Finite Group Databasebeta

← #15865629

Subgroups — #15865629

Order 3328.

1198 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×#11718893C211yesmaximal, characteristic
2C2×M16×Dic13C211yesmaximal, characteristic, complemented
3#13117327C211yesmaximal, characteristic
4C2×#11718646C211yesmaximal, characteristic, complemented
5#13115535C212yesmaximal, complemented
6#13115528C212yesmaximal, complemented
7#13115521C212yesmaximal, complemented
8#13115514C212yesmaximal, complemented
9#13114556C211yesmaximal, characteristic, complemented
10#13114555C211yesmaximal, characteristic, complemented
11C2×C13×#417C211yesmaximal, characteristic
12C22×C13×#41C2211yescharacteristic
13C22×C13×M16C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C2×#88180C2211yescharacteristic
16C22×C4⋊Dic13—21no—
17C22×C4⋊Dic13C2211yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19C22×#6099C2211yescharacteristic
20C13×#417C418yescomplemented
21C2×C13×#103C2212yes—
22C2×C13×#102C2212yes—
23#11718893—24no—
24#11718891C2211yescharacteristic
25#11718891C2211yescharacteristic
26C2×#88075C2212yes—
27#11718882C2211yescharacteristic
28#11718882C2211yescharacteristic
29#11718881—21no—
30#11718881—21no—
31C2×#88072C2212yes—
32C4×C22⋊Dic13—21no—
33C4×C22⋊Dic13—21no—
34C2×#88071—21no—
35M16×Dic13—24no—
36C2×#88055C2212yes—
37C2×C8×Dic13C2212yes—
38#11718646—21no—
39#11718646—21no—
40#11718646C2212yes—
41#11718646C2212yes—
42#11718143C2212yes—
43#11718143C2212yes—
44#11718121C2212yes—
45#11718121C2212yes—
46C2×C13×#41—22no—
47C2×C13×#41—22no—
48C2×C13×#41—22no—
49C2×C13×#41—22no—
50C2×C13×#41—22no—
51C2×C13×#41C2×C414yes—
52C22×Q8×C13C2311yescharacteristic
53C22×D4×C13—21no—
54C22×D4×C13C2311yescharacteristic
55C23×C4×C13—21no—
56C23×C4×C13—21no—
57C23×C4×C13C2311yescharacteristic
58C23×Dic13—21no—
59C23×Dic13D411yescharacteristic
60C23×Dic13D411yescharacteristic
61C2×C13×M16—22no—
62C2×C13×M16—22no—
63C2×C13×M16C2×C414yes—
64C22×C8×C13C2312yes—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—22no—
67C2×C13×C4⋊C4C2×C414yes—
68C2×C13×C4⋊C4C2312yes—
69C2×C22⋊Dic13—21no—
70C2×C22⋊Dic13—21no—
71C2×C22⋊Dic13—21no—
72C2×C22⋊Dic13—21no—
73C2×C22⋊Dic13—21no—
74#88180—22no—
75#88180—22no—
76C2×C4⋊Dic13—22no—
77C2×C4⋊Dic13—22no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13—44no—
82C2×C4⋊Dic13C2312yes—
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4⋊Dic13C2311yescharacteristic
85C2×C4×Dic13—22no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13—21no—
88C2×C4×Dic13D411yescharacteristic
89C2×C4×Dic13D411yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×#6099—22no—
93C2×#6099—22no—
94C2×#6099D414yes—
95C22×#1055C2312yes—
96C13×#103C2×C418yescomplemented
97C13×#102C2×C418yescomplemented
98#88075—24no—
99#88072—24no—
100#88071—22no—
101#88071—22no—
102#88071—21no—
103#88071—21no—
104#88071—21no—
105#88071—21no—
106#88055—24no—
107C8×Dic13—24no—
108#21093—131nomaximal, Sylow
109C24×C13—21no—
110C24×C13C2×D411yescharacteristic
111C13×#41—28no—
112C13×#41—28no—
113C13×#41—44no—
114C13×#41—44no—
115C2×Q8×C13—21no—
116C2×Q8×C13—21no—
117C2×Q8×C13—21no—
118C2×Q8×C13—21no—
119C2×Q8×C13—21no—
120C2×Q8×C13C22×C412yes—
121C2×D4×C13—24no—
122C2×D4×C13—21no—
123C2×D4×C13—21no—
124C2×D4×C13—21no—
125C2×D4×C13—21no—
126C2×D4×C13—21no—
127C2×D4×C13—42no—
128C2×D4×C13—42no—
129C2×D4×C13C22×C412yes—
130C22×C4×C13—24no—
131C22×C4×C13—21no—
132C22×C4×C13—22no—
133C22×C4×C13—24no—
134C22×C4×C13—42no—
135C22×C4×C13—42no—
136C22×C4×C13—42no—
137C22×C4×C13—42no—
138C22×C4×C13C22×C412yes—
139C22×C4×C13C2×D411yescharacteristic
140C22×C4×C13C2×D412yes—
141C22×C4×C13#4112yes—
142C22×C4×C13#4112yes—
143C22×C4×C13#4112yes—
144C22×C4×C13#4112yes—
145C22×C4×C13C2411yescharacteristic
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—21no—
151C22×Dic13—21no—
152C22×Dic13—22no—
153C22×Dic13—41no—
154C22×Dic13—41no—
155C22×Dic13—42no—
156C22×Dic13—42no—
157C22×Dic13—42no—
158C22×Dic13C2×D411yescharacteristic
159C22×Dic13C2×D411yescharacteristic
160C22×Dic13C2×D411yescharacteristic
161C22×Dic13C2×D411yescharacteristic
162C13×M16—28no—
163C2×C8×C13—22no—
164C2×C8×C13—22no—
165C2×C8×C13C22×C414yes—
166C13×C4⋊C4—24no—
167C13×C4⋊C4—22no—
168C13×C4⋊C4C22×C414yes—
169C22⋊Dic13—42no—
170C22⋊Dic13—42no—
171C22⋊Dic13—42no—
172C22⋊Dic13—42no—
173C22⋊Dic13—42no—
174C4⋊Dic13—22no—
175C4⋊Dic13—22no—
176C4⋊Dic13—21no—
177C4⋊Dic13—21no—
178C4⋊Dic13—21no—
179C4⋊Dic13—21no—
180C4⋊Dic13—44no—
181C4⋊Dic13—42no—
182C4⋊Dic13—44no—
183C4×Dic13—22no—
184C4×Dic13—22no—
185C4×Dic13—21no—
186C4×Dic13—21no—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—42no—
190#6099—28no—
191C2×#1055—22no—
192C2×#1055—22no—
193C2×#1055C2×D414yes—
194C2×#417—131no—
195C2×#417D1311yescharacteristic
196C2×C4×M16—131no—
197#2297—131no—
198C2×#389—131no—
199#1936—132no—
200#1935—132no—
201#1934—132no—
202#1933—132no—
203#1736—131no—
204#1736—131no—
205C23×C13—21no—
206C23×C13—21no—
207C23×C13—21no—
208C23×C13—21no—
209C23×C13—22no—
210C23×C13—41no—
211C23×C13—41no—
212C23×C13—41no—
213C23×C13—41no—
214C23×C13—41no—
215C23×C13—41no—
216C23×C13C4×D412yes—
217C23×C13#12112yes—
218C23×C13#12412yes—
219C23×C13C22×D411yescharacteristic
220Q8×C13—22no—
221Q8×C13—22no—
222Q8×C13—41no—
223Q8×C13—41no—
224D4×C13—22no—
225D4×C13—22no—
226D4×C13—48no—
227D4×C13—41no—
228D4×C13—41no—
229C2×C4×C13—24no—
230C2×C4×C13—22no—
231C2×C4×C13—48no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—42no—
235C2×C4×C13—42no—
236C2×C4×C13—48no—
237C2×C4×C13#11714yes—
238C2×C4×C13C4×D414yes—
239C2×C4×C13C4×D412yes—
240C2×C4×C13#12112yes—
241C2×C4×C13#12314yes—
242C2×C4×C13#12412yes—
243C2×C4×C13#12614yes—
244C2×C4×C13C23×C411yescharacteristic
245C2×C4×C13C22×D411yescharacteristic
246C2×C4×C13C2×#4111yescharacteristic
247C2×C4×C13C2×#4111yescharacteristic
248C2×C4×C13C2×#4111yescharacteristic
249C2×C4×C13C2×#4111yescharacteristic
250C2×Dic13—21no—
251C2×Dic13—21no—
252C2×Dic13—21no—
253C2×Dic13—21no—
254C2×Dic13—21no—
255C2×Dic13—21no—
256C2×Dic13—21no—
257C2×Dic13—21no—
258C2×Dic13—21no—
259C2×Dic13—21no—
260C2×Dic13—44no—
261C2×Dic13—44no—
262C2×Dic13—42no—
263C2×Dic13—41no—
264C2×Dic13—41no—
265C2×Dic13—42no—
266C2×Dic13—41no—
267C2×Dic13—41no—
268C2×Dic13—41no—
269C2×Dic13—41no—
270C2×Dic13—42no—
271C2×Dic13—84no—
272C2×Dic13AGL(1,ℤ/8)12yes—
273C2×Dic13#13712yes—
274C8×C13—24no—
275#1055—24no—
276C22×#41C2×D1311yescharacteristic
277C22×M16—131no—
278C22×M16C2×D1311yescharacteristic
279C2×#117—131no—
280C22×C4⋊C4—131no—
281C22×C4⋊C4—261no—
282C22×C4⋊C4C2×D1311yescharacteristic
283C22×C42—131no—
284#417—264no—
285#417Dic1318yescomplemented
286C2×#103—132no—
287C2×#103C2×D1312yes—
288C2×#102—132no—
289C2×#102C2×D1312yes—
290C4×M16—264no—
291C2×#97—132no—
292C2×C4×C8—132no—
293#391—131no—
294#391—131no—
295#390—131no—
296#390—131no—
297#389—132no—
298#389—132no—
299#389—261no—
300#389—261no—
301#386—261no—
302#386—261no—
303C4×C22⋊C4—261no—
304C4×C22⋊C4—261no—
305C2×#95—261no—
306#340—132no—
307#340—132no—
308#340—132no—
309#340—132no—
310C22×C13—22no—
311C22×C13—22no—
312C22×C13—22no—
313C22×C13—22no—
314C22×C13—22no—
315C22×C13—22no—
316C22×C13—42no—
317C22×C13—42no—
318C22×C13—42no—
319C22×C13—81no—
320C22×C13—81no—
321C22×C13—81no—
322C22×C13—81no—
323C22×C13#38814yes—
324C22×C13C2×C4×D411yescharacteristic
325C22×C13C2×#12111yescharacteristic
326C22×C13C2×#12411yescharacteristic
327C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C13C2×#13711yescharacteristic
329C22×C13#57511yescharacteristic
330C22×C13#57511yescharacteristic
331C4×C13—44no—
332C4×C13—42no—
333C4×C13#38814yes—
334C4×C13C2×#11711yescharacteristic
335C4×C13C2×C4×D411yescharacteristic
336C4×C13C2×#12311yescharacteristic
337C4×C13C2×#12611yescharacteristic
338Dic13—22no—
339Dic13—22no—
340Dic13—41no—
341Dic13—41no—
342Dic13—81no—
343Dic13—81no—
344Dic13—82no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C2×Dic1314yes—
351C22×Q8C22×D1311yescharacteristic
352C22×D4—21no—
353C22×D4C22×D1311yescharacteristic
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—131no—
357C23×C4—131no—
358C23×C4—261no—
359C23×C4C22×D1311yescharacteristic
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—134no—
363C2×M16—262no—
364C2×M16—262no—
365C2×M16C2×Dic1314yes—
366C22×C8—132no—
367C22×C8C22×D1312yes—
368#117—262no—
369#117—262no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—132no—
373C2×C4⋊C4—131no—
374C2×C4⋊C4—131no—
375C2×C4⋊C4—262no—
376C2×C4⋊C4—262no—
377C2×C4⋊C4—262no—
378C2×C4⋊C4—262no—
379C2×C4⋊C4—261no—
380C2×C4⋊C4—524no—
381C2×C4⋊C4C2×Dic1314yes—
382C2×C4⋊C4C22×D1312yes—
383C2×C22⋊C4—261no—
384C2×C22⋊C4—261no—
385C2×C22⋊C4—261no—
386C2×C22⋊C4—261no—
387C2×C22⋊C4—261no—
388C2×C42—131no—
389C2×C42—131no—
390C2×C42—131no—
391C2×C42—131no—
392C2×C42—262no—
393C2×C42—262no—
394C2×C42—261no—
395#103—264no—
396#103C2×Dic1318yescomplemented
397#102—264no—
398#102C2×Dic1318yescomplemented
399#97—264no—
400C4×C8—264no—
401#95—262no—
402#95—262no—
403#95—261no—
404#95—261no—
405#95—261no—
406#95—261no—
407C2×C13—24no—
408C2×C13—82no—
409C2×C13C2×#38811yescharacteristic
410C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
411C2×C13C4×#13711yescharacteristic
412C2×C13#305511yescharacteristic
413C2×C13#305611yescharacteristic
414C2×C13#314111yescharacteristic
415C2×C13#314211yescharacteristic
416C24—21no—
417C24D4×D1311yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C22×Dic1312yes—
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4C22×Dic1312yes—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—131no—
446C22×C4—131no—
447C22×C4—131no—
448C22×C4—131no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—262no—
456C22×C4—521no—
457C22×C4—521no—
458C22×C4—522no—
459C22×C4—522no—
460C22×C4—522no—
461C22×C4#612712yes—
462C22×C4#612712yes—
463C22×C4D4×D1312yes—
464C22×C4D4×D1311yescharacteristic
465C22×C4#612912yes—
466C22×C4#612912yes—
467C22×C4C22×Dic1312yes—
468C22×C4C23×D1311yescharacteristic
469M16—28no—
470M16—268no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—134no—
474C2×C8—262no—
475C2×C8—262no—
476C2×C8C22×Dic1314yes—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—262no—
480C4⋊C4—261no—
481C4⋊C4—261no—
482C4⋊C4—262no—
483C4⋊C4—261no—
484C4⋊C4—261no—
485C4⋊C4—524no—
486C4⋊C4—524no—
487C4⋊C4—522no—
488C4⋊C4C22×Dic1314yes—
489C22⋊C4—522no—
490C22⋊C4—522no—
491C22⋊C4—522no—
492C22⋊C4—522no—
493C22⋊C4—522no—
494C42—262no—
495C42—262no—
496C42—261no—
497C42—261no—
498C42—261no—
499C42—261no—
500C42—522no—
501C13#2109311yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—22no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23#8813812yes—
514C23#8813912yes—
515C23D4×Dic1312yes—
516C23C2×D4×D1311yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—48no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—48no—
534C2×C4—132no—
535C2×C4—132no—
536C2×C4—261no—
537C2×C4—261no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—261no—
542C2×C4—261no—
543C2×C4—261no—
544C2×C4—261no—
545C2×C4—261no—
546C2×C4—524no—
547C2×C4—524no—
548C2×C4—522no—
549C2×C4—521no—
550C2×C4—521no—
551C2×C4—521no—
552C2×C4—522no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—522no—
557C2×C4—1044no—
558C2×C4#8813314yes—
559C2×C4#8813714yes—
560C2×C4#8813812yes—
561C2×C4#8813912yes—
562C2×C4#8818014yes—
563C2×C4D4×Dic1314yes—
564C2×C4D4×Dic1312yes—
565C2×C4C2×#612711yescharacteristic
566C2×C4C2×#612711yescharacteristic
567C2×C4C2×D4×D1311yescharacteristic
568C2×C4C2×#612911yescharacteristic
569C2×C4C2×#612911yescharacteristic
570C2×C4C23×Dic1311yescharacteristic
571C8—24no—
572C8—264no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22#1171861214yes—
587C22C2×#8813811yescharacteristic
588C22C2×#8813911yescharacteristic
589C22D13×AGL(1,ℤ/8)11yescharacteristic
590C22#1171947811yescharacteristic
591C22D13×#13711yescharacteristic
592C22#1171948211yescharacteristic
593C22C2×D4×Dic1311yescharacteristic
594C4—44no—
595C4—42no—
596C4—262no—
597C4—262no—
598C4—521no—
599C4—521no—
600C4—1042no—
601C4—1041no—
602C4—1041no—
603C4#1171861214yes—
604C4C2×#8813311yescharacteristic
605C4C2×#8813711yescharacteristic
606C4C2×#8818011yescharacteristic
607C4C2×D4×Dic1311yescharacteristic
608C2—24no—
609C2—82no—
610C2C2×#1171861211yescharacteristic
611C2AGL(1,ℤ/8)×Dic1311yescharacteristic
612C2#137×Dic1311yescharacteristic
613C2#1312445811yescharacteristic
614C2#1312446911yescharacteristic
615C2#1312522711yescharacteristic
616C2#1312523411yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#417C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenterC2×#417D13
hypocenterC13#21093
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×M16×Dic13C2semidirect
4. C2×#11718646C2semidirect
5. #13115535C2semidirect
6. #13115528C2semidirect
7. #13115521C2semidirect
8. #13115514C2semidirect
9. #13114556C2semidirect
10. #13114555C2semidirect
20. C13×#417C4semidirect
96. C13×#103C2×C4semidirect
97. C13×#102C2×C4semidirect
285. #417Dic13semidirect
396. #103C2×Dic13semidirect
398. #102C2×Dic13semidirect
501. C13#21093semidirect

Maximal subgroups

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

subgroupindex
C2×#117188932
C2×M16×Dic132
#131173272
C2×#117186462
#131155352
#131155282
#131155212
#131155142
#131145562
#131145552
C2×C13×#4172
#2109313