Finite Group Databasebeta

← #15784896

Subgroups — #15784896

Order 3328.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88180C211yesmaximal, characteristic
2C2×#11718637C211yesmaximal, characteristic, complemented
3C2×#11718638C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Dic13C211yesmaximal, characteristic
5#13113079C214yesmaximal
6#13113067C214yesmaximal, complemented
7#13113065C211yesmaximal, characteristic, complemented
8#13113065C211yesmaximal, characteristic, complemented
9C2×C13×#380C211yesmaximal, characteristic
10C24×C4×C13C2211yescharacteristic
11C22×C13×C4⋊C4C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C22⋊C4C2212yes—
14C22×C22⋊Dic13C2212yes—
15C22×C22⋊Dic13C2212yes—
16C2×#88180—24no—
17C2×#88180—24no—
18C2×#88180—24no—
19C22×C4⋊Dic13C2211yescharacteristic
20C22×C4⋊Dic13C2211yescharacteristic
21C22×#6101C2214yes—
22C22×C4×Dic13C2212yes—
23C22×C4×Dic13C2211yescharacteristic
24C22×C4×Dic13C2211yescharacteristic
25C13×#380—24no—
26C2×C13×#95C2212yes—
27C4×C22⋊Dic13C418yescomplemented
28C2×#88071C2214yes—
29C2×#88071C2214yes—
30C2×#88071C2214yes—
31C2×#88071C2212yes—
32#11718638—24no—
33#11718637—24no—
34C25×C13C2311yescharacteristic
35C23×C4×C13—22no—
36C23×C4×C13—22no—
37C23×C4×C13—22no—
38C23×C4×C13—22no—
39C23×C4×C13C2×C414yes—
40C23×C4×C13D414yes—
41C23×C4×C13Q814yes—
42C23×C4×C13C2312yes—
43C23×C4×C13C2312yes—
44C23×C4×C13C2311yescharacteristic
45C23×C4×C13C2311yescharacteristic
46C23×Dic13C2312yes—
47C23×Dic13C2312yes—
48C23×Dic13C2312yes—
49C23×Dic13C2312yes—
50C2×C13×C4⋊C4—22no—
51C2×C13×C4⋊C4—22no—
52C2×C13×C4⋊C4—22no—
53C2×C13×C4⋊C4—22no—
54C2×C13×C4⋊C4—22no—
55C2×C13×C4⋊C4—22no—
56C2×C13×C22⋊C4—24no—
57C2×C13×C22⋊C4—44no—
58C2×C22⋊Dic13—24no—
59C2×C22⋊Dic13—24no—
60C2×C22⋊Dic13—24no—
61C2×C22⋊Dic13—24no—
62C2×C22⋊Dic13—24no—
63C2×C22⋊Dic13C2×C418yes—
64#88180—232no—
65C2×C4⋊Dic13—22no—
66C2×C4⋊Dic13—22no—
67C2×C4⋊Dic13—22no—
68C2×C4⋊Dic13—22no—
69C2×C4⋊Dic13D414yes—
70C2×C4⋊Dic13D414yes—
71C2×#6101—28no—
72C2×#6101—28no—
73C2×#6101—44no—
74C2×C4×Dic13—24no—
75C2×C4×Dic13—22no—
76C2×C4×Dic13—22no—
77C2×C4×Dic13—22no—
78C2×C4×Dic13—22no—
79C2×C4×Dic13—24no—
80C2×C4×Dic13—24no—
81C2×C4×Dic13C2×C414yes—
82C2×C4×Dic13C2×C414yes—
83C13×#95—24no—
84#88071—28no—
85#88071—28no—
86#88071—44no—
87#88071C2×C418yes—
88#13879—131nomaximal, Sylow
89C24×C13—21no—
90C24×C13—21no—
91C24×C13—22no—
92C24×C13—22no—
93C24×C13—21no—
94C24×C13—21no—
95C24×C13—22no—
96C24×C13—22no—
97C24×C13C22×C412yes—
98C24×C13C2×D412yes—
99C24×C13C2×Q812yes—
100C24×C13C2411yescharacteristic
101C22×C4×C13—21no—
102C22×C4×C13—21no—
103C22×C4×C13—21no—
104C22×C4×C13—21no—
105C22×C4×C13—28no—
106C22×C4×C13—28no—
107C22×C4×C13—28no—
108C22×C4×C13—28no—
109C22×C4×C13—28no—
110C22×C4×C13—28no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—24no—
116C22×C4×C13—24no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—24no—
120C22×C4×C13—24no—
121C22×C4×C13C4⋊C4116yescomplemented
122C22×C4×C13C22×C412yes—
123C22×C4×C13C22×C412yes—
124C22×C4×C13C2×D412yes—
125C22×C4×C13C2×D412yes—
126C22×C4×C13C2×D412yes—
127C22×C4×C13C2×D412yes—
128C22×C4×C13C2×Q812yes—
129C22×C4×C13C2×Q812yes—
130C22×C4×C13#4112yes—
131C22×C4×C13#4112yes—
132C22×Dic13—22no—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—24no—
142C22×Dic13—22no—
143C22×Dic13—24no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—24no—
148C22×Dic13—22no—
149C22×Dic13—24no—
150C22×Dic13C22×C414yes—
151C22×Dic13C22×C414yes—
152C22×Dic13C2×D414yes—
153C22×Dic13C2×D414yes—
154C22×Dic13#4114yes—
155C22×Dic13#4114yes—
156C13×C4⋊C4—44no—
157C13×C4⋊C4—44no—
158C13×C22⋊C4—48no—
159C22⋊Dic13—216no—
160C22⋊Dic13—216no—
161C4⋊Dic13—28no—
162C4⋊Dic13—28no—
163#6101—416no—
164C4×Dic13—216no—
165C4×Dic13—28no—
166C4×Dic13—28no—
167C22×#117—131no—
168C2×#384—131no—
169C2×#382—131no—
170C2×#380D1311yescharacteristic
171C2×C4×C22⋊C4—131no—
172#1439—134no—
173#1436—134no—
174C4⋊#95—131no—
175C4⋊#95—131no—
176C23×C13—24no—
177C23×C13—22no—
178C23×C13—22no—
179C23×C13—24no—
180C23×C13—22no—
181C23×C13—22no—
182C23×C13—22no—
183C23×C13—22no—
184C23×C13—22no—
185C23×C13—22no—
186C23×C13—24no—
187C23×C13—22no—
188C23×C13—22no—
189C23×C13—24no—
190C23×C13—42no—
191C23×C13—42no—
192C23×C13—42no—
193C23×C13—42no—
194C23×C13—42no—
195C23×C13—42no—
196C23×C13—42no—
197C23×C13—42no—
198C23×C13C2×C4⋊C414yes—
199C23×C13C23×C411yescharacteristic
200C23×C13C22×D411yescharacteristic
201C23×C13C22×D411yescharacteristic
202C23×C13C22×D411yescharacteristic
203C23×C13C22×Q811yescharacteristic
204C23×C13C2×#4112yes—
205C23×C13C2×#4112yes—
206C23×C13C2×#4111yescharacteristic
207C23×C13C2×#4111yescharacteristic
208C23×C13H(5,𝔽2)12yes—
209C23×C132−1+412yes—
210C2×C4×C13—22no—
211C2×C4×C13—22no—
212C2×C4×C13—22no—
213C2×C4×C13—22no—
214C2×C4×C13—22no—
215C2×C4×C13—22no—
216C2×C4×C13—232no—
217C2×C4×C13—42no—
218C2×C4×C13—42no—
219C2×C4×C13—42no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—44no—
224C2×C4×C13—44no—
225C2×C4×C13—44no—
226C2×C4×C13—44no—
227C2×C4×C13C2×C4⋊C414yes—
228C2×C4×C13C2×C4⋊C414yes—
229C2×C4×C13C4×D414yes—
230C2×C4×C13C4×D414yes—
231C2×C4×C13#12014yes—
232C2×C4×C13C22⋊Q814yes—
233C2×C4×C13C22⋊Q814yes—
234C2×C4×C13#12314yes—
235C2×Dic13—24no—
236C2×Dic13—24no—
237C2×Dic13—24no—
238C2×Dic13—24no—
239C2×Dic13—24no—
240C2×Dic13—24no—
241C2×Dic13—24no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—28no—
247C2×Dic13—28no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250C2×Dic13—42no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—44no—
254C2×Dic13—42no—
255C2×Dic13—44no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258C2×Dic13#11718yescomplemented
259C24×C4C2×D1311yescharacteristic
260C2×#117—264no—
261C2×#117—264no—
262C2×#117—264no—
263C22×C4⋊C4—131no—
264C22×C4⋊C4—131no—
265C22×C4⋊C4—134no—
266C22×C4⋊C4C2×D1311yescharacteristic
267C22×C4⋊C4C2×D1311yescharacteristic
268C22×C22⋊C4—132no—
269C22×C22⋊C4—132no—
270C22×C22⋊C4C2×D1312yes—
271C22×C42—131no—
272C22×C42—131no—
273C22×C42—132no—
274#384—264no—
275#382—264no—
276#380—24no—
277C4×C22⋊C4—138no—
278C2×#95—132no—
279C2×#95—134no—
280C2×#95—134no—
281C2×#95—134no—
282C2×#95C2×D1312yes—
283C22×C13—24no—
284C22×C13—24no—
285C22×C13—24no—
286C22×C13—44no—
287C22×C13—44no—
288C22×C13—42no—
289C22×C13—42no—
290C22×C13—44no—
291C22×C13—44no—
292C22×C13—42no—
293C22×C13—42no—
294C22×C13C22×C4⋊C411yescharacteristic
295C22×C13C2×#11712yes—
296C22×C13C2×C4×D411yescharacteristic
297C22×C13C2×C4×D411yescharacteristic
298C22×C13#52012yes—
299C22×C13C2×#12011yescharacteristic
300C22×C13C2×C22⋊Q811yescharacteristic
301C22×C13C2×C22⋊Q811yescharacteristic
302C22×C13C2×#12311yescharacteristic
303C22×C13#53211yescharacteristic
304C22×C13#53211yescharacteristic
305C22×C13#53312yes—
306C22×C13#53511yescharacteristic
307C22×C13#53611yescharacteristic
308C22×C13#54412yes—
309C22×C13#54612yes—
310C22×C13#54712yes—
311C22×C13#54812yes—
312C22×C13#55012yes—
313C22×C13#55212yes—
314C22×C13#55312yes—
315C22×C13#55512yes—
316C22×C13#55612yes—
317C4×C13—24no—
318C4×C13—82no—
319C4×C13—82no—
320C4×C13#38518yes—
321Dic13—28no—
322Dic13—44no—
323Dic13—44no—
324Dic13—44no—
325C25C22×D1311yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—22no—
330C23×C4—132no—
331C23×C4—132no—
332C23×C4—132no—
333C23×C4—132no—
334C23×C4Dic2614yes—
335C23×C4C4×D1314yes—
336C23×C4C22⋊D1314yes—
337C23×C4C22×D1312yes—
338C23×C4C22×D1312yes—
339C23×C4C22×D1311yescharacteristic
340C23×C4C22×D1311yescharacteristic
341#117—2632no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—134no—
349C2×C4⋊C4—134no—
350C2×C4⋊C4—262no—
351C2×C4⋊C4—262no—
352C2×C4⋊C4—262no—
353C2×C4⋊C4—262no—
354C2×C4⋊C4—268no—
355C2×C4⋊C4—268no—
356C2×C4⋊C4—524no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—138no—
360C2×C22⋊C4—264no—
361C2×C22⋊C4—264no—
362C2×C22⋊C4—264no—
363C2×C22⋊C4—264no—
364C2×C22⋊C4—264no—
365C2×C42—134no—
366C2×C42—134no—
367C2×C42—264no—
368C2×C42—264no—
369C2×C42—262no—
370C2×C42—262no—
371C2×C42—262no—
372C2×C42—262no—
373C2×C42—264no—
374#95—24no—
375#95—138no—
376#95—268no—
377#95—268no—
378#95—524no—
379C2×C13—44no—
380C2×C13#228511yescharacteristic
381C2×C13C2×#38511yescharacteristic
382C2×C13#231711yescharacteristic
383C2×C13#235212yes—
384C2×C13#235412yes—
385C2×C13#243111yescharacteristic
386C2×C13#243211yescharacteristic
387C2×C13#243912yes—
388C2×C13#244112yes—
389C2×C13#249911yescharacteristic
390C2×C13#250011yescharacteristic
391C24—21no—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24C2×Dic2612yes—
400C24C2×C4×D1312yes—
401C24C2×C22⋊D1312yes—
402C24C23×D1311yescharacteristic
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—134no—
424C22×C4—134no—
425C22×C4—134no—
426C22×C4—134no—
427C22×C4—134no—
428C22×C4—134no—
429C22×C4—262no—
430C22×C4—262no—
431C22×C4—262no—
432C22×C4—262no—
433C22×C4—262no—
434C22×C4—262no—
435C22×C4—262no—
436C22×C4—262no—
437C22×C4—262no—
438C22×C4—264no—
439C22×C4—262no—
440C22×C4—264no—
441C22×C4—264no—
442C22×C4—262no—
443C22×C4—262no—
444C22×C4—264no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4#6101116yescomplemented
448C22×C4C2×Dic2612yes—
449C22×C4C2×Dic2612yes—
450C22×C4C2×C4×D1312yes—
451C22×C4C2×C4×D1312yes—
452C22×C4D4×D1312yes—
453C22×C4D4×D1312yes—
454C22×C4#612912yes—
455C22×C4#612912yes—
456C22×C4C2×C22⋊D1312yes—
457C22×C4C2×C22⋊D1312yes—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—268no—
461C4⋊C4—268no—
462C4⋊C4—5216no—
463C22⋊C4—48no—
464C22⋊C4—2616no—
465C22⋊C4—2616no—
466C42—2616no—
467C42—268no—
468C42—268no—
469C13#1387911yescharacteristic, complemented, Sylow
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×#610114yes—
493C23C22×Dic2611yescharacteristic
494C23C22×C4×D1311yescharacteristic
495C23C2×D4×D1311yescharacteristic
496C23C2×D4×D1311yescharacteristic
497C23C2×#612911yescharacteristic
498C23C2×#612911yescharacteristic
499C23#8825112yes—
500C23#8825412yes—
501C23#41×D1312yes—
502C23#41×D1312yes—
503C23C22×C22⋊D1311yescharacteristic
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—232no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—138no—
522C2×C4—264no—
523C2×C4—264no—
524C2×C4—264no—
525C2×C4—264no—
526C2×C4—264no—
527C2×C4—264no—
528C2×C4—264no—
529C2×C4—264no—
530C2×C4—264no—
531C2×C4—264no—
532C2×C4—264no—
533C2×C4—268no—
534C2×C4—268no—
535C2×C4—522no—
536C2×C4—522no—
537C2×C4—522no—
538C2×C4—522no—
539C2×C4—522no—
540C2×C4—524no—
541C2×C4—522no—
542C2×C4—522no—
543C2×C4—524no—
544C2×C4—522no—
545C2×C4#8813214yes—
546C2×C4#8813214yes—
547C2×C4#8813514yes—
548C2×C4#8813514yes—
549C2×C4C2×#610114yes—
550C2×C4C2×#610114yes—
551C2×C4#8818914yes—
552C2×C4#8819114yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—44no—
563C22—42no—
564C22C2×#8813211yescharacteristic
565C22C2×#8813211yescharacteristic
566C22C2×#8813511yescharacteristic
567C22C2×#8813511yescharacteristic
568C22#1171920512yes—
569C22#1171920912yes—
570C22#1171922212yes—
571C22D13×#11712yes—
572C22#1171930412yes—
573C22#1171931712yes—
574C22#1171931912yes—
575C22#1171932112yes—
576C22#1171933612yes—
577C22#1171933812yes—
578C22#1171934412yes—
579C22#1171934612yes—
580C22C22×#610111yescharacteristic
581C22C2×#8818911yescharacteristic
582C22C2×#8819111yescharacteristic
583C22#1171951011yescharacteristic
584C22#1171952111yescharacteristic
585C22#1171953711yescharacteristic
586C22#1171953711yescharacteristic
587C4—24no—
588C4—82no—
589C4—82no—
590C4—268no—
591C4—524no—
592C4—524no—
593C4—524no—
594C4#1171860718yes—
595C2—44no—
596C2#1311723911yescharacteristic
597C2C2×#1171860711yescharacteristic
598C2#1311746011yescharacteristic
599C2#1311775812yes—
600C2#1311778012yes—
601C2#1311845611yescharacteristic
602C2#1311846511yescharacteristic
603C2#1311852812yes—
604C2#1311855012yes—
605C2#1311929011yescharacteristic
606C2#1311929911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#380C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenterC2×#380D13
hypocenterC13#13879
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#11718637C2semidirect
3. C2×#11718638C2semidirect
6. #13113067C2semidirect
7. #13113065C2semidirect
8. #13113065C2semidirect
27. C4×C22⋊Dic13C4semidirect
121. C22×C4×C13C4⋊C4semidirect
258. C2×Dic13#117semidirect
447. C22×C4#6101semidirect
469. C13#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#881802
C2×#117186372
C2×#117186382
C2×C4×C22⋊Dic132
#131130792
#131130672
#131130652
C2×C13×#3802
#1387913