Finite Group Databasebeta

← #15793710

Subgroups — #15793710

Order 3328.

1442 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
1C2×#11718608C211yesmaximal, characteristic
2#13113106C212yesmaximal, complemented
3#13113095C212yesmaximal, complemented
4#13113091C212yesmaximal, complemented
5#13113091C212yesmaximal, complemented
6#13113079C212yesmaximal
7#13113058C211yesmaximal, characteristic
8#13113058C212yesmaximal
9C13×#1439C211yesmaximal, characteristic
10C24×Dic13C2211yescharacteristic
11C23×Dic26C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C22⋊C4C2212yes—
14C22×C13×C22⋊C4C2211yescharacteristic
15C22×C22⋊Dic13C2211yescharacteristic
16C22×C22⋊Dic13C2212yes—
17C22×#6101C2212yes—
18C2×C13×#95C2211yescharacteristic
19C2×C13×#95C2212yes—
20C2×#88071C2212yes—
21C2×#88071C2212yes—
22C2×#88071C2212yes—
23C2×#88071C2212yes—
24C2×#88071C2212yes—
25C2×#88071C2212yes—
26C2×#88071C2212yes—
27C2×#88071C2212yes—
28C2×#88071C2212yes—
29C2×#88071C2211yescharacteristic
30C2×#88071C2212yes—
31#11718608C418yescomplemented
32C25×C13C2311yescharacteristic
33C23×C4×C13C2312yes—
34C23×C4×C13C2312yes—
35C23×C4×C13C2311yescharacteristic
36C23×C4×C13C2311yescharacteristic
37C23×Dic13—42no—
38C23×Dic13—41no—
39C23×Dic13—41no—
40C23×Dic13—42no—
41C23×Dic13Q814yescomplemented
42C23×Dic13C2312yes—
43C23×Dic13C2312yes—
44C23×Dic13C2312yes—
45C23×Dic13C2312yes—
46C22×Dic26—41no—
47C22×Dic26—41no—
48C22×Dic26—42no—
49C22×Dic26—42no—
50C22×Dic26C2×C414yescomplemented
51C2×C13×C4⋊C4—42no—
52C2×C13×C4⋊C4—41no—
53C2×C13×C22⋊C4—24no—
54C2×C13×C22⋊C4—41no—
55C2×C13×C22⋊C4—41no—
56C2×C13×C22⋊C4—42no—
57C2×C13×C22⋊C4—42no—
58C2×C13×C22⋊C4C2×C414yes—
59C2×C22⋊Dic13—22no—
60C2×C22⋊Dic13—42no—
61C2×C22⋊Dic13—42no—
62C2×C22⋊Dic13—41no—
63C2×C22⋊Dic13—41no—
64C2×C22⋊Dic13C2×C418yes—
65C2×#6101—42no—
66C2×#6101—42no—
67C2×#6101—42no—
68C13×#95—41no—
69C13×#95—42no—
70#88071—42no—
71#88071—42no—
72#88071—42no—
73#88071—42no—
74#88071—42no—
75#88071—42no—
76#88071—42no—
77#88071—42no—
78#88071—42no—
79#88071C2×C414yescomplemented
80#88071C2×C418yescomplemented
81#14635—131nomaximal, Sylow
82C24×C13—22no—
83C24×C13—21no—
84C24×C13—22no—
85C24×C13—21no—
86C24×C13—21no—
87C24×C13—22no—
88C24×C13—21no—
89C24×C13—22no—
90C24×C13C22×C412yes—
91C24×C13C2×Q812yes—
92C24×C13#4112yes—
93C24×C13C2411yescharacteristic
94C22×C4×C13—22no—
95C22×C4×C13—22no—
96C22×C4×C13—21no—
97C22×C4×C13—22no—
98C22×C4×C13—21no—
99C22×C4×C13—21no—
100C22×C4×C13—21no—
101C22×C4×C13—22no—
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—22no—
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13C22×C412yes—
117C22×C4×C13C22×C412yes—
118C22×Dic13—22no—
119C22×Dic13—22no—
120C22×Dic13—22no—
121C22×Dic13—22no—
122C22×Dic13—22no—
123C22×Dic13—22no—
124C22×Dic13—22no—
125C22×Dic13—22no—
126C22×Dic13—22no—
127C22×Dic13—22no—
128C22×Dic13—22no—
129C22×Dic13—22no—
130C22×Dic13—22no—
131C22×Dic13—22no—
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—22no—
142C22×Dic13—22no—
143C22×Dic13—48no—
144C22×Dic13—44no—
145C22×Dic13—44no—
146C22×Dic13—48no—
147C22×Dic13—44no—
148C22×Dic13C22×C414yes—
149C22×Dic13C22×C414yes—
150C22×Dic13C2×Q814yes—
151C2×Dic26—48no—
152C2×Dic26—44no—
153C2×Dic26—44no—
154C2×Dic26—48no—
155C2×Dic26—44no—
156C13×C4⋊C4—44no—
157C13×C22⋊C4—44no—
158C13×C22⋊C4—48no—
159C22⋊Dic13—48no—
160C22⋊Dic13—44no—
161#6101—48no—
162C2×#393—131no—
163#1446—132no—
164#1442—132no—
165#1442—132no—
166#1442—132no—
167#1439—132no—
168#1439D1311yescharacteristic
169C22⋊#95—132no—
170C22⋊#95—131no—
171C23×C13—24no—
172C23×C13—22no—
173C23×C13—22no—
174C23×C13—24no—
175C23×C13—22no—
176C23×C13—24no—
177C23×C13—22no—
178C23×C13—24no—
179C23×C13—22no—
180C23×C13—24no—
181C23×C13—22no—
182C23×C13—24no—
183C23×C13—41no—
184C23×C13—42no—
185C23×C13—41no—
186C23×C13—42no—
187C23×C13—41no—
188C23×C13—42no—
189C23×C13—42no—
190C23×C13—42no—
191C23×C13—41no—
192C23×C13—42no—
193C23×C13C4×Q814yes—
194C23×C13C23×C411yescharacteristic
195C23×C13C22×Q811yescharacteristic
196C23×C13C2×#4111yescharacteristic
197C23×C13H(5,𝔽2)11yescharacteristic
198C23×C13H(5,𝔽2)12yes—
199C23×C13H(5,𝔽2)11yescharacteristic
200C23×C13H(5,𝔽2)12yes—
201C23×C13H(5,𝔽2)12yes—
202C23×C13H(5,𝔽2)11yescharacteristic
203C23×C132−1+411yescharacteristic
204C23×C132−1+412yes—
205C2×C4×C13—22no—
206C2×C4×C13—22no—
207C2×C4×C13—24no—
208C2×C4×C13—24no—
209C2×C4×C13—41no—
210C2×C4×C13—42no—
211C2×C4×C13—41no—
212C2×C4×C13—41no—
213C2×C4×C13—42no—
214C2×C4×C13—41no—
215C2×C4×C13—42no—
216C2×C4×C13—42no—
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—42no—
224C2×C4×C13—42no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—42no—
228C2×C4×C13—42no—
229C2×Dic13—24no—
230C2×Dic13—24no—
231C2×Dic13—24no—
232C2×Dic13—24no—
233C2×Dic13—24no—
234C2×Dic13—24no—
235C2×Dic13—24no—
236C2×Dic13—24no—
237C2×Dic13—24no—
238C2×Dic13—416no—
239C2×Dic13—42no—
240C2×Dic13—42no—
241C2×Dic13—42no—
242C2×Dic13—42no—
243C2×Dic13—42no—
244C2×Dic13—42no—
245C2×Dic13—42no—
246C2×Dic13—42no—
247C2×Dic13—42no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250C2×Dic13—42no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13—42no—
255C2×Dic13—42no—
256C2×Dic13—42no—
257C2×Dic13—42no—
258Dic26—416no—
259C23×Q8—131no—
260C24×C4—131no—
261C22×C4⋊C4—132no—
262C22×C4⋊C4C2×D1311yescharacteristic
263C22×C22⋊C4—132no—
264C22×C22⋊C4—131no—
265C22×C22⋊C4C2×D1312yes—
266C22×C22⋊C4C2×D1311yescharacteristic
267#393—138no—
268C2×#95—131no—
269C2×#95—132no—
270C2×#95—132no—
271C2×#95—132no—
272C2×#95—132no—
273C2×#95—132no—
274C2×#95—132no—
275C2×#95—132no—
276C2×#95—132no—
277C2×#95—132no—
278C2×#95—132no—
279C2×#95C2×D1311yescharacteristic
280C2×#95C2×D1312yes—
281C22×C13—24no—
282C22×C13—28no—
283C22×C13—42no—
284C22×C13—44no—
285C22×C13—42no—
286C22×C13—42no—
287C22×C13—44no—
288C22×C13—42no—
289C22×C13—42no—
290C22×C13—44no—
291C22×C13—42no—
292C22×C13C2×C4×Q811yescharacteristic
293C22×C13#51811yescharacteristic
294C22×C13#51812yes—
295C22×C13#52011yescharacteristic
296C22×C13#52012yes—
297C22×C13#53812yes—
298C22×C13#53811yescharacteristic
299C22×C13#53911yescharacteristic
300C22×C13#53912yes—
301C22×C13#54312yes—
302C22×C13#54311yescharacteristic
303C22×C13#54411yescharacteristic
304C22×C13#54412yes—
305C22×C13#56011yescharacteristic
306C22×C13#56012yes—
307C22×C13H(3,𝔽4)11yescharacteristic
308C22×C13H(3,𝔽4)12yes—
309C22×C13H(3,𝔽4)11yescharacteristic
310C22×C13#56212yes—
311C22×C13#56212yes—
312C22×C13#56212yes—
313C22×C13#56311yescharacteristic
314C22×C13#56312yes—
315C4×C13—42no—
316C4×C13—42no—
317C4×C13—82no—
318C4×C13—82no—
319Dic13—44no—
320Dic13—44no—
321Dic13—44no—
322Dic13—82no—
323C25C22×D1311yescharacteristic
324C22×Q8—134no—
325C22×Q8—522no—
326C22×Q8—521no—
327C22×Q8—521no—
328C22×Q8—522no—
329C23×C4—132no—
330C23×C4—132no—
331C23×C4—132no—
332C23×C4—134no—
333C23×C4—132no—
334C23×C4—521no—
335C23×C4—522no—
336C23×C4—521no—
337C23×C4—522no—
338C23×C4C22×D1312yes—
339C23×C4C22×D1312yes—
340C23×C4C22×D1311yescharacteristic
341C23×C4C22×D1311yescharacteristic
342C2×C4⋊C4—42no—
343C2×C4⋊C4—41no—
344C2×C4⋊C4—522no—
345C2×C4⋊C4—522no—
346C2×C4⋊C4—522no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—41no—
349C2×C22⋊C4—41no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—138no—
353C2×C22⋊C4—262no—
354C2×C22⋊C4—522no—
355C2×C22⋊C4—522no—
356C2×C22⋊C4—521no—
357C2×C22⋊C4—521no—
358C2×C22⋊C4C4×D1314yes—
359#95—41no—
360#95—42no—
361#95—134no—
362#95—138no—
363#95—522no—
364#95—522no—
365#95—522no—
366#95—522no—
367#95—522no—
368#95—522no—
369#95—522no—
370#95—522no—
371#95—522no—
372C2×C13—44no—
373C2×C13#232911yescharacteristic
374C2×C13#232912yes—
375C2×C13#237511yescharacteristic
376C2×C13#237912yes—
377C2×C13#237911yescharacteristic
378C2×C13#282411yescharacteristic
379C2×C13#282412yes—
380C2×C13#283511yescharacteristic
381C2×C13#284011yescharacteristic
382C2×C13#284112yes—
383C2×C13#284111yescharacteristic
384C24—21no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—22no—
390C24—21no—
391C24—22no—
392C24C2×C4×D1312yes—
393C24#612912yes—
394C24Q8×D1312yes—
395C24C23×D1311yescharacteristic
396C2×Q8—524no—
397C2×Q8—528no—
398C2×Q8—524no—
399C2×Q8—528no—
400C2×Q8—524no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
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—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—134no—
424C22×C4—134no—
425C22×C4—134no—
426C22×C4—262no—
427C22×C4—262no—
428C22×C4—262no—
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—262no—
439C22×C4—262no—
440C22×C4—262no—
441C22×C4—262no—
442C22×C4—262no—
443C22×C4—262no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—262no—
448C22×C4—262no—
449C22×C4—262no—
450C22×C4—262no—
451C22×C4—528no—
452C22×C4—524no—
453C22×C4—524no—
454C22×C4—524no—
455C22×C4—528no—
456C22×C4C2×C4×D1312yes—
457C22×C4C2×C4×D1312yes—
458C4⋊C4—44no—
459C4⋊C4—528no—
460C22⋊C4—44no—
461C22⋊C4—48no—
462C22⋊C4—528no—
463C22⋊C4—524no—
464C13#1463511yescharacteristic, complemented, Sylow
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—41no—
485C23—42no—
486C23—42no—
487C23#8814114yes—
488C23C22×C4×D1311yescharacteristic
489C23C2×#612911yescharacteristic
490C23#8825111yescharacteristic
491C23#8825112yes—
492C23#8825111yescharacteristic
493C23#8825112yes—
494C23C2×Q8×D1311yescharacteristic
495C23#8825611yescharacteristic
496C23#8825612yes—
497C23#8825711yescharacteristic
498C23#8825712yes—
499Q8—5216no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
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—5216no—
534C2×C4—522no—
535C2×C4—522no—
536C2×C4—522no—
537C2×C4—522no—
538C2×C4—522no—
539C2×C4—522no—
540C2×C4—522no—
541C2×C4—522no—
542C2×C4—522no—
543C2×C4—522no—
544C2×C4—522no—
545C2×C4—522no—
546C2×C4—522no—
547C2×C4—522no—
548C2×C4—522no—
549C2×C4—522no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4—522no—
553C22—24no—
554C22—28no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—42no—
564C22#1171919111yescharacteristic
565C22C2×#8814111yescharacteristic
566C22#1171922811yescharacteristic
567C22#1171925012yes—
568C22#1171925112yes—
569C22#1171929812yes—
570C22#1171929911yescharacteristic
571C22#1171930711yescharacteristic
572C22#1171932211yescharacteristic
573C22#1171932512yes—
574C22#1171932612yes—
575C22#1171932612yes—
576C22#1171932811yescharacteristic
577C22#1171932812yes—
578C22#1171933912yes—
579C22#1171933911yescharacteristic
580C22#1171935611yescharacteristic
581C22#1171937012yes—
582C22#1171937212yes—
583C22#1171938811yescharacteristic
584C22#1171938812yes—
585C22#1171941712yes—
586C22#1171942511yescharacteristic
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—524no—
592C4—524no—
593C4—524no—
594C4—1042no—
595C2—44no—
596C2#1311756611yescharacteristic
597C2#1311756712yes—
598C2#1311797311yescharacteristic
599C2#1311800512yes—
600C2#1311800611yescharacteristic
601C2#1312291811yescharacteristic
602C2#1312291912yes—
603C2#1312302611yescharacteristic
604C2#1312308811yescharacteristic
605C2#1312309712yes—
606C2#1312309811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC13×#1439C2
socle, cosocleC24×C13C24
prime radicalC25×C13C23
hypercenter#1439D13
hypocenterC13#14635
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. #13113106C2semidirect
3. #13113095C2semidirect
4. #13113091C2semidirect
5. #13113091C2semidirect
31. #11718608C4semidirect
41. C23×Dic13Q8semidirect
50. C22×Dic26C2×C4semidirect
79. #88071C2×C4semidirect
80. #88071C2×C4semidirect
464. C13#14635semidirect

Maximal subgroups

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

subgroupindex
C2×#117186082
#131131062
#131130952
#131130912
#131130792
#131130582
C13×#14392
#1463513