Finite Group Databasebeta

← #15865980

Subgroups — #15865980

Order 3328.

922 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718687C211yesmaximal, characteristic
2#13117300C211yesmaximal, characteristic
3#13115463C211yesmaximal, characteristic
4#13115456C211yesmaximal, characteristic
5#13115428C211yesmaximal, characteristic, complemented
6#13115407C211yesmaximal, characteristic, complemented
7#13115191C211yesmaximal, characteristic
8#13115191C211yesmaximal, characteristic
9#13115104C211yesmaximal, characteristic, complemented
10#13115090C211yesmaximal, characteristic, complemented
11#13114734C211yesmaximal, characteristic, complemented
12#13114666C211yesmaximal, characteristic, complemented
13#13114664C211yesmaximal, characteristic
14#13114585C211yesmaximal, characteristic, complemented
15C42×Q8×C13C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×#6106C2211yescharacteristic
20C2×C4×D52C2211yescharacteristic
21C2×#88126—21no—
22C4×C13×C4⋊C4—21no—
23C4×C13×C4⋊C4—21no—
24C4×C13×C4⋊C4—21no—
25C4×C13×C4⋊C4C2211yescharacteristic
26C4×C13×C4⋊C4C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C43×C13—21no—
29C43×C13C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33#11718687C418yescomplemented
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88049C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
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×Q8×C13C2311yescharacteristic
61C22×D52C2311yescharacteristic
62C22×C4×D13—21no—
63C4×Q8×C13—22no—
64C4×Q8×C13—22no—
65C4×Q8×C13—22no—
66C4×Q8×C13—22no—
67C4×Q8×C13—22no—
68C4×Q8×C13C2×C414yes—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C42×C13—21no—
76C2×C42×C13—21no—
77C2×C42×C13—21no—
78C2×C42×C13—21no—
79C2×C42×C13—21no—
80C2×C42×C13—21no—
81C2×C42×C13D411yescharacteristic
82C2×C42×C13D411yescharacteristic
83C2×C42×C13D411yescharacteristic
84C2×C42×C13D411yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×C42×C13C2311yescharacteristic
88C2×#6106—22no—
89C2×#6106—22no—
90C2×#6106C2×C414yes—
91C2×#6103—21no—
92C2×#6103—21no—
93C2×#6103—21no—
94C2×#6103—21no—
95C2×#6103—21no—
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4⋊Dic13C2311yescharacteristic
99C2×#6101—21no—
100C2×#6101—21no—
101C2×C4×Dic13—21no—
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C4×D52—22no—
107C4×D52C2×C414yes—
108#88126—44no—
109C13×#95—21no—
110C13×#95—21no—
111C13×#95—21no—
112#88075—22no—
113#88075—22no—
114#88075C2×C414yes—
115#88071—21no—
116#88071—21no—
117#88071—21no—
118#88049—22no—
119#88049—22no—
120#88049C2×C414yes—
121#88048—22no—
122#88047—22no—
123#88047C2×C414yes—
124#88043—22no—
125#88043C2×C414yes—
126#21124—131nomaximal, Sylow
127C23×D13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13—21no—
131C2×Q8×C13C22×C412yes—
132C2×Q8×C13#4112yes—
133C2×Q8×C13#4112yes—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2×D411yescharacteristic
144C22×C4×C13C2411yescharacteristic
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13—21no—
148C2×D52—21no—
149C2×D52—21no—
150C2×D52—21no—
151C2×D52—21no—
152C2×D52—21no—
153C2×D52C22×C412yes—
154C2×C4×D13—24no—
155C2×C4×D13—42no—
156C2×C4×D13—42no—
157C13×C4⋊C4—24no—
158C13×C4⋊C4—24no—
159C13×C4⋊C4—24no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—21no—
163C13×C4⋊C4C22×C412yes—
164C13×C4⋊C4#4112yes—
165C13×C4⋊C4#4112yes—
166C42×C13—24no—
167C42×C13—24no—
168C42×C13—24no—
169C42×C13—22no—
170C42×C13—22no—
171C42×C13—21no—
172C42×C13—21no—
173C42×C13—21no—
174C42×C13—21no—
175C42×C13—41no—
176C42×C13—41no—
177C42×C13—42no—
178C42×C13—42no—
179C42×C13—42no—
180C42×C13C22×C412yes—
181C42×C13#4112yes—
182#6106—28no—
183#6103—24no—
184#6103—42no—
185#6103—42no—
186#6103—42no—
187#6103—42no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13C22×C412yes—
194#6101—42no—
195#6101—42no—
196C4×Dic13—42no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055C22×C412yes—
208C2×#1055C22×C412yes—
209C2×#440—131no—
210#2293—131no—
211C42×Q8D1311yescharacteristic, complemented
212#1926—131no—
213#1925—131no—
214#1921—131no—
215#1918—131no—
216#1880—131no—
217#1880—131no—
218#1865—131no—
219#1863—131no—
220#1776—131no—
221#1763—131no—
222#1762—131no—
223#1745—131no—
224C23×C13C22×D411yescharacteristic
225C22×D13—22no—
226C22×D13—41no—
227C22×D13—41no—
228C22×D13—41no—
229C22×D13—41no—
230C22×D13—41no—
231C22×D13—41no—
232Q8×C13—22no—
233Q8×C13—22no—
234Q8×C13—22no—
235Q8×C13#11714yescomplemented
236C2×C4×C13—22no—
237C2×C4×C13—21no—
238C2×C4×C13—21no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—22no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—22no—
251C2×C4×C13—22no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13C4×D412yes—
263C2×C4×C13C4×D412yes—
264C2×C4×C13AGL(1,ℤ/8)12yes—
265C2×C4×C13#13712yes—
266C2×C4×C13C23×C411yescharacteristic
267C2×C4×C13C2×#4111yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×Dic13—22no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281D52—22no—
282D52—22no—
283D52—41no—
284D52—41no—
285C4×D13—48no—
286#1055—22no—
287#1055—22no—
288#1055—41no—
289#1055—41no—
290C22×SD16—131no—
291C2×C4×Q8C2×D1311yescharacteristic
292C2×C4×Q8C2×D1311yescharacteristic
293C2×C4×Q8C2×D1311yescharacteristic
294C2×C4×D4—131no—
295C2×#117—261no—
296#440—138no—
297C2×#107—131no—
298C2×#107—131no—
299C2×#106—131no—
300C2×#103—131no—
301C2×#103—131no—
302C2×#103—131no—
303C2×#102—131no—
304C2×#102—131no—
305C2×#102—131no—
306C2×#97—131no—
307C2×#97—131no—
308#390—131no—
309#390—131no—
310#389—131no—
311#389—131no—
312#388—261no—
313#388—261no—
314C4×C4⋊C4—21no—
315C4×C4⋊C4—21no—
316C4×C4⋊C4—21no—
317C4×C4⋊C4—131no—
318C4×C4⋊C4C2×D1311yescharacteristic
319C4×C4⋊C4C2×D1311yescharacteristic
320C4×C4⋊C4C2×D1311yescharacteristic
321C4×C22⋊C4—261no—
322C43—21no—
323C43C2×D1311yescharacteristic
324#340—131no—
325#340—131no—
326#340—131no—
327#340—131no—
328#340—131no—
329#340—131no—
330#336—131no—
331#336—131no—
332#336—131no—
333#336—131no—
334C22×C13C2×C4×D411yescharacteristic
335C22×C13#52511yescharacteristic
336C22×C13#52511yescharacteristic
337C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
338C22×C13C2×#13711yescharacteristic
339C22×C13#57511yescharacteristic
340C22×C13#57511yescharacteristic
341C2×D13—42no—
342C2×D13—42no—
343C2×D13—42no—
344C2×D13—81no—
345C2×D13—81no—
346C2×D13—81no—
347C2×D13—81no—
348C4×C13—22no—
349C4×C13—22no—
350C4×C13—22no—
351C4×C13—22no—
352C4×C13—24no—
353C4×C13—41no—
354C4×C13—42no—
355C4×C13—41no—
356C4×C13—42no—
357C4×C13—42no—
358C4×C13C2×#11711yescharacteristic
359C4×C13C4×#4111yescharacteristic
360C4×C13#52911yescharacteristic
361C4×C13#52911yescharacteristic
362Dic13—42no—
363Dic13—81no—
364Dic13—81no—
365C22×Q8C22×D1311yescharacteristic
366C22×D4—131no—
367C23×C4—261no—
368C2×SD16—134no—
369C2×SD16—262no—
370C2×SD16—262no—
371C22×C8—131no—
372C22×C8—131no—
373C22×C8—131no—
374C22×C8—131no—
375C4×Q8—22no—
376C4×Q8—22no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×Q8—22no—
380C4×Q8C4×D1314yes—
381C4×D4—134no—
382C4×D4—262no—
383#117—524no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—131no—
388C2×C4⋊C4—131no—
389C2×C4⋊C4—131no—
390C2×C4⋊C4—261no—
391C2×C4⋊C4—261no—
392C2×C4⋊C4C22×D1311yescharacteristic
393C2×C4⋊C4C22×D1311yescharacteristic
394C2×C4⋊C4C22×D1311yescharacteristic
395C2×C22⋊C4—261no—
396C2×C22⋊C4—261no—
397C2×C22⋊C4—261no—
398C2×C22⋊C4—261no—
399C2×C22⋊C4—261no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42—261no—
407C2×C42C22⋊D1311yescharacteristic
408C2×C42C22⋊D1311yescharacteristic
409C2×C42C22⋊D1311yescharacteristic
410C2×C42C22⋊D1311yescharacteristic
411C2×C42C22×D1311yescharacteristic
412C2×C42C22×D1311yescharacteristic
413C2×C42C22×D1311yescharacteristic
414#107—134no—
415#107—262no—
416#106—262no—
417#103—134no—
418#103—262no—
419#103—262no—
420#102—134no—
421#102—262no—
422#102—262no—
423#97—134no—
424#97—262no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—261no—
429#95—261no—
430#95—261no—
431C2×C13#229011yescharacteristic
432C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
433C2×C13C4×#13711yescharacteristic
434C2×C13#311111yescharacteristic
435C2×C13#311211yescharacteristic
436C2×C13#311511yescharacteristic
437C2×C13#311711yescharacteristic
438D13—82no—
439C24—261no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×C4×D1312yes—
444C2×Q8#612712yes—
445C2×Q8#612712yes—
446C2×D4—132no—
447C2×D4—261no—
448C2×D4—261no—
449C2×D4—261no—
450C2×D4—261no—
451C2×D4—261no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—261no—
457C22×C4—264no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—522no—
461C22×C4—522no—
462C22×C4C2×C22⋊D1311yescharacteristic
463C22×C4C2×C22⋊D1311yescharacteristic
464C22×C4C2×C22⋊D1311yescharacteristic
465C22×C4C2×C22⋊D1311yescharacteristic
466C22×C4C2×C22⋊D1311yescharacteristic
467C22×C4C2×C22⋊D1311yescharacteristic
468C22×C4C23×D1311yescharacteristic
469SD16—268no—
470C2×C8—132no—
471C2×C8—132no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C2×C8—261no—
477C2×C8—261no—
478C2×C8—261no—
479C2×C8—261no—
480C2×C8—261no—
481C2×C8—261no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—132no—
489C4⋊C4—261no—
490C4⋊C4—261no—
491C4⋊C4—261no—
492C4⋊C4—261no—
493C4⋊C4—261no—
494C4⋊C4—522no—
495C4⋊C4—522no—
496C4⋊C4C2×C4×D1312yes—
497C4⋊C4#612712yes—
498C4⋊C4#612712yes—
499C22⋊C4—264no—
500C22⋊C4—522no—
501C22⋊C4—522no—
502C22⋊C4—522no—
503C22⋊C4—522no—
504C42—24no—
505C42—24no—
506C42—24no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—41no—
514C42—41no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—522no—
519C42C2×C4×D1312yes—
520C42#612712yes—
521C13#2112411yescharacteristic, complemented, Sylow
522C23—262no—
523C23—521no—
524C23—521no—
525C23—521no—
526C23—521no—
527C23—521no—
528C23—521no—
529C23C22×C22⋊D1311yescharacteristic
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8#8812614yescomplemented
534D4—262no—
535D4—262no—
536D4—521no—
537D4—521no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—262no—
565C2×C4—528no—
566C2×C4—521no—
567C2×C4—521no—
568C2×C4—521no—
569C2×C4—521no—
570C2×C4—521no—
571C2×C4—521no—
572C2×C4—521no—
573C2×C4—521no—
574C2×C4C4×C22⋊D1312yes—
575C2×C4C4×C22⋊D1312yes—
576C2×C4#8819612yes—
577C2×C4#8820312yes—
578C2×C4C22×C4×D1311yescharacteristic
579C2×C4C2×#612711yescharacteristic
580C2×C4C2×#612711yescharacteristic
581C2×C4C2×#612711yescharacteristic
582C2×C4C2×#612711yescharacteristic
583C2×C4C2×#612711yescharacteristic
584C8—262no—
585C8—262no—
586C8—521no—
587C8—521no—
588C22—522no—
589C22—522no—
590C22—522no—
591C22—1041no—
592C22—1041no—
593C22—1041no—
594C22—1041no—
595C22C2×C4×C22⋊D1311yescharacteristic
596C22#1171949611yescharacteristic
597C22#1171949611yescharacteristic
598C22C2×#8819611yescharacteristic
599C22C2×#8820311yescharacteristic
600C22#1171952711yescharacteristic
601C22#1171952711yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—24no—
606C4—22no—
607C4—42no—
608C4—41no—
609C4—42no—
610C4—41no—
611C4—42no—
612C4—522no—
613C4—1041no—
614C4—1041no—
615C4C2×#8812611yescharacteristic
616C4C4×#612711yescharacteristic
617C4#1171918411yescharacteristic
618C4#1171918411yescharacteristic
619C2—1042no—
620C2#1311727911yescharacteristic
621C2C4×#8820311yescharacteristic
622C2C4×#8819611yescharacteristic
623C2#1312494511yescharacteristic
624C2#1312495811yescharacteristic
625C2#1312498711yescharacteristic
626C2#1312500711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC42×Q8×C13C2
socleC23×C13C22×D4
prime radicalC22×D52C23
cosocleC22×C4×C13C24
hypercenterC42×Q8D13
hypocenterC13#21124
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C43 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
5. #13115428C2semidirect
6. #13115407C2semidirect
9. #13115104C2semidirect
10. #13115090C2semidirect
11. #13114734C2semidirect
12. #13114666C2semidirect
14. #13114585C2semidirect
15. C42×Q8×C13C2semidirect
33. #11718687C4semidirect
211. C42×Q8D13semidirect
235. Q8×C13#117semidirect
521. C13#21124semidirect
533. Q8#88126semidirect

Maximal subgroups

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

subgroupindex
C2×#117186872
#131173002
#131154632
#131154562
#131154282
#131154072
#131151912
#131151042
#131150902
#131147342
#131146662
#131146642
#131145852
C42×Q8×C132
#2112413