Finite Group Databasebeta

← #15873526

Subgroups — #15873526

Order 3328.

762 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718351C211yesmaximal, characteristic
2#13117461C211yesmaximal, characteristic
3#13117446C211yesmaximal, characteristic
4#13115605C211yesmaximal, characteristic, complemented
5#13115601C211yesmaximal, characteristic, complemented
6#13115596C211yesmaximal, characteristic, complemented
7#13115591C211yesmaximal, characteristic, complemented
8#13115034C211yesmaximal, characteristic
9#13115032C211yesmaximal, characteristic
10#13115027C211yesmaximal, characteristic
11#13115022C211yesmaximal, characteristic
12#13115005C211yesmaximal, characteristic, complemented
13#13114969C211yesmaximal, characteristic, complemented
14#13114964C211yesmaximal, characteristic, complemented
15C13×#1846C211yesmaximal, characteristic, complemented
16C2×#88146C2211yescharacteristic
17C2×C4⋊C4×D13C2211yescharacteristic
18C2×#88126C2211yescharacteristic
19C2×C13×#107C2211yescharacteristic
20C2×C13×#107C2211yescharacteristic
21C2×C13×#106C2211yescharacteristic
22C2×C13×#106C2211yescharacteristic
23C2×C4×C8×C13C2211yescharacteristic
24C13×#382C2211yescharacteristic
25C13×#382C2211yescharacteristic
26C2×#88060C2211yescharacteristic
27C2×#88060C2211yescharacteristic
28C2×#88054C2211yescharacteristic
29C2×#88054C2211yescharacteristic
30#11718648C2211yescharacteristic
31#11718648C2211yescharacteristic
32#11718638C2211yescharacteristic
33#11718637C2211yescharacteristic
34C4⋊C4×Dic13C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C2×#88048C2211yescharacteristic
37C2×#88048C2211yescharacteristic
38C2×#88047C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40#11718596C2211yescharacteristic
41#11718596C2211yescharacteristic
42C2×#88043C2211yescharacteristic
43#11718351C418yescomplemented
44#11718331—21no—
45#11718331—21no—
46#11718329—21no—
47#11718329—21no—
48#11718316—21no—
49#11718316—21no—
50#11718308—21no—
51#11718308—21no—
52#11718143C2211yescharacteristic
53#11718143C2211yescharacteristic
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×C4×D13—21no—
61C22×C4×D13C2311yescharacteristic
62C22×C8×C13C2311yescharacteristic
63C22×C8×C13C2311yescharacteristic
64C2×C13×C4⋊C4C2311yescharacteristic
65C2×C13×C4⋊C4C2311yescharacteristic
66C2×C13×C4⋊C4C2311yescharacteristic
67C2×C13×C4⋊C4C2311yescharacteristic
68C2×C42×C13C2311yescharacteristic
69C2×#6103—21no—
70C2×#6103—21no—
71C2×#6103—21no—
72C2×#6103D411yescharacteristic
73C2×#6103D411yescharacteristic
74C2×C4⋊Dic13—21no—
75C2×C4⋊Dic13—21no—
76C2×C4⋊Dic13—21no—
77C2×C4⋊Dic13—21no—
78C2×C4⋊Dic13C2311yescharacteristic
79C2×C4⋊Dic13C2311yescharacteristic
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×#6101—21no—
83C2×#6101—21no—
84C2×#6101—21no—
85C2×#6101D411yescharacteristic
86C2×#6101D411yescharacteristic
87C2×C4×Dic13—21no—
88C2×C4×Dic13—21no—
89C2×C4×Dic13—21no—
90C2×C4×Dic13C2311yescharacteristic
91C22×#1055C2311yescharacteristic
92C22×#1055C2311yescharacteristic
93#88146—41no—
94#88146—41no—
95C4⋊C4×D13—41no—
96C4⋊C4×D13—41no—
97#88126C2×C414yes—
98#88126Q814yes—
99C13×#107—41no—
100C13×#107—41no—
101C13×#106—41no—
102C13×#106—41no—
103C4×C8×C13C2×C414yescomplemented
104C13×#95—21no—
105C13×#95—21no—
106C13×#95—21no—
107C13×#95—21no—
108#88071—21no—
109#88071—21no—
110#88071—21no—
111#88071—21no—
112#88071—21no—
113#88071—21no—
114#88060C2×C414yes—
115#88060C2×C414yes—
116#88054C2×C414yescomplemented
117#88054C2×C414yescomplemented
118#88048—41no—
119#88048—41no—
120#88047—41no—
121#88047—41no—
122#88043C2×C414yescomplemented
123#21992—131nomaximal, Sylow
124C23×D13C2×D411yescharacteristic
125C22×C4×C13—21no—
126C22×C4×C13—21no—
127C22×C4×C13—21no—
128C22×C4×C13—21no—
129C22×C4×C13C2×D411yescharacteristic
130C22×C4×C13C2×D411yescharacteristic
131C22×C4×C13C2411yescharacteristic
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13C2×D411yescharacteristic
137C22×Dic13C2×D411yescharacteristic
138C22×Dic13C2×D411yescharacteristic
139C2×C4×D13—21no—
140C2×C4×D13—21no—
141C2×C4×D13—21no—
142C2×C4×D13—21no—
143C2×C4×D13—42no—
144C2×C4×D13—81no—
145C2×C4×D13—81no—
146C2×C4×D13C22×C412yes—
147C2×C4×D13C2×Q812yes—
148C2×C8×C13—21no—
149C2×C8×C13—21no—
150C2×C8×C13—21no—
151C2×C8×C13—21no—
152C2×C8×C13C22×C412yes—
153C2×C8×C13C22×C412yes—
154C13×C4⋊C4—21no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—21no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—21no—
161C13×C4⋊C4—21no—
162C42×C13C22×C412yes—
163C42×C13C2×Q812yes—
164#6103—42no—
165#6103—81no—
166#6103—81no—
167#6103C4⋊C414yes—
168#6103C4⋊C414yes—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—21no—
172C4⋊Dic13—21no—
173C4⋊Dic13—21no—
174C4⋊Dic13—21no—
175C4⋊Dic13—21no—
176C4⋊Dic13—21no—
177C4⋊Dic13—81no—
178C4⋊Dic13—81no—
179C4⋊Dic13—81no—
180C4⋊Dic13—81no—
181#6101—42no—
182#6101—81no—
183#6101—81no—
184#6101C4⋊C414yes—
185#6101C4⋊C414yes—
186C4×Dic13—42no—
187C4×Dic13—81no—
188C4×Dic13—81no—
189C4×Dic13C22×C412yes—
190C4×Dic13C2×Q812yes—
191C2×#1055—21no—
192C2×#1055—21no—
193C2×#1055—21no—
194C2×#1055—21no—
195C2×#1055C22×C412yes—
196C2×#1055C22×C412yes—
197C2×#433—131no—
198#2317—131no—
199#2314—131no—
200#1949—131no—
201#1949—131no—
202#1948—131no—
203#1947—131no—
204#1854—131no—
205#1854—131no—
206#1853—131no—
207#1852—131no—
208#1847—131no—
209#1846D1311yescharacteristic, complemented
210#1836—131no—
211#1834—131no—
212C23×C13C22×D411yescharacteristic
213C22×D13—21no—
214C22×D13—21no—
215C22×D13—21no—
216C22×D13—21no—
217C22×D13—21no—
218C22×D13—21no—
219C22×D13C2×C4⋊C412yes—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—41no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—41no—
231C2×C4×C13—41no—
232C2×C4×C13—41no—
233C2×C4×C13—41no—
234C2×C4×C13—41no—
235C2×C4×C13—41no—
236C2×C4×C13C2×C4⋊C412yes—
237C2×C4×C13C2×C4⋊C412yes—
238C2×C4×C13C23×C411yescharacteristic
239C2×C4×C13C22×Q811yescharacteristic
240C2×C4×C13H(5,𝔽2)11yescharacteristic
241C2×C4×C13H(5,𝔽2)11yescharacteristic
242C2×C4×C132−1+411yescharacteristic
243C2×C4×C132−1+411yescharacteristic
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—21no—
247C2×Dic13—21no—
248C2×Dic13—21no—
249C2×Dic13—21no—
250C2×Dic13—41no—
251C2×Dic13—41no—
252C2×Dic13—41no—
253C2×Dic13—41no—
254C2×Dic13—41no—
255C2×Dic13—41no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—41no—
259C2×Dic13—41no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13C2×C4⋊C412yes—
263C2×Dic13C2×C4⋊C412yes—
264C2×Dic13C2×C4⋊C412yes—
265C4×D13—22no—
266C4×D13—22no—
267C4×D13—22no—
268C4×D13—22no—
269C4×D13—84no—
270C8×C13—22no—
271C8×C13—22no—
272#1055—22no—
273#1055—22no—
274C2×#117—131no—
275C2×#117—131no—
276C22×C4⋊C4—131no—
277#433—138no—
278C2×#107—131no—
279C2×#107—131no—
280C2×#107C2×D1311yescharacteristic
281C2×#107C2×D1311yescharacteristic
282C2×#106—131no—
283C2×#106—131no—
284C2×#106C2×D1311yescharacteristic
285C2×#106C2×D1311yescharacteristic
286C2×C4⋊C8—131no—
287C2×C4⋊C8—131no—
288C2×C22⋊C8—131no—
289C2×C22⋊C8—131no—
290C2×#97—131no—
291C2×C4×C8C2×D1311yescharacteristic
292#389—261no—
293#389—261no—
294#388—261no—
295#388—261no—
296#387—261no—
297#387—261no—
298#385—261no—
299#385—261no—
300#384—131no—
301#384—131no—
302#384—131no—
303#382—131no—
304#382C2×D1311yescharacteristic
305#382C2×D1311yescharacteristic
306#380—131no—
307#380—131no—
308C4×C4⋊C4—131no—
309C4×C4⋊C4—131no—
310#340—131no—
311#340—131no—
312#340—131no—
313#340—131no—
314#340—131no—
315#340—131no—
316#340—131no—
317#340—131no—
318C22×C13C22×C4⋊C411yescharacteristic
319C22×C13#53511yescharacteristic
320C22×C13#53611yescharacteristic
321C22×C13#57611yescharacteristic
322C22×C13#57711yescharacteristic
323C22×C13#57711yescharacteristic
324C22×C13#57811yescharacteristic
325C2×D13—22no—
326C2×D13—22no—
327C2×D13—22no—
328C2×D13—41no—
329C2×D13—41no—
330C2×D13—41no—
331C2×D13—41no—
332C4×C13—22no—
333C4×C13—22no—
334C4×C13—81no—
335C4×C13—81no—
336C4×C13—81no—
337C4×C13—81no—
338C4×C13#52011yescharacteristic
339C4×C13#52011yescharacteristic
340C4×C13#54411yescharacteristic
341C4×C13#54411yescharacteristic
342Dic13—22no—
343Dic13—22no—
344Dic13—22no—
345Dic13—81no—
346Dic13—81no—
347Dic13—81no—
348Dic13—81no—
349C23×C4—131no—
350C23×C4—261no—
351C22×C8—131no—
352C22×C8—131no—
353C22×C8C22×D1311yescharacteristic
354C22×C8C22×D1311yescharacteristic
355#117—134no—
356#117—134no—
357#117—521no—
358#117—521no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—131no—
361C2×C4⋊C4—131no—
362C2×C4⋊C4—131no—
363C2×C4⋊C4—131no—
364C2×C4⋊C4—131no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—261no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4—261no—
369C2×C4⋊C4—261no—
370C2×C4⋊C4—261no—
371C2×C4⋊C4—261no—
372C2×C4⋊C4—521no—
373C2×C4⋊C4—521no—
374C2×C4⋊C4C22×D1311yescharacteristic
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C4⋊C4C22×D1311yescharacteristic
377C2×C4⋊C4C22×D1311yescharacteristic
378C2×C22⋊C4—131no—
379C2×C22⋊C4—131no—
380C2×C22⋊C4—261no—
381C2×C22⋊C4—261no—
382C2×C22⋊C4—261no—
383C2×C42—131no—
384C2×C42—261no—
385C2×C42—261no—
386C2×C42—261no—
387C2×C42C22×D1311yescharacteristic
388#107—41no—
389#107—41no—
390#107—521no—
391#107—521no—
392#106—41no—
393#106—41no—
394#106—521no—
395#106—521no—
396C4⋊C8—134no—
397C4⋊C8—134no—
398C22⋊C8—134no—
399C22⋊C8—134no—
400#97—134no—
401C4×C8C4×D1314yescomplemented
402#95—21no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—261no—
407#95—261no—
408#95—261no—
409#95—261no—
410#95—261no—
411#95—261no—
412C2×C13#231711yescharacteristic
413C2×C13#291211yescharacteristic
414C2×C13#291311yescharacteristic
415C2×C13#322411yescharacteristic
416C2×C13#322511yescharacteristic
417C2×C13#324111yescharacteristic
418C2×C13#324211yescharacteristic
419D13—42no—
420C24—131no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—131no—
426C22×C4—131no—
427C22×C4—131no—
428C22×C4—132no—
429C22×C4—132no—
430C22×C4—261no—
431C22×C4—261no—
432C22×C4—261no—
433C22×C4—261no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4—261no—
437C22×C4—261no—
438C22×C4—522no—
439C22×C4—1041no—
440C22×C4—1041no—
441C22×C4D4×D1311yescharacteristic
442C22×C4D4×D1311yescharacteristic
443C22×C4C23×D1311yescharacteristic
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—132no—
449C2×C8—132no—
450C2×C8—261no—
451C2×C8—261no—
452C2×C8—261no—
453C2×C8—261no—
454C2×C8C2×C4×D1312yes—
455C2×C8C2×C4×D1312yes—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—134no—
465C4⋊C4—134no—
466C4⋊C4—261no—
467C4⋊C4—261no—
468C4⋊C4—261no—
469C4⋊C4—261no—
470C4⋊C4—261no—
471C4⋊C4—261no—
472C4⋊C4—261no—
473C4⋊C4—261no—
474C4⋊C4—522no—
475C4⋊C4—1041no—
476C4⋊C4—1041no—
477C4⋊C4—1041no—
478C4⋊C4—1041no—
479C4⋊C4—1041no—
480C4⋊C4—1041no—
481C22⋊C4—134no—
482C22⋊C4—134no—
483C22⋊C4—522no—
484C22⋊C4—1041no—
485C22⋊C4—1041no—
486C42—132no—
487C42—132no—
488C42—522no—
489C42—1041no—
490C42—1041no—
491C42C2×C4×D1312yes—
492C42Q8×D1312yes—
493C13#2199211yescharacteristic, complemented, Sylow
494C23—132no—
495C23—261no—
496C23—261no—
497C23—261no—
498C23—261no—
499C23—261no—
500C23—261no—
501C23C2×D4×D1311yescharacteristic
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—132no—
519C2×C4—132no—
520C2×C4—132no—
521C2×C4—261no—
522C2×C4—261no—
523C2×C4—261no—
524C2×C4—261no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—262no—
528C2×C4—262no—
529C2×C4—262no—
530C2×C4—262no—
531C2×C4—521no—
532C2×C4—521no—
533C2×C4—521no—
534C2×C4—521no—
535C2×C4—521no—
536C2×C4—521no—
537C2×C4—521no—
538C2×C4—521no—
539C2×C4—521no—
540C2×C4—521no—
541C2×C4—521no—
542C2×C4—521no—
543C2×C4—1044no—
544C2×C4C4⋊C4×D1312yes—
545C2×C4C4⋊C4×D1312yes—
546C2×C4C22×C4×D1311yescharacteristic
547C2×C4#8825111yescharacteristic
548C2×C4#8825111yescharacteristic
549C2×C4C2×Q8×D1311yescharacteristic
550C2×C4#8825411yescharacteristic
551C2×C4#8825411yescharacteristic
552C8—22no—
553C8—22no—
554C8—262no—
555C8—262no—
556C22—262no—
557C22—262no—
558C22—262no—
559C22—521no—
560C22—521no—
561C22—521no—
562C22—521no—
563C22C2×C4⋊C4×D1311yescharacteristic
564C22#1171942011yescharacteristic
565C22#1171942911yescharacteristic
566C22#1171947411yescharacteristic
567C22#1171947511yescharacteristic
568C22#1171947511yescharacteristic
569C22#1171947611yescharacteristic
570C4—22no—
571C4—22no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4—262no—
577C4—262no—
578C4—262no—
579C4—1041no—
580C4—1041no—
581C4—1041no—
582C4—1041no—
583C4#1171920911yescharacteristic
584C4#1171920911yescharacteristic
585C4#1171943411yescharacteristic
586C4#1171943411yescharacteristic
587C2—522no—
588C2#1311745711yescharacteristic
589C2#1312342811yescharacteristic
590C2#1312343311yescharacteristic
591C2#1312592311yescharacteristic
592C2#1312593811yescharacteristic
593C2#1312609211yescharacteristic
594C2#1312610711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1846C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1846D13
hypocenterC13#21992
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1846

Complements

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

NH ≅ G/Ntype
4. #13115605C2semidirect
5. #13115601C2semidirect
6. #13115596C2semidirect
7. #13115591C2semidirect
12. #13115005C2semidirect
13. #13114969C2semidirect
14. #13114964C2semidirect
15. C13×#1846C2semidirect
43. #11718351C4semidirect
103. C4×C8×C13C2×C4semidirect
116. #88054C2×C4semidirect
117. #88054C2×C4semidirect
122. #88043C2×C4semidirect
209. #1846D13semidirect
401. C4×C8C4×D13semidirect
493. C13#21992semidirect

Maximal subgroups

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

subgroupindex
C2×#117183512
#131174612
#131174462
#131156052
#131156012
#131155962
#131155912
#131150342
#131150322
#131150272
#131150222
#131150052
#131149692
#131149642
C13×#18462
#2199213