Finite Group Databasebeta

← #15789297

Subgroups — #15789297

Order 3328.

2142 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88151C211yesmaximal, characteristic
2C22×#88126C211yesmaximal, characteristic
3C2×#11718590C211yesmaximal, characteristic, complemented
4C2×#11718635C211yesmaximal, characteristic, complemented
5C2×#11718648C211yesmaximal, characteristic
6#13113088C214yesmaximal, complemented
7#13113084C214yesmaximal
8#13113065C211yesmaximal, characteristic, complemented
9C13×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D13C2211yescharacteristic
11C23×Dic26C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C22×#6103C2212yes—
16C22×#6103C2212yes—
17C22×#6103C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×C4⋊Dic13C2212yes—
20C22×#6101C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C2×#88151—24no—
23C2×#88151—24no—
24C2×#88151C418yescomplemented
25C2×#88126—24no—
26C2×#88126—24no—
27C2×#88126—24no—
28C2×C13×#95C2214yes—
29C2×#88071C2212yes—
30C2×#88071C2214yes—
31C2×#88071C2214yes—
32C2×#88071C2214yes—
33#11718648—24no—
34#11718635—24no—
35#11718590—24no—
36C24×D13C2311yescharacteristic
37C23×C4×C13C2312yes—
38C23×C4×C13C2312yes—
39C23×C4×C13C2312yes—
40C23×C4×C13C2311yescharacteristic
41C23×Dic13C2312yes—
42C23×Dic13C2312yes—
43C23×Dic13C2312yes—
44C23×Dic13C2311yescharacteristic
45C22×C4×D13—22no—
46C22×C4×D13—22no—
47C22×C4×D13—22no—
48C22×C4×D13—22no—
49C22×C4×D13C2×C414yes—
50C22×C4×D13D414yes—
51C22×C4×D13D414yes—
52C22×Dic26—22no—
53C22×Dic26—22no—
54C22×Dic26—22no—
55C22×Dic26—22no—
56C22×Dic26—22no—
57C22×Dic26—22no—
58C22×Dic26C2×C414yes—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4—22no—
61C2×C13×C4⋊C4—22no—
62C2×C13×C4⋊C4—22no—
63C2×C13×C4⋊C4—22no—
64C2×C13×C4⋊C4C2×C414yes—
65C2×C42×C13—22no—
66C2×C42×C13—22no—
67C2×C42×C13D414yes—
68C2×#6103—24no—
69C2×#6103—24no—
70C2×#6103—24no—
71C2×#6103—24no—
72C2×#6103—24no—
73C2×#6103—24no—
74C2×#6103—44no—
75C2×#6103C2×C418yes—
76C2×C4⋊Dic13—24no—
77C2×C4⋊Dic13—24no—
78C2×C4⋊Dic13—22no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13—22no—
81C2×C4⋊Dic13C2×C418yes—
82C2×#6101—24no—
83C2×#6101—24no—
84C2×#6101—24no—
85C2×C4×Dic13—22no—
86C2×C4×Dic13—22no—
87C2×C4×Dic13D414yes—
88#88151—232no—
89#88126—232no—
90C13×#95—44no—
91#88071—28no—
92#88071—28no—
93#88071—24no—
94#88071—44no—
95#14258—131nomaximal, Sylow
96C24×C13C2411yescharacteristic
97C23×D13—21no—
98C23×D13—21no—
99C23×D13—21no—
100C23×D13—21no—
101C23×D13—22no—
102C23×D13—22no—
103C23×D13—22no—
104C23×D13—22no—
105C23×D13C22×C412yes—
106C23×D13C2×D412yes—
107C23×D13C2×D412yes—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—24no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—24no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—22no—
120C22×C4×C13—22no—
121C22×C4×C13—24no—
122C22×C4×C13—24no—
123C22×C4×C13C22×C414yes—
124C22×C4×C13C22×C412yes—
125C22×C4×C13C2×D414yes—
126C22×C4×C13C2×D412yes—
127C22×C4×C13C2×D412yes—
128C22×C4×C13C2×D412yes—
129C22×C4×C13C2×D412yes—
130C22×C4×C13#4112yes—
131C22×C4×C13#4112yes—
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—22no—
145C22×Dic13—22no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—24no—
152C22×Dic13—24no—
153C22×Dic13C22×C414yes—
154C22×Dic13C22×C412yes—
155C22×Dic13C2×D414yes—
156C22×Dic13C2×D412yes—
157C22×Dic13C2×D412yes—
158C2×C4×D13—28no—
159C2×C4×D13—28no—
160C2×C4×D13—28no—
161C2×C4×D13—28no—
162C2×C4×D13—28no—
163C2×C4×D13—28no—
164C2×C4×D13C22⋊C4116yes—
165C2×Dic26—28no—
166C2×Dic26—28no—
167C2×Dic26—28no—
168C2×Dic26—44no—
169C2×Dic26—44no—
170C2×Dic26—44no—
171C2×Dic26—44no—
172C13×C4⋊C4—28no—
173C13×C4⋊C4—44no—
174C42×C13—28no—
175#6103—216no—
176#6103—216no—
177#6103—48no—
178C4⋊Dic13—216no—
179C4⋊Dic13—44no—
180#6101—216no—
181C4×Dic13—28no—
182C22×C22⋊Q8—131no—
183C22×#117—131no—
184C2×#391—131no—
185C2×#391—131no—
186C2×#380—131no—
187#1441—134no—
188#1440—134no—
189C4⋊#95—131no—
190C4⋊#95D1311yescharacteristic, complemented
191C23×C13C23×C411yescharacteristic
192C23×C13C22×D411yescharacteristic
193C23×C13C22×D411yescharacteristic
194C23×C13C22×D411yescharacteristic
195C23×C13C22×D411yescharacteristic
196C23×C13C2×#4111yescharacteristic
197C23×C13C2×#4111yescharacteristic
198C23×C13H(5,𝔽2)12yes—
199C23×C132−1+412yes—
200C23×C132−1+412yes—
201C23×C132−1+412yes—
202C22×D13—22no—
203C22×D13—22no—
204C22×D13—24no—
205C22×D13—24no—
206C22×D13—22no—
207C22×D13—22no—
208C22×D13—22no—
209C22×D13—22no—
210C22×D13—22no—
211C22×D13—24no—
212C22×D13—22no—
213C22×D13—22no—
214C22×D13—22no—
215C22×D13—24no—
216C22×D13—42no—
217C22×D13—42no—
218C22×D13—42no—
219C22×D13—42no—
220C22×D13—42no—
221C22×D13—42no—
222C22×D13—42no—
223C22×D13—42no—
224C22×D13C2×C22⋊C414yes—
225C2×C4×C13—24no—
226C2×C4×C13—28no—
227C2×C4×C13—24no—
228C2×C4×C13—24no—
229C2×C4×C13—24no—
230C2×C4×C13—42no—
231C2×C4×C13—42no—
232C2×C4×C13—42no—
233C2×C4×C13—42no—
234C2×C4×C13—44no—
235C2×C4×C13—42no—
236C2×C4×C13—42no—
237C2×C4×C13—42no—
238C2×C4×C13—42no—
239C2×C4×C13—42no—
240C2×C4×C13—44no—
241C2×C4×C13—44no—
242C2×C4×C13C2×C22⋊C414yes—
243C2×C4×C13C4×D414yes—
244C2×C4×C13C4×D414yes—
245C2×C4×C13#12014yes—
246C2×C4×C13#12114yes—
247C2×C4×C13#12114yes—
248C2×C4×C13#12314yes—
249C2×Dic13—22no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—22no—
253C2×Dic13—22no—
254C2×Dic13—22no—
255C2×Dic13—24no—
256C2×Dic13—24no—
257C2×Dic13—24no—
258C2×Dic13—24no—
259C2×Dic13—24no—
260C2×Dic13—24no—
261C2×Dic13—42no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268C2×Dic13—42no—
269C2×Dic13—42no—
270C2×Dic13—42no—
271C2×Dic13—42no—
272C2×Dic13—44no—
273C2×Dic13—44no—
274C2×Dic13C2×C22⋊C414yes—
275C4×D13—232no—
276Dic26—416no—
277C23×Q8—131no—
278C24×C4—131no—
279C2×C22⋊Q8—138no—
280C2×C22⋊Q8—264no—
281C2×C22⋊Q8—264no—
282C2×#117—264no—
283C2×#117—264no—
284C2×#117—264no—
285C22×C4⋊C4—132no—
286C22×C4⋊C4—132no—
287C22×C4⋊C4—131no—
288C22×C4⋊C4C2×D1311yescharacteristic
289C22×C4⋊C4C2×D1311yescharacteristic
290C22×C22⋊C4—132no—
291C22×C22⋊C4—132no—
292C22×C22⋊C4—132no—
293C22×C42—131no—
294C22×C42C2×D1311yescharacteristic
295#391—264no—
296#391—264no—
297#380—264no—
298C2×#95—134no—
299C2×#95—134no—
300C2×#95—132no—
301C2×#95—134no—
302C2×#95C2×D1314yes—
303C22×C13C22×C22⋊C411yescharacteristic
304C22×C13C2×C4×D411yescharacteristic
305C22×C13C2×C4×D411yescharacteristic
306C22×C13#51912yes—
307C22×C13#52012yes—
308C22×C13C2×#12011yescharacteristic
309C22×C13C2×#12111yescharacteristic
310C22×C13C2×#12111yescharacteristic
311C22×C13C2×#12311yescharacteristic
312C22×C13#53511yescharacteristic
313C22×C13#53611yescharacteristic
314C22×C13#53612yes—
315C22×C13#53612yes—
316C22×C13#53612yes—
317C22×C13#53611yescharacteristic
318C22×C13#53611yescharacteristic
319C22×C13#53712yes—
320C22×C13#53712yes—
321C22×C13#53712yes—
322C22×C13#54112yes—
323C22×C13#54112yes—
324C22×C13#54212yes—
325C22×C13#54212yes—
326C2×D13—24no—
327C2×D13—24no—
328C2×D13—24no—
329C2×D13—44no—
330C2×D13—44no—
331C2×D13—44no—
332C2×D13—42no—
333C2×D13—42no—
334C2×D13—42no—
335C2×D13—42no—
336C2×D13—44no—
337C4×C13—44no—
338C4×C13—44no—
339C4×C13—82no—
340C4×C13#38618yes—
341Dic13—24no—
342Dic13—44no—
343Dic13—44no—
344Dic13—82no—
345C25—131no—
346C22×Q8—134no—
347C22×Q8—262no—
348C22×Q8—262no—
349C22×Q8—262no—
350C22×Q8—262no—
351C22×Q8—262no—
352C22×Q8—262no—
353C23×C4—131no—
354C23×C4—134no—
355C23×C4—134no—
356C23×C4—134no—
357C23×C4—132no—
358C23×C4—132no—
359C23×C4—132no—
360C23×C4—262no—
361C23×C4—262no—
362C23×C4—262no—
363C23×C4—262no—
364C23×C4C22×D1312yes—
365C23×C4C22×D1312yes—
366C23×C4C22×D1312yes—
367C23×C4C22×D1311yescharacteristic
368C22⋊Q8—2632no—
369#117—2632no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—138no—
376C2×C4⋊C4—264no—
377C2×C4⋊C4—264no—
378C2×C4⋊C4—264no—
379C2×C4⋊C4—264no—
380C2×C4⋊C4—264no—
381C2×C4⋊C4—262no—
382C2×C4⋊C4—262no—
383C2×C4⋊C4—262no—
384C2×C4⋊C4C4×D1314yes—
385C2×C22⋊C4—138no—
386C2×C22⋊C4—264no—
387C2×C22⋊C4—264no—
388C2×C22⋊C4—264no—
389C2×C22⋊C4—264no—
390C2×C22⋊C4—264no—
391C2×C22⋊C4—264no—
392C2×C22⋊C4—524no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—134no—
396C2×C42—262no—
397C2×C42—262no—
398C2×C42D5214yes—
399#95—44no—
400#95—264no—
401#95—268no—
402#95—268no—
403#95—524no—
404C2×C13C2×#38611yescharacteristic
405C2×C13#231011yescharacteristic
406C2×C13#231111yescharacteristic
407C2×C13#232312yes—
408C2×C13#232312yes—
409C2×C13#261612yes—
410C2×C13#262011yescharacteristic
411C2×C13#262011yescharacteristic
412C2×C13#262111yescharacteristic
413C2×C13#262111yescharacteristic
414C2×C13#262512yes—
415D13—44no—
416C24—132no—
417C24—132no—
418C24—132no—
419C24—261no—
420C24—261no—
421C24—261no—
422C24—262no—
423C24—261no—
424C24—262no—
425C24—262no—
426C24—262no—
427C24C23×D1311yescharacteristic
428C2×Q8—268no—
429C2×Q8—268no—
430C2×Q8—268no—
431C2×Q8—524no—
432C2×Q8—524no—
433C2×Q8—524no—
434C2×Q8—524no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—132no—
451C22×C4—132no—
452C22×C4—132no—
453C22×C4—1316no—
454C22×C4—134no—
455C22×C4—134no—
456C22×C4—261no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—268no—
461C22×C4—268no—
462C22×C4—268no—
463C22×C4—268no—
464C22×C4—268no—
465C22×C4—268no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—262no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—262no—
474C22×C4—262no—
475C22×C4—262no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—262no—
479C22×C4—262no—
480C22×C4—264no—
481C22×C4—262no—
482C22×C4—264no—
483C22×C4C2×C4×D1314yes—
484C22×C4C2×C4×D1312yes—
485C22×C4C2×D5214yes—
486C22×C4C2×D5212yes—
487C22×C4#612712yes—
488C22×C4D4×D1312yes—
489C22×C4D4×D1312yes—
490C22×C4D4×D1312yes—
491C22×C4#613112yes—
492C4⋊C4—28no—
493C4⋊C4—44no—
494C4⋊C4—2616no—
495C4⋊C4—2616no—
496C4⋊C4—524no—
497C22⋊C4—2616no—
498C22⋊C4—2616no—
499C22⋊C4—528no—
500C42—28no—
501C42—268no—
502C13#1425811yescharacteristic, complemented, Sylow
503C23—134no—
504C23—262no—
505C23—262no—
506C23—262no—
507C23—262no—
508C23—262no—
509C23—262no—
510C23—262no—
511C23—264no—
512C23—264no—
513C23—264no—
514C23—264no—
515C23—262no—
516C23—262no—
517C23—262no—
518C23—522no—
519C23—522no—
520C23—522no—
521C23—522no—
522C23—522no—
523C23—522no—
524C23—522no—
525C23—522no—
526C23C22×C4×D1311yescharacteristic
527C23C22×D5211yescharacteristic
528C23C2×#612711yescharacteristic
529C23C2×D4×D1311yescharacteristic
530C23C2×D4×D1311yescharacteristic
531C23C2×D4×D1311yescharacteristic
532C23#8825112yes—
533C23C2×#613111yescharacteristic
534C23#8825412yes—
535C23#8825712yes—
536C23#8825712yes—
537Q8—5216no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—134no—
556C2×C4—262no—
557C2×C4—262no—
558C2×C4—262no—
559C2×C4—262no—
560C2×C4—262no—
561C2×C4—262no—
562C2×C4—2632no—
563C2×C4—264no—
564C2×C4—264no—
565C2×C4—264no—
566C2×C4—264no—
567C2×C4—264no—
568C2×C4—264no—
569C2×C4—522no—
570C2×C4—522no—
571C2×C4—522no—
572C2×C4—522no—
573C2×C4—522no—
574C2×C4—522no—
575C2×C4—522no—
576C2×C4—522no—
577C2×C4—522no—
578C2×C4—524no—
579C2×C4—522no—
580C2×C4—524no—
581C2×C4—522no—
582C2×C4C4×D5214yes—
583C2×C4C22⋊C4×D1314yes—
584C2×C4#8813614yes—
585C2×C4#8813814yes—
586C2×C4#8814714yes—
587C2×C4#8814814yes—
588C2×C4#8814914yes—
589C22—264no—
590C22—264no—
591C22—264no—
592C22—524no—
593C22—524no—
594C22—524no—
595C22—522no—
596C22—522no—
597C22—522no—
598C22—522no—
599C22—524no—
600C22C2×C4×D5211yescharacteristic
601C22C2×C22⋊C4×D1311yescharacteristic
602C22C2×#8813611yescharacteristic
603C22C2×#8813811yescharacteristic
604C22C2×#8814711yescharacteristic
605C22C2×#8814811yescharacteristic
606C22C2×#8814911yescharacteristic
607C22#1171921512yes—
608C22#1171923112yes—
609C22#1171925112yes—
610C22#1171926612yes—
611C22#1171927712yes—
612C22#1171928812yes—
613C22#1171931312yes—
614C22#1171931312yes—
615C22#1171933312yes—
616C22#1171933312yes—
617C22#1171938011yescharacteristic
618C22#1171938011yescharacteristic
619C22#1171940212yes—
620C22#1171942011yescharacteristic
621C22#1171942911yescharacteristic
622C22#1171943812yes—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—264no—
627C4—524no—
628C4—524no—
629C4—1042no—
630C4#1171832618yes—
631C2—524no—
632C2C2×#1171832611yescharacteristic
633C2#1311741911yescharacteristic
634C2#1311742611yescharacteristic
635C2#1311751612yes—
636C2#1311751912yes—
637C2#1312056712yes—
638C2#1312059611yescharacteristic
639C2#1312059911yescharacteristic
640C2#1312060511yescharacteristic
641C2#1312060711yescharacteristic
642C2#1312063812yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×C4⋊#95C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC4⋊#95D13
hypocenterC13#14258
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
3. C2×#11718590C2semidirect
4. C2×#11718635C2semidirect
6. #13113088C2semidirect
8. #13113065C2semidirect
9. C13×C4⋊#95C2semidirect
24. C2×#88151C4semidirect
190. C4⋊#95D13semidirect
502. C13#14258semidirect

Maximal subgroups

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

subgroupindex
C22×#881512
C22×#881262
C2×#117185902
C2×#117186352
C2×#117186482
#131130882
#131130842
#131130652
C13×C4⋊#952
#1425813