Finite Group Databasebeta

← #15799560

Subgroups — #15799560

Order 3328.

2006 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718337C214yesmaximal, complemented
2C2×#11718600C211yesmaximal, characteristic
3C2×#11718649C211yesmaximal, characteristic
4C2×#11718649C211yesmaximal, characteristic
5C2×#11718634C211yesmaximal, characteristic, complemented
6C2×#11718637C211yesmaximal, characteristic, complemented
7C2×C4×#6101C211yesmaximal, characteristic, complemented
8#13113080C214yesmaximal, complemented
9C13×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D52C2211yescharacteristic
11C22×C13×C4⋊C4C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C42×C13C2211yescharacteristic
14C22×#6103C2212yes—
15C22×#6103C2214yes—
16C22×#6103C2214yes—
17C22×#6103C2212yes—
18C22×C4⋊Dic13C2211yescharacteristic
19C22×#6101C2212yes—
20C22×#6101C2214yes—
21C22×#6101C2212yes—
22C22×C4×Dic13C2211yescharacteristic
23C22×C4×Dic13C2211yescharacteristic
24C2×C13×#95C2214yes—
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27#11718649—24no—
28#11718649—24no—
29#11718637—24no—
30#11718634—24no—
31#11718600—24no—
32C4×#6101—24no—
33#11718337—216no—
34C24×D13C2312yes—
35C23×C4×C13C2312yes—
36C23×C4×C13C2312yes—
37C23×C4×C13C2312yes—
38C23×C4×C13C2311yescharacteristic
39C23×Dic13C2311yescharacteristic
40C23×Dic13C2312yes—
41C23×Dic13C2311yescharacteristic
42C23×Dic13C2312yes—
43C22×D52—22no—
44C22×D52—22no—
45C22×D52—22no—
46C22×D52—22no—
47C22×D52—22no—
48C22×D52—22no—
49C22×D52Q814yes—
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×C42×C13—22no—
57C2×C42×C13—22no—
58C2×C42×C13—22no—
59C2×#6103—28no—
60C2×#6103—28no—
61C2×#6103—28no—
62C2×#6103—28no—
63C2×#6103—24no—
64C2×#6103—28no—
65C2×#6103—24no—
66C2×#6103—28no—
67C2×#6103—44no—
68C2×#6103—44no—
69C2×C4⋊Dic13—22no—
70C2×C4⋊Dic13—22no—
71C2×C4⋊Dic13D414yes—
72C2×#6101—24no—
73C2×#6101—24no—
74C2×#6101—28no—
75C2×#6101—28no—
76C2×#6101—24no—
77C2×#6101—28no—
78C2×#6101—44no—
79C2×#6101D418yescomplemented
80C2×C4×Dic13—22no—
81C2×C4×Dic13—22no—
82C2×C4×Dic13—22no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13D414yes—
85C2×C4×Dic13D414yes—
86C13×#95—28no—
87#88071—24no—
88#88071—24no—
89#15134—131nomaximal, Sylow
90C24×C13C2411yescharacteristic
91C23×D13—24no—
92C23×D13—22no—
93C23×D13—22no—
94C23×D13—22no—
95C23×D13—24no—
96C23×D13—22no—
97C23×D13—22no—
98C23×D13—22no—
99C23×D13—24no—
100C23×D13—24no—
101C23×D13C2×Q814yes—
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—24no—
110C22×C4×C13—24no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—24no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—24no—
118C22×C4×C13C2×D414yes—
119C22×C4×C13C2×D412yes—
120C22×C4×C13C2×D412yes—
121C22×C4×C13C2×D412yes—
122C22×C4×C13C2×Q812yes—
123C22×C4×C13#4112yes—
124C22×C4×C13#4112yes—
125C22×C4×C13#4112yes—
126C22×Dic13—21no—
127C22×Dic13—22no—
128C22×Dic13—21no—
129C22×Dic13—22no—
130C22×Dic13—22no—
131C22×Dic13—22no—
132C22×Dic13—22no—
133C22×Dic13—21no—
134C22×Dic13—22no—
135C22×Dic13—24no—
136C22×Dic13—21no—
137C22×Dic13—22no—
138C22×Dic13—24no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13C2×D412yes—
142C22×Dic13C2×D414yes—
143C22×Dic13C2×D414yes—
144C22×Dic13C2×D414yes—
145C22×Dic13C2×D412yes—
146C22×Dic13C2×D414yes—
147C22×Dic13#4114yes—
148C2×D52—28no—
149C2×D52—28no—
150C2×D52—28no—
151C2×D52—44no—
152C2×D52—44no—
153C2×D52—44no—
154C2×D52—44no—
155C13×C4⋊C4—44no—
156C13×C4⋊C4—44no—
157C42×C13—44no—
158#6103—416no—
159#6103—48no—
160#6103—416no—
161#6103—48no—
162C4⋊Dic13—28no—
163#6101—216no—
164#6101—416no—
165#6101—48no—
166C4×Dic13—28no—
167C4×Dic13—28no—
168C2×#396—134no—
169C2×#390—131no—
170C2×#390—131no—
171C2×#390—131no—
172C2×#389—131no—
173C2×#384—131no—
174C2×C4×C4⋊C4—131no—
175#1439—134no—
176C4⋊#95D1311yescharacteristic, complemented
177C23×C13C22×D411yescharacteristic
178C23×C13C22×D411yescharacteristic
179C23×C13C22×D411yescharacteristic
180C23×C13C22×D412yes—
181C23×C13C22×Q811yescharacteristic
182C23×C13C2×#4111yescharacteristic
183C23×C13C2×#4111yescharacteristic
184C23×C13C2×#4111yescharacteristic
185C23×C13C2×#4112yes—
186C23×C13H(5,𝔽2)12yes—
187C23×C132−1+412yes—
188C22×D13—24no—
189C22×D13—24no—
190C22×D13—24no—
191C22×D13—28no—
192C22×D13—28no—
193C22×D13—44no—
194C22×D13—42no—
195C22×D13—42no—
196C22×D13—42no—
197C22×D13—42no—
198C22×D13—42no—
199C22×D13—42no—
200C22×D13—44no—
201C22×D13—42no—
202C22×D13—42no—
203C22×D13—44no—
204C22×D13—44no—
205C22×D13—44no—
206C22×D13—44no—
207C22×D13—44no—
208C22×D13—44no—
209C22×D13—44no—
210C22×D13—44no—
211C2×C4×C13—24no—
212C2×C4×C13—24no—
213C2×C4×C13—24no—
214C2×C4×C13—42no—
215C2×C4×C13—42no—
216C2×C4×C13—42no—
217C2×C4×C13—42no—
218C2×C4×C13—44no—
219C2×C4×C13—42no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—44no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—44no—
227C2×C4×C13—44no—
228C2×C4×C13#12114yes—
229C2×C4×C13#12114yes—
230C2×C4×C13#12114yes—
231C2×C4×C13C22⋊Q814yes—
232C2×C4×C13C22⋊Q814yes—
233C2×C4×C13C22⋊Q814yes—
234C2×C4×C13#12614yes—
235C2×Dic13—22no—
236C2×Dic13—24no—
237C2×Dic13—24no—
238C2×Dic13—24no—
239C2×Dic13—22no—
240C2×Dic13—24no—
241C2×Dic13—24no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—22no—
246C2×Dic13—24no—
247C2×Dic13—22no—
248C2×Dic13—22no—
249C2×Dic13—28no—
250C2×Dic13—42no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13—44no—
255C2×Dic13#12118yes—
256C2×Dic13#12714yes—
257D52—416no—
258C23×D4—131no—
259C22×C4⋊C4—132no—
260C22×C4⋊C4—131no—
261C22×C4⋊C4—132no—
262C22×C4⋊C4—134no—
263C22×C4⋊C4C2×D1311yescharacteristic
264C22×C4⋊C4C2×D1311yescharacteristic
265C22×C22⋊C4—132no—
266C22×C22⋊C4—134no—
267C22×C22⋊C4—134no—
268C22×C22⋊C4—132no—
269C22×C42—131no—
270C22×C42—131no—
271C22×C42C2×D1311yescharacteristic
272#396—2616no—
273#390—264no—
274#390—264no—
275#390—264no—
276#389—264no—
277#384—264no—
278C4×C4⋊C4—264no—
279C2×#95—132no—
280C2×#95—132no—
281C2×#95C2×D1314yes—
282C22×C13C2×#12111yescharacteristic
283C22×C13C2×#12111yescharacteristic
284C22×C13C2×#12111yescharacteristic
285C22×C13C2×#12112yes—
286C22×C13C2×C22⋊Q811yescharacteristic
287C22×C13C2×C22⋊Q811yescharacteristic
288C22×C13C2×C22⋊Q811yescharacteristic
289C22×C13C2×#12611yescharacteristic
290C22×C13C2×#12711yescharacteristic
291C22×C13#53212yes—
292C22×C13#53211yescharacteristic
293C22×C13#53511yescharacteristic
294C22×C13#53611yescharacteristic
295C22×C13#53712yes—
296C22×C13#54212yes—
297C22×C13D4212yes—
298C22×C13#54712yes—
299C22×C13#54812yes—
300C22×C13D4×Q812yes—
301C22×C13#55012yes—
302C22×C13#55312yes—
303C22×C13#55412yes—
304C22×C13#55612yes—
305C2×D13—44no—
306C2×D13—44no—
307C2×D13—48no—
308C2×D13—48no—
309C2×D13—44no—
310C2×D13—82no—
311C2×D13—82no—
312C2×D13—84no—
313C2×D13—84no—
314C2×D13—82no—
315C2×D13—82no—
316C4×C13—44no—
317C4×C13—82no—
318C4×C13—82no—
319C4×C13#39618yescomplemented
320Dic13—24no—
321Dic13—28no—
322Dic13—42no—
323Dic13—44no—
324C25—132no—
325C22×D4—134no—
326C22×D4—262no—
327C22×D4—262no—
328C22×D4—262no—
329C22×D4—262no—
330C22×D4—262no—
331C22×D4—262no—
332C23×C4—131no—
333C23×C4—132no—
334C23×C4—132no—
335C23×C4—131no—
336C23×C4C22×D1312yes—
337C23×C4C22×D1312yes—
338C23×C4C22×D1312yes—
339C23×C4C22×D1311yescharacteristic
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—138no—
347C2×C4⋊C4—134no—
348C2×C4⋊C4—264no—
349C2×C4⋊C4—264no—
350C2×C4⋊C4—262no—
351C2×C4⋊C4—262no—
352C2×C4⋊C4—268no—
353C2×C4⋊C4—264no—
354C2×C4⋊C4—268no—
355C2×C4⋊C4—268no—
356C2×C4⋊C4—524no—
357C2×C22⋊C4—268no—
358C2×C22⋊C4—268no—
359C2×C22⋊C4—268no—
360C2×C22⋊C4—264no—
361C2×C22⋊C4—264no—
362C2×C22⋊C4—268no—
363C2×C22⋊C4—268no—
364C2×C22⋊C4—268no—
365C2×C22⋊C4—524no—
366C2×C22⋊C4—524no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—134no—
371C2×C42—134no—
372C2×C42—262no—
373C2×C42—262no—
374C2×C42—262no—
375C2×C42—262no—
376#95—28no—
377#95—264no—
378#95—264no—
379C2×C13C2×#39611yescharacteristic
380C2×C13#241612yes—
381C2×C13#245212yes—
382C2×C13#251212yes—
383C2×C13#253812yes—
384C2×C13#254111yescharacteristic
385C2×C13#254211yescharacteristic
386C2×C13#254511yescharacteristic
387C2×C13#256611yescharacteristic
388C2×C13#262011yescharacteristic
389C2×C13#285611yescharacteristic
390D13—84no—
391C24—134no—
392C24—262no—
393C24—262no—
394C24—262no—
395C24—264no—
396C24—262no—
397C24—262no—
398C24—262no—
399C24—264no—
400C24—264no—
401C24—264no—
402C24C23×D1311yescharacteristic
403C2×D4—268no—
404C2×D4—268no—
405C2×D4—268no—
406C2×D4—524no—
407C2×D4—524no—
408C2×D4—524no—
409C2×D4—524no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—134no—
427C22×C4—132no—
428C22×C4—134no—
429C22×C4—134no—
430C22×C4—134no—
431C22×C4—132no—
432C22×C4—134no—
433C22×C4—262no—
434C22×C4—261no—
435C22×C4—261no—
436C22×C4—262no—
437C22×C4—262no—
438C22×C4—262no—
439C22×C4—262no—
440C22×C4—261no—
441C22×C4—262no—
442C22×C4—261no—
443C22×C4—262no—
444C22×C4—264no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—264no—
448C22×C4#612712yes—
449C22×C4#612712yes—
450C22×C4#612712yes—
451C22×C4D4×D1314yes—
452C22×C4D4×D1312yes—
453C22×C4D4×D1312yes—
454C22×C4D4×D1312yes—
455C22×C4Q8×D1312yes—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—2616no—
459C4⋊C4—268no—
460C4⋊C4—528no—
461C4⋊C4—5216no—
462C22⋊C4—5216no—
463C22⋊C4—5216no—
464C22⋊C4—528no—
465C22⋊C4—528no—
466C42—44no—
467C42—268no—
468C42—268no—
469C13#1513411yescharacteristic, complemented, Sylow
470C23—268no—
471C23—264no—
472C23—264no—
473C23—264no—
474C23—268no—
475C23—522no—
476C23—522no—
477C23—522no—
478C23—522no—
479C23—522no—
480C23—524no—
481C23—524no—
482C23—524no—
483C23—524no—
484C23—524no—
485C23—524no—
486C23—524no—
487C23—524no—
488C23—524no—
489C23—524no—
490C23—522no—
491C23—522no—
492C23—522no—
493C23C2×#612711yescharacteristic
494C23C2×#612711yescharacteristic
495C23C2×#612711yescharacteristic
496C23C2×D4×D1311yescharacteristic
497C23C2×D4×D1312yes—
498C23C2×D4×D1311yescharacteristic
499C23C2×D4×D1311yescharacteristic
500C23C2×Q8×D1311yescharacteristic
501C23#8825412yes—
502C23#41×D1312yes—
503C23#8825612yes—
504D4—5216no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—138no—
523C2×C4—134no—
524C2×C4—264no—
525C2×C4—262no—
526C2×C4—264no—
527C2×C4—264no—
528C2×C4—264no—
529C2×C4—264no—
530C2×C4—264no—
531C2×C4—264no—
532C2×C4—264no—
533C2×C4—262no—
534C2×C4—262no—
535C2×C4—262no—
536C2×C4—264no—
537C2×C4—268no—
538C2×C4—262no—
539C2×C4—524no—
540C2×C4—522no—
541C2×C4—522no—
542C2×C4—522no—
543C2×C4—522no—
544C2×C4#8813014yes—
545C2×C4#8813814yes—
546C2×C4#8813814yes—
547C2×C4#8813814yes—
548C2×C4#8815014yes—
549C2×C4#8815014yes—
550C2×C4#8815014yes—
551C22—528no—
552C22—524no—
553C22—524no—
554C22—524no—
555C22—528no—
556C22—1042no—
557C22—1044no—
558C22—1042no—
559C22—1042no—
560C22—1042no—
561C22—1044no—
562C22C2×#8813011yescharacteristic
563C22C2×#8813811yescharacteristic
564C22C2×#8813811yescharacteristic
565C22C2×#8813811yescharacteristic
566C22#1171921212yes—
567C22C2×#8815011yescharacteristic
568C22C2×#8815011yescharacteristic
569C22C2×#8815011yescharacteristic
570C22#1171923512yes—
571C22#1171925312yes—
572C22#1171925812yes—
573C22#1171927312yes—
574C22#1171928712yes—
575C22D13×#12112yes—
576C22#1171931412yes—
577C22#1171933412yes—
578C22#1171933812yes—
579C22#1171937311yescharacteristic
580C22#1171937911yescharacteristic
581C22#1171939512yes—
582C22D13×#12711yescharacteristic
583C22#1171942911yescharacteristic
584C22#1171943212yes—
585C4—44no—
586C4—82no—
587C4—82no—
588C4—268no—
589C4—264no—
590C4—524no—
591C4—522no—
592C4#1171833718yescomplemented
593C2—1044no—
594C2C2×#1171833711yescharacteristic
595C2#1311827812yes—
596C2#1311868712yes—
597C2#1311941612yes—
598C2#1311975712yes—
599C2#1311978111yescharacteristic
600C2#1311978811yescharacteristic
601C2#1311981911yescharacteristic
602C2#1312003511yescharacteristic
603C2#1312060011yescharacteristic
604C2#1312318011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutator, socle, cosocleC24×C13C24
FittingC13×C4⋊#95C2
prime radicalC23×D52C22
hypercenterC4⋊#95D13
hypocenterC13#15134
LCSG ▷ C24×C13 ▷ C13
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
1. C2×#11718337C2semidirect
5. C2×#11718634C2semidirect
6. C2×#11718637C2semidirect
7. C2×C4×#6101C2semidirect
8. #13113080C2semidirect
9. C13×C4⋊#95C2semidirect
79. C2×#6101D4semidirect
176. C4⋊#95D13semidirect
319. C4×C13#396semidirect
469. C13#15134semidirect
592. C4#11718337semidirect

Maximal subgroups

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

subgroupindex
C2×#117183372
C2×#117186002
C2×#117186492
C2×#117186342
C2×#117186372
C2×C4×#61012
#131130802
C13×C4⋊#952
#1513413