Finite Group Databasebeta

← #16022017

Subgroups — #16022017

Order 3328.

2518 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719240C211yesmaximal, characteristic
2#13117632C214yesmaximal, complemented
3#13117599C214yesmaximal, complemented
4#13117575C211yesmaximal, characteristic, complemented
5#13117559C211yesmaximal, characteristic, complemented
6#13117525C212yesmaximal, complemented
7#13117474C212yesmaximal, complemented
8#13117398C211yesmaximal, characteristic, complemented
9#13117391C211yesmaximal, characteristic, complemented
10#13117357C214yesmaximal, complemented
11#13117348C214yesmaximal, complemented
12#13117342C212yesmaximal, complemented
13C4×#88147C212yesmaximal, complemented
14#117×Dic13C211yesmaximal, characteristic, complemented
15C13×#2314C211yesmaximal, characteristic, complemented
16C2×C13×#117C2211yescharacteristic
17C2×#88184C2212yes—
18C2×#88180C2211yescharacteristic
19C22×C4×Dic13C2211yescharacteristic
20#11719240C4116yescomplemented
21C2×#88148C2214yes—
22C2×#88147C2212yes—
23C2×#88147C2212yes—
24C2×#88147C2214yes—
25C2×#88138C2214yes—
26C2×#88135C2214yes—
27C2×#88128C2211yescharacteristic
28C2×C4×D52C2212yes—
29C2×C42×D13C2214yes—
30C2×#88124C2211yescharacteristic
31C13×#388C2214yes—
32C13×#387C2214yes—
33C13×#384C2211yescharacteristic
34C13×#382C2211yescharacteristic
35C13×#380C2212yes—
36C4×C13×C4⋊C4C2212yes—
37#11718649C2212yes—
38#11718649C2212yes—
39#11718649C2214yes—
40#11718639C2214yes—
41C4⋊C4×Dic13C2212yes—
42C4⋊C4×Dic13C2212yes—
43C4⋊C4×Dic13C2212yes—
44C4⋊C4×Dic13C2212yes—
45#11718634C2212yes—
46#11718634C2212yes—
47#11718633C2214yes—
48#11718619C2214yes—
49#11718618C2214yes—
50#11718617C2214yes—
51C22⋊C4×Dic13C2212yes—
52C22⋊C4×Dic13C2212yes—
53#11718601C2214yes—
54#11718601C2214yes—
55C4×#6103C2214yes—
56#11718593C2211yescharacteristic
57#11718593C2211yescharacteristic
58C42×Dic13C2211yescharacteristic
59C42×Dic13C2211yescharacteristic
60#11718331C2214yes—
61#11718330C2214yes—
62#11718329C2214yes—
63#11718328C2214yes—
64#11718327C2214yes—
65#11718327C2214yes—
66#11718327C2214yes—
67#11718326C2214yes—
68#11718326C2214yes—
69#11718325C2214yes—
70#11718313C2214yes—
71#11718310C2214yes—
72C23×C4×C13C2311yescharacteristic
73C22×C22⋊D13C2314yes—
74C23×Dic13C2311yescharacteristic
75C23×Dic13C2311yescharacteristic
76C22×D52C2312yes—
77C22×D52C2314yes—
78C22×C4×D13C2314yes—
79C22×C4×D13C2314yes—
80C22×C4×D13C2314yes—
81C13×#117C2×C418yescomplemented
82C2×C13×C4⋊C4C2312yes—
83C2×C13×C4⋊C4C2312yes—
84C2×C13×C4⋊C4C2314yes—
85C2×C13×C4⋊C4C2312yes—
86C2×C13×C22⋊C4C2314yes—
87C2×C13×C22⋊C4C2312yes—
88C2×C42×C13C2314yes—
89C2×C42×C13C2311yescharacteristic
90C2×C42×C13C2311yescharacteristic
91C2×C22⋊Dic13C2312yes—
92#88184C2×C4116yescomplemented
93C2×#6103C2314yes—
94C2×#6103C2314yes—
95C2×#6103C2314yes—
96C2×#6103C2314yes—
97C2×#6103C2314yes—
98C2×#6103C2314yes—
99C2×#6103C2314yes—
100C2×#6103C2314yes—
101#88180—24no—
102C2×C4⋊Dic13C2312yes—
103C2×C4⋊Dic13C2312yes—
104C2×#6101C2314yes—
105C2×#6101C2314yes—
106C2×#6101C2314yes—
107C2×C4×Dic13—24no—
108C2×C4×Dic13C2312yes—
109C2×C4×Dic13C2314yes—
110C2×C4×Dic13C2312yes—
111C2×C4×Dic13C2311yescharacteristic
112C2×C4×Dic13C2311yescharacteristic
113C2×C4×Dic13C2311yescharacteristic
114C2×C4×Dic13C2311yescharacteristic
115C2×C4×Dic13C2314yes—
116C2×C4×Dic13C2312yes—
117C2×C4×Dic13C2312yes—
118C2×C4×Dic13C2311yescharacteristic
119C2×C4×Dic13C2311yescharacteristic
120#88148C2×C4132yescomplemented
121#88147—28no—
122#88147—28no—
123#88147—48no—
124#88138C2×C4132yescomplemented
125#88135—48no—
126#88128C2×C418yescomplemented
127C4×D52C2×C4116yescomplemented
128C42×D13—48no—
129#88124C2×C418yescomplemented
130C13×#95C2314yes—
131C13×#95C2314yes—
132C13×#95C2314yes—
133#88071C2314yes—
134#88071C2314yes—
135#88071C2312yes—
136#88071C2314yes—
137#88071C2312yes—
138#88071C2312yes—
139#88071C2314yes—
140#88071C2312yes—
141C4×#540—131nomaximal, Sylow
142C24×C13C2411yescharacteristic
143C23×D13C2414yes—
144C22×C4×C13—22no—
145C22×C4×C13—22no—
146C22×C4×C13C22×C414yes—
147C22×C4×C13C2414yes—
148C22×C4×C13C2414yes—
149C22×C4×C13C2412yes—
150C22×C4×C13C2412yes—
151C22×C4×C13C2411yescharacteristic
152C22×C4×C13C2411yescharacteristic
153C2×C22⋊D13—44no—
154C2×C22⋊D13—44no—
155C2×C22⋊D13C22×C4116yes—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—24no—
159C22×Dic13—42no—
160C22×Dic13C2411yescharacteristic
161C22×Dic13C2412yes—
162C22×Dic13C2411yescharacteristic
163C22×Dic13C2414yes—
164C22×Dic13C2411yescharacteristic
165C22×Dic13C2412yes—
166C22×Dic13C2411yescharacteristic
167C2×D52—24no—
168C2×D52—24no—
169C2×D52—44no—
170C2×D52—44no—
171C2×D52C22×C418yes—
172C2×D52C22×C4116yes—
173C2×C4×D13—28no—
174C2×C4×D13—28no—
175C2×C4×D13—28no—
176C2×C4×D13—28no—
177C2×C4×D13—44no—
178C2×C4×D13—44no—
179C2×C4×D13—44no—
180C2×C4×D13—44no—
181C2×C4×D13C22×C4116yes—
182C13×C4⋊C4—24no—
183C13×C4⋊C4—24no—
184C13×C4⋊C4—44no—
185C13×C4⋊C4C22×C418yes—
186C13×C22⋊C4—44no—
187C13×C22⋊C4C22×C418yes—
188C42×C13—44no—
189C42×C13C22×C414yes—
190C42×C13C22×C414yes—
191C22⋊Dic13—24no—
192#6103—28no—
193#6103—28no—
194#6103—44no—
195#6103—44no—
196#6103—44no—
197#6103—44no—
198#6103C22×C4116yes—
199#6103C22×C4116yes—
200C4⋊Dic13—24no—
201C4⋊Dic13C22×C418yes—
202#6101—44no—
203#6101—44no—
204#6101C22×C4116yes—
205C4×Dic13—22no—
206C4×Dic13—22no—
207C4×Dic13—22no—
208C4×Dic13—22no—
209C4×Dic13—24no—
210C4×Dic13—28no—
211C4×Dic13—24no—
212C4×Dic13—44no—
213C4×Dic13—42no—
214C4×Dic13—42no—
215C4×Dic13#4114yes—
216C4×Dic13#4114yes—
217C2×#540—131no—
218#2335—134no—
219#2333—134no—
220#2330—131no—
221#2328—131no—
222#2324—132no—
223#2319—132no—
224#2314D1311yescharacteristic, complemented
225C4×#127—131no—
226C4×#125—131no—
227C4×#123—134no—
228C4×#121—134no—
229C4×#121—132no—
230C42×D4—132no—
231C4×#117—131no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—22no—
235C23×C13—22no—
236C23×C13C23×C412yes—
237C23×C13C2511yescharacteristic
238C22×D13—24no—
239C22×D13—24no—
240C22×D13—24no—
241C22×D13—24no—
242C22×D13—24no—
243C22×D13—24no—
244C22×D13C23×C418yes—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—22no—
248C2×C4×C13—22no—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—28no—
252C2×C4×C13—24no—
253C2×C4×C13—24no—
254C2×C4×C13—24no—
255C2×C4×C13—24no—
256C2×C4×C13—24no—
257C2×C4×C13—24no—
258C2×C4×C13C23×C414yes—
259C2×C4×C13C23×C412yes—
260C2×C4×C13C23×C414yes—
261C2×C4×C13C23×C412yes—
262C2×C4×C13C2×#4112yes—
263C2×C4×C13C2×#4112yes—
264C22⋊D13—416no—
265C2×Dic13—22no—
266C2×Dic13—24no—
267C2×Dic13—22no—
268C2×Dic13—22no—
269C2×Dic13—24no—
270C2×Dic13—21no—
271C2×Dic13—21no—
272C2×Dic13—22no—
273C2×Dic13—22no—
274C2×Dic13—22no—
275C2×Dic13—22no—
276C2×Dic13—22no—
277C2×Dic13—44no—
278C2×Dic13—44no—
279C2×Dic13C23×C418yes—
280C2×Dic13C2×#4112yes—
281C2×Dic13C2×#4114yes—
282C2×Dic13C2×#4112yes—
283C2×Dic13H(5,𝔽2)14yes—
284D52—216no—
285D52—416no—
286C4×D13—416no—
287C4×D13—416no—
288C4×D13—416no—
289#540—1316no—
290C2×#127—131no—
291C2×#125—131no—
292C2×#123—134no—
293C2×#121—132no—
294C2×#121—134no—
295C2×C4×D4—132no—
296C2×C4×D4—132no—
297C2×C4×D4—134no—
298C2×C4×D4—134no—
299C2×C4×D4—132no—
300C2×#117—131no—
301C2×#117C2×D1311yescharacteristic
302C22×C42—131no—
303C22×C42—134no—
304#390—132no—
305#390—134no—
306#390—132no—
307#390—134no—
308#389—132no—
309#389—132no—
310#389—134no—
311#388—134no—
312#388—134no—
313#388—134no—
314#388—134no—
315#388C2×D1314yes—
316#387—134no—
317#387—134no—
318#387C2×D1314yes—
319#386—134no—
320#386—134no—
321#386—134no—
322#386—134no—
323#385—134no—
324#384C2×D1311yescharacteristic
325#382C2×D1311yescharacteristic
326#381—134no—
327#380—134no—
328#380C2×D1312yes—
329C4×C4⋊C4—132no—
330C4×C4⋊C4—132no—
331C4×C4⋊C4—132no—
332C4×C4⋊C4—132no—
333C4×C4⋊C4—134no—
334C4×C4⋊C4C2×D1312yes—
335C4×C22⋊C4—132no—
336C4×C22⋊C4—132no—
337C4×C22⋊C4—134no—
338C4×C22⋊C4—134no—
339C4×C22⋊C4—134no—
340C4×C22⋊C4—134no—
341#376—131no—
342#376—131no—
343C43—131no—
344C43—131no—
345C22×C13—22no—
346C22×C13—24no—
347C22×C13—41no—
348C22×C13—41no—
349C22×C13—42no—
350C22×C13C24×C411yescharacteristic
351C22×C13C22×#4111yescharacteristic
352C22×C13C22×#4111yescharacteristic
353C22×C13C2×H(5,𝔽2)12yes—
354C22×C13#58512yes—
355C2×D13—28no—
356C2×D13—28no—
357C2×D13—28no—
358C2×D13—44no—
359C2×D13—44no—
360C2×D13—44no—
361C2×D13—44no—
362C4×C13—22no—
363C4×C13—24no—
364C4×C13—24no—
365C4×C13—44no—
366C4×C13—44no—
367C4×C13C4×#4114yes—
368Dic13—28no—
369Dic13—22no—
370Dic13—24no—
371Dic13—41no—
372Dic13—41no—
373Dic13—42no—
374Dic13#54014yescomplemented
375C22×D4—134no—
376C22×D4—132no—
377C22×D4—134no—
378C23×C4—134no—
379C23×C4—131no—
380C23×C4—131no—
381C23×C4—134no—
382C23×C4—134no—
383C23×C4C22×D1311yescharacteristic
384#127—138no—
385#125—138no—
386#123—1332no—
387#121—1316no—
388#121—1332no—
389C4×D4—1316no—
390C4×D4—268no—
391C4×D4—268no—
392C4×D4—528no—
393C4×D4—528no—
394#117—264no—
395#117C4×D1318yescomplemented
396C2×C4⋊C4—132no—
397C2×C4⋊C4—134no—
398C2×C4⋊C4—132no—
399C2×C4⋊C4—134no—
400C2×C4⋊C4—134no—
401C2×C4⋊C4C22×D1312yes—
402C2×C4⋊C4C22×D1312yes—
403C2×C4⋊C4C22×D1314yes—
404C2×C4⋊C4C22×D1312yes—
405C2×C22⋊C4—134no—
406C2×C22⋊C4—134no—
407C2×C22⋊C4—132no—
408C2×C22⋊C4—134no—
409C2×C22⋊C4—134no—
410C2×C22⋊C4—134no—
411C2×C22⋊C4—134no—
412C2×C22⋊C4—134no—
413C2×C22⋊C4—134no—
414C2×C22⋊C4C22×D1314yes—
415C2×C22⋊C4C22×D1312yes—
416C2×C42—131no—
417C2×C42—131no—
418C2×C42—131no—
419C2×C42—132no—
420C2×C42—134no—
421C2×C42—132no—
422C2×C42—131no—
423C2×C42—131no—
424C2×C42—131no—
425C2×C42—132no—
426C2×C42—132no—
427C2×C42—134no—
428C2×C42—264no—
429C2×C42—528no—
430C2×C42C22×D1314yes—
431C2×C42C22×D1311yescharacteristic
432C2×C42C22×D1311yescharacteristic
433#95—132no—
434#95—132no—
435#95—132no—
436#95—132no—
437#95—134no—
438#95—134no—
439#95—134no—
440#95—134no—
441#95C22×D1314yes—
442#95C22×D1314yes—
443#95C22×D1314yes—
444C2×C13—42no—
445C2×C13C2×C4×#4111yescharacteristic
446C2×C13C4×H(5,𝔽2)12yes—
447C2×C13C2×#54011yescharacteristic
448C2×C13#345511yescharacteristic
449C2×C13#346012yes—
450D13—48no—
451C24—134no—
452C24C23×D1311yescharacteristic
453C2×D4—1316no—
454C2×D4—138no—
455C2×D4—1316no—
456C2×D4—264no—
457C2×D4—264no—
458C2×D4—524no—
459C2×D4—524no—
460C2×D4—524no—
461C2×D4—524no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—131no—
465C22×C4—131no—
466C22×C4—132no—
467C22×C4—134no—
468C22×C4—1316no—
469C22×C4—131no—
470C22×C4—132no—
471C22×C4—131no—
472C22×C4—268no—
473C22×C4—268no—
474C22×C4—268no—
475C22×C4—268no—
476C22×C4—262no—
477C22×C4—262no—
478C22×C4—264no—
479C22×C4—524no—
480C22×C4—524no—
481C22×C4—524no—
482C22×C4—524no—
483C22×C4—522no—
484C22×C4C2×C4×D1314yes—
485C22×C4C23×D1314yes—
486C22×C4C23×D1314yes—
487C22×C4C23×D1312yes—
488C22×C4C23×D1312yes—
489C22×C4C23×D1311yescharacteristic
490C22×C4C23×D1311yescharacteristic
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—44no—
494C4⋊C4—138no—
495C4⋊C4—1316no—
496C4⋊C4—264no—
497C4⋊C4—524no—
498C4⋊C4—524no—
499C4⋊C4C2×C4×D1318yes—
500C22⋊C4—44no—
501C22⋊C4—1316no—
502C22⋊C4—1316no—
503C22⋊C4—268no—
504C22⋊C4—268no—
505C22⋊C4—264no—
506C22⋊C4—524no—
507C22⋊C4—524no—
508C22⋊C4—524no—
509C22⋊C4—524no—
510C22⋊C4C2×C4×D1318yes—
511C42—44no—
512C42—134no—
513C42—134no—
514C42—262no—
515C42—262no—
516C42—264no—
517C42—268no—
518C42—264no—
519C42—262no—
520C42—262no—
521C42—524no—
522C42—522no—
523C42—522no—
524C42C2×C4×D1314yes—
525C42C2×C4×D1314yes—
526C13C4×#54011yescharacteristic, complemented, Sylow
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—138no—
532C23—264no—
533C23—264no—
534C23—264no—
535C23—264no—
536C23—264no—
537C23—264no—
538C23C22×C4×D1312yes—
539C23C24×D1311yescharacteristic
540D4—2616no—
541D4—5216no—
542D4—5216no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—132no—
557C2×C4—134no—
558C2×C4—138no—
559C2×C4—132no—
560C2×C4—134no—
561C2×C4—262no—
562C2×C4—264no—
563C2×C4—262no—
564C2×C4—262no—
565C2×C4—264no—
566C2×C4—261no—
567C2×C4—261no—
568C2×C4—262no—
569C2×C4—262no—
570C2×C4—262no—
571C2×C4—262no—
572C2×C4—262no—
573C2×C4—5216no—
574C2×C4—5216no—
575C2×C4—5216no—
576C2×C4—524no—
577C2×C4—524no—
578C2×C4C22×C4×D1314yes—
579C2×C4C22×C4×D1314yes—
580C2×C4C22×C4×D1312yes—
581C2×C4C22×C4×D1312yes—
582C2×C4#41×D1312yes—
583C2×C4#41×D1312yes—
584C22—22no—
585C22—24no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—268no—
590C22—268no—
591C22—268no—
592C22—524no—
593C22—524no—
594C22—524no—
595C22—524no—
596C22C23×C4×D1311yescharacteristic
597C22C2×#41×D1311yescharacteristic
598C22C2×#41×D1311yescharacteristic
599C22D13×H(5,𝔽2)12yes—
600C22#1171965212yes—
601C4—22no—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—44no—
606C4—134no—
607C4—262no—
608C4—264no—
609C4—268no—
610C4—521no—
611C4—522no—
612C4—521no—
613C4#1171922714yes—
614C2—42no—
615C2—528no—
616C2C2×#1171922711yescharacteristic
617C2#1312804212yes—
618C2D13×#54011yescharacteristic
619C2#1312822711yescharacteristic
620C2#1312825612yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2314C2
socle, cosocleC23×C13C25
prime radicalC2×#11719240C2
hypercenter#2314D13
hypocenterC13C4×#540
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #13117632C2semidirect
3. #13117599C2semidirect
4. #13117575C2semidirect
5. #13117559C2semidirect
6. #13117525C2semidirect
7. #13117474C2semidirect
8. #13117398C2semidirect
9. #13117391C2semidirect
10. #13117357C2semidirect
11. #13117348C2semidirect
12. #13117342C2semidirect
13. C4×#88147C2semidirect
14. #117×Dic13C2semidirect
15. C13×#2314C2semidirect
20. #11719240C4semidirect
81. C13×#117C2×C4semidirect
92. #88184C2×C4semidirect
120. #88148C2×C4semidirect
124. #88138C2×C4semidirect
126. #88128C2×C4semidirect
127. C4×D52C2×C4semidirect
129. #88124C2×C4semidirect
224. #2314D13semidirect
374. Dic13#540semidirect
395. #117C4×D13semidirect
526. C13C4×#540semidirect

Maximal subgroups

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

subgroupindex
C2×#117192402
#131176322
#131175992
#131175752
#131175592
#131175252
#131174742
#131173982
#131173912
#131173572
#131173482
#131173422
C4×#881472
#117×Dic132
C13×#23142
C4×#54013