Finite Group Databasebeta

← #15872021

Subgroups — #15872021

Order 3328.

1326 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719530C211yesmaximal, characteristic
2C2×#11718895C211yesmaximal, characteristic
3#13115267C212yesmaximal, complemented
4#13115257C212yesmaximal, complemented
5#13115148C212yesmaximal, complemented
6#13115143C212yesmaximal, complemented
7#13114928C212yesmaximal, complemented
8#13114770C212yesmaximal, complemented
9C13×#2311C211yesmaximal, characteristic
10C22×C13×#41C2211yescharacteristic
11C23×Dic26C2211yescharacteristic
12C2×C13×#117C2211yescharacteristic
13C22×C13×C4⋊C4C2211yescharacteristic
14#11719530—24no—
15#11719530C418yescomplemented
16C2×#88198—21no—
17C2×#88190—21no—
18C2×#88189—21no—
19C2×#88189—21no—
20C2×#88180C2211yescharacteristic
21C2×#88178C2212yes—
22C2×#88178C2212yes—
23C22×#6099C2211yescharacteristic
24C23×#1055C2211yescharacteristic
25C13×#391—21no—
26C13×#390—21no—
27C13×#386—21no—
28C13×#386—21no—
29C13×#380C2212yes—
30C13×#380C2212yes—
31#11718895—24no—
32C2×C22⋊#1055C2212yes—
33C2×C22⋊#1055C2212yes—
34C2×#88050C2212yes—
35C2×#88050C2212yes—
36C2×#88050C2212yes—
37C2×#88050C2212yes—
38#11718121C2212yes—
39#11718121C2212yes—
40#11718121C2212yes—
41#11718121C2212yes—
42C2×C13×#41—41no—
43C2×C13×#41—41no—
44C2×C13×#41—41no—
45C2×C13×#41—41no—
46C2×C13×#41C2×C414yes—
47C2×C13×#41D414yes—
48C22×Q8×C13D411yescharacteristic
49C22×D4×C13D411yescharacteristic
50C22×D4×C13D411yescharacteristic
51C22×D4×C13D411yescharacteristic
52C23×C4×C13—21no—
53C23×C4×C13C2311yescharacteristic
54C23×C4×C13C2311yescharacteristic
55C23×C4×C13C2311yescharacteristic
56C23×Dic13—21no—
57C22×Dic26—21no—
58C22×Dic26—22no—
59C22×Dic26—41no—
60C22×Dic26—41no—
61C22×Dic26—41no—
62C22×Dic26—41no—
63C22×Dic26C2×C414yescomplemented
64C22×Dic26C2312yes—
65C13×#117—41no—
66C13×#117—41no—
67C2×C13×C4⋊C4—21no—
68C2×C13×C4⋊C4—41no—
69C2×C13×C4⋊C4—41no—
70C2×C13×C4⋊C4C2312yes—
71C2×C13×C4⋊C4C2312yes—
72C2×C13×C22⋊C4—21no—
73C2×C13×C22⋊C4—22no—
74C2×C13×C22⋊C4—22no—
75C2×C42×C13—21no—
76C2×C22⋊Dic13—22no—
77C2×C22⋊Dic13—22no—
78C2×C22⋊Dic13—21no—
79#88198—28no—
80#88190—28no—
81#88189—28no—
82#88189—28no—
83#88180—22no—
84#88180C2×C414yescomplemented
85C2×C4⋊Dic13C2312yes—
86#88178—24no—
87#88178—24no—
88#88178C2×C418yes—
89#88178C2×C418yescomplemented
90C2×#6101—22no—
91C2×#6101—22no—
92C2×C4×Dic13—21no—
93C2×#6099—22no—
94C2×#6099—41no—
95C2×#6099—41no—
96C22×#1055—22no—
97C22×#1055—41no—
98C22×#1055—41no—
99C22×#1055C2312yes—
100C22×#1055C2312yes—
101C13×#95—22no—
102C13×#95—22no—
103C22⋊#1055—24no—
104C22⋊#1055—24no—
105#88050—42no—
106#88050—42no—
107#88050—42no—
108#88050—42no—
109#21771—131nomaximal, Sylow
110C24×C13C2×D411yescharacteristic
111C24×C13C2×D411yescharacteristic
112C24×C13C2×D411yescharacteristic
113C13×#41—44no—
114C13×#41—44no—
115C13×#41—44no—
116C13×#41—44no—
117C2×Q8×C13—41no—
118C2×Q8×C13—41no—
119C2×Q8×C13C22⋊C414yes—
120C2×D4×C13—41no—
121C2×D4×C13—41no—
122C2×D4×C13—41no—
123C2×D4×C13—41no—
124C2×D4×C13—41no—
125C2×D4×C13—41no—
126C2×D4×C13C22⋊C414yes—
127C2×D4×C13C22⋊C414yes—
128C2×D4×C13C22⋊C414yes—
129C22×C4×C13—21no—
130C22×C4×C13—21no—
131C22×C4×C13—21no—
132C22×C4×C13—21no—
133C22×C4×C13—21no—
134C22×C4×C13—21no—
135C22×C4×C13—21no—
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13—21no—
141C22×C4×C13—22no—
142C22×C4×C13—22no—
143C22×C4×C13—42no—
144C22×C4×C13—82no—
145C22×C4×C13C22×C412yes—
146C22×C4×C13C22×C412yes—
147C22×C4×C13C22×C412yes—
148C22×C4×C13C2×D412yes—
149C22×C4×C13C2×D412yes—
150C22×C4×C13C2×D412yes—
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—24no—
156C22×Dic13—22no—
157C22×Dic13—22no—
158C22×Dic13—42no—
159C22×Dic13—42no—
160C2×Dic26—22no—
161C2×Dic26—22no—
162C2×Dic26—22no—
163C2×Dic26—22no—
164C2×Dic26—24no—
165C2×Dic26—22no—
166C2×Dic26—44no—
167C2×Dic26—44no—
168C2×Dic26—42no—
169C2×Dic26—42no—
170C2×Dic26—44no—
171C2×Dic26—44no—
172C2×Dic26C22×C414yes—
173C13×C4⋊C4—22no—
174C13×C4⋊C4—22no—
175C13×C4⋊C4—22no—
176C13×C4⋊C4—22no—
177C13×C4⋊C4—42no—
178C13×C22⋊C4—42no—
179C13×C22⋊C4—82no—
180C13×C22⋊C4—82no—
181C42×C13—42no—
182C22⋊Dic13—28no—
183C22⋊Dic13—28no—
184C22⋊Dic13—24no—
185C4⋊Dic13—22no—
186C4⋊Dic13C22×C414yes—
187#6101—28no—
188#6101—28no—
189C4×Dic13—24no—
190#6099—44no—
191C2×#1055—22no—
192C2×#1055—22no—
193C2×#1055—22no—
194C2×#1055—22no—
195C2×#1055—22no—
196C2×#1055—22no—
197C2×#1055—44no—
198C2×#536—131no—
199C2×#408—131no—
200#2311D1311yescharacteristic
201#1894—132no—
202#1892—132no—
203#1873—132no—
204#1872—132no—
205#1825—132no—
206#1787—132no—
207C23×C13—21no—
208C23×C13—21no—
209C23×C13—21no—
210C23×C13—21no—
211C23×C13—21no—
212C23×C13—21no—
213C23×C13—22no—
214C23×C13—22no—
215C23×C13—22no—
216C23×C13—22no—
217C23×C13—22no—
218C23×C13—22no—
219C23×C13C2×C22⋊C412yes—
220C23×C13C2×C22⋊C412yes—
221C23×C13C2×C22⋊C412yes—
222C23×C13C22×D411yescharacteristic
223Q8×C13—44no—
224D4×C13—44no—
225D4×C13—44no—
226D4×C13—44no—
227C2×C4×C13—21no—
228C2×C4×C13—21no—
229C2×C4×C13—21no—
230C2×C4×C13—21no—
231C2×C4×C13—21no—
232C2×C4×C13—21no—
233C2×C4×C13—22no—
234C2×C4×C13—22no—
235C2×C4×C13—22no—
236C2×C4×C13—22no—
237C2×C4×C13—22no—
238C2×C4×C13—22no—
239C2×C4×C13—22no—
240C2×C4×C13—22no—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—22no—
244C2×C4×C13—22no—
245C2×C4×C13—42no—
246C2×C4×C13—42no—
247C2×C4×C13—42no—
248C2×C4×C13—42no—
249C2×C4×C13—42no—
250C2×C4×C13—42no—
251C2×C4×C13—84no—
252C2×C4×C13C2×C22⋊C412yes—
253C2×C4×C13C2×C22⋊C412yes—
254C2×C4×C13C2×C22⋊C412yes—
255C2×C4×C13C23×C411yescharacteristic
256C2×C4×C13C22×D411yescharacteristic
257C2×C4×C13H(5,𝔽2)11yescharacteristic
258C2×C4×C13H(5,𝔽2)11yescharacteristic
259C2×C4×C13H(5,𝔽2)11yescharacteristic
260C2×C4×C13H(5,𝔽2)11yescharacteristic
261C2×Dic13—24no—
262C2×Dic13—24no—
263C2×Dic13—48no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13—42no—
268Dic26—24no—
269Dic26—24no—
270Dic26—48no—
271Dic26—42no—
272Dic26—42no—
273#1055—42no—
274#1055—42no—
275C22×#41C2×D1311yescharacteristic
276C23×Q8—131no—
277C22×M16—131no—
278C23×C8—131no—
279#536—138no—
280#536—264no—
281C2×C4⋊Q8—261no—
282C2×#124—261no—
283C2×#123—261no—
284C2×#123—261no—
285C2×C22⋊Q8—132no—
286C2×C22⋊Q8—132no—
287C2×#117—131no—
288C2×#117C2×D1311yescharacteristic
289C22×C4⋊C4C2×D1311yescharacteristic
290C2×#103—132no—
291C2×#103—132no—
292C2×#103—132no—
293C2×#103—132no—
294#408—264no—
295C2×C22⋊C8—132no—
296C2×C22⋊C8—132no—
297#391—21no—
298#390—21no—
299#386—21no—
300#386—21no—
301#380C2×D1312yes—
302#380C2×D1312yes—
303#340—132no—
304#340—132no—
305#340—132no—
306#340—132no—
307C22×C13—22no—
308C22×C13—22no—
309C22×C13—22no—
310C22×C13—24no—
311C22×C13—24no—
312C22×C13—24no—
313C22×C13—41no—
314C22×C13—41no—
315C22×C13—41no—
316C22×C13—42no—
317C22×C13—42no—
318C22×C13—42no—
319C22×C13—41no—
320C22×C13—41no—
321C22×C13—41no—
322C22×C13C22×C22⋊C411yescharacteristic
323C22×C13#53411yescharacteristic
324C22×C13#53411yescharacteristic
325C22×C13#57712yes—
326C22×C13#57812yes—
327C4×C13—22no—
328C4×C13—22no—
329C4×C13—22no—
330C4×C13—82no—
331C4×C13—82no—
332C4×C13#51811yescharacteristic
333C4×C13#51811yescharacteristic
334C4×C13#53411yescharacteristic
335C4×C13#53411yescharacteristic
336Dic13—44no—
337Dic13—44no—
338C2×#41—41no—
339C2×#41—41no—
340C2×#41—41no—
341C2×#41—41no—
342C2×#41C4×D1314yes—
343C2×#41D5214yes—
344C22×Q8—134no—
345C22×Q8—132no—
346C22×Q8—262no—
347C22×Q8—261no—
348C22×Q8—521no—
349C22×Q8—521no—
350C22×Q8—521no—
351C22×Q8—521no—
352C22×Q8C22⋊D1311yescharacteristic
353C22×D4C22⋊D1311yescharacteristic
354C22×D4C22⋊D1311yescharacteristic
355C22×D4C22⋊D1311yescharacteristic
356C23×C4—21no—
357C23×C4—261no—
358C23×C4C22×D1311yescharacteristic
359C23×C4C22×D1311yescharacteristic
360C23×C4C22×D1311yescharacteristic
361C2×M16—262no—
362C2×M16—521no—
363C2×M16—521no—
364C22×C8—132no—
365C22×C8—132no—
366C22×C8—262no—
367C22×C8—521no—
368C22×C8—521no—
369C4⋊Q8—268no—
370#124—268no—
371#123—268no—
372#123—268no—
373C22⋊Q8—138no—
374C22⋊Q8—138no—
375C22⋊Q8—264no—
376C22⋊Q8—264no—
377#117—41no—
378#117—41no—
379#117—134no—
380#117—262no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—41no—
384C2×C4⋊C4—132no—
385C2×C4⋊C4—262no—
386C2×C4⋊C4—262no—
387C2×C4⋊C4C22×D1312yes—
388C2×C4⋊C4C22×D1312yes—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—261no—
393C2×C22⋊C4—262no—
394C2×C22⋊C4—262no—
395C2×C42—21no—
396C2×C42—261no—
397#103—522no—
398#103—522no—
399#103—522no—
400#103—522no—
401C22⋊C8—264no—
402C22⋊C8—264no—
403#95—22no—
404#95—22no—
405C2×C13—42no—
406C2×C13—42no—
407C2×C13—42no—
408C2×C13#230811yescharacteristic
409C2×C13#289812yes—
410C2×C13#320011yescharacteristic
411C2×C13#320112yes—
412C2×C13#320211yescharacteristic
413C24C2×C22⋊D1311yescharacteristic
414C24C2×C22⋊D1311yescharacteristic
415C24C2×C22⋊D1311yescharacteristic
416#41—44no—
417#41—44no—
418#41—44no—
419#41—44no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—134no—
423C2×Q8—264no—
424C2×Q8—262no—
425C2×Q8—262no—
426C2×Q8—262no—
427C2×Q8—262no—
428C2×Q8—262no—
429C2×Q8—524no—
430C2×Q8—524no—
431C2×Q8—522no—
432C2×Q8—522no—
433C2×Q8—524no—
434C2×Q8—524no—
435C2×Q8#610314yes—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4#610314yes—
443C2×D4#610314yes—
444C2×D4#610314yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—82no—
461C22×C4—264no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—522no—
465C22×C4—522no—
466C22×C4C2×C4×D1312yes—
467C22×C4C2×C4×D1312yes—
468C22×C4C2×C4×D1312yes—
469C22×C4C2×D5212yes—
470C22×C4C2×D5212yes—
471C22×C4C2×D5212yes—
472C22×C4C2×C22⋊D1311yescharacteristic
473C22×C4C2×C22⋊D1311yescharacteristic
474C22×C4C2×C22⋊D1311yescharacteristic
475C22×C4C23×D1311yescharacteristic
476M16—524no—
477C2×C8—262no—
478C2×C8—262no—
479C2×C8—262no—
480C2×C8—262no—
481C2×C8—262no—
482C2×C8—262no—
483C2×C8—524no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—42no—
489C4⋊C4—134no—
490C4⋊C4—268no—
491C4⋊C4—268no—
492C4⋊C4—262no—
493C22⋊C4—42no—
494C22⋊C4—82no—
495C22⋊C4—82no—
496C22⋊C4—264no—
497C22⋊C4—268no—
498C22⋊C4—268no—
499C42—42no—
500C42—264no—
501C13#2177111yescharacteristic, complemented, Sylow
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23C2×#610312yes—
515C23C2×#610312yes—
516C23C2×#610312yes—
517C23C22×C22⋊D1311yescharacteristic
518Q8—44no—
519Q8—264no—
520Q8—264no—
521Q8—528no—
522Q8—522no—
523Q8—522no—
524D4—44no—
525D4—44no—
526D4—44no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—264no—
553C2×C4—264no—
554C2×C4—528no—
555C2×C4—522no—
556C2×C4—522no—
557C2×C4—522no—
558C2×C4—522no—
559C2×C4C2×#610312yes—
560C2×C4C2×#610312yes—
561C2×C4C2×#610312yes—
562C2×C4C22×C4×D1311yescharacteristic
563C2×C4C22×D5211yescharacteristic
564C2×C4#8825111yescharacteristic
565C2×C4#8825111yescharacteristic
566C2×C4#8825111yescharacteristic
567C2×C4#8825111yescharacteristic
568C8—522no—
569C8—522no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C22×#610311yescharacteristic
586C22#1171950911yescharacteristic
587C22#1171950911yescharacteristic
588C22#1171954112yes—
589C22#1171954312yes—
590C4—22no—
591C4—22no—
592C4—22no—
593C4—82no—
594C4—82no—
595C4—524no—
596C4—524no—
597C4#1171919111yescharacteristic
598C4#1171919111yescharacteristic
599C4#1171919811yescharacteristic
600C4#1171919811yescharacteristic
601C2—42no—
602C2—42no—
603C2—42no—
604C2#1311741111yescharacteristic
605C2#1312336612yes—
606C2#1312570811yescharacteristic
607C2#1312571212yes—
608C2#1312572711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2311C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenter#2311D13
hypocenterC13#21771
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. #13115267C2semidirect
4. #13115257C2semidirect
5. #13115148C2semidirect
6. #13115143C2semidirect
7. #13114928C2semidirect
8. #13114770C2semidirect
15. #11719530C4semidirect
63. C22×Dic26C2×C4semidirect
84. #88180C2×C4semidirect
89. #88178C2×C4semidirect
501. C13#21771semidirect

Maximal subgroups

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

subgroupindex
C2×#117195302
C2×#117188952
#131152672
#131152572
#131151482
#131151432
#131149282
#131147702
C13×#23112
#2177113