#1055—41no—
307C22×Q16C2×D1311yescharacteristic
308C22×SD16C2×D1311yescharacteristic
309C2×#124C2×D1311yescharacteristic
310C2×C4×Q8—131no—
311C2×C4×Q8—131no—
312C2×C4×Q8—131no—
313C2×C4×Q8C2×D1311yescharacteristic
314C2×C4×D4—131no—
315#464Dic1318yescomplemented
316C4×Q16—264no—
317C4×SD16—264no—
318C2×#107—131no—
319C2×#106—131no—
320C2×C4⋊C8C2×D1311yescharacteristic
321C2×#103—131no—
322C2×#103—131no—
323C2×#103—131no—
324C2×#103—131no—
325C2×#103C2×D1311yescharacteristic
326C2×#102—131no—
327C2×#102—131no—
328C2×#102C2×D1311yescharacteristic
329C2×C4×C8—131no—
330C2×C4×C8—131no—
331C2×C4×C8—131no—
332#391—131no—
333#390—131no—
334#389—131no—
335#389—131no—
336#389—261no—
337#389—261no—
338#388—261no—
339#388—261no—
340#385—261no—
341#382—131no—
342#382—261no—
343C4×C4⋊C4—131no—
344C4×C4⋊C4—131no—
345C4×C4⋊C4—261no—
346C4×C22⋊C4—261no—
347#340—131no—
348#340—131no—
349#340—131no—
350#340—131no—
351#336—131no—
352#336—131no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—42no—
356C22×C13—81no—
357C22×C13—81no—
358C22×C13—81no—
359C22×C13—81no—
360C22×C13C2×C4×D411yescharacteristic
361C22×C13D4211yescharacteristic
362C22×C13#54811yescharacteristic
363C22×C13C2×#13511yescharacteristic
364C22×C13C2×#13511yescharacteristic
365C22×C13#57711yescharacteristic
366C22×C13#57811yescharacteristic
367C4×C13—24no—
368C4×C13—22no—
369C4×C13—22no—
370C4×C13—42no—
371C4×C13—42no—
372C4×C13C2×C4×D411yescharacteristic
373C4×C13#52011yescharacteristic
374C4×C13#54711yescharacteristic
375C4×C13#55311yescharacteristic
376Dic13—22no—
377Dic13—22no—
378Dic13—41no—
379Dic13—42no—
380Dic13—41no—
381Dic13—81no—
382Dic13—81no—
383Dic13—81no—
384C22×Q8C22×D1311yescharacteristic
385C22×Q8C22×D1311yescharacteristic
386C22×D4C22×D1311yescharacteristic
387C23×C4—261no—
388C2×Q16—22no—
389C2×Q16—22no—
390C2×Q16C2×Dic1314yescomplemented
391C2×SD16—22no—
392C2×SD16—22no—
393C2×SD16C2×Dic1314yes—
394C22×C8—131no—
395C22×C8—131no—
396C22×C8C22×D1311yescharacteristic
397C22×C8C22×D1311yescharacteristic
398#124—22no—
399#124C2×Dic1314yes—
400C4×Q8—22no—
401C4×Q8—262no—
402C4×Q8—262no—
403C4×Q8—262no—
404C4×Q8—521no—
405C4×Q8—521no—
406C4×Q8—521no—
407C4×Q8C2×Dic1314yescomplemented
408C4×D4—262no—
409C4×D4—521no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—131no—
412C2×C4⋊C4—131no—
413C2×C4⋊C4—131no—
414C2×C4⋊C4—131no—
415C2×C4⋊C4—131no—
416C2×C4⋊C4—131no—
417C2×C4⋊C4—261no—
418C2×C4⋊C4—261no—
419C2×C4⋊C4—261no—
420C2×C4⋊C4—261no—
421C2×C4⋊C4—261no—
422C2×C4⋊C4C22×D1311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—261no—
426C2×C22⋊C4—261no—
427C2×C22⋊C4—261no—
428C2×C42—21no—
429C2×C42—131no—
430C2×C42—131no—
431C2×C42—261no—
432C2×C42—261no—
433C2×C42—261no—
434C2×C42—261no—
435C2×C42C22×D1311yescharacteristic
436#107—262no—
437#106—262no—
438C4⋊C8C2×Dic1314yescomplemented
439#103—134no—
440#103—262no—
441#103—262no—
442#103—521no—
443#103C2×Dic1314yescomplemented
444#102—262no—
445#102—521no—
446#102C2×Dic1314yes—
447C4×C8—262no—
448C4×C8—262no—
449C4×C8—262no—
450#95—131no—
451#95—131no—
452#95—261no—
453#95—261no—
454#95—261no—
455#95—261no—
456#95—261no—
457#95—261no—
458#95—261no—
459C2×C13—82no—
460C2×C13#236111yescharacteristic
461C2×C13C4×#13511yescharacteristic
462C2×C13#294711yescharacteristic
463C2×C13#328211yescharacteristic
464C2×C13#328511yescharacteristic
465C2×C13#336411yescharacteristic
466C2×C13#336511yescharacteristic
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8D4×D1312yes—
477C2×Q8#612912yes—
478C2×Q8C22×Dic1312yes—
479C2×Q8C22×Dic1312yes—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4C22×Dic1312yes—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—131no—
490C22×C4—131no—
491C22×C4—131no—
492C22×C4—131no—
493C22×C4—261no—
494C22×C4—261no—
495C22×C4—261no—
496C22×C4—261no—
497C22×C4—522no—
498C22×C4—522no—
499C22×C4—1041no—
500C22×C4D4×D1311yescharacteristic
501C22×C4D4×D1311yescharacteristic
502C22×C4C23×D1311yescharacteristic
503Q16—28no—
504SD16—28no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—132no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C2×C8—261no—
514C2×C8—261no—
515C2×C8C22×Dic1312yes—
516C2×C8C22×Dic1312yes—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—132no—
520C4⋊C4—262no—
521C4⋊C4—262no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—261no—
527C4⋊C4—261no—
528C4⋊C4—261no—
529C4⋊C4—522no—
530C4⋊C4—522no—
531C4⋊C4—522no—
532C4⋊C4—522no—
533C4⋊C4—522no—
534C4⋊C4C22×Dic1312yes—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—522no—
538C22⋊C4—522no—
539C22⋊C4—1041no—
540C42—24no—
541C42—261no—
542C42—261no—
543C42—261no—
544C42—261no—
545C42—522no—
546C42—522no—
547C42—522no—
548C42—1041no—
549C42#612912yes—
550C42C22×Dic1312yes—
551C13#2300011yescharacteristic, complemented, Sylow
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D1311yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8D4×Dic1314yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—261no—
596C2×C4—261no—
597C2×C4—262no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4—521no—
609C2×C4—1044no—
610C2×C4D4×Dic1312yes—
611C2×C4D4×Dic1312yes—
612C2×C4C2×D4×D1311yescharacteristic
613C2×C4C2×#612911yescharacteristic
614C2×C4C2×#612911yescharacteristic
615C2×C4#8825611yescharacteristic
616C2×C4#8825711yescharacteristic
617C2×C4C23×Dic1311yescharacteristic
618C8—22no—
619C8—22no—
620C8—262no—
621C8—521no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22#1171930311yescharacteristic
630C22#1171931411yescharacteristic
631C22D13×#13511yescharacteristic
632C22D13×#13511yescharacteristic
633C22#1171948311yescharacteristic
634C22#1171948411yescharacteristic
635C22C2×D4×Dic1311yescharacteristic
636C4—22no—
637C4—22no—
638C4—24no—
639C4—42no—
640C4—42no—
641C4—262no—
642C4—262no—
643C4—521no—
644C4—522no—
645C4—521no—
646C4—1041no—
647C4—1041no—
648C4—1041no—
649C4#1171931611yescharacteristic
650C4#1171932311yescharacteristic
651C4C2×D4×Dic1311yescharacteristic
652C4#1171953211yescharacteristic
653C2—82no—
654C2#1311785611yescharacteristic
655C2#135×Dic1311yescharacteristic
656C2#1312368011yescharacteristic
657C2#1312653811yescharacteristic
658C2#1312657211yescharacteristic
659C2#1312751611yescharacteristic
660C2#1312752711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#464C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#464D13
hypocenterC13#23000
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
1. C2×Q16×Dic13C2semidirect
4. #13117757C2semidirect
6. #13115606C2semidirect
8. #13115514C2semidirect
10. #13115179C2semidirect
11. #13115106C2semidirect
13. #13114701C2semidirect
14. C4×#88075C2semidirect
24. C13×#464C4semidirect
68. C2×C13×Q16C2×C4semidirect
77. C4×Q8×C13C2×C4semidirect
114. C13×C4⋊C8C2×C4semidirect
115. C13×#103C2×C4semidirect
315. #464Dic13semidirect
390. C2×Q16C2×Dic13semidirect
407. C4×Q8C2×Dic13semidirect
438. C4⋊C8C2×Dic13semidirect
443. #103C2×Dic13semidirect
551. C13#23000semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×Dic132
C2×SD16×Dic132
#131179112
#131177572
#131156482
#131156062
#131155212
#131155142
#131151932
#131151792
#131151062
#131150852
#131147012
C4×#880752
C2×C13×#4642
#2300013