Finite Group Databasebeta

← #15866040

Subgroups — #15866040

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718687C211yesmaximal, characteristic
2#13117480C211yesmaximal, 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
12C4×#88075C211yesmaximal, characteristic, complemented
13C4×#88049C211yesmaximal, characteristic
14#13114588C211yesmaximal, characteristic, complemented
15C13×#2276C211yesmaximal, characteristic, complemented
16C2×C4×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×#6106C2211yescharacteristic
20C2×C4×D52C2211yescharacteristic
21C4×C13×C4⋊C4—21no—
22C4×C13×C4⋊C4—21no—
23C4×C13×C4⋊C4—21no—
24C4×C13×C4⋊C4C2211yescharacteristic
25C4×C13×C4⋊C4C2211yescharacteristic
26C4×C13×C4⋊C4C2211yescharacteristic
27C13×#376—21no—
28C13×#376C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88075C2211yescharacteristic
32#11718687C418yescomplemented
33C2×#88049C2211yescharacteristic
34C2×#88049C2211yescharacteristic
35C2×#88049C2211yescharacteristic
36C2×#88048C2211yescharacteristic
37C2×#88047C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39#11718600C2211yescharacteristic
40#11718600C2211yescharacteristic
41C4×C4⋊Dic13C2211yescharacteristic
42#11718589C2211yescharacteristic
43#11718589C2211yescharacteristic
44C2×#88043C2211yescharacteristic
45C2×C4×#1055C2211yescharacteristic
46#11718330—21no—
47#11718330—21no—
48#11718328—21no—
49#11718325—21no—
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×C13C2311yescharacteristic
82C2×C42×C13C2311yescharacteristic
83C2×C42×C13C2311yescharacteristic
84C2×#6106—22no—
85C2×#6106—22no—
86C2×#6106C2×C414yes—
87C2×#6103—21no—
88C2×#6103—21no—
89C2×#6103—21no—
90C2×#6103—21no—
91C2×#6103—21no—
92C2×C4⋊Dic13C2311yescharacteristic
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×#6101—21no—
96C2×#6101—21no—
97C2×C4×Dic13—21no—
98C22×#1055C2311yescharacteristic
99C22×#1055C2311yescharacteristic
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C4×D52—22no—
103C4×D52C2×C414yes—
104C13×#95—21no—
105C13×#95—21no—
106C13×#95—21no—
107C13×#95D411yescharacteristic
108C13×#95D411yescharacteristic
109C13×#95D411yescharacteristic
110C13×#95D411yescharacteristic
111#88075—22no—
112#88075—22no—
113#88075C2×C414yes—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117#88049—22no—
118#88049—22no—
119#88049C2×C414yes—
120#88048—22no—
121#88047—22no—
122#88047C2×C414yes—
123#88043C2×C414yes—
124C4×#1055—22no—
125#21128—131nomaximal, Sylow
126C23×D13—21no—
127C2×Q8×C13—21no—
128C2×Q8×C13—21no—
129C2×Q8×C13—21no—
130C2×Q8×C13C22×C412yes—
131C2×Q8×C13#4112yes—
132C2×Q8×C13#4112yes—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13C2×D411yescharacteristic
138C22×C4×C13C2×D411yescharacteristic
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2×D411yescharacteristic
143C22×C4×C13C2411yescharacteristic
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C2×D52—21no—
148C2×D52—21no—
149C2×D52—21no—
150C2×D52—21no—
151C2×D52—21no—
152C2×D52C22×C412yes—
153C2×C4×D13—24no—
154C2×C4×D13—81no—
155C2×C4×D13—81no—
156C13×C4⋊C4—24no—
157C13×C4⋊C4—24no—
158C13×C4⋊C4—24no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4C22×C412yes—
163C13×C4⋊C4#4112yes—
164C13×C4⋊C4#4112yes—
165C42×C13—24no—
166C42×C13—24no—
167C42×C13—24no—
168C42×C13—21no—
169C42×C13—21no—
170C42×C13—21no—
171C42×C13—21no—
172C42×C13—81no—
173C42×C13—81no—
174C42×C13—81no—
175C42×C13C22×C412yes—
176C42×C13#4112yes—
177#6106—28no—
178#6103—24no—
179#6103—81no—
180#6103—81no—
181#6103—81no—
182#6103—81no—
183C4⋊Dic13—21no—
184C4⋊Dic13—21no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13C22×C412yes—
189#6101—81no—
190#6101—81no—
191C4×Dic13—81no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055—21no—
196C2×#1055—21no—
197C2×#1055—21no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055C22×C412yes—
203C2×#1055C22×C412yes—
204C2×#440—131no—
205#2319—131no—
206#2276D1311yescharacteristic, complemented
207#1926—131no—
208#1925—131no—
209#1921—131no—
210#1918—131no—
211#1880—131no—
212#1880—131no—
213#1865—131no—
214#1863—131no—
215#1776—131no—
216C4×#103—131no—
217C4×#102—131no—
218#1746—131no—
219C23×C13C22×D411yescharacteristic
220C22×D13—22no—
221C22×D13—41no—
222C22×D13—41no—
223C22×D13—41no—
224C22×D13—41no—
225C22×D13—41no—
226C22×D13—41no—
227Q8×C13—22no—
228Q8×C13—22no—
229Q8×C13—22no—
230Q8×C13#11714yes—
231C2×C4×C13—22no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—21no—
235C2×C4×C13—21no—
236C2×C4×C13—21no—
237C2×C4×C13—21no—
238C2×C4×C13—22no—
239C2×C4×C13—21no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—21no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13C4×D412yes—
260C2×C4×C13C4×D412yes—
261C2×C4×C13C23×C411yescharacteristic
262C2×C4×C13C2×#4111yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×C4×C13C2×#4111yescharacteristic
266C2×C4×C13C2×#4111yescharacteristic
267C2×Dic13—22no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—41no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276D52—22no—
277D52—22no—
278D52—41no—
279D52—41no—
280C4×D13—84no—
281#1055—22no—
282#1055—22no—
283#1055—41no—
284#1055—41no—
285C22×SD16—131no—
286C2×C4×Q8C2×D1311yescharacteristic
287C2×C4×Q8C2×D1311yescharacteristic
288C2×C4×Q8C2×D1311yescharacteristic
289C2×C4×D4—131no—
290#440—138no—
291C2×#107—131no—
292C2×#107—131no—
293C2×#106—131no—
294C2×#103—131no—
295C2×#103—131no—
296C2×#103—131no—
297C2×#102—131no—
298C2×#102—131no—
299C2×#102—131no—
300C2×#97—131no—
301C2×C4×C8—131no—
302#390—131no—
303#390—131no—
304#389—131no—
305#389—131no—
306#388—261no—
307#388—261no—
308#381—261no—
309#380—261no—
310C4×C4⋊C4—21no—
311C4×C4⋊C4—21no—
312C4×C4⋊C4—21no—
313C4×C4⋊C4—131no—
314C4×C4⋊C4C2×D1311yescharacteristic
315C4×C4⋊C4C2×D1311yescharacteristic
316C4×C4⋊C4C2×D1311yescharacteristic
317#376—21no—
318#376C2×D1311yescharacteristic
319#340—131no—
320#340—131no—
321#340—131no—
322#340—131no—
323#340—131no—
324#340—131no—
325#336—131no—
326#336—131no—
327#336—131no—
328#336—131no—
329C22×C13C2×C4×D411yescharacteristic
330C22×C13#52511yescharacteristic
331C22×C13#52511yescharacteristic
332C22×C13#57611yescharacteristic
333C22×C13#57711yescharacteristic
334C22×C13#57711yescharacteristic
335C22×C13#57811yescharacteristic
336C2×D13—42no—
337C2×D13—42no—
338C2×D13—42no—
339C2×D13—81no—
340C2×D13—81no—
341C2×D13—81no—
342C2×D13—81no—
343C4×C13—22no—
344C4×C13—22no—
345C4×C13—24no—
346C4×C13—41no—
347C4×C13—42no—
348C4×C13—41no—
349C4×C13—41no—
350C4×C13—42no—
351C4×C13—41no—
352C4×C13—42no—
353C4×C13C2×#11711yescharacteristic
354C4×C13C4×#4111yescharacteristic
355C4×C13#52911yescharacteristic
356C4×C13#52911yescharacteristic
357Dic13—42no—
358Dic13—81no—
359Dic13—81no—
360C22×Q8C22×D1311yescharacteristic
361C22×D4—131no—
362C23×C4—261no—
363C2×SD16—134no—
364C2×SD16—262no—
365C2×SD16—262no—
366C22×C8—131no—
367C22×C8—131no—
368C22×C8—131no—
369C22×C8—131no—
370C4×Q8—22no—
371C4×Q8—22no—
372C4×Q8—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×Q8C4×D1314yes—
376C4×D4—134no—
377C4×D4—262no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—131no—
382C2×C4⋊C4—131no—
383C2×C4⋊C4—131no—
384C2×C4⋊C4—261no—
385C2×C4⋊C4—261no—
386C2×C4⋊C4C22×D1311yescharacteristic
387C2×C4⋊C4C22×D1311yescharacteristic
388C2×C4⋊C4C22×D1311yescharacteristic
389C2×C22⋊C4—261no—
390C2×C22⋊C4—261no—
391C2×C22⋊C4—261no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—261no—
394C2×C42—21no—
395C2×C42—21no—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—261no—
401C2×C42C22×D1311yescharacteristic
402C2×C42C22×D1311yescharacteristic
403C2×C42C22×D1311yescharacteristic
404#107—134no—
405#107—262no—
406#106—262no—
407#103—134no—
408#103—262no—
409#103—262no—
410#102—134no—
411#102—262no—
412#102—262no—
413#97—134no—
414C4×C8—262no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—261no—
419#95—261no—
420#95—261no—
421#95C22⋊D1311yescharacteristic
422#95C22⋊D1311yescharacteristic
423#95C22⋊D1311yescharacteristic
424#95C22⋊D1311yescharacteristic
425C2×C13#229011yescharacteristic
426C2×C13#294611yescharacteristic
427C2×C13#294711yescharacteristic
428C2×C13#311911yescharacteristic
429C2×C13#312111yescharacteristic
430C2×C13#312411yescharacteristic
431C2×C13#312611yescharacteristic
432D13—82no—
433C24—261no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×C4×D1312yes—
438C2×Q8#612712yes—
439C2×Q8#612712yes—
440C2×D4—132no—
441C2×D4—261no—
442C2×D4—261no—
443C2×D4—261no—
444C2×D4—261no—
445C2×D4—261no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—261no—
451C22×C4—264no—
452C22×C4—261no—
453C22×C4—261no—
454C22×C4—1041no—
455C22×C4—1041no—
456C22×C4C2×C22⋊D1311yescharacteristic
457C22×C4C2×C22⋊D1311yescharacteristic
458C22×C4C2×C22⋊D1311yescharacteristic
459C22×C4C2×C22⋊D1311yescharacteristic
460C22×C4C2×C22⋊D1311yescharacteristic
461C22×C4C2×C22⋊D1311yescharacteristic
462C22×C4C23×D1311yescharacteristic
463SD16—268no—
464C2×C8—132no—
465C2×C8—132no—
466C2×C8—261no—
467C2×C8—261no—
468C2×C8—261no—
469C2×C8—261no—
470C2×C8—261no—
471C2×C8—261no—
472C2×C8—261no—
473C2×C8—261no—
474C2×C8—261no—
475C2×C8—261no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—132no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—261no—
486C4⋊C4—261no—
487C4⋊C4—261no—
488C4⋊C4—1041no—
489C4⋊C4—1041no—
490C4⋊C4C2×C4×D1312yes—
491C4⋊C4#612712yes—
492C4⋊C4#612712yes—
493C22⋊C4—264no—
494C22⋊C4—1041no—
495C22⋊C4—1041no—
496C22⋊C4—1041no—
497C22⋊C4—1041no—
498C42—24no—
499C42—24no—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—81no—
506C42—81no—
507C42—81no—
508C42—1041no—
509C42C2×C4×D1312yes—
510C42#612712yes—
511C13#2112811yescharacteristic, complemented, Sylow
512C23—262no—
513C23—521no—
514C23—521no—
515C23—521no—
516C23—521no—
517C23—521no—
518C23—521no—
519C23C22×C22⋊D1311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—22no—
523Q8#8812614yes—
524D4—262no—
525D4—262no—
526D4—521no—
527D4—521no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—262no—
557C2×C4—521no—
558C2×C4—521no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—521no—
563C2×C4—521no—
564C2×C4—521no—
565C2×C4—1044no—
566C2×C4C4×C22⋊D1312yes—
567C2×C4C4×C22⋊D1312yes—
568C2×C4C22×C4×D1311yescharacteristic
569C2×C4C2×#612711yescharacteristic
570C2×C4C2×#612711yescharacteristic
571C2×C4C2×#612711yescharacteristic
572C2×C4C2×#612711yescharacteristic
573C2×C4C2×#612711yescharacteristic
574C8—262no—
575C8—262no—
576C8—521no—
577C8—521no—
578C22—522no—
579C22—522no—
580C22—522no—
581C22—1041no—
582C22—1041no—
583C22—1041no—
584C22—1041no—
585C22C2×C4×C22⋊D1311yescharacteristic
586C22#1171949611yescharacteristic
587C22#1171949611yescharacteristic
588C22#1171954011yescharacteristic
589C22#1171954211yescharacteristic
590C22#1171954211yescharacteristic
591C22#1171954311yescharacteristic
592C4—22no—
593C4—24no—
594C4—22no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—522no—
603C4—1041no—
604C4—1041no—
605C4C2×#8812611yescharacteristic
606C4C4×#612711yescharacteristic
607C4#1171918411yescharacteristic
608C4#1171918411yescharacteristic
609C2—1042no—
610C2#1311727911yescharacteristic
611C2#1312366411yescharacteristic
612C2#1312367911yescharacteristic
613C2#1312503511yescharacteristic
614C2#1312506511yescharacteristic
615C2#1312509811yescharacteristic
616C2#1312512711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2276C2
socleC23×C13C22×D4
prime radicalC22×D52C23
cosocleC22×C4×C13C24
hypercenter#2276D13
hypocenterC13#21128
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ #376 ◁ #2276

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. C4×#88075C2semidirect
14. #13114588C2semidirect
15. C13×#2276C2semidirect
32. #11718687C4semidirect
206. #2276D13semidirect
511. C13#21128semidirect

Maximal subgroups

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

subgroupindex
C2×#117186872
#131174802
#131154632
#131154562
#131154282
#131154072
#131151912
#131151042
#131150902
#131147342
C4×#880752
C4×#880492
#131145882
C13×#22762
#2112813