Finite Group Databasebeta

← #15878565

Subgroups — #15878565

Order 3328.

1530 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719256C211yesmaximal, characteristic
2C2×#11718673C211yesmaximal, characteristic, complemented
3C2×#11718627C212yesmaximal, complemented
4#13115759C211yesmaximal, characteristic, complemented
5#13115637C211yesmaximal, characteristic, complemented
6#13115267C212yesmaximal, complemented
7#13115143C212yesmaximal, complemented
8#13114833C211yesmaximal, characteristic, complemented
9#13114774C212yesmaximal, complemented
10C4×#88049C211yesmaximal, characteristic, complemented
11D4×C13×C4⋊C4C211yesmaximal, characteristic, complemented
12C22×#6127C2212yes—
13C2×C4×D4×C13—21no—
14C2×C4×D4×C13C2211yescharacteristic
15C22×C13×C4⋊C4C2212yes—
16C2×#88192—21no—
17C2×#88178C2212yes—
18C22×#6099C2212yes—
19#11719256—24no—
20#11719256C418yescomplemented
21C2×C4⋊C4×D13—21no—
22C2×#88137—22no—
23C2×C4×D52C2211yescharacteristic
24C2×#88123C2211yescharacteristic
25C13×#389—21no—
26C13×#385—22no—
27C13×#384C2211yescharacteristic
28C13×#380C2212yes—
29C4×C13×C4⋊C4C2211yescharacteristic
30C2×C22⋊#1055C2212yes—
31#11718673—24no—
32C2×#88050C2212yes—
33C2×#88050C2212yes—
34C2×#88050C2212yes—
35C2×#88050C2212yes—
36#11718627—28no—
37C2×#88049C2212yes—
38C2×#88049C2211yescharacteristic
39C2×#88049C2211yescharacteristic
40C2×C4⋊#1055C2211yescharacteristic
41C2×C4×#1055C2211yescharacteristic
42#11718121C2211yescharacteristic
43#11718121C2212yes—
44#11718121C2211yescharacteristic
45C22×D4×C13D411yescharacteristic
46C23×C4×C13—22no—
47C23×C4×C13C2312yes—
48C22×C22⋊D13—22no—
49C2×#6127—24no—
50C2×#6127—24no—
51C2×#6127—24no—
52C2×#6127—42no—
53C2×#6127—42no—
54C2×#6127C2×C418yescomplemented
55C22×D52C2311yescharacteristic
56C22×C4×D13—21no—
57C22×C4×D13—22no—
58C22×Dic26C2312yes—
59C4×D4×C13—44no—
60C4×D4×C13C2×C414yes—
61C4×D4×C13D414yes—
62C2×C13×C4⋊C4—24no—
63C2×C13×C4⋊C4—21no—
64C2×C13×C4⋊C4—22no—
65C2×C13×C4⋊C4—21no—
66C2×C13×C4⋊C4—42no—
67C2×C13×C4⋊C4D411yescharacteristic
68C2×C13×C4⋊C4C2311yescharacteristic
69C2×C13×C4⋊C4C2312yes—
70C2×C13×C4⋊C4C2311yescharacteristic
71C2×C13×C22⋊C4—22no—
72C2×C13×C22⋊C4D412yes—
73C2×C42×C13—21no—
74C2×C42×C13C2311yescharacteristic
75C2×C22⋊Dic13—22no—
76#88192—28no—
77C2×#6103—21no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13C2311yescharacteristic
81#88178—24no—
82#88178C2×C418yescomplemented
83C2×#6101—22no—
84C2×#6099—24no—
85C2×#6099—24no—
86C2×#6099—42no—
87C22×#1055C2312yes—
88C22×#1055C2311yescharacteristic
89C22×#1055C2311yescharacteristic
90C4⋊C4×D13—28no—
91#88137—216no—
92C4×D52—22no—
93C4×D52C2×C414yescomplemented
94#88123—22no—
95#88123C2×C414yescomplemented
96C13×#95—21no—
97C13×#95—22no—
98C22⋊#1055—24no—
99#88050—24no—
100#88050—24no—
101#88050—42no—
102#88050—42no—
103#88049—24no—
104#88049—22no—
105#88049D414yescomplemented
106C4⋊#1055—22no—
107C4×#1055—22no—
108#22431—131nomaximal, Sylow
109C24×C13C2×D412yes—
110C23×D13—21no—
111C2×D4×C13—22no—
112C2×D4×C13—22no—
113C2×D4×C13C22⋊C414yes—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—22no—
118C22×C4×C13—21no—
119C22×C4×C13—21no—
120C22×C4×C13—22no—
121C22×C4×C13—44no—
122C22×C4×C13—44no—
123C22×C4×C13—82no—
124C22×C4×C13C22×C414yes—
125C22×C4×C13C2×D414yes—
126C22×C4×C13C2×D412yes—
127C22×C4×C13C2×D411yescharacteristic
128C22×C4×C13C2×D411yescharacteristic
129C22×C4×C13C2411yescharacteristic
130C2×C22⋊D13—28no—
131C2×C22⋊D13—44no—
132C2×C22⋊D13—44no—
133C22×Dic13—21no—
134C22×Dic13—22no—
135#6127—216no—
136#6127—48no—
137#6127—48no—
138#6127—48no—
139C2×D52—21no—
140C2×D52—21no—
141C2×D52—21no—
142C2×D52C22×C412yes—
143C2×D52C2×D412yes—
144C2×D52#4112yes—
145C2×C4×D13—24no—
146C2×C4×D13—28no—
147C2×C4×D13—44no—
148C2×C4×D13—44no—
149C2×C4×D13—42no—
150C2×C4×D13—42no—
151C2×Dic26—22no—
152C2×Dic26—22no—
153C2×Dic26—22no—
154C2×Dic26—22no—
155C2×Dic26—22no—
156C2×Dic26C22×C414yes—
157C13×C4⋊C4—21no—
158C13×C4⋊C4—22no—
159C13×C4⋊C4—21no—
160C13×C4⋊C4—22no—
161C13×C4⋊C4—21no—
162C13×C4⋊C4—44no—
163C13×C4⋊C4—42no—
164C13×C4⋊C4—81no—
165C13×C4⋊C4C22⋊C414yes—
166C13×C4⋊C4C2×D412yes—
167C13×C22⋊C4—44no—
168C13×C22⋊C4C22⋊C418yes—
169C42×C13—42no—
170C42×C13C22×C412yes—
171C42×C13C2×D412yes—
172C22⋊Dic13—28no—
173#6103—24no—
174C4⋊Dic13—24no—
175C4⋊Dic13—24no—
176C4⋊Dic13—21no—
177C4⋊Dic13C22×C412yes—
178#6101—28no—
179#6099—48no—
180C2×#1055—21no—
181C2×#1055—21no—
182C2×#1055—21no—
183C2×#1055—22no—
184C2×#1055—22no—
185C2×#1055—21no—
186C2×#1055—22no—
187C2×#1055—21no—
188C2×#1055C2×D412yes—
189C2×#547—131no—
190C2×#436—131no—
191C2×#417—132no—
192D4×C4⋊C4D1311yescharacteristic, complemented
193#1981—131no—
194#1956—131no—
195#1894—132no—
196#1872—132no—
197#1802—131no—
198#1788—132no—
199C4×#102—131no—
200C23×C13—22no—
201C23×C13—22no—
202C23×C13—22no—
203C23×C13—22no—
204C23×C13—22no—
205C23×C13—22no—
206C23×C13C2×C22⋊C414yes—
207C23×C13C22×D411yescharacteristic
208C22×D13—22no—
209C22×D13—22no—
210C22×D13—22no—
211C22×D13—41no—
212C22×D13—41no—
213C22×D13—41no—
214C22×D13—41no—
215D4×C13—28no—
216C2×C4×C13—22no—
217C2×C4×C13—22no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—24no—
221C2×C4×C13—24no—
222C2×C4×C13—24no—
223C2×C4×C13—24no—
224C2×C4×C13—22no—
225C2×C4×C13—41no—
226C2×C4×C13—41no—
227C2×C4×C13—42no—
228C2×C4×C13—41no—
229C2×C4×C13—41no—
230C2×C4×C13—42no—
231C2×C4×C13—42no—
232C2×C4×C13—41no—
233C2×C4×C13—88no—
234C2×C4×C13C2×C22⋊C414yes—
235C2×C4×C13C2×C22⋊C412yes—
236C2×C4×C13C2×C22⋊C412yes—
237C2×C4×C13AGL(1,ℤ/8)12yes—
238C2×C4×C13#13712yes—
239C2×C4×C13C23×C411yescharacteristic
240C2×C4×C13C22×D411yescharacteristic
241C2×C4×C13C22×D411yescharacteristic
242C2×C4×C13C2×#4111yescharacteristic
243C2×C4×C13H(5,𝔽2)11yescharacteristic
244C2×C4×C13H(5,𝔽2)11yescharacteristic
245C22⋊D13—416no—
246C2×Dic13—22no—
247C2×Dic13—24no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250C2×Dic13—41no—
251C2×Dic13—41no—
252D52—22no—
253D52—22no—
254D52—22no—
255D52C4×D414yescomplemented
256C4×D13—416no—
257C4×D13—48no—
258Dic26—24no—
259Dic26—24no—
260Dic26—42no—
261Dic26—42no—
262#1055—22no—
263#1055—42no—
264#1055—41no—
265C22×#41—132no—
266C22×M16—132no—
267#547—138no—
268#547—264no—
269C2×C4⋊Q8—131no—
270C2×#123—262no—
271C2×C22⋊Q8—132no—
272C2×#121—261no—
273C2×C4×D4—21no—
274C2×C4×D4—131no—
275C2×C4×D4C2×D1311yescharacteristic
276C22×C4⋊C4—261no—
277C22×C4⋊C4C2×D1312yes—
278#436—264no—
279C2×C4⋊C8—131no—
280#417—268no—
281C2×#103—132no—
282C2×#103—132no—
283C2×#103—132no—
284C2×#103—132no—
285C2×#102—131no—
286C2×#102—132no—
287C2×#102—131no—
288C2×C22⋊C8—132no—
289C2×C4×C8—131no—
290#389—21no—
291#385—22no—
292#384C2×D1311yescharacteristic
293#380C2×D1312yes—
294C4×C4⋊C4C2×D1311yescharacteristic
295#340—132no—
296#340—131no—
297#340—131no—
298C22×C13—24no—
299C22×C13—24no—
300C22×C13—24no—
301C22×C13—42no—
302C22×C13—42no—
303C22×C13—42no—
304C22×C13—42no—
305C22×C13C22×C22⋊C411yescharacteristic
306C22×C13D4211yescharacteristic
307C22×C13#54611yescharacteristic
308C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
309C22×C13C2×#13711yescharacteristic
310C22×C13#57711yescharacteristic
311C22×C13#57811yescharacteristic
312C2×D13—24no—
313C2×D13—42no—
314C2×D13—42no—
315C2×D13—42no—
316C2×D13—42no—
317C2×D13—42no—
318C2×D13—42no—
319C4×C13—24no—
320C4×C13—22no—
321C4×C13—42no—
322C4×C13—81no—
323C4×C13—82no—
324C4×C13#41714yescomplemented
325C4×C13C2×C4×D411yescharacteristic
326C4×C13#51811yescharacteristic
327C4×C13D4211yescharacteristic
328C4×C13#54611yescharacteristic
329Dic13—44no—
330Dic13—42no—
331C2×#41—138no—
332C2×#41—264no—
333C2×#41—264no—
334C2×#41—264no—
335C2×#41—522no—
336C2×#41—522no—
337C22×Q8—132no—
338C22×D4—131no—
339C22×D4—262no—
340C22×D4C22⋊D1311yescharacteristic
341C23×C4—22no—
342C23×C4—261no—
343C23×C4—262no—
344C23×C4C22×D1312yes—
345C2×M16—264no—
346C2×M16—264no—
347C2×M16—522no—
348C22×C8—131no—
349C22×C8—132no—
350C22×C8—131no—
351C4⋊Q8—134no—
352C4⋊Q8—262no—
353#123—2616no—
354C22⋊Q8—138no—
355C22⋊Q8—264no—
356#121—268no—
357C4×D4—44no—
358C4×D4—134no—
359C4×D4—262no—
360C4×D4C4×D1314yes—
361C4×D4D5214yes—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—42no—
367C2×C4⋊C4—131no—
368C2×C4⋊C4—261no—
369C2×C4⋊C4—262no—
370C2×C4⋊C4—261no—
371C2×C4⋊C4—268no—
372C2×C4⋊C4C22⋊D1311yescharacteristic
373C2×C4⋊C4C22×D1312yes—
374C2×C4⋊C4C22×D1311yescharacteristic
375C2×C4⋊C4C22×D1311yescharacteristic
376C2×C22⋊C4—22no—
377C2×C22⋊C4—262no—
378C2×C22⋊C4—261no—
379C2×C22⋊C4C22⋊D1312yes—
380C2×C42—21no—
381C2×C42C22×D1311yescharacteristic
382C4⋊C8—262no—
383#103—264no—
384#103—264no—
385#103—522no—
386#103—522no—
387#102—134no—
388#102—264no—
389#102—262no—
390C22⋊C8—264no—
391C4×C8—262no—
392#95—21no—
393#95—22no—
394C2×C13—44no—
395C2×C13D4×C22⋊C411yescharacteristic
396C2×C13C2×#41711yescharacteristic
397C2×C13#289811yescharacteristic
398C2×C13#327211yescharacteristic
399C2×C13#327611yescharacteristic
400C2×C13#333011yescharacteristic
401C2×C13#333111yescharacteristic
402D13—44no—
403C24—261no—
404C24C2×C22⋊D1312yes—
405#41—2616no—
406#41—528no—
407#41—528no—
408#41—528no—
409C2×Q8—134no—
410C2×Q8—262no—
411C2×Q8—262no—
412C2×Q8—262no—
413C2×Q8—262no—
414C2×Q8—262no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—132no—
418C2×D4—132no—
419C2×D4—132no—
420C2×D4—268no—
421C2×D4—261no—
422C2×D4—261no—
423C2×D4—261no—
424C2×D4—524no—
425C2×D4—524no—
426C2×D4#610314yes—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—82no—
437C22×C4—261no—
438C22×C4—262no—
439C22×C4—264no—
440C22×C4—268no—
441C22×C4—524no—
442C22×C4—524no—
443C22×C4—522no—
444C22×C4—522no—
445C22×C4C2×C4×D1314yes—
446C22×C4C2×D5214yes—
447C22×C4C2×C22⋊D1311yescharacteristic
448C22×C4C2×C22⋊D1312yes—
449C22×C4C2×C22⋊D1311yescharacteristic
450C22×C4C23×D1311yescharacteristic
451M16—528no—
452C2×C8—132no—
453C2×C8—261no—
454C2×C8—261no—
455C2×C8—261no—
456C2×C8—262no—
457C2×C8—262no—
458C2×C8—261no—
459C2×C8—261no—
460C2×C8—262no—
461C4⋊C4—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—132no—
470C4⋊C4—264no—
471C4⋊C4—268no—
472C4⋊C4—264no—
473C4⋊C4—261no—
474C4⋊C4#610314yes—
475C4⋊C4D4×D1312yes—
476C22⋊C4—44no—
477C22⋊C4—268no—
478C22⋊C4—264no—
479C22⋊C4#610318yes—
480C42—42no—
481C42C2×C4×D1312yes—
482C42C2×D5212yes—
483C13#2243111yescharacteristic, complemented, Sylow
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—262no—
491C23—262no—
492C23—262no—
493C23—521no—
494C23—521no—
495C23—521no—
496C23—521no—
497C23C2×#610314yes—
498C23C22×C22⋊D1311yescharacteristic
499Q8—264no—
500Q8—264no—
501Q8—522no—
502Q8—522no—
503D4—28no—
504D4—134no—
505D4—262no—
506D4—262no—
507D4—262no—
508D4—5216no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—88no—
527C2×C4—262no—
528C2×C4—264no—
529C2×C4—5216no—
530C2×C4—521no—
531C2×C4—522no—
532C2×C4—521no—
533C2×C4—522no—
534C2×C4—528no—
535C2×C4C2×#610312yes—
536C2×C4C2×#610314yes—
537C2×C4C2×#610312yes—
538C2×C4#8818612yes—
539C2×C4#8819612yes—
540C2×C4C22×C4×D1311yescharacteristic
541C2×C4C22×D5211yescharacteristic
542C2×C4C2×D4×D1311yescharacteristic
543C2×C4#8825111yescharacteristic
544C2×C4#41×D1311yescharacteristic
545C2×C4#8825611yescharacteristic
546C8—262no—
547C8—522no—
548C8—521no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—264no—
557C22—522no—
558C22—522no—
559C22—522no—
560C22—522no—
561C22—522no—
562C22—522no—
563C22C22×#610311yescharacteristic
564C22C2×#8818611yescharacteristic
565C22D4×C22⋊D1311yescharacteristic
566C22C2×#8819611yescharacteristic
567C22#1171953411yescharacteristic
568C22#1171954111yescharacteristic
569C22#1171954311yescharacteristic
570C4—22no—
571C4—24no—
572C4—42no—
573C4—82no—
574C4—81no—
575C4—522no—
576C4—524no—
577C4#1171862714yescomplemented
578C4C4×D4×D1311yescharacteristic
579C4#1171925011yescharacteristic
580C4D4×D5211yescharacteristic
581C4#1171925911yescharacteristic
582C2—44no—
583C2—524no—
584C2D4×#610311yescharacteristic
585C2C2×#1171862711yescharacteristic
586C2#1312336611yescharacteristic
587C2#1312643211yescharacteristic
588C2#1312648211yescharacteristic
589C2#1312713511yescharacteristic
590C2#1312714611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingD4×C13×C4⋊C4C2
socleC23×C13C22×D4
prime radicalC2×#11719256C2
cosocleC22×C4×C13C24
hypercenterD4×C4⋊C4D13
hypocenterC13#22431
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11718673C2semidirect
3. C2×#11718627C2semidirect
4. #13115759C2semidirect
5. #13115637C2semidirect
6. #13115267C2semidirect
7. #13115143C2semidirect
8. #13114833C2semidirect
9. #13114774C2semidirect
10. C4×#88049C2semidirect
11. D4×C13×C4⋊C4C2semidirect
20. #11719256C4semidirect
54. C2×#6127C2×C4semidirect
82. #88178C2×C4semidirect
93. C4×D52C2×C4semidirect
95. #88123C2×C4semidirect
105. #88049D4semidirect
192. D4×C4⋊C4D13semidirect
255. D52C4×D4semidirect
324. C4×C13#417semidirect
483. C13#22431semidirect
577. C4#11718627semidirect

Maximal subgroups

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

subgroupindex
C2×#117192562
C2×#117186732
C2×#117186272
#131157592
#131156372
#131152672
#131151432
#131148332
#131147742
C4×#880492
D4×C13×C4⋊C42
#2243113