Finite Group Databasebeta

← #15874138

Subgroups — #15874138

Order 3328.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#121×Dic13C211yesmaximal, characteristic
2C2×#11718891C211yesmaximal, characteristic, complemented
3#13115719C212yesmaximal, complemented
4#13115528C212yesmaximal, complemented
5#13115479C212yesmaximal, complemented
6#13115172C212yesmaximal, complemented
7#13115154C211yesmaximal, characteristic
8#13115026C211yesmaximal, characteristic, complemented
9#13114771C211yesmaximal, characteristic, complemented
10#13114600C211yesmaximal, characteristic, complemented
11C2×C13×#448C211yesmaximal, characteristic
12C23×Q8×C13C2211yescharacteristic
13C22×C13×SD16C2212yes—
14C2×C13×#121C2211yescharacteristic
15C2×D4×Dic13—21no—
16C2×D4×Dic13C2211yescharacteristic
17C22×C4⋊Dic13C2211yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19C13×#448C418yescomplemented
20C2×C13×#103C2212yes—
21C2×C13×C22⋊C8C2211yescharacteristic
22#11718891—21no—
23#11718891—21no—
24#11718891C2212yes—
25#11718891C2212yes—
26C2×#88075C2212yes—
27C2×#88075C2212yes—
28#11718882C2211yescharacteristic
29#11718881—21no—
30C2×#88072C2212yes—
31#11718873C2211yescharacteristic
32C2×#88071—21no—
33C2×#88071—21no—
34C2×C22⋊#1055C2211yescharacteristic
35C2×#88058C2212yes—
36C2×#88055C2212yes—
37#11718646C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39#11718612—21no—
40C22⋊C4×Dic13—21no—
41#11718143C2212yes—
42#11718142C2212yes—
43#11718121C2212yes—
44C22×Q8×C13—21no—
45C22×Q8×C13—22no—
46C22×Q8×C13—42no—
47C22×Q8×C13—42no—
48C22×Q8×C13C2×C414yescomplemented
49C22×Q8×C13C2312yes—
50C22×D4×C13—21no—
51C22×D4×C13C2311yescharacteristic
52C23×C4×C13—21no—
53C23×C4×C13C2311yescharacteristic
54C23×Dic13—21no—
55C23×Dic13—21no—
56C23×Dic13D411yescharacteristic
57C23×Dic13D411yescharacteristic
58C2×C13×SD16—42no—
59C2×C13×SD16—42no—
60C2×C13×SD16C2×C418yescomplemented
61C22×C8×C13C2312yes—
62C13×#121—22no—
63C13×#121C2×C414yes—
64C2×C13×C4⋊C4C2311yescharacteristic
65C2×C13×C22⋊C4—21no—
66C2×C22⋊Dic13—21no—
67C2×C22⋊Dic13—21no—
68C2×C22⋊Dic13—21no—
69C2×C22⋊Dic13—21no—
70D4×Dic13—42no—
71D4×Dic13—44no—
72C2×C4⋊Dic13—21no—
73C2×C4⋊Dic13—21no—
74C2×C4⋊Dic13—42no—
75C2×C4⋊Dic13C2312yes—
76C2×C4⋊Dic13C2311yescharacteristic
77C2×C4⋊Dic13C2311yescharacteristic
78C2×C4×Dic13—21no—
79C2×C4×Dic13—42no—
80C2×C4×Dic13D411yescharacteristic
81C2×C4×Dic13D411yescharacteristic
82C2×C4×Dic13C2311yescharacteristic
83C2×C4×Dic13C2311yescharacteristic
84C22×#1055C2312yes—
85C13×#103C2×C418yescomplemented
86C13×C22⋊C8C2×C414yescomplemented
87#88075—42no—
88#88075—42no—
89#88072—42no—
90#88071—21no—
91#88071—21no—
92#88071—21no—
93#88071—22no—
94#88071—22no—
95#88071—22no—
96#88071—22no—
97C22⋊#1055—22no—
98#88058—42no—
99#88055—42no—
100#22047—131nomaximal, Sylow
101C24×C13—21no—
102C24×C13C2×D411yescharacteristic
103C2×Q8×C13—24no—
104C2×Q8×C13—22no—
105C2×Q8×C13—22no—
106C2×Q8×C13—22no—
107C2×Q8×C13—22no—
108C2×Q8×C13—22no—
109C2×Q8×C13—48no—
110C2×Q8×C13—42no—
111C2×Q8×C13—42no—
112C2×Q8×C13—48no—
113C2×Q8×C13C22×C414yes—
114C2×D4×C13—24no—
115C2×D4×C13—21no—
116C2×D4×C13—22no—
117C2×D4×C13—22no—
118C2×D4×C13—42no—
119C2×D4×C13—42no—
120C2×D4×C13C22×C412yes—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13—24no—
124C22×C4×C13—22no—
125C22×C4×C13—21no—
126C22×C4×C13—42no—
127C22×C4×C13—42no—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13#4112yes—
131C22×C4×C13C2411yescharacteristic
132C22×Dic13—22no—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—21no—
137C22×Dic13—21no—
138C22×Dic13—41no—
139C22×Dic13—41no—
140C22×Dic13—41no—
141C22×Dic13—42no—
142C22×Dic13—42no—
143C22×Dic13—42no—
144C22×Dic13—42no—
145C22×Dic13—42no—
146C22×Dic13—42no—
147C22×Dic13C2×D411yescharacteristic
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C22×Dic13C2×D411yescharacteristic
151C13×SD16—48no—
152C2×C8×C13—22no—
153C2×C8×C13—22no—
154C2×C8×C13C22×C414yes—
155C13×C4⋊C4—21no—
156C13×C4⋊C4C22×C412yes—
157C13×C22⋊C4—24no—
158C22⋊Dic13—42no—
159C22⋊Dic13—42no—
160C22⋊Dic13—42no—
161C22⋊Dic13—42no—
162C4⋊Dic13—22no—
163C4⋊Dic13—22no—
164C4⋊Dic13—22no—
165C4⋊Dic13—22no—
166C4⋊Dic13—42no—
167C4⋊Dic13—42no—
168C4×Dic13—22no—
169C4×Dic13—22no—
170C4×Dic13—41no—
171C4×Dic13—41no—
172C4×Dic13—42no—
173C2×#1055—22no—
174C2×#1055—22no—
175C2×#1055—22no—
176C2×#448D1311yescharacteristic
177C4×#121—131no—
178C2×#391—131no—
179#1973—132no—
180#1935—132no—
181#1928—132no—
182#1877—132no—
183#1874—131no—
184#1852—131no—
185#1787—131no—
186#1749—131no—
187C23×C13—21no—
188C23×C13—21no—
189C23×C13—21no—
190C23×C13—21no—
191C23×C13—22no—
192C23×C13—41no—
193C23×C13—41no—
194C23×C13—41no—
195C23×C13—41no—
196C23×C13—41no—
197C23×C13—41no—
198C23×C13C4×D412yes—
199C23×C13AGL(1,ℤ/8)12yes—
200C23×C13#13712yes—
201C23×C13C22×D411yescharacteristic
202Q8×C13—24no—
203Q8×C13—24no—
204Q8×C13—48no—
205Q8×C13—42no—
206Q8×C13—42no—
207D4×C13—24no—
208D4×C13—48no—
209D4×C13—42no—
210C2×C4×C13—21no—
211C2×C4×C13—21no—
212C2×C4×C13—24no—
213C2×C4×C13—24no—
214C2×C4×C13—22no—
215C2×C4×C13—24no—
216C2×C4×C13—48no—
217C2×C4×C13—42no—
218C2×C4×C13—42no—
219C2×C4×C13—41no—
220C2×C4×C13—41no—
221C2×C4×C13C4×D412yes—
222C2×C4×C13C23×C411yescharacteristic
223C2×C4×C13C2×#4111yescharacteristic
224C2×C4×C13H(5,𝔽2)12yes—
225C2×C4×C13H(5,𝔽2)12yes—
226C2×Dic13—21no—
227C2×Dic13—21no—
228C2×Dic13—21no—
229C2×Dic13—21no—
230C2×Dic13—21no—
231C2×Dic13—21no—
232C2×Dic13—21no—
233C2×Dic13—21no—
234C2×Dic13—21no—
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—44no—
238C2×Dic13—44no—
239C2×Dic13—41no—
240C2×Dic13—42no—
241C2×Dic13—41no—
242C2×Dic13—41no—
243C2×Dic13—41no—
244C2×Dic13—42no—
245C2×Dic13—41no—
246C2×Dic13—41no—
247C2×Dic13—42no—
248C2×Dic13—84no—
249C2×Dic13—84no—
250C2×Dic13#13712yes—
251C8×C13—24no—
252#1055—42no—
253C23×Q8C2×D1311yescharacteristic
254C22×SD16C2×D1312yes—
255C2×#121C2×D1311yescharacteristic
256C2×C4×D4—131no—
257C2×C4×D4—261no—
258C22×C4⋊C4—131no—
259C22×C42—131no—
260#448Dic1318yescomplemented
261C2×#107—132no—
262C2×#103—132no—
263C2×#103—132no—
264C2×#103C2×D1312yes—
265C2×#102—132no—
266C2×C22⋊C8—131no—
267C2×C22⋊C8C2×D1311yescharacteristic
268C2×#97—132no—
269#391—132no—
270#391—132no—
271#391—261no—
272#391—261no—
273#390—131no—
274#389—131no—
275#388—261no—
276#386—261no—
277#380—131no—
278C4×C4⋊C4—131no—
279C4×C22⋊C4—261no—
280C2×#95—261no—
281C2×#95—261no—
282#340—132no—
283#340—132no—
284#336—132no—
285C22×C13—22no—
286C22×C13—22no—
287C22×C13—22no—
288C22×C13—22no—
289C22×C13—22no—
290C22×C13—22no—
291C22×C13—42no—
292C22×C13—42no—
293C22×C13—42no—
294C22×C13—81no—
295C22×C13—81no—
296C22×C13—81no—
297C22×C13—81no—
298C22×C13#44014yes—
299C22×C13C2×C4×D411yescharacteristic
300C22×C13#53411yescharacteristic
301C22×C13#53511yescharacteristic
302C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C13C2×#13711yescharacteristic
304C22×C13C2×#13711yescharacteristic
305C22×C13#57511yescharacteristic
306C4×C13—22no—
307C4×C13—44no—
308C4×C13—42no—
309C4×C13#51812yes—
310C4×C13#53812yes—
311Dic13—22no—
312Dic13—41no—
313Dic13—41no—
314Dic13—41no—
315Dic13—81no—
316Dic13—81no—
317Dic13—82no—
318C22×Q8—21no—
319C22×Q8—22no—
320C22×Q8—42no—
321C22×Q8—42no—
322C22×Q8C2×Dic1314yescomplemented
323C22×Q8C22×D1312yes—
324C22×D4—21no—
325C22×D4C22×D1311yescharacteristic
326C23×C4—21no—
327C23×C4—131no—
328C23×C4—131no—
329C23×C4—261no—
330C23×C4—261no—
331C23×C4C22×D1311yescharacteristic
332C2×SD16—42no—
333C2×SD16—42no—
334C2×SD16C2×Dic1318yescomplemented
335C22×C8—132no—
336C22×C8C22×D1312yes—
337#121—22no—
338#121C2×Dic1314yes—
339C4×D4—522no—
340C4×D4—524no—
341C2×C4⋊C4—132no—
342C2×C4⋊C4—131no—
343C2×C4⋊C4—131no—
344C2×C4⋊C4—261no—
345C2×C4⋊C4—261no—
346C2×C4⋊C4—522no—
347C2×C4⋊C4C22×D1311yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—261no—
350C2×C22⋊C4—261no—
351C2×C22⋊C4—261no—
352C2×C22⋊C4—261no—
353C2×C42—131no—
354C2×C42—131no—
355C2×C42—131no—
356C2×C42—131no—
357C2×C42—261no—
358C2×C42—522no—
359#107—522no—
360#103—522no—
361#103—522no—
362#103C2×Dic1318yescomplemented
363#102—522no—
364C22⋊C8—262no—
365C22⋊C8C2×Dic1314yescomplemented
366#97—522no—
367#95—262no—
368#95—262no—
369#95—262no—
370#95—262no—
371#95—261no—
372#95—261no—
373#95—261no—
374C2×C13—24no—
375C2×C13—82no—
376C2×C13#232111yescharacteristic
377C2×C13C2×#44011yescharacteristic
378C2×C13C4×#13711yescharacteristic
379C2×C13#319311yescharacteristic
380C2×C13#319611yescharacteristic
381C2×C13#321311yescharacteristic
382C2×C13#321811yescharacteristic
383C24—21no—
384C24D4×D1311yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—48no—
392C2×Q8—42no—
393C2×Q8—42no—
394C2×Q8—48no—
395C2×Q8C22×Dic1314yes—
396C2×D4—24no—
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4C22×Dic1312yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—131no—
411C22×C4—131no—
412C22×C4—131no—
413C22×C4—131no—
414C22×C4—262no—
415C22×C4—262no—
416C22×C4—262no—
417C22×C4—262no—
418C22×C4—261no—
419C22×C4—261no—
420C22×C4—522no—
421C22×C4—521no—
422C22×C4—521no—
423C22×C4—521no—
424C22×C4—522no—
425C22×C4—522no—
426C22×C4—522no—
427C22×C4—522no—
428C22×C4—522no—
429C22×C4D4×D1311yescharacteristic
430C22×C4#612912yes—
431C22×C4C22×Dic1312yes—
432C22×C4C23×D1311yescharacteristic
433SD16—48no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—262no—
437C2×C8—262no—
438C2×C8—262no—
439C2×C8C22×Dic1314yes—
440C4⋊C4—21no—
441C4⋊C4—262no—
442C4⋊C4—262no—
443C4⋊C4—262no—
444C4⋊C4—262no—
445C4⋊C4—522no—
446C4⋊C4—522no—
447C4⋊C4C22×Dic1312yes—
448C22⋊C4—24no—
449C22⋊C4—522no—
450C22⋊C4—522no—
451C22⋊C4—522no—
452C22⋊C4—522no—
453C42—262no—
454C42—262no—
455C42—521no—
456C42—521no—
457C42—522no—
458C13#2204711yescharacteristic, complemented, Sylow
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23#8816812yes—
471C23#8816912yes—
472C23D4×Dic1312yes—
473C23C2×D4×D1311yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—24no—
480D4—48no—
481D4—42no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—48no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—132no—
494C2×C4—261no—
495C2×C4—261no—
496C2×C4—261no—
497C2×C4—261no—
498C2×C4—261no—
499C2×C4—261no—
500C2×C4—261no—
501C2×C4—261no—
502C2×C4—261no—
503C2×C4—261no—
504C2×C4—261no—
505C2×C4—524no—
506C2×C4—524no—
507C2×C4—521no—
508C2×C4—521no—
509C2×C4—522no—
510C2×C4—521no—
511C2×C4—521no—
512C2×C4—522no—
513C2×C4—521no—
514C2×C4—522no—
515C2×C4—521no—
516C2×C4—1044no—
517C2×C4—1044no—
518C2×C4D4×Dic1312yes—
519C2×C4C2×#612911yescharacteristic
520C2×C4#8825112yes—
521C2×C4#8825112yes—
522C2×C4C23×Dic1311yescharacteristic
523C8—24no—
524C8—522no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22#1171882514yes—
539C22#1171929411yescharacteristic
540C22#1171929711yescharacteristic
541C22C2×#8816811yescharacteristic
542C22C2×#8816911yescharacteristic
543C22D13×#13711yescharacteristic
544C22#1171948211yescharacteristic
545C22C2×D4×Dic1311yescharacteristic
546C4—22no—
547C4—42no—
548C4—44no—
549C4—262no—
550C4—521no—
551C4—521no—
552C4—521no—
553C4—1041no—
554C4—1041no—
555C4—1042no—
556C4#1171929812yes—
557C4#1171950312yes—
558C2—24no—
559C2—82no—
560C2#1311749711yescharacteristic
561C2C2×#1171882511yescharacteristic
562C2#137×Dic1311yescharacteristic
563C2#1312564011yescharacteristic
564C2#1312566311yescharacteristic
565C2#1312582211yescharacteristic
566C2#1312586111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#448C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
cosocleC22×C4×C13C24
hypercenterC2×#448D13
hypocenterC13#22047
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. C2×#11718891C2semidirect
3. #13115719C2semidirect
4. #13115528C2semidirect
5. #13115479C2semidirect
6. #13115172C2semidirect
8. #13115026C2semidirect
9. #13114771C2semidirect
10. #13114600C2semidirect
19. C13×#448C4semidirect
48. C22×Q8×C13C2×C4semidirect
60. C2×C13×SD16C2×C4semidirect
85. C13×#103C2×C4semidirect
86. C13×C22⋊C8C2×C4semidirect
260. #448Dic13semidirect
322. C22×Q8C2×Dic13semidirect
334. C2×SD16C2×Dic13semidirect
362. #103C2×Dic13semidirect
365. C22⋊C8C2×Dic13semidirect
458. C13#22047semidirect

Maximal subgroups

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

subgroupindex
#121×Dic132
C2×#117188912
#131157192
#131155282
#131154792
#131151722
#131151542
#131150262
#131147712
#131146002
C2×C13×#4482
#2204713