Finite Group Databasebeta

← #15870260

Subgroups — #15870260

Order 3328.

900 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718696C211yesmaximal, characteristic
2#13117192C211yesmaximal, characteristic, complemented
3#13115453C211yesmaximal, characteristic, complemented
4#13115453C211yesmaximal, characteristic, complemented
5#13115425C211yesmaximal, characteristic, complemented
6#13115411C211yesmaximal, characteristic, complemented
7#13115154C211yesmaximal, characteristic
8#13115023C211yesmaximal, characteristic, complemented
9#13114928C211yesmaximal, characteristic, complemented
10#13114734C211yesmaximal, characteristic, complemented
11C4×#88047C211yesmaximal, characteristic, complemented
12#13114661C211yesmaximal, characteristic, complemented
13C4×#88072C211yesmaximal, characteristic, complemented
14#13114602C211yesmaximal, characteristic, complemented
15C13×#2324C211yesmaximal, characteristic
16C2×C13×#121C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C2×C13×#117C2211yescharacteristic
19C2×#88180C2211yescharacteristic
20C22×C4⋊Dic13C2211yescharacteristic
21C13×#390—21no—
22C13×#390C2211yescharacteristic
23C13×#389C2211yescharacteristic
24C13×#388—21no—
25C13×#386—21no—
26C13×#385—21no—
27C13×#380C2211yescharacteristic
28C4×C13×C4⋊C4C2211yescharacteristic
29C2×#88072C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C2×C22⋊#1055C2211yescharacteristic
34C2×C22⋊#1055C2211yescharacteristic
35#11718696C418yescomplemented
36C4⋊C4×Dic13—21no—
37C4⋊C4×Dic13—21no—
38C2×#88048C2211yescharacteristic
39C2×#88047C2211yescharacteristic
40C2×#88047C2211yescharacteristic
41C2×#88047C2211yescharacteristic
42C22⋊C4×Dic13—21no—
43C22⋊C4×Dic13—21no—
44C4×C4⋊Dic13C2211yescharacteristic
45C4×C4⋊Dic13C2211yescharacteristic
46C4×C4⋊Dic13C2211yescharacteristic
47C4×C4⋊Dic13C2211yescharacteristic
48C2×#88043C2211yescharacteristic
49C2×C4×#1055C2211yescharacteristic
50#11718121C2211yescharacteristic
51#11718121C2211yescharacteristic
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60C22×D4×C13—21no—
61C22×D4×C13C2311yescharacteristic
62C23×C4×C13—21no—
63C23×C4×C13C2311yescharacteristic
64C23×Dic13—21no—
65C13×#121—22no—
66C13×#121C2×C414yes—
67C4×D4×C13—22no—
68C4×D4×C13—22no—
69C13×#117—22no—
70C13×#117—22no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4D411yescharacteristic
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C22⋊C4—21no—
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C13×C22⋊C4D411yescharacteristic
83C2×C13×C22⋊C4D411yescharacteristic
84C2×C42×C13—21no—
85C2×C42×C13C2311yescharacteristic
86C2×C42×C13C2311yescharacteristic
87C2×C22⋊Dic13—21no—
88#88180—22no—
89#88180C2×C414yescomplemented
90C2×C4⋊Dic13—21no—
91C2×C4⋊Dic13—22no—
92C2×C4⋊Dic13—22no—
93C2×C4⋊Dic13C2311yescharacteristic
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—21no—
101C2×C4×Dic13—21no—
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C22×#1055C2311yescharacteristic
105C22×#1055C2311yescharacteristic
106C13×#95—21no—
107C13×#95—21no—
108C13×#95—21no—
109#88072—22no—
110#88072—22no—
111#88072C2×C414yescomplemented
112#88072C2×C414yescomplemented
113#88071—21no—
114#88071—21no—
115#88071—21no—
116#88071—21no—
117C22⋊#1055—22no—
118C22⋊#1055C2×C414yescomplemented
119#88048C2×C414yescomplemented
120#88047—22no—
121#88047—22no—
122#88047C2×C414yescomplemented
123#88043—22no—
124C4×#1055—22no—
125#21590—131nomaximal, Sylow
126C24×C13—21no—
127C24×C13C2×D411yescharacteristic
128C2×D4×C13—24no—
129C2×D4×C13—21no—
130C2×D4×C13—21no—
131C2×D4×C13—21no—
132C2×D4×C13—21no—
133C2×D4×C13—21no—
134C2×D4×C13—81no—
135C2×D4×C13—81no—
136C2×D4×C13C22×C412yes—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—21no—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—42no—
145C22×C4×C13—81no—
146C22×C4×C13—81no—
147C22×C4×C13C22×C412yes—
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13#4112yes—
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C22×Dic13—24no—
160C22×Dic13—81no—
161C22×Dic13—81no—
162C13×C4⋊C4—22no—
163C13×C4⋊C4—22no—
164C13×C4⋊C4—21no—
165C13×C4⋊C4—21no—
166C13×C4⋊C4—21no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—81no—
170C13×C4⋊C4C22×C412yes—
171C13×C22⋊C4—24no—
172C13×C22⋊C4—22no—
173C13×C22⋊C4—22no—
174C13×C22⋊C4—42no—
175C13×C22⋊C4—81no—
176C13×C22⋊C4—81no—
177C13×C22⋊C4—81no—
178C42×C13—21no—
179C42×C13—21no—
180C42×C13—81no—
181C42×C13#4112yes—
182C42×C13#4112yes—
183C22⋊Dic13—24no—
184C4⋊Dic13—24no—
185C4⋊Dic13—21no—
186C4⋊Dic13—21no—
187C4⋊Dic13—21no—
188C4⋊Dic13—21no—
189C4⋊Dic13C22×C412yes—
190C4⋊Dic13C22×C412yes—
191C4⋊Dic13#4112yes—
192C4⋊Dic13#4112yes—
193C4×Dic13—24no—
194C4×Dic13—81no—
195C4×Dic13—81no—
196C4×Dic13—81no—
197C4×Dic13—81no—
198C2×#1055—21no—
199C2×#1055—21no—
200C2×#1055—21no—
201C2×#1055—21no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055C22×C412yes—
209C2×#1055C22×C412yes—
210C2×#482—131no—
211#2324D1311yescharacteristic
212#2274—131no—
213#1925—131no—
214#1925—131no—
215#1921—131no—
216#1919—131no—
217#1874—131no—
218#1852—131no—
219#1825—131no—
220#1776—131no—
221C4×#107—131no—
222#1762—131no—
223C4×#102—131no—
224#1750—131no—
225C23×C13—21no—
226C23×C13—21no—
227C23×C13—21no—
228C23×C13—21no—
229C23×C13—21no—
230C23×C13—21no—
231C23×C13—22no—
232C23×C13—41no—
233C23×C13—41no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13C4×D412yes—
239C23×C13C22×D411yescharacteristic
240D4×C13—22no—
241D4×C13—22no—
242D4×C13—41no—
243D4×C13—41no—
244D4×C13—84no—
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—22no—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—21no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—41no—
268C2×C4×C13—41no—
269C2×C4×C13—41no—
270C2×C4×C13—84no—
271C2×C4×C13C4×D412yes—
272C2×C4×C13#12312yes—
273C2×C4×C13#12312yes—
274C2×C4×C13C23×C411yescharacteristic
275C2×C4×C13C2×#4111yescharacteristic
276C2×C4×C13C2×#4111yescharacteristic
277C2×C4×C13C2×#4111yescharacteristic
278C2×C4×C13C2×#4111yescharacteristic
279C2×C4×C13C2×#4111yescharacteristic
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—84no—
293#1055—22no—
294#1055—22no—
295#1055—41no—
296#1055—41no—
297C2×#121C2×D1311yescharacteristic
298C2×C4×D4C2×D1311yescharacteristic
299C2×#117—131no—
300C2×#117C2×D1311yescharacteristic
301C22×C4⋊C4—131no—
302#482—138no—
303C2×#107—131no—
304C2×#107—131no—
305C2×#107—131no—
306C2×#106—131no—
307C2×#102—131no—
308C2×#102—131no—
309C2×#102—131no—
310C2×#102—131no—
311C2×C22⋊C8—131no—
312C2×C22⋊C8—131no—
313C2×#97—131no—
314C2×C4×C8—131no—
315#390—21no—
316#390C2×D1311yescharacteristic
317#389C2×D1311yescharacteristic
318#388—21no—
319#386—21no—
320#385—21no—
321#380C2×D1311yescharacteristic
322C4×C4⋊C4—131no—
323C4×C4⋊C4—131no—
324C4×C4⋊C4—131no—
325C4×C4⋊C4—131no—
326C4×C4⋊C4—261no—
327C4×C4⋊C4—261no—
328C4×C4⋊C4C2×D1311yescharacteristic
329C4×C22⋊C4—261no—
330C4×C22⋊C4—261no—
331#340—131no—
332#340—131no—
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#340—131no—
338#340—131no—
339#336—131no—
340#336—131no—
341C22×C13—22no—
342C22×C13—22no—
343C22×C13—22no—
344C22×C13—41no—
345C22×C13—41no—
346C22×C13—41no—
347C22×C13—41no—
348C22×C13—42no—
349C22×C13—42no—
350C22×C13—42no—
351C22×C13—81no—
352C22×C13—81no—
353C22×C13—81no—
354C22×C13—81no—
355C22×C13C2×C4×D411yescharacteristic
356C22×C13C2×#12311yescharacteristic
357C22×C13#52511yescharacteristic
358C22×C13#57611yescharacteristic
359C22×C13#57611yescharacteristic
360C22×C13#57711yescharacteristic
361C22×C13#57811yescharacteristic
362C4×C13—22no—
363C4×C13—22no—
364C4×C13—22no—
365C4×C13—41no—
366C4×C13—41no—
367C4×C13—42no—
368C4×C13—81no—
369C4×C13—81no—
370C4×C13C4×#4111yescharacteristic
371C4×C13C4×#4111yescharacteristic
372C4×C13#52911yescharacteristic
373C4×C13#52911yescharacteristic
374Dic13—42no—
375Dic13—42no—
376Dic13—42no—
377Dic13—42no—
378C22×D4—21no—
379C22×D4C22×D1311yescharacteristic
380C23×C4—21no—
381C23×C4—261no—
382C23×C4C22×D1311yescharacteristic
383C22×C8—131no—
384C22×C8—131no—
385C22×C8—131no—
386C22×C8—131no—
387#121—22no—
388#121C4×D1314yes—
389C4×D4—22no—
390C4×D4—22no—
391#117—22no—
392#117—22no—
393#117—134no—
394#117—262no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—262no—
401C2×C4⋊C4—262no—
402C2×C4⋊C4—261no—
403C2×C4⋊C4C22⋊D1311yescharacteristic
404C2×C4⋊C4C22⋊D1311yescharacteristic
405C2×C4⋊C4C22×D1311yescharacteristic
406C2×C4⋊C4C22×D1311yescharacteristic
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—261no—
414C2×C22⋊C4C22⋊D1311yescharacteristic
415C2×C22⋊C4C22⋊D1311yescharacteristic
416C2×C42—21no—
417C2×C42—261no—
418C2×C42—261no—
419C2×C42—261no—
420C2×C42—261no—
421C2×C42—261no—
422C2×C42C22×D1311yescharacteristic
423C2×C42C22×D1311yescharacteristic
424#107—134no—
425#107—262no—
426#107—262no—
427#106—134no—
428#102—134no—
429#102—134no—
430#102—262no—
431#102—262no—
432C22⋊C8—134no—
433C22⋊C8—262no—
434#97—262no—
435C4×C8—262no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—261no—
440#95—261no—
441#95—261no—
442#95—261no—
443C2×C13—42no—
444C2×C13—82no—
445C2×C13C4×#12311yescharacteristic
446C2×C13#294611yescharacteristic
447C2×C13#295111yescharacteristic
448C2×C13#309611yescharacteristic
449C2×C13#309811yescharacteristic
450C2×C13#312111yescharacteristic
451C2×C13#312211yescharacteristic
452C24—21no—
453C24C2×C22⋊D1311yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4C2×C4×D1312yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4—261no—
477C22×C4—264no—
478C22×C4—1041no—
479C22×C4—1041no—
480C22×C4C2×C4×D1312yes—
481C22×C4#612712yes—
482C22×C4C2×C22⋊D1311yescharacteristic
483C22×C4C2×C22⋊D1311yescharacteristic
484C22×C4C2×C22⋊D1311yescharacteristic
485C22×C4C2×C22⋊D1311yescharacteristic
486C22×C4C2×C22⋊D1311yescharacteristic
487C22×C4C23×D1311yescharacteristic
488C2×C8—132no—
489C2×C8—132no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8—261no—
499C2×C8—261no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—81no—
508C4⋊C4—132no—
509C4⋊C4—132no—
510C4⋊C4—132no—
511C4⋊C4—132no—
512C4⋊C4—264no—
513C4⋊C4—261no—
514C4⋊C4—261no—
515C4⋊C4—261no—
516C4⋊C4—261no—
517C4⋊C4C2×C4×D1312yes—
518C22⋊C4—24no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4—81no—
525C22⋊C4—264no—
526C42—21no—
527C42—21no—
528C42—81no—
529C42—264no—
530C42—1041no—
531C42—1041no—
532C42—1041no—
533C42—1041no—
534C42#612912yes—
535C42#612912yes—
536C13#2159011yescharacteristic, complemented, Sylow
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—22no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23C4×C22⋊D1312yes—
551C23C22×C22⋊D1311yescharacteristic
552D4—22no—
553D4—22no—
554D4—41no—
555D4—41no—
556D4—84no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—84no—
583C2×C4—262no—
584C2×C4—262no—
585C2×C4—262no—
586C2×C4—262no—
587C2×C4—521no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—521no—
591C2×C4—521no—
592C2×C4—521no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—1044no—
596C2×C4C4×C22⋊D1312yes—
597C2×C4#8818912yes—
598C2×C4#8818912yes—
599C2×C4C22×C4×D1311yescharacteristic
600C2×C4C2×#612711yescharacteristic
601C2×C4C2×#612911yescharacteristic
602C2×C4C2×#612911yescharacteristic
603C2×C4#41×D1311yescharacteristic
604C2×C4#41×D1311yescharacteristic
605C8—262no—
606C8—262no—
607C8—521no—
608C8—521no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22C2×C4×C22⋊D1311yescharacteristic
624C22C2×#8818911yescharacteristic
625C22#1171953311yescharacteristic
626C22#1171954011yescharacteristic
627C22#1171954011yescharacteristic
628C22#1171954111yescharacteristic
629C22#1171954311yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—522no—
639C4—522no—
640C4—522no—
641C4—522no—
642C4C4×#612911yescharacteristic
643C4C4×#612911yescharacteristic
644C4#1171924311yescharacteristic
645C4#1171924311yescharacteristic
646C2—42no—
647C2—82no—
648C2C4×#8818911yescharacteristic
649C2#1312366811yescharacteristic
650C2#1312370911yescharacteristic
651C2#1312478811yescharacteristic
652C2#1312481011yescharacteristic
653C2#1312505911yescharacteristic
654C2#1312507411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2324C2
socleC23×C13C22×D4
prime radicalC2×C13×#121C22
cosocleC22×C4×C13C24
hypercenter#2324D13
hypocenterC13#21590
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2324

Complements

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

NH ≅ G/Ntype
2. #13117192C2semidirect
3. #13115453C2semidirect
4. #13115453C2semidirect
5. #13115425C2semidirect
6. #13115411C2semidirect
8. #13115023C2semidirect
9. #13114928C2semidirect
10. #13114734C2semidirect
11. C4×#88047C2semidirect
12. #13114661C2semidirect
13. C4×#88072C2semidirect
14. #13114602C2semidirect
35. #11718696C4semidirect
89. #88180C2×C4semidirect
111. #88072C2×C4semidirect
112. #88072C2×C4semidirect
118. C22⋊#1055C2×C4semidirect
119. #88048C2×C4semidirect
122. #88047C2×C4semidirect
536. C13#21590semidirect

Maximal subgroups

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

subgroupindex
C2×#117186962
#131171922
#131154532
#131154252
#131154112
#131151542
#131150232
#131149282
#131147342
C4×#880472
#131146612
C4×#880722
#131146022
C13×#23242
#2159013