Finite Group Databasebeta

← #15864279

Subgroups — #15864279

Order 3328.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88186C211yesmaximal, characteristic
2C2×#11718627C211yesmaximal, characteristic, complemented
3C2×#11718876C211yesmaximal, characteristic, complemented
4#13117326C211yesmaximal, characteristic, complemented
5#13115184C214yesmaximal, complemented
6#13115170C214yesmaximal, complemented
7#13114555C211yesmaximal, characteristic, complemented
8#13114530C211yesmaximal, characteristic, complemented
9C22×C4×D4×C13C211yesmaximal, characteristic, complemented
10C23×D4×C13C2211yescharacteristic
11C24×C4×C13—21no—
12C22×#6127C2211yescharacteristic
13C2×C4×D4×C13—24no—
14C2×C4×D4×C13—24no—
15C2×C4×D4×C13—22no—
16C2×C4×D4×C13C2214yes—
17C22×C13×C4⋊C4C2211yescharacteristic
18C22×C13×C22⋊C4—21no—
19C22×C42×C13C2211yescharacteristic
20C22×#6105C2212yes—
21C2×#88186—24no—
22C2×#88186—24no—
23C2×#88186C418yescomplemented
24C22×#6104C2212yes—
25C2×#88180C2211yescharacteristic
26C22×#6099C2211yescharacteristic
27C22×#6099C2211yescharacteristic
28#11718876—24no—
29C2×#88072C2214yes—
30#11718874—21no—
31#11718874—21no—
32C2×#88050C2212yescomplemented
33#11718627—24no—
34C2×#88049C2212yescomplemented
35#11718600C2212yescomplemented
36C4×#6103—22no—
37#11718590C2212yescomplemented
38#11718121C2214yescomplemented
39#11718116C2214yescomplemented
40C25×C13—21no—
41C22×D4×C13—21no—
42C22×D4×C13—22no—
43C22×D4×C13—22no—
44C22×D4×C13—22no—
45C22×D4×C13C2×C414yes—
46C22×D4×C13D414yes—
47C22×D4×C13D414yes—
48C22×D4×C13C2312yes—
49C23×C4×C13—28no—
50C23×C4×C13—21no—
51C23×C4×C13—24no—
52C23×C4×C13—48no—
53C23×C4×C13D411yescharacteristic
54C23×C4×C13D411yescharacteristic
55C23×C4×C13C2311yescharacteristic
56C22×C22⋊D13—21no—
57C2×#6127—22no—
58C2×#6127—22no—
59C2×#6127—22no—
60C2×#6127—22no—
61C2×#6127—22no—
62C2×#6127C2×C414yescomplemented
63C22×D52C2311yescharacteristic
64C22×C4×D13—21no—
65C22×Dic26C2311yescharacteristic
66C4×D4×C13—216no—
67C4×D4×C13—28no—
68C4×D4×C13—28no—
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—22no—
71C2×C13×C4⋊C4D414yes—
72C2×C13×C4⋊C4C2312yes—
73C2×C13×C22⋊C4—28no—
74C2×C13×C22⋊C4—24no—
75C2×C42×C13—22no—
76C2×C42×C13—22no—
77C2×C42×C13D412yes—
78C2×C42×C13C2312yes—
79C2×C22⋊Dic13—21no—
80C2×#6105—24no—
81C2×#6105—24no—
82C2×#6105C2×C418yescomplemented
83#88186—232no—
84C2×#6104—24no—
85C2×#6104—24no—
86C2×#6104C2×C418yescomplemented
87C2×#6103—22no—
88C2×#6103—22no—
89#88180—22no—
90#88180—22no—
91C2×C4⋊Dic13C2312yes—
92C2×C4×Dic13—21no—
93C2×#6099—22no—
94C2×#6099—22no—
95C2×#6099—22no—
96C2×#6099—22no—
97C2×#6099—22no—
98C2×#6099C2×C414yescomplemented
99C22×#1055C2312yes—
100C22×#1055C2312yes—
101#88072—28no—
102#88071—22no—
103#88071—22no—
104#88050—24no—
105#88049—24no—
106#20925—131nomaximal, Sylow
107C24×C13—24no—
108C24×C13—24no—
109C24×C13—24no—
110C24×C13—22no—
111C24×C13—44no—
112C24×C13—44no—
113C24×C13C2×D411yescharacteristic
114C23×D13—21no—
115C2×D4×C13—28no—
116C2×D4×C13—24no—
117C2×D4×C13—24no—
118C2×D4×C13—28no—
119C2×D4×C13—24no—
120C2×D4×C13—28no—
121C2×D4×C13—22no—
122C2×D4×C13—22no—
123C2×D4×C13—22no—
124C2×D4×C13C22⋊C4116yescomplemented
125C2×D4×C13C22×C414yes—
126C2×D4×C13C2×D414yes—
127C2×D4×C13C2×D414yes—
128C22×C4×C13—22no—
129C22×C4×C13—24no—
130C22×C4×C13—216no—
131C22×C4×C13—22no—
132C22×C4×C13—24no—
133C22×C4×C13—22no—
134C22×C4×C13—42no—
135C22×C4×C13—432no—
136C22×C4×C13—48no—
137C22×C4×C13—44no—
138C22×C4×C13—48no—
139C22×C4×C13C22×C412yes—
140C22×C4×C13C2×D412yes—
141C22×C4×C13C2×D412yes—
142C22×C4×C13C2×D412yes—
143C22×C4×C13C2×D412yes—
144C22×C4×C13C2×D411yescharacteristic
145C22×C4×C13C2×D412yes—
146C22×C4×C13#4112yes—
147C22×C4×C13#4112yes—
148C22×C4×C13C2411yescharacteristic
149C2×C22⋊D13—24no—
150C2×C22⋊D13—42no—
151C2×C22⋊D13—42no—
152C22×Dic13—21no—
153C22×Dic13—22no—
154#6127—28no—
155#6127—28no—
156#6127—44no—
157#6127—44no—
158C2×D52—21no—
159C2×D52—21no—
160C2×D52—21no—
161C2×D52—21no—
162C2×D52—21no—
163C2×D52C22×C412yescomplemented
164C2×C4×D13—24no—
165C2×C4×D13—42no—
166C2×C4×D13—42no—
167C2×Dic26—21no—
168C2×Dic26—21no—
169C2×Dic26—21no—
170C2×Dic26—21no—
171C2×Dic26—21no—
172C2×Dic26C22×C412yescomplemented
173C13×C4⋊C4—24no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4C2×D414yes—
176C13×C22⋊C4—216no—
177C42×C13—24no—
178C42×C13—22no—
179C42×C13—22no—
180C22⋊Dic13—42no—
181#6105—216no—
182#6104—216no—
183#6103—44no—
184#6103—44no—
185C4⋊Dic13—22no—
186C4⋊Dic13—22no—
187C4×Dic13—42no—
188#6099—28no—
189#6099—44no—
190C2×#1055—22no—
191C2×#1055—22no—
192C2×#1055—22no—
193C2×#1055—22no—
194C2×#1055—22no—
195C2×#1055C22×C414yescomplemented
196C22×AGL(1,ℤ/8)—131no—
197C22×C4×D4D1311yescharacteristic, complemented
198C2×#417—131no—
199C2×#418—131no—
200#2297—131no—
201#1879—134no—
202#1877—134no—
203#1736—131no—
204#1728—131no—
205C23×C13—22no—
206C23×C13—22no—
207C23×C13—24no—
208C23×C13—24no—
209C23×C13—24no—
210C23×C13—216no—
211C23×C13—48no—
212C23×C13—416no—
213C23×C13—48no—
214C23×C13—416no—
215C23×C13—44no—
216C23×C13—44no—
217C23×C13C4×D412yes—
218C23×C13#12012yes—
219C23×C13#12112yes—
220C23×C13C22×D411yescharacteristic
221C22×D13—22no—
222C22×D13—41no—
223C22×D13—41no—
224C22×D13—42no—
225C22×D13—42no—
226D4×C13—24no—
227D4×C13—24no—
228D4×C13—24no—
229D4×C13—216no—
230D4×C13C2×C22⋊C418yes—
231C2×C4×C13—28no—
232C2×C4×C13—22no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—24no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—432no—
239C2×C4×C13—44no—
240C2×C4×C13—48no—
241C2×C4×C13—44no—
242C2×C4×C13C2×C22⋊C414yes—
243C2×C4×C13C4×D414yes—
244C2×C4×C13C4×D412yes—
245C2×C4×C13#12012yes—
246C2×C4×C13#12114yes—
247C2×C4×C13#12112yes—
248C2×C4×C13#12314yes—
249C2×C4×C13AGL(1,ℤ/8)14yescomplemented
250C2×C4×C13C23×C411yescharacteristic
251C2×C4×C13C22×D411yescharacteristic
252C2×C4×C13C22×D411yescharacteristic
253C2×C4×C13C22×D411yescharacteristic
254C2×C4×C13C2×#4111yescharacteristic
255C2×C4×C13C2×#4111yescharacteristic
256C22⋊D13—48no—
257C2×Dic13—22no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—42no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263D52—22no—
264D52—22no—
265D52—41no—
266D52—41no—
267C4×D13—48no—
268Dic26—22no—
269Dic26—22no—
270Dic26—41no—
271Dic26—41no—
272#1055—24no—
273#1055—42no—
274C22×#41—131no—
275C23×D4C2×D1311yescharacteristic
276C24×C4—21no—
277C2×AGL(1,ℤ/8)—138no—
278C2×AGL(1,ℤ/8)—264no—
279C2×AGL(1,ℤ/8)—264no—
280C22×SD16—132no—
281C22×D8—132no—
282C22×M16—131no—
283C22×M16—131no—
284C2×C4×D4—24no—
285C2×C4×D4—24no—
286C2×C4×D4—22no—
287C2×C4×D4C2×D1314yes—
288C2×#117—131no—
289C22×C4⋊C4C2×D1311yescharacteristic
290C22×C22⋊C4—21no—
291C22×C42C2×D1311yescharacteristic
292#418—264no—
293#417—264no—
294C2×#103—132no—
295C2×#102—134no—
296C2×#102—132no—
297#391—132no—
298#390—132no—
299#386—261no—
300#386—261no—
301C4×C22⋊C4—262no—
302#340—134no—
303#336—134no—
304C22×C13—24no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13—24no—
308C22×C13—28no—
309C22×C13—432no—
310C22×C13—48no—
311C22×C13—44no—
312C22×C13—44no—
313C22×C13—48no—
314C22×C13#38614yes—
315C22×C13C2×C4×D411yescharacteristic
316C22×C13C2×#12011yescharacteristic
317C22×C13C2×#12111yescharacteristic
318C22×C13C2×AGL(1,ℤ/8)12yes—
319C22×C13#57512yes—
320C2×D13—44no—
321C2×D13—42no—
322C2×D13—81no—
323C2×D13—81no—
324C2×D13—82no—
325C4×C13—24no—
326C4×C13—44no—
327C4×C13—42no—
328C4×C13#38614yes—
329C4×C13C22×C22⋊C411yescharacteristic
330C4×C13C2×C4×D411yescharacteristic
331C4×C13C2×#12111yescharacteristic
332C4×C13C2×#12311yescharacteristic
333Dic13—42no—
334Dic13—82no—
335C25—21no—
336C2×#41—134no—
337C2×#41—262no—
338C2×#41—262no—
339C2×#41—262no—
340C2×#41—262no—
341C2×#41—262no—
342C22×Q8—131no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—21no—
346C22×D4—22no—
347C22×D4—131no—
348C22×D4—261no—
349C22×D4C4×D1314yes—
350C22×D4D5214yes—
351C22×D4C22⋊D1314yes—
352C22×D4C22×D1312yes—
353C23×C4—28no—
354C23×C4—21no—
355C23×C4—24no—
356C23×C4—48no—
357C23×C4—261no—
358C23×C4C22⋊D1311yescharacteristic
359C23×C4C22⋊D1311yescharacteristic
360C23×C4C22×D1311yescharacteristic
361AGL(1,ℤ/8)—2632no—
362C2×SD16—138no—
363C2×SD16—264no—
364C2×SD16—264no—
365C2×D8—138no—
366C2×D8—264no—
367C2×D8—264no—
368C2×M16—134no—
369C2×M16—262no—
370C2×M16—262no—
371C2×M16—262no—
372C2×M16—262no—
373C2×M16—262no—
374C22×C8—132no—
375C22×C8—132no—
376C4×D4—216no—
377C4×D4—28no—
378C4×D4—28no—
379#117—262no—
380#117—262no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—132no—
384C2×C4⋊C4C22⋊D1314yes—
385C2×C4⋊C4C22×D1312yes—
386C2×C22⋊C4—28no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—261no—
389C2×C22⋊C4—262no—
390C2×C22⋊C4—262no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—261no—
394C2×C42C22⋊D1312yes—
395C2×C42C22×D1312yes—
396#103—264no—
397#102—268no—
398#102—264no—
399#95—262no—
400#95—262no—
401C2×C13—24no—
402C2×C13—48no—
403C2×C13C2×#38611yescharacteristic
404C2×C13C4×AGL(1,ℤ/8)12yes—
405C2×C13#300311yescharacteristic
406C2×C13#300511yescharacteristic
407C2×C13#303612yes—
408D13—82no—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—44no—
414C24—44no—
415C24—261no—
416C24C2×C22⋊D1311yescharacteristic
417#41—268no—
418#41—268no—
419#41—524no—
420#41—524no—
421C2×Q8—132no—
422C2×Q8—261no—
423C2×Q8—261no—
424C2×Q8—261no—
425C2×Q8—261no—
426C2×Q8—261no—
427C2×D4—28no—
428C2×D4—24no—
429C2×D4—28no—
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—132no—
437C2×D4—264no—
438C2×D4—261no—
439C2×D4—261no—
440C2×D4—261no—
441C2×D4—261no—
442C2×D4—261no—
443C2×D4—522no—
444C2×D4—522no—
445C2×D4#6103116yescomplemented
446C2×D4C2×C4×D1314yes—
447C2×D4C2×D5214yes—
448C2×D4C2×C22⋊D1314yes—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—216no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—432no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—261no—
461C22×C4—264no—
462C22×C4—262no—
463C22×C4—522no—
464C22×C4—522no—
465C22×C4C2×C4×D1312yes—
466C22×C4C2×D5212yes—
467C22×C4#612712yes—
468C22×C4#612712yes—
469C22×C4C2×C22⋊D1312yes—
470C22×C4C2×C22⋊D1312yes—
471C22×C4C2×C22⋊D1312yes—
472C22×C4C2×C22⋊D1311yescharacteristic
473C22×C4C2×C22⋊D1312yes—
474C22×C4C23×D1311yescharacteristic
475SD16—2616no—
476D8—2616no—
477M16—268no—
478M16—524no—
479C2×C8—134no—
480C2×C8—262no—
481C2×C8—262no—
482C2×C8—262no—
483C2×C8—262no—
484C2×C8—262no—
485C4⋊C4—24no—
486C4⋊C4—22no—
487C4⋊C4—262no—
488C4⋊C4—262no—
489C4⋊C4C2×C22⋊D1314yes—
490C22⋊C4—216no—
491C22⋊C4—524no—
492C22⋊C4—522no—
493C22⋊C4—524no—
494C42—24no—
495C42—22no—
496C42—22no—
497C42—522no—
498C13#2092511yescharacteristic, complemented, Sylow
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—216no—
505C23—48no—
506C23—416no—
507C23—416no—
508C23—48no—
509C23—44no—
510C23—44no—
511C23—262no—
512C23—521no—
513C23—522no—
514C23—522no—
515C23—521no—
516C23C4×C22⋊D1312yes—
517C23#8818412yes—
518C23#8820712yes—
519C23C22×C22⋊D1311yescharacteristic
520Q8—262no—
521Q8—262no—
522Q8—521no—
523Q8—521no—
524D4—24no—
525D4—24no—
526D4—24no—
527D4—216no—
528D4—262no—
529D4—262no—
530D4—528no—
531D4—521no—
532D4—521no—
533D4C2×#610318yes—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—432no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—262no—
546C2×C4—521no—
547C2×C4—521no—
548C2×C4—528no—
549C2×C4—522no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4C2×#610314yes—
553C2×C4C4×C22⋊D1314yes—
554C2×C4C4×C22⋊D1312yes—
555C2×C4#8818314yes—
556C2×C4#8818414yes—
557C2×C4#8818412yes—
558C2×C4#8818614yescomplemented
559C2×C4#8820712yes—
560C2×C4C22×C4×D1311yescharacteristic
561C2×C4C22×D5211yescharacteristic
562C2×C4C2×#612711yescharacteristic
563C2×C4C2×#612711yescharacteristic
564C2×C4C22×C22⋊D1311yescharacteristic
565C2×C4C22×C22⋊D1311yescharacteristic
566C8—264no—
567C8—522no—
568C22—24no—
569C22—24no—
570C22—22no—
571C22—22no—
572C22—28no—
573C22—432no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—524no—
579C22—522no—
580C22—1042no—
581C22—1041no—
582C22—1041no—
583C22#1171887414yes—
584C22C2×C4×C22⋊D1311yescharacteristic
585C22C2×#8818411yescharacteristic
586C22C2×#8818612yes—
587C22#1171952712yes—
588C22C2×#8820711yescharacteristic
589C4—24no—
590C4—44no—
591C4—42no—
592C4—522no—
593C4—1042no—
594C4#1171887414yes—
595C4C22×#610311yescharacteristic
596C4C2×C4×C22⋊D1311yescharacteristic
597C4C2×#8818311yescharacteristic
598C4C2×#8818411yescharacteristic
599C2—24no—
600C2—48no—
601C2—1042no—
602C2C2×#1171887411yescharacteristic
603C2C4×#8818612yes—
604C2#1312406311yescharacteristic
605C2#1312407811yescharacteristic
606C2#1312430012yes—
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C4×D4×C13C2
socleC23×C13C22×D4
prime radicalC22×#88186C2
cosocleC22×C4×C13C24
hypercenterC22×C4×D4D13
hypocenterC13#20925
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#11718627C2semidirect
3. C2×#11718876C2semidirect
4. #13117326C2semidirect
5. #13115184C2semidirect
6. #13115170C2semidirect
7. #13114555C2semidirect
8. #13114530C2semidirect
9. C22×C4×D4×C13C2semidirect
23. C2×#88186C4semidirect
32. C2×#88050C22semidirect
34. C2×#88049C22semidirect
35. #11718600C22semidirect
37. #11718590C22semidirect
38. #11718121C22semidirect
39. #11718116C22semidirect
62. C2×#6127C2×C4semidirect
82. C2×#6105C2×C4semidirect
86. C2×#6104C2×C4semidirect
98. C2×#6099C2×C4semidirect
124. C2×D4×C13C22⋊C4semidirect
163. C2×D52C22×C4semidirect
172. C2×Dic26C22×C4semidirect
195. C2×#1055C22×C4semidirect
197. C22×C4×D4D13semidirect
249. C2×C4×C13AGL(1,ℤ/8)semidirect
445. C2×D4#6103semidirect
498. C13#20925semidirect
558. C2×C4#88186semidirect

Maximal subgroups

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

subgroupindex
C22×#881862
C2×#117186272
C2×#117188762
#131173262
#131151842
#131151702
#131145552
#131145302
C22×C4×D4×C132
#2092513