Finite Group Databasebeta

← #15861414

Subgroups — #15861414

Order 3328.

1206 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718679C211yesmaximal, characteristic
2C2×#11718418C211yesmaximal, characteristic
3C2×#11718419C211yesmaximal, characteristic
4C4×#88141C211yesmaximal, characteristic, complemented
5C4×D4×Dic13C211yesmaximal, characteristic
6#13114734C211yesmaximal, characteristic, complemented
7#13114732C211yesmaximal, characteristic, complemented
8#13114728C211yesmaximal, characteristic, complemented
9#13114667C211yesmaximal, characteristic, complemented
10#13114665C211yesmaximal, characteristic, complemented
11#13114663C211yesmaximal, characteristic
12#13114655C211yesmaximal, characteristic, complemented
13#102×Dic13C211yesmaximal, characteristic
14#97×Dic13C211yesmaximal, characteristic, complemented
15C13×#1762C211yesmaximal, characteristic
16C2×C4×D4×C13C2211yescharacteristic
17C2×D4×Dic13C2211yescharacteristic
18C2×D4×Dic13C2211yescharacteristic
19C22×#6105C2211yescharacteristic
20C22×C4×Dic13—21no—
21C2×#88141C2211yescharacteristic
22C2×#88141C2211yescharacteristic
23C2×C4×Dic26C2211yescharacteristic
24C2×C13×#102C2211yescharacteristic
25C2×C13×#102C2211yescharacteristic
26C2×C13×#97C2211yescharacteristic
27C4×C13×C4⋊C4C2211yescharacteristic
28C2×#88072C2211yescharacteristic
29C4×C22⋊Dic13—21no—
30C2×#88056C2211yescharacteristic
31C2×#88056C2211yescharacteristic
32C2×#88055C2211yescharacteristic
33C2×#88055C2211yescharacteristic
34C2×C8×Dic13C2211yescharacteristic
35C2×C8×Dic13C2211yescharacteristic
36#11718679C418yescomplemented
37C4⋊C4×Dic13C2211yescharacteristic
38C4⋊C4×Dic13C2211yescharacteristic
39C4⋊C4×Dic13C2211yescharacteristic
40C4⋊C4×Dic13C2211yescharacteristic
41C2×#88050C2211yescharacteristic
42C2×#88048C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44C2×#88047C2211yescharacteristic
45C22⋊C4×Dic13—21no—
46C22⋊C4×Dic13—21no—
47C4×C4⋊Dic13C2211yescharacteristic
48C4×#6101—21no—
49C42×Dic13—21no—
50C42×Dic13C2211yescharacteristic
51C2×#88043C2211yescharacteristic
52#11718419C418yescomplemented
53#11718418C418yescomplemented
54#11718310—21no—
55#11718310—21no—
56C13×#340C2211yescharacteristic
57C13×#340C2211yescharacteristic
58#11718143C2211yescharacteristic
59#11718143C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C22×D4×C13C2311yescharacteristic
63C23×C4×C13—21no—
64C23×Dic13—21no—
65C23×Dic13—21no—
66C22×Dic26C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C4×D4×C13—22no—
70C4×D4×C13C2×C414yes—
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C22⋊C4—21no—
75C2×C42×C13—21no—
76C2×C42×C13C2311yescharacteristic
77C2×C22⋊Dic13—21no—
78C2×C22⋊Dic13—21no—
79D4×Dic13—22no—
80D4×Dic13—22no—
81D4×Dic13C2×C414yes—
82D4×Dic13C2×C414yes—
83C2×#6105C2×C414yes—
84C2×#6105C2×C414yes—
85C2×#6105C2×C414yes—
86C2×C4⋊Dic13C2311yescharacteristic
87C2×C4⋊Dic13C2311yescharacteristic
88C2×C4⋊Dic13C2311yescharacteristic
89C2×#6101—21no—
90C2×#6101—21no—
91C2×#6101—21no—
92C2×C4×Dic13—21no—
93C2×C4×Dic13—21no—
94C2×C4×Dic13—21no—
95C2×C4×Dic13—21no—
96C2×C4×Dic13—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C2×C4×Dic13—44no—
101C2×C4×Dic13D411yescharacteristic
102C2×C4×Dic13D411yescharacteristic
103C2×C4×Dic13D411yescharacteristic
104C2×C4×Dic13D411yescharacteristic
105C2×C4×Dic13C2311yescharacteristic
106C2×C4×Dic13C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109#88141—22no—
110#88141—22no—
111#88141C2×C414yescomplemented
112#88141C2×C414yescomplemented
113C4×Dic26—22no—
114C4×Dic26C2×C414yescomplemented
115C13×#102C2×C414yes—
116C13×#102C2×C414yes—
117C13×#97—22no—
118C13×#95—21no—
119#88072C2×C414yes—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88056C2×C414yescomplemented
126#88056C2×C414yescomplemented
127#88055—22no—
128#88055C2×C414yescomplemented
129C8×Dic13—22no—
130C8×Dic13C2×C414yescomplemented
131#88050C2×C414yescomplemented
132#88048C2×C414yescomplemented
133#88047C2×C414yescomplemented
134#88047C2×C414yescomplemented
135#88043C2×C414yescomplemented
136C4×#440—131nomaximal, Sylow
137C24×C13—21no—
138C2×D4×C13—21no—
139C2×D4×C13—21no—
140C2×D4×C13—21no—
141C2×D4×C13C22×C412yes—
142C2×D4×C13C22×C412yes—
143C2×D4×C13C22×C412yes—
144C22×C4×C13—21no—
145C22×C4×C13—21no—
146C22×C4×C13—21no—
147C22×C4×C13—24no—
148C22×C4×C13—42no—
149C22×C4×C13—42no—
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2411yescharacteristic
153C22×Dic13—21no—
154C22×Dic13—21no—
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—24no—
158C22×Dic13—24no—
159C22×Dic13—42no—
160C22×Dic13—42no—
161C22×Dic13—42no—
162C22×Dic13—42no—
163C22×Dic13C2×D411yescharacteristic
164C22×Dic13C2×D411yescharacteristic
165C22×Dic13C2×D411yescharacteristic
166C22×Dic13C2×D411yescharacteristic
167C2×Dic26—21no—
168C2×Dic26—21no—
169C2×Dic26—21no—
170C2×Dic26C22×C412yes—
171C2×Dic26C22×C412yes—
172C2×Dic26C22×C412yes—
173C2×C8×C13—21no—
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13C22×C412yes—
178C2×C8×C13C22×C412yes—
179C13×C4⋊C4—21no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4C22×C412yes—
183C13×C4⋊C4C22×C412yes—
184C13×C4⋊C4C22×C412yes—
185C13×C22⋊C4—24no—
186C42×C13—42no—
187C42×C13C22×C412yes—
188C42×C13#4112yes—
189C22⋊Dic13—24no—
190C22⋊Dic13—24no—
191#6105C42116yescomplemented
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13C22×C412yes—
196C4⋊Dic13C22×C412yes—
197C4⋊Dic13C22×C412yes—
198#6101—24no—
199#6101—24no—
200#6101—24no—
201C4×Dic13—24no—
202C4×Dic13—24no—
203C4×Dic13—24no—
204C4×Dic13—22no—
205C4×Dic13—22no—
206C4×Dic13—41no—
207C4×Dic13—41no—
208C4×Dic13—42no—
209C4×Dic13—42no—
210C4×Dic13—42no—
211C4×Dic13—42no—
212C4×Dic13—42no—
213C4×Dic13C22×C412yes—
214C4×Dic13C22×C412yes—
215C4×Dic13#4112yes—
216C4×Dic13#4112yes—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055C22×C412yes—
220C2×#1055C22×C412yes—
221C2×#1055C22×C412yes—
222C2×#1055C22×C412yes—
223C2×#440—131no—
224C2×#440—131no—
225C2×C4×SD16—131no—
226C42×Q8—131no—
227C42×D4—131no—
228#1776—131no—
229#1776—131no—
230C4×#107—131no—
231#1763—131no—
232#1763—131no—
233#1762—131no—
234#1762D1311yescharacteristic
235C4×#103—131no—
236C4×#102—131no—
237C4×#97—131no—
238C23×C13—22no—
239C23×C13—22no—
240C23×C13—22no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13—41no—
244C23×C13—41no—
245C23×C13C22×D411yescharacteristic
246D4×C13—22no—
247D4×C13—22no—
248D4×C13—22no—
249D4×C13C2×C4214yes—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—22no—
255C2×C4×C13—22no—
256C2×C4×C13—22no—
257C2×C4×C13—48no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13C4×D412yes—
265C2×C4×C13C4×D412yes—
266C2×C4×C13C23×C411yescharacteristic
267C2×C4×C13C23×C411yescharacteristic
268C2×C4×C13C23×C411yescharacteristic
269C2×C4×C13C2×#4111yescharacteristic
270C2×C4×C13C2×#4111yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×Dic13—22no—
273C2×Dic13—22no—
274C2×Dic13—22no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—22no—
282C2×Dic13—22no—
283C2×Dic13—22no—
284C2×Dic13—48no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—48no—
292C2×Dic13C4×D412yes—
293C2×Dic13C4×D412yes—
294C2×Dic13C4×D412yes—
295C2×Dic13C4×D412yes—
296C2×Dic13AGL(1,ℤ/8)12yes—
297C2×Dic13#13712yes—
298Dic26—22no—
299Dic26—22no—
300Dic26—22no—
301Dic26C2×C4214yescomplemented
302C8×C13—22no—
303C8×C13—22no—
304#1055—22no—
305#1055C2×C4214yescomplemented
306C22×SD16—131no—
307C2×C4×Q8—131no—
308C2×C4×Q8—131no—
309C2×C4×Q8—131no—
310C2×C4×D4—131no—
311C2×C4×D4—131no—
312C2×C4×D4C2×D1311yescharacteristic
313C22×C42—261no—
314#440—138no—
315#440—138no—
316C4×SD16—138no—
317C2×#107—131no—
318C2×#107—131no—
319C2×#106—131no—
320C2×#103—131no—
321C2×#103—131no—
322C2×#103—131no—
323C2×#102—131no—
324C2×#102C2×D1311yescharacteristic
325C2×#102C2×D1311yescharacteristic
326C2×#97—131no—
327C2×#97—131no—
328C2×#97—131no—
329C2×#97C2×D1311yescharacteristic
330C2×C4×C8—131no—
331C2×C4×C8—131no—
332C4×C4⋊C4—131no—
333C4×C4⋊C4—131no—
334C4×C4⋊C4—131no—
335C4×C4⋊C4—131no—
336C4×C4⋊C4—131no—
337C4×C4⋊C4—261no—
338C4×C4⋊C4—261no—
339C4×C4⋊C4—261no—
340C4×C4⋊C4C2×D1311yescharacteristic
341C4×C22⋊C4—261no—
342C4×C22⋊C4—261no—
343C4×C22⋊C4—261no—
344C43—131no—
345C43—261no—
346#340—131no—
347#340—131no—
348#340—131no—
349#340—131no—
350#340C2×D1311yescharacteristic
351#340C2×D1311yescharacteristic
352C22×C13—24no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—42no—
356C22×C13—42no—
357C22×C13—42no—
358C22×C13—42no—
359C22×C13C2×C4×D411yescharacteristic
360C22×C13C2×C4×D411yescharacteristic
361C22×C13C2×C4×D411yescharacteristic
362C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
363C22×C13C2×#13711yescharacteristic
364C22×C13#57511yescharacteristic
365C22×C13#57511yescharacteristic
366C4×C13—22no—
367C4×C13—22no—
368C4×C13—42no—
369C4×C13—42no—
370C4×C13—42no—
371C4×C13C22×C4211yescharacteristic
372C4×C13C4×#4111yescharacteristic
373C4×C13C4×#4111yescharacteristic
374C4×C13C4×#4111yescharacteristic
375Dic13—22no—
376Dic13—22no—
377Dic13—22no—
378Dic13—24no—
379Dic13—42no—
380Dic13—42no—
381Dic13—42no—
382Dic13#44014yescomplemented
383C22×Q8—131no—
384C22×D4C22×D1311yescharacteristic
385C23×C4—21no—
386C23×C4—261no—
387C23×C4—261no—
388C2×SD16—134no—
389C2×SD16—134no—
390C2×SD16—134no—
391C22×C8—131no—
392C22×C8—131no—
393C22×C8C22×D1311yescharacteristic
394C22×C8C22×D1311yescharacteristic
395C4×Q8—134no—
396C4×Q8—134no—
397C4×Q8—134no—
398C4×Q8—262no—
399C4×Q8—262no—
400C4×Q8—262no—
401C4×D4—22no—
402C4×D4—134no—
403C4×D4—134no—
404C4×D4—262no—
405C4×D4—262no—
406C4×D4C4×D1314yes—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4—261no—
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C4⋊C4C22×D1311yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—261no—
418C2×C22⋊C4—261no—
419C2×C42—21no—
420C2×C42—131no—
421C2×C42—131no—
422C2×C42—131no—
423C2×C42—131no—
424C2×C42—131no—
425C2×C42—131no—
426C2×C42—261no—
427C2×C42—261no—
428C2×C42—261no—
429C2×C42—261no—
430C2×C42—261no—
431C2×C42—261no—
432C2×C42—261no—
433C2×C42—261no—
434C2×C42—524no—
435C2×C42C22×D1311yescharacteristic
436#107—134no—
437#107—134no—
438#106—134no—
439#103—134no—
440#103—134no—
441#103—134no—
442#102—134no—
443#102C4×D1314yes—
444#102C4×D1314yes—
445#97—22no—
446#97—134no—
447#97—134no—
448#97—262no—
449C4×C8—134no—
450C4×C8—262no—
451#95—21no—
452#95—261no—
453#95—261no—
454#95—261no—
455#95—261no—
456#95—261no—
457C2×C13—44no—
458C2×C13C42×D411yescharacteristic
459C2×C13C2×#44011yescharacteristic
460C2×C13#294311yescharacteristic
461C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
462C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
463C2×C13C4×#13711yescharacteristic
464C2×C13C4×#13711yescharacteristic
465C24—21no—
466C2×Q8—132no—
467C2×Q8—132no—
468C2×Q8—132no—
469C2×Q8—261no—
470C2×Q8—261no—
471C2×Q8—261no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4C2×C4×D1312yes—
476C2×D4C2×C4×D1312yes—
477C2×D4C2×C4×D1312yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—131no—
487C22×C4—131no—
488C22×C4—264no—
489C22×C4—264no—
490C22×C4—261no—
491C22×C4—261no—
492C22×C4—261no—
493C22×C4—261no—
494C22×C4—522no—
495C22×C4—522no—
496C22×C4—522no—
497C22×C4—522no—
498C22×C4D4×D1311yescharacteristic
499C22×C4D4×D1311yescharacteristic
500C22×C4C23×D1311yescharacteristic
501SD16—1316no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—132no—
507C2×C8—132no—
508C2×C8—132no—
509C2×C8—132no—
510C2×C8—261no—
511C2×C8—261no—
512C2×C8C2×C4×D1312yes—
513C2×C8C2×C4×D1312yes—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—132no—
518C4⋊C4—132no—
519C4⋊C4—132no—
520C4⋊C4—264no—
521C4⋊C4—264no—
522C4⋊C4—264no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4C2×C4×D1312yes—
527C4⋊C4C2×C4×D1312yes—
528C4⋊C4C2×C4×D1312yes—
529C22⋊C4—24no—
530C22⋊C4—264no—
531C22⋊C4—264no—
532C42—42no—
533C42—132no—
534C42—132no—
535C42—132no—
536C42—132no—
537C42—262no—
538C42—262no—
539C42—264no—
540C42—264no—
541C42—264no—
542C42—521no—
543C42—521no—
544C42—522no—
545C42—522no—
546C42—522no—
547C42—522no—
548C42—522no—
549C42C2×C4×D1312yes—
550C42#612912yes—
551C13C4×#44011yescharacteristic, complemented, Sylow
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D1311yescharacteristic
560Q8—134no—
561Q8—262no—
562Q8—262no—
563Q8—262no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4C42×D1314yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—48no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—132no—
583C2×C4—132no—
584C2×C4—132no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—132no—
588C2×C4—262no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—261no—
592C2×C4—261no—
593C2×C4—261no—
594C2×C4—261no—
595C2×C4—262no—
596C2×C4—262no—
597C2×C4—262no—
598C2×C4—262no—
599C2×C4—262no—
600C2×C4—528no—
601C2×C4—528no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—521no—
606C2×C4—521no—
607C2×C4—521no—
608C2×C4#8813512yes—
609C2×C4#8813512yes—
610C2×C4C22×C4×D1311yescharacteristic
611C2×C4C22×C4×D1311yescharacteristic
612C2×C4C22×C4×D1311yescharacteristic
613C2×C4C2×#612911yescharacteristic
614C2×C4#41×D1311yescharacteristic
615C2×C4#41×D1311yescharacteristic
616C8—22no—
617C8—22no—
618C8—134no—
619C8—262no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C2×#8813511yescharacteristic
628C22C4×D4×D1311yescharacteristic
629C22C4×D4×D1311yescharacteristic
630C22D13×AGL(1,ℤ/8)11yescharacteristic
631C22#1171947811yescharacteristic
632C22#1171947811yescharacteristic
633C22D13×#13711yescharacteristic
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—134no—
640C4—262no—
641C4—262no—
642C4—262no—
643C4—264no—
644C4—522no—
645C4—522no—
646C4—522no—
647C4C2×C42×D1311yescharacteristic
648C4#1171922711yescharacteristic
649C4C4×#612911yescharacteristic
650C4C4×#612911yescharacteristic
651C2—44no—
652C2C4×#8813511yescharacteristic
653C2D13×#44011yescharacteristic
654C2#1312363311yescharacteristic
655C2#1312364011yescharacteristic
656C2#1312364711yescharacteristic
657C2#1312365811yescharacteristic
658C2#1312366011yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC4×C13C22×C42
FrattiniC22×C4C23×D13
FittingC13×#1762C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenter#1762D13
hypocenterC13C4×#440
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
4. C4×#88141C2semidirect
6. #13114734C2semidirect
7. #13114732C2semidirect
8. #13114728C2semidirect
9. #13114667C2semidirect
10. #13114665C2semidirect
12. #13114655C2semidirect
14. #97×Dic13C2semidirect
36. #11718679C4semidirect
52. #11718419C4semidirect
53. #11718418C4semidirect
111. #88141C2×C4semidirect
112. #88141C2×C4semidirect
114. C4×Dic26C2×C4semidirect
125. #88056C2×C4semidirect
126. #88056C2×C4semidirect
128. #88055C2×C4semidirect
130. C8×Dic13C2×C4semidirect
131. #88050C2×C4semidirect
132. #88048C2×C4semidirect
133. #88047C2×C4semidirect
134. #88047C2×C4semidirect
135. #88043C2×C4semidirect
191. #6105C42semidirect
301. Dic26C2×C42semidirect
305. #1055C2×C42semidirect
382. Dic13#440semidirect
551. C13C4×#440semidirect

Maximal subgroups

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

subgroupindex
C2×#117186792
C2×#117184182
C2×#117184192
C4×#881412
C4×D4×Dic132
#131147342
#131147322
#131147282
#131146672
#131146652
#131146632
#131146552
#102×Dic132
#97×Dic132
C13×#17622
C4×#44013