Finite Group Databasebeta

← #15799638

Subgroups — #15799638

Order 3328.

2214 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88142C211yesmaximal, characteristic, complemented
2C2×#11718334C214yesmaximal, complemented
3C2×#11718600C211yesmaximal, characteristic
4C2×#11718649C211yesmaximal, characteristic
5C2×#11718649C211yesmaximal, characteristic
6#13113080C214yesmaximal, complemented
7#13113065C211yesmaximal, characteristic, complemented
8C4×#88071C211yesmaximal, characteristic, complemented
9C13×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D52C2211yescharacteristic
11C23×Dic26C2212yes—
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14C22×C42×C13C2211yescharacteristic
15C22×#6103C2212yes—
16C22×#6103C2212yes—
17C22×#6103C2214yes—
18C22×#6103C2214yes—
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6101C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C22×C4×Dic13C2211yescharacteristic
23C2×#88142—24no—
24C2×#88142—24no—
25C2×#88142—24no—
26C2×C13×#95C2214yes—
27C2×#88071C2214yes—
28C2×#88071C2212yes—
29C2×#88071C2212yes—
30#11718649—24no—
31#11718649—24no—
32#11718600—24no—
33#11718334—216no—
34C24×D13C2312yes—
35C23×C4×C13C2312yes—
36C23×C4×C13C2312yes—
37C23×C4×C13C2312yes—
38C23×C4×C13C2311yescharacteristic
39C23×Dic13C2312yes—
40C23×Dic13C2312yes—
41C23×Dic13C2311yescharacteristic
42C23×Dic13C2311yescharacteristic
43C22×D52—22no—
44C22×D52—22no—
45C22×D52—22no—
46C22×D52—22no—
47C22×D52—22no—
48C22×D52—22no—
49C22×D52Q814yescomplemented
50C22×Dic26—24no—
51C22×Dic26—24no—
52C22×Dic26—24no—
53C22×Dic26—28no—
54C22×Dic26—44no—
55C2×C13×C4⋊C4—22no—
56C2×C13×C4⋊C4—22no—
57C2×C13×C4⋊C4—22no—
58C2×C13×C4⋊C4—22no—
59C2×C13×C4⋊C4—22no—
60C2×C13×C4⋊C4D414yes—
61C2×C42×C13—22no—
62C2×C42×C13—22no—
63C2×C42×C13—22no—
64C2×#6103—28no—
65C2×#6103—28no—
66C2×#6103—28no—
67C2×#6103—28no—
68C2×#6103—28no—
69C2×#6103—24no—
70C2×#6103—24no—
71C2×#6103—28no—
72C2×#6103—44no—
73C2×#6103—44no—
74C2×C4⋊Dic13—22no—
75C2×C4⋊Dic13—22no—
76C2×C4⋊Dic13D414yes—
77C2×#6101—24no—
78C2×#6101—24no—
79C2×#6101—24no—
80C2×C4×Dic13—22no—
81C2×C4×Dic13—22no—
82C2×C4×Dic13—22no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13—22no—
85C2×C4×Dic13D414yes—
86#88142—232no—
87C13×#95—28no—
88#88071—28no—
89#88071—24no—
90#88071D418yescomplemented
91#15144—131nomaximal, Sylow
92C24×C13C2411yescharacteristic
93C23×D13—24no—
94C23×D13—22no—
95C23×D13—22no—
96C23×D13—22no—
97C23×D13—24no—
98C23×D13—22no—
99C23×D13—22no—
100C23×D13—22no—
101C23×D13—24no—
102C23×D13—24no—
103C23×D13C2×Q814yes—
104C22×C4×C13—24no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
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—24no—
115C22×C4×C13—22no—
116C22×C4×C13—24no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13C2×D414yes—
120C22×C4×C13C2×D412yes—
121C22×C4×C13C2×D414yes—
122C22×C4×C13C2×D412yes—
123C22×C4×C13C2×D412yes—
124C22×C4×C13C2×Q812yes—
125C22×C4×C13#4112yes—
126C22×C4×C13#4112yes—
127C22×C4×C13#4112yes—
128C22×Dic13—21no—
129C22×Dic13—21no—
130C22×Dic13—21no—
131C22×Dic13—21no—
132C22×Dic13—22no—
133C22×Dic13—22no—
134C22×Dic13—24no—
135C22×Dic13—22no—
136C22×Dic13—22no—
137C22×Dic13—22no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—22no—
142C22×Dic13—24no—
143C22×Dic13—22no—
144C22×Dic13C2×D414yes—
145C22×Dic13C2×D414yes—
146C22×Dic13C2×D412yes—
147C22×Dic13C2×D414yes—
148C22×Dic13C2×D412yes—
149C22×Dic13#4114yes—
150C2×D52—28no—
151C2×D52—28no—
152C2×D52—28no—
153C2×D52—44no—
154C2×D52—44no—
155C2×D52—44no—
156C2×D52—44no—
157C2×Dic26—216no—
158C2×Dic26—416no—
159C2×Dic26—416no—
160C2×Dic26—48no—
161C2×Dic26—48no—
162C13×C4⋊C4—28no—
163C13×C4⋊C4—44no—
164C42×C13—44no—
165#6103—416no—
166#6103—416no—
167#6103—48no—
168#6103—48no—
169C4⋊Dic13—28no—
170#6101—216no—
171C4×Dic13—28no—
172C4×Dic13—44no—
173C22×C4⋊Q8—131no—
174C2×#393—134no—
175C2×#390—131no—
176C2×#390—131no—
177C2×#390—131no—
178#1439—134no—
179C4⋊#95—131no—
180C4⋊#95D1311yescharacteristic, complemented
181C4×#95—131no—
182C23×C13C22×D411yescharacteristic
183C23×C13C22×D411yescharacteristic
184C23×C13C22×D411yescharacteristic
185C23×C13C22×D412yes—
186C23×C13C22×Q811yescharacteristic
187C23×C13C2×#4111yescharacteristic
188C23×C13C2×#4111yescharacteristic
189C23×C13C2×#4111yescharacteristic
190C23×C13C2×#4112yes—
191C23×C13H(5,𝔽2)12yes—
192C23×C132−1+412yes—
193C22×D13—24no—
194C22×D13—24no—
195C22×D13—24no—
196C22×D13—28no—
197C22×D13—28no—
198C22×D13—42no—
199C22×D13—42no—
200C22×D13—42no—
201C22×D13—42no—
202C22×D13—42no—
203C22×D13—42no—
204C22×D13—44no—
205C22×D13—44no—
206C22×D13—42no—
207C22×D13—42no—
208C22×D13—44no—
209C22×D13—44no—
210C22×D13—44no—
211C22×D13—44no—
212C22×D13—44no—
213C22×D13—44no—
214C22×D13—44no—
215C22×D13—44no—
216C2×C4×C13—24no—
217C2×C4×C13—24no—
218C2×C4×C13—24no—
219C2×C4×C13—28no—
220C2×C4×C13—24no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—44no—
226C2×C4×C13—42no—
227C2×C4×C13—42no—
228C2×C4×C13—42no—
229C2×C4×C13—42no—
230C2×C4×C13—42no—
231C2×C4×C13—44no—
232C2×C4×C13—44no—
233C2×C4×C13#12014yes—
234C2×C4×C13C22⋊Q814yes—
235C2×C4×C13C22⋊Q814yes—
236C2×C4×C13C22⋊Q814yes—
237C2×C4×C13#12414yes—
238C2×C4×C13#12414yes—
239C2×C4×C13#12414yes—
240C2×Dic13—22no—
241C2×Dic13—22no—
242C2×Dic13—22no—
243C2×Dic13—22no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—24no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—24no—
251C2×Dic13—28no—
252C2×Dic13—24no—
253C2×Dic13—42no—
254C2×Dic13—42no—
255C2×Dic13—42no—
256C2×Dic13—42no—
257C2×Dic13—44no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13#12414yes—
261C2×Dic13#12714yes—
262D52—416no—
263Dic26—432no—
264C23×Q8—132no—
265C23×D4—131no—
266C2×C4⋊Q8—264no—
267C2×C4⋊Q8—264no—
268C2×C4⋊Q8—264no—
269C22×C4⋊C4—131no—
270C22×C4⋊C4—132no—
271C22×C4⋊C4C2×D1311yescharacteristic
272C22×C4⋊C4C2×D1311yescharacteristic
273C22×C22⋊C4—132no—
274C22×C22⋊C4—132no—
275C22×C22⋊C4—134no—
276C22×C22⋊C4—134no—
277C22×C42—131no—
278C22×C42—131no—
279C22×C42C2×D1311yescharacteristic
280#393—2616no—
281#390—264no—
282#390—264no—
283#390—264no—
284C2×#95—132no—
285C2×#95—134no—
286C2×#95—132no—
287C2×#95C2×D1314yes—
288C22×C13C2×#12011yescharacteristic
289C22×C13C2×C22⋊Q811yescharacteristic
290C22×C13C2×C22⋊Q811yescharacteristic
291C22×C13C2×C22⋊Q811yescharacteristic
292C22×C13C2×#12411yescharacteristic
293C22×C13C2×#12411yescharacteristic
294C22×C13C2×#12411yescharacteristic
295C22×C13C2×#12411yescharacteristic
296C22×C13C2×#12711yescharacteristic
297C22×C13#53212yes—
298C22×C13#53212yes—
299C22×C13#53511yescharacteristic
300C22×C13#53611yescharacteristic
301C22×C13#54212yes—
302C22×C13#54212yes—
303C22×C13D4212yes—
304C22×C13D4212yes—
305C22×C13#54712yes—
306C22×C13#54812yes—
307C22×C13#54812yes—
308C22×C13D4×Q812yes—
309C22×C13#55412yes—
310C22×C13#55512yes—
311C2×D13—44no—
312C2×D13—44no—
313C2×D13—48no—
314C2×D13—48no—
315C2×D13—44no—
316C2×D13—82no—
317C2×D13—82no—
318C2×D13—82no—
319C2×D13—82no—
320C2×D13—84no—
321C2×D13—84no—
322C4×C13—44no—
323C4×C13—44no—
324C4×C13—82no—
325C4×C13#39318yescomplemented
326Dic13—24no—
327Dic13—24no—
328Dic13—44no—
329Dic13—44no—
330C25—132no—
331C22×Q8—264no—
332C22×Q8—264no—
333C22×Q8—268no—
334C22×Q8—264no—
335C22×Q8—524no—
336C22×D4—134no—
337C22×D4—262no—
338C22×D4—262no—
339C22×D4—262no—
340C22×D4—262no—
341C22×D4—262no—
342C22×D4—262no—
343C23×C4—131no—
344C23×C4—131no—
345C23×C4—132no—
346C23×C4—132no—
347C23×C4C22×D1312yes—
348C23×C4C22×D1312yes—
349C23×C4C22×D1312yes—
350C23×C4C22×D1311yescharacteristic
351C4⋊Q8—2632no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—134no—
358C2×C4⋊C4—264no—
359C2×C4⋊C4—264no—
360C2×C4⋊C4—264no—
361C2×C4⋊C4—262no—
362C2×C4⋊C4—262no—
363C2×C4⋊C4D5214yes—
364C2×C22⋊C4—268no—
365C2×C22⋊C4—268no—
366C2×C22⋊C4—264no—
367C2×C22⋊C4—264no—
368C2×C22⋊C4—268no—
369C2×C22⋊C4—268no—
370C2×C22⋊C4—268no—
371C2×C22⋊C4—268no—
372C2×C22⋊C4—524no—
373C2×C22⋊C4—524no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—134no—
378C2×C42—262no—
379C2×C42—262no—
380C2×C42—262no—
381C2×C42—262no—
382C2×C42—262no—
383#95—28no—
384#95—138no—
385#95—264no—
386#95—268no—
387C2×C13C2×#39311yescharacteristic
388C2×C13#243311yescharacteristic
389C2×C13#243611yescharacteristic
390C2×C13#245212yes—
391C2×C13#245212yes—
392C2×C13#249112yes—
393C2×C13#249112yes—
394C2×C13#254211yescharacteristic
395C2×C13#256311yescharacteristic
396C2×C13#264011yescharacteristic
397C2×C13#285811yescharacteristic
398D13—84no—
399C24—134no—
400C24—262no—
401C24—262no—
402C24—262no—
403C24—264no—
404C24—262no—
405C24—262no—
406C24—262no—
407C24—264no—
408C24—264no—
409C24—264no—
410C24C23×D1311yescharacteristic
411C2×Q8—2616no—
412C2×Q8—5216no—
413C2×Q8—528no—
414C2×Q8—5216no—
415C2×Q8—528no—
416C2×D4—268no—
417C2×D4—268no—
418C2×D4—268no—
419C2×D4—524no—
420C2×D4—524no—
421C2×D4—524no—
422C2×D4—524no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—134no—
439C22×C4—132no—
440C22×C4—132no—
441C22×C4—134no—
442C22×C4—134no—
443C22×C4—134no—
444C22×C4—261no—
445C22×C4—261no—
446C22×C4—261no—
447C22×C4—261no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—262no—
452C22×C4—262no—
453C22×C4—262no—
454C22×C4—264no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—262no—
458C22×C4—262no—
459C22×C4—264no—
460C22×C4C2×D5214yes—
461C22×C4C2×D5212yes—
462C22×C4#612712yes—
463C22×C4#612712yes—
464C22×C4D4×D1314yes—
465C22×C4D4×D1312yes—
466C22×C4D4×D1312yes—
467C22×C4#612912yes—
468C22×C4Q8×D1312yes—
469C4⋊C4—28no—
470C4⋊C4—44no—
471C4⋊C4—2616no—
472C4⋊C4—268no—
473C22⋊C4—5216no—
474C22⋊C4—5216no—
475C22⋊C4—528no—
476C22⋊C4—528no—
477C42—44no—
478C42—268no—
479C42—524no—
480C13#1514411yescharacteristic, complemented, Sylow
481C23—268no—
482C23—264no—
483C23—264no—
484C23—264no—
485C23—268no—
486C23—522no—
487C23—522no—
488C23—522no—
489C23—522no—
490C23—522no—
491C23—524no—
492C23—524no—
493C23—524no—
494C23—524no—
495C23—524no—
496C23—524no—
497C23—524no—
498C23—524no—
499C23—524no—
500C23—524no—
501C23—522no—
502C23—522no—
503C23—522no—
504C23C22×D5211yescharacteristic
505C23C2×#612711yescharacteristic
506C23C2×#612711yescharacteristic
507C23C2×D4×D1311yescharacteristic
508C23C2×D4×D1312yes—
509C23C2×D4×D1311yescharacteristic
510C23C2×#612911yescharacteristic
511C23C2×Q8×D1311yescharacteristic
512C23#8825412yes—
513C23#41×D1312yes—
514C23#8825612yes—
515Q8—5232no—
516D4—5216no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—134no—
535C2×C4—134no—
536C2×C4—262no—
537C2×C4—262no—
538C2×C4—262no—
539C2×C4—264no—
540C2×C4—264no—
541C2×C4—264no—
542C2×C4—264no—
543C2×C4—262no—
544C2×C4—264no—
545C2×C4—264no—
546C2×C4—264no—
547C2×C4—264no—
548C2×C4—268no—
549C2×C4—522no—
550C2×C4—522no—
551C2×C4—524no—
552C2×C4—522no—
553C2×C4—522no—
554C2×C4—522no—
555C2×C4—522no—
556C2×C4#8812914yes—
557C2×C4#8813614yes—
558C2×C4#8813914yes—
559C2×C4#8813914yes—
560C2×C4#8815014yes—
561C2×C4#8815014yes—
562C2×C4#8815114yes—
563C22—524no—
564C22—524no—
565C22—524no—
566C22—528no—
567C22—528no—
568C22—1042no—
569C22—1042no—
570C22—1042no—
571C22—1042no—
572C22—1044no—
573C22—1044no—
574C22C2×#8812911yescharacteristic
575C22C2×#8813611yescharacteristic
576C22C2×#8813911yescharacteristic
577C22C2×#8813911yescharacteristic
578C22#1171921212yes—
579C22C2×#8815011yescharacteristic
580C22C2×#8815011yescharacteristic
581C22C2×#8815111yescharacteristic
582C22#1171923712yes—
583C22#1171925312yes—
584C22D4×D5212yes—
585C22#1171928112yes—
586C22#1171928712yes—
587C22#1171930112yes—
588C22#1171931412yes—
589C22#1171933812yes—
590C22#1171934812yes—
591C22D13×#12411yescharacteristic
592C22#1171937911yescharacteristic
593C22#1171939512yes—
594C22D13×#12711yescharacteristic
595C22#1171942911yescharacteristic
596C22#1171943212yes—
597C4—44no—
598C4—44no—
599C4—82no—
600C4—264no—
601C4—264no—
602C4—524no—
603C4—524no—
604C4#1171833418yescomplemented
605C2—1044no—
606C2C2×#1171833411yescharacteristic
607C2#1311847211yescharacteristic
608C2#1311849311yescharacteristic
609C2#1311868412yes—
610C2#1311868712yes—
611C2#1311919312yes—
612C2#1311919512yes—
613C2#1311978811yescharacteristic
614C2#1312000811yescharacteristic
615C2#1312077511yescharacteristic
616C2#1312319011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, cosocleC24×C13C24
FittingC13×C4⋊#95C2
prime radicalC23×D52C22
hypercenterC4⋊#95D13
hypocenterC13#15144
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C22×#88142C2semidirect
2. C2×#11718334C2semidirect
6. #13113080C2semidirect
7. #13113065C2semidirect
8. C4×#88071C2semidirect
9. C13×C4⋊#95C2semidirect
49. C22×D52Q8semidirect
90. #88071D4semidirect
180. C4⋊#95D13semidirect
325. C4×C13#393semidirect
480. C13#15144semidirect
604. C4#11718334semidirect

Maximal subgroups

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

subgroupindex
C22×#881422
C2×#117183342
C2×#117186002
C2×#117186492
#131130802
#131130652
C4×#880712
C13×C4⋊#952
#1514413