Finite Group Databasebeta

← #15865950

Subgroups — #15865950

Order 3328.

1042 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718675C211yesmaximal, characteristic
2#13117300C211yesmaximal, characteristic, complemented
3#13115463C211yesmaximal, characteristic, complemented
4#13115456C211yesmaximal, characteristic, complemented
5#13115421C211yesmaximal, characteristic, complemented
6#13115414C211yesmaximal, characteristic, complemented
7#13115184C211yesmaximal, characteristic
8#13115184C211yesmaximal, characteristic
9#13115097C211yesmaximal, characteristic, complemented
10#13115083C211yesmaximal, characteristic, complemented
11#13114734C211yesmaximal, characteristic, complemented
12#13114664C211yesmaximal, characteristic, complemented
13#13114661C211yesmaximal, characteristic, complemented
14#13114585C211yesmaximal, characteristic, complemented
15C42×D4×C13C211yesmaximal, characteristic, complemented
16C2×C4×D4×C13C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C22×C42×C13—21no—
20C22×#6104C2211yescharacteristic
21C2×C4×D52C2211yescharacteristic
22C2×#88126—21no—
23C4×C13×C4⋊C4C2211yescharacteristic
24C4×C13×C4⋊C4C2211yescharacteristic
25C4×C13×C4⋊C4C2211yescharacteristic
26C4×C13×C22⋊C4—21no—
27C4×C13×C22⋊C4—21no—
28C4×C13×C22⋊C4—21no—
29C43×C13C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718675C418yescomplemented
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88049C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88048C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40#11718601—21no—
41#11718601—21no—
42#11718600C2211yescharacteristic
43#11718600C2211yescharacteristic
44C4×#6103—21no—
45C4×C4⋊Dic13C2211yescharacteristic
46#11718589C2211yescharacteristic
47#11718589C2211yescharacteristic
48C2×#88043C2211yescharacteristic
49C2×#88043C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718116C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60C22×D4×C13C2311yescharacteristic
61C23×C4×C13—21no—
62C23×C4×C13—21no—
63C23×C4×C13—21no—
64C22×D52C2311yescharacteristic
65C22×C4×D13—21no—
66C4×D4×C13—22no—
67C4×D4×C13—22no—
68C4×D4×C13—22no—
69C4×D4×C13—22no—
70C4×D4×C13—22no—
71C4×D4×C13C2×C414yes—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C22⋊C4—21no—
76C2×C13×C22⋊C4—21no—
77C2×C13×C22⋊C4—21no—
78C2×C42×C13—21no—
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13—44no—
82C2×C42×C13D411yescharacteristic
83C2×C42×C13D411yescharacteristic
84C2×C42×C13D411yescharacteristic
85C2×C42×C13D411yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×C42×C13C2311yescharacteristic
89C2×#6104—22no—
90C2×#6104—22no—
91C2×#6104C2×C414yes—
92C2×#6103—21no—
93C2×#6103—21no—
94C2×#6103—21no—
95C2×#6103—21no—
96C2×#6103—21no—
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×#6101—21no—
101C2×#6101—21no—
102C2×C4×Dic13—21no—
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C22×#1055C2311yescharacteristic
107C4×D52—22no—
108C4×D52C2×C414yescomplemented
109#88126—44no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95—21no—
113#88072—22no—
114#88072—22no—
115#88072C2×C414yes—
116#88071—21no—
117#88071—21no—
118#88071—21no—
119#88049—22no—
120#88049—22no—
121#88049C2×C414yescomplemented
122#88048—22no—
123#88048C2×C414yescomplemented
124#88047—22no—
125#88043—22no—
126#88043C2×C414yescomplemented
127#21122—131nomaximal, Sylow
128C24×C13—21no—
129C23×D13—21no—
130C2×D4×C13—21no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13C22×C412yes—
134C2×D4×C13#4112yes—
135C2×D4×C13#4112yes—
136C22×C4×C13—24no—
137C22×C4×C13—24no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—24no—
142C22×C4×C13—42no—
143C22×C4×C13—42no—
144C22×C4×C13—42no—
145C22×C4×C13—42no—
146C22×C4×C13—42no—
147C22×C4×C13—42no—
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52—21no—
163C2×D52C22×C412yes—
164C2×C4×D13—24no—
165C2×C4×D13—42no—
166C2×C4×D13—42no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4C22×C412yes—
171C13×C4⋊C4#4112yes—
172C13×C4⋊C4#4112yes—
173C13×C22⋊C4—24no—
174C13×C22⋊C4—24no—
175C13×C22⋊C4—24no—
176C42×C13—22no—
177C42×C13—22no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—21no—
181C42×C13—21no—
182C42×C13—41no—
183C42×C13—41no—
184C42×C13—42no—
185C42×C13—42no—
186C42×C13—42no—
187C42×C13C22×C412yes—
188C42×C13#4112yes—
189#6104—28no—
190#6103—24no—
191#6103—42no—
192#6103—42no—
193#6103—42no—
194#6103—42no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—21no—
199C4⋊Dic13—21no—
200C4⋊Dic13C22×C412yes—
201#6101—42no—
202#6101—42no—
203C4×Dic13—42no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055C22×C412yes—
215C2×#1055C22×C412yes—
216C2×#442—131no—
217#2293—131no—
218C42×D4D1311yescharacteristic, complemented
219#1926—131no—
220#1925—131no—
221#1920—131no—
222#1919—131no—
223#1879—131no—
224#1879—131no—
225#1864—131no—
226#1862—131no—
227#1776—131no—
228#1762—131no—
229#1762—131no—
230#1745—131no—
231C23×C13—22no—
232C23×C13—22no—
233C23×C13—22no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13C22×D411yescharacteristic
239C22×D13—22no—
240C22×D13—41no—
241C22×D13—41no—
242C22×D13—41no—
243C22×D13—41no—
244C22×D13—41no—
245C22×D13—41no—
246D4×C13—22no—
247D4×C13—22no—
248D4×C13—22no—
249D4×C13#11714yescomplemented
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—22no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—22no—
262C2×C4×C13—22no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—48no—
268C2×C4×C13—48no—
269C2×C4×C13—48no—
270C2×C4×C13—41no—
271C2×C4×C13—41no—
272C2×C4×C13—41no—
273C2×C4×C13—41no—
274C2×C4×C13—41no—
275C2×C4×C13—41no—
276C2×C4×C13C4×D412yes—
277C2×C4×C13C4×D412yes—
278C2×C4×C13AGL(1,ℤ/8)12yes—
279C2×C4×C13AGL(1,ℤ/8)12yes—
280C2×C4×C13C23×C411yescharacteristic
281C2×C4×C13C2×#4111yescharacteristic
282C2×C4×C13C2×#4111yescharacteristic
283C2×C4×C13C2×#4111yescharacteristic
284C2×C4×C13C2×#4111yescharacteristic
285C2×C4×C13C2×#4111yescharacteristic
286C2×Dic13—22no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295D52—22no—
296D52—22no—
297D52—41no—
298D52—41no—
299C4×D13—48no—
300#1055—22no—
301#1055—22no—
302#1055—41no—
303#1055—41no—
304C22×D8—131no—
305C2×C4×D4—131no—
306C2×C4×D4C2×D1311yescharacteristic
307C2×C4×D4C2×D1311yescharacteristic
308C2×C4×D4C2×D1311yescharacteristic
309C2×#117—261no—
310C22×C42—21no—
311#442—138no—
312C2×#107—131no—
313C2×#106—131no—
314C2×#106—131no—
315C2×#102—131no—
316C2×#102—131no—
317C2×#102—131no—
318C2×#102—131no—
319C2×#102—131no—
320C2×#102—131no—
321C2×#97—131no—
322C2×#97—131no—
323#390—131no—
324#390—131no—
325#389—131no—
326#389—131no—
327#388—261no—
328#388—261no—
329C4×C4⋊C4—131no—
330C4×C4⋊C4C2×D1311yescharacteristic
331C4×C4⋊C4C2×D1311yescharacteristic
332C4×C4⋊C4C2×D1311yescharacteristic
333C4×C22⋊C4—21no—
334C4×C22⋊C4—21no—
335C4×C22⋊C4—21no—
336C4×C22⋊C4—261no—
337C43C2×D1311yescharacteristic
338#340—131no—
339#340—131no—
340#340—131no—
341#340—131no—
342#340—131no—
343#340—131no—
344#336—131no—
345#336—131no—
346#336—131no—
347#336—131no—
348C22×C13—24no—
349C22×C13—42no—
350C22×C13—42no—
351C22×C13—42no—
352C22×C13—42no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13C2×C4×D411yescharacteristic
356C22×C13#52511yescharacteristic
357C22×C13#52511yescharacteristic
358C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
359C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
360C22×C13#57511yescharacteristic
361C22×C13#57511yescharacteristic
362C2×D13—42no—
363C2×D13—42no—
364C2×D13—42no—
365C2×D13—81no—
366C2×D13—81no—
367C2×D13—81no—
368C2×D13—81no—
369C4×C13—22no—
370C4×C13—22no—
371C4×C13—22no—
372C4×C13—22no—
373C4×C13—42no—
374C4×C13—41no—
375C4×C13—42no—
376C4×C13—41no—
377C4×C13—42no—
378C4×C13C2×#11711yescharacteristic
379C4×C13C4×#4111yescharacteristic
380C4×C13#52911yescharacteristic
381C4×C13#52911yescharacteristic
382Dic13—42no—
383Dic13—81no—
384Dic13—81no—
385C22×D4—131no—
386C22×D4C22×D1311yescharacteristic
387C23×C4—21no—
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—261no—
391C2×D8—134no—
392C2×D8—262no—
393C2×D8—262no—
394C22×C8—131no—
395C22×C8—131no—
396C22×C8—131no—
397C22×C8—131no—
398C4×D4—22no—
399C4×D4—22no—
400C4×D4—22no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—134no—
404C4×D4—262no—
405C4×D4C4×D1314yes—
406#117—524no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4C22×D1311yescharacteristic
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C22⋊C4—21no—
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—261no—
419C2×C22⋊C4—261no—
420C2×C22⋊C4—261no—
421C2×C22⋊C4—261no—
422C2×C22⋊C4—261no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—21no—
426C2×C42—44no—
427C2×C42—261no—
428C2×C42C22⋊D1311yescharacteristic
429C2×C42C22⋊D1311yescharacteristic
430C2×C42C22⋊D1311yescharacteristic
431C2×C42C22⋊D1311yescharacteristic
432C2×C42C22×D1311yescharacteristic
433C2×C42C22×D1311yescharacteristic
434C2×C42C22×D1311yescharacteristic
435#107—262no—
436#106—134no—
437#106—262no—
438#102—134no—
439#102—134no—
440#102—262no—
441#102—262no—
442#102—262no—
443#102—262no—
444#97—134no—
445#97—262no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—261no—
450#95—261no—
451#95—261no—
452C2×C13—44no—
453C2×C13#229011yescharacteristic
454C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
455C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
456C2×C13#310911yescharacteristic
457C2×C13#311111yescharacteristic
458C2×C13#311411yescharacteristic
459C2×C13#311511yescharacteristic
460D13—82no—
461C24—21no—
462C24—261no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—132no—
467C2×D4—261no—
468C2×D4—261no—
469C2×D4—261no—
470C2×D4—261no—
471C2×D4—261no—
472C2×D4C2×C4×D1312yes—
473C2×D4#612712yes—
474C2×D4#612712yes—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—24no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—261no—
488C22×C4—264no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—522no—
492C22×C4—522no—
493C22×C4C2×C22⋊D1311yescharacteristic
494C22×C4C2×C22⋊D1311yescharacteristic
495C22×C4C2×C22⋊D1311yescharacteristic
496C22×C4C2×C22⋊D1311yescharacteristic
497C22×C4C2×C22⋊D1311yescharacteristic
498C22×C4C2×C22⋊D1311yescharacteristic
499C22×C4C23×D1311yescharacteristic
500D8—268no—
501C2×C8—132no—
502C2×C8—132no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8—261no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8—261no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—132no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—522no—
523C4⋊C4—522no—
524C4⋊C4C2×C4×D1312yes—
525C4⋊C4#612712yes—
526C4⋊C4#612712yes—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—264no—
531C22⋊C4—522no—
532C22⋊C4—522no—
533C22⋊C4—522no—
534C22⋊C4—522no—
535C42—22no—
536C42—22no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—41no—
543C42—42no—
544C42—42no—
545C42—42no—
546C42—522no—
547C42C2×C4×D1312yes—
548C42#612712yes—
549C13#2112211yescharacteristic, complemented, Sylow
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—262no—
558C23—521no—
559C23—521no—
560C23—521no—
561C23—521no—
562C23—521no—
563C23—521no—
564C23C22×C22⋊D1311yescharacteristic
565D4—22no—
566D4—22no—
567D4—22no—
568D4—262no—
569D4—262no—
570D4—521no—
571D4—521no—
572D4#8812614yescomplemented
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—48no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—262no—
600C2×C4—528no—
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×C4C4×C22⋊D1312yes—
610C2×C4C4×C22⋊D1312yes—
611C2×C4#8818612yes—
612C2×C4#8820312yes—
613C2×C4C22×C4×D1311yescharacteristic
614C2×C4C2×#612711yescharacteristic
615C2×C4C2×#612711yescharacteristic
616C2×C4C2×#612711yescharacteristic
617C2×C4C2×#612711yescharacteristic
618C2×C4C2×#612711yescharacteristic
619C8—262no—
620C8—262no—
621C8—521no—
622C8—521no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—522no—
631C22—522no—
632C22—522no—
633C22—1041no—
634C22—1041no—
635C22—1041no—
636C22—1041no—
637C22C2×C4×C22⋊D1311yescharacteristic
638C22#1171949611yescharacteristic
639C22#1171949611yescharacteristic
640C22C2×#8818611yescharacteristic
641C22C2×#8820311yescharacteristic
642C22#1171952711yescharacteristic
643C22#1171952711yescharacteristic
644C4—22no—
645C4—22no—
646C4—22no—
647C4—22no—
648C4—42no—
649C4—42no—
650C4—41no—
651C4—42no—
652C4—41no—
653C4—522no—
654C4—1041no—
655C4—1041no—
656C4C2×#8812611yescharacteristic
657C4C4×#612711yescharacteristic
658C4#1171918411yescharacteristic
659C4#1171918411yescharacteristic
660C2—44no—
661C2—1042no—
662C2#1311727911yescharacteristic
663C2C4×#8820311yescharacteristic
664C2C4×#8818611yescharacteristic
665C2#1312492711yescharacteristic
666C2#1312494511yescharacteristic
667C2#1312497411yescharacteristic
668C2#1312498711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC42×D4×C13C2
socleC23×C13C22×D4
prime radicalC22×#6104C22
cosocleC22×C4×C13C24
hypercenterC42×D4D13
hypocenterC13#21122
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C43 ◁ C42×D4

Complements

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

NH ≅ G/Ntype
2. #13117300C2semidirect
3. #13115463C2semidirect
4. #13115456C2semidirect
5. #13115421C2semidirect
6. #13115414C2semidirect
9. #13115097C2semidirect
10. #13115083C2semidirect
11. #13114734C2semidirect
12. #13114664C2semidirect
13. #13114661C2semidirect
14. #13114585C2semidirect
15. C42×D4×C13C2semidirect
33. #11718675C4semidirect
108. C4×D52C2×C4semidirect
121. #88049C2×C4semidirect
123. #88048C2×C4semidirect
126. #88043C2×C4semidirect
218. C42×D4D13semidirect
249. D4×C13#117semidirect
549. C13#21122semidirect
572. D4#88126semidirect

Maximal subgroups

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

subgroupindex
C2×#117186752
#131173002
#131154632
#131154562
#131154212
#131154142
#131151842
#131150972
#131150832
#131147342
#131146642
#131146612
#131145852
C42×D4×C132
#2112213