Finite Group Databasebeta

← #15867438

Subgroups — #15867438

Order 3328.

1250 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719222C211yesmaximal, characteristic
2C2×#11718779C211yesmaximal, characteristic
3C2×C4⋊C4×Dic13C211yesmaximal, characteristic
4#13115792C212yesmaximal, complemented
5#13115790C212yesmaximal, complemented
6#13115786C212yesmaximal, complemented
7#13115782C212yesmaximal, complemented
8#13114557C211yesmaximal, characteristic
9#13114556C211yesmaximal, characteristic
10#13114555C211yesmaximal, characteristic
11C13×#1736C211yesmaximal, characteristic
12C22×C13×M16C2211yescharacteristic
13C2×C13×#117C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C2×#88180C2211yescharacteristic
16C22×C4⋊Dic13C2211yescharacteristic
17C22×C4×Dic13—21no—
18C22×C4×Dic13C2211yescharacteristic
19C22×#6099C2211yescharacteristic
20#11719222—24no—
21#11719222C418yescomplemented
22C2×#88144C2211yescharacteristic
23C2×#88144C2211yescharacteristic
24C2×#88142C2211yescharacteristic
25C2×#88142C2211yescharacteristic
26C2×#88132—21no—
27C2×#88132—21no—
28C2×C4×Dic26—21no—
29C2×C4×Dic26—21no—
30C2×#88071—21no—
31#11718779—24no—
32C2×#88054C2212yes—
33C2×#88054C2212yes—
34C4⋊C4×Dic13—21no—
35C4⋊C4×Dic13—21no—
36C4⋊C4×Dic13C2212yes—
37C4⋊C4×Dic13C2212yes—
38C13×#340C2212yes—
39C13×#340C2212yes—
40#11718143C2212yes—
41#11718143C2212yes—
42#11718121C2212yes—
43#11718121C2212yes—
44#11718121C2212yes—
45#11718121C2212yes—
46C23×C4×C13—21no—
47C23×C4×C13C2311yescharacteristic
48C23×Dic13—21no—
49C23×Dic13D411yescharacteristic
50C23×Dic13D411yescharacteristic
51C22×Dic26—21no—
52C22×Dic26—21no—
53C2×C13×M16—22no—
54C2×C13×M16—22no—
55C2×C13×M16—22no—
56C22×C8×C13C2312yes—
57C13×#117—22no—
58C13×#117C2×C414yes—
59C2×C13×C4⋊C4—21no—
60C2×C13×C4⋊C4—22no—
61C2×C13×C4⋊C4Q814yescomplemented
62C2×C13×C4⋊C4C2311yescharacteristic
63C2×C13×C4⋊C4C2311yescharacteristic
64C2×C13×C4⋊C4C2312yes—
65C2×C13×C22⋊C4—21no—
66C2×C42×C13—21no—
67C2×C22⋊Dic13—21no—
68#88180—22no—
69#88180C2×C414yes—
70C2×C4⋊Dic13—21no—
71C2×C4⋊Dic13—22no—
72C2×C4⋊Dic13—21no—
73C2×C4⋊Dic13—21no—
74C2×C4⋊Dic13Q814yescomplemented
75C2×C4⋊Dic13C2311yescharacteristic
76C2×C4⋊Dic13C2311yescharacteristic
77C2×C4⋊Dic13C2312yes—
78C2×#6101—21no—
79C2×#6101—21no—
80C2×#6101—21no—
81C2×#6101—21no—
82C2×C4×Dic13—21no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13—22no—
85C2×C4×Dic13—44no—
86C2×C4×Dic13C2×C414yes—
87C2×C4×Dic13D411yescharacteristic
88C2×C4×Dic13D411yescharacteristic
89C2×C4×Dic13D414yes—
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×#6099—22no—
93C2×#6099—22no—
94C2×#6099—22no—
95C22×#1055C2312yes—
96#88144—22no—
97#88144—22no—
98#88144C2×C414yescomplemented
99#88144C2×C414yescomplemented
100#88142—22no—
101#88142—22no—
102#88142C2×C414yescomplemented
103#88142C2×C414yescomplemented
104#88132—28no—
105#88132—28no—
106C4×Dic26—28no—
107C4×Dic26—28no—
108#88071—22no—
109#88071—22no—
110#88054—24no—
111#88054—24no—
112#21312—131nomaximal, Sylow
113C24×C13C2×D411yescharacteristic
114C22×C4×C13—24no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—21no—
118C22×C4×C13—42no—
119C22×C4×C13—42no—
120C22×C4×C13C22×C412yes—
121C22×C4×C13C2×D411yescharacteristic
122C22×C4×C13C2×D412yes—
123C22×C4×C13C2×Q812yes—
124C22×C4×C13C2×Q812yes—
125C22×C4×C13#4112yes—
126C22×C4×C13#4112yes—
127C22×C4×C13C2411yescharacteristic
128C22×Dic13—22no—
129C22×Dic13—22no—
130C22×Dic13—24no—
131C22×Dic13—22no—
132C22×Dic13—21no—
133C22×Dic13—21no—
134C22×Dic13—41no—
135C22×Dic13—41no—
136C22×Dic13—42no—
137C22×Dic13—42no—
138C22×Dic13C22⋊C414yes—
139C22×Dic13C22⋊C414yes—
140C22×Dic13C2×D411yescharacteristic
141C22×Dic13C2×D411yescharacteristic
142C22×Dic13C2×D411yescharacteristic
143C22×Dic13C2×D411yescharacteristic
144C2×Dic26—24no—
145C2×Dic26—24no—
146C2×Dic26—42no—
147C2×Dic26—42no—
148C2×Dic26—42no—
149C2×Dic26—42no—
150C13×M16—44no—
151C2×C8×C13—22no—
152C2×C8×C13—22no—
153C2×C8×C13—22no—
154C13×C4⋊C4—24no—
155C13×C4⋊C4—21no—
156C13×C4⋊C4—21no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4C22×C412yes—
159C13×C4⋊C4C22×C412yes—
160C13×C4⋊C4C2×Q814yes—
161C13×C22⋊C4—24no—
162C42×C13—24no—
163C22⋊Dic13—24no—
164C4⋊Dic13—24no—
165C4⋊Dic13—21no—
166C4⋊Dic13—21no—
167C4⋊Dic13—22no—
168C4⋊Dic13—24no—
169C4⋊Dic13—24no—
170C4⋊Dic13C22×C412yes—
171C4⋊Dic13C22×C412yes—
172C4⋊Dic13C2×Q814yes—
173#6101—24no—
174#6101—24no—
175#6101—24no—
176#6101—24no—
177C4×Dic13—24no—
178C4×Dic13—44no—
179C4×Dic13—44no—
180C4×Dic13C22⋊C414yes—
181C4×Dic13C22⋊C414yes—
182C4×Dic13C22×C412yes—
183C4×Dic13C22×C412yes—
184C4×Dic13C2×D412yes—
185C4×Dic13C2×D412yes—
186#6099—44no—
187C2×#1055—22no—
188C2×#1055—22no—
189C2×#1055—22no—
190C2×#533—131no—
191C2×#423—131no—
192C2×C4×C4⋊C4—131no—
193#1989—132no—
194#1989—132no—
195#1988—132no—
196#1987—132no—
197#1736—131no—
198#1736—131no—
199#1736—131no—
200#1736D1311yescharacteristic
201C23×C13—21no—
202C23×C13—21no—
203C23×C13—21no—
204C23×C13—21no—
205C23×C13C2×C22⋊C412yes—
206C23×C13C22⋊Q812yes—
207C23×C13C22⋊Q812yes—
208C23×C13C22×D411yescharacteristic
209C2×C4×C13—22no—
210C2×C4×C13—22no—
211C2×C4×C13—24no—
212C2×C4×C13—48no—
213C2×C4×C13—42no—
214C2×C4×C13—41no—
215C2×C4×C13—41no—
216C2×C4×C13—41no—
217C2×C4×C13—41no—
218C2×C4×C13—42no—
219C2×C4×C13C2×C22⋊C412yes—
220C2×C4×C13C4×Q814yes—
221C2×C4×C13C4×Q814yes—
222C2×C4×C13C22⋊Q812yes—
223C2×C4×C13C22⋊Q812yes—
224C2×C4×C13#12414yes—
225C2×C4×C13C4⋊Q814yes—
226C2×C4×C13C23×C411yescharacteristic
227C2×C4×C13C22×D411yescharacteristic
228C2×C4×C13C22×Q811yescharacteristic
229C2×C4×C13C22×Q811yescharacteristic
230C2×C4×C13C2×#4111yescharacteristic
231C2×C4×C13C2×#4111yescharacteristic
232C2×Dic13—21no—
233C2×Dic13—21no—
234C2×Dic13—21no—
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—21no—
238C2×Dic13—28no—
239C2×Dic13—22no—
240C2×Dic13—22no—
241C2×Dic13—24no—
242C2×Dic13—44no—
243C2×Dic13—48no—
244C2×Dic13—42no—
245C2×Dic13—41no—
246C2×Dic13—41no—
247C2×Dic13—41no—
248C2×Dic13—41no—
249C2×Dic13—42no—
250C2×Dic13C2×C22⋊C412yes—
251C2×Dic13C2×C22⋊C412yes—
252C2×Dic13C2×C22⋊C412yes—
253C2×Dic13C2×C22⋊C412yes—
254C2×Dic13AGL(1,ℤ/8)12yes—
255C2×Dic13#13712yes—
256Dic26—48no—
257Dic26—48no—
258C8×C13—42no—
259#1055—42no—
260C22×M16—131no—
261C22×M16C2×D1311yescharacteristic
262#533—138no—
263#533—264no—
264C2×C4⋊Q8—131no—
265C2×C4⋊Q8—131no—
266C2×#125—131no—
267C2×#125—131no—
268C2×C22⋊Q8—261no—
269C2×C22⋊Q8—261no—
270C2×C4×Q8—261no—
271C2×C4×Q8—261no—
272C2×#117—131no—
273C2×#117C2×D1311yescharacteristic
274C22×C4⋊C4—131no—
275C22×C4⋊C4C2×D1311yescharacteristic
276C22×C42—131no—
277C22×C42—261no—
278#423—264no—
279C2×C4⋊C8—132no—
280C2×C4⋊C8—132no—
281C4×C4⋊C4—132no—
282C4×C4⋊C4—132no—
283C4×C4⋊C4—261no—
284C4×C4⋊C4—261no—
285C2×#95—261no—
286#340—132no—
287#340—132no—
288#340—132no—
289#340—132no—
290#340—132no—
291#340—132no—
292#340C2×D1312yes—
293#340C2×D1312yes—
294C22×C13—22no—
295C22×C13—22no—
296C22×C13—22no—
297C22×C13—22no—
298C22×C13—22no—
299C22×C13—22no—
300C22×C13#39114yes—
301C22×C13C22×C22⋊C411yescharacteristic
302C22×C13C2×C22⋊Q811yescharacteristic
303C22×C13C2×C22⋊Q811yescharacteristic
304C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
305C22×C13C2×#13711yescharacteristic
306C22×C13#57511yescharacteristic
307C22×C13#57511yescharacteristic
308C4×C13—44no—
309C4×C13—42no—
310C4×C13—42no—
311C4×C13#39114yes—
312C4×C13C2×C4×Q811yescharacteristic
313C4×C13C2×C4×Q811yescharacteristic
314C4×C13C2×#12411yescharacteristic
315C4×C13C2×C4⋊Q811yescharacteristic
316Dic13—22no—
317Dic13—22no—
318Dic13—22no—
319Dic13—44no—
320Dic13—42no—
321Dic13—42no—
322Dic13#41714yes—
323C22×Q8—261no—
324C22×Q8—261no—
325C23×C4—21no—
326C23×C4—131no—
327C23×C4—131no—
328C23×C4—261no—
329C23×C4C22×D1311yescharacteristic
330C2×M16—22no—
331C2×M16—22no—
332C2×M16—22no—
333C2×M16—262no—
334C2×M16—262no—
335C2×M16—262no—
336C22×C8—132no—
337C22×C8C22×D1312yes—
338C4⋊Q8—134no—
339C4⋊Q8—134no—
340C4⋊Q8—262no—
341C4⋊Q8—262no—
342#125—134no—
343#125—134no—
344#125—262no—
345#125—262no—
346C22⋊Q8—268no—
347C22⋊Q8—268no—
348C4×Q8—268no—
349C4×Q8—268no—
350#117—22no—
351#117—134no—
352#117—262no—
353#117C4×D1314yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—134no—
357C2×C4⋊C4—132no—
358C2×C4⋊C4—131no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—261no—
361C2×C4⋊C4—261no—
362C2×C4⋊C4—261no—
363C2×C4⋊C4—261no—
364C2×C4⋊C4—261no—
365C2×C4⋊C4—261no—
366C2×C4⋊C4—262no—
367C2×C4⋊C4—261no—
368C2×C4⋊C4Dic2614yescomplemented
369C2×C4⋊C4C22×D1312yes—
370C2×C4⋊C4C22×D1311yescharacteristic
371C2×C4⋊C4C22×D1311yescharacteristic
372C2×C22⋊C4—21no—
373C2×C22⋊C4—261no—
374C2×C42—21no—
375C2×C42—134no—
376C2×C42—131no—
377C2×C42—131no—
378C2×C42—134no—
379C2×C42—131no—
380C2×C42—131no—
381C2×C42—261no—
382C2×C42—262no—
383C2×C42—262no—
384C2×C42—524no—
385C4⋊C8—264no—
386C4⋊C8—264no—
387#95—262no—
388#95—262no—
389C2×C13—24no—
390C2×C13C2×#39111yescharacteristic
391C2×C13C2×#41711yescharacteristic
392C2×C13#289611yescharacteristic
393C2×C13#307711yescharacteristic
394C2×C13#307711yescharacteristic
395C2×C13#307811yescharacteristic
396C2×C13#307811yescharacteristic
397C24D4×D1311yescharacteristic
398C2×Q8—264no—
399C2×Q8—264no—
400C2×Q8—522no—
401C2×Q8—522no—
402C2×Q8—522no—
403C2×Q8—522no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—131no—
411C22×C4—134no—
412C22×C4—131no—
413C22×C4—131no—
414C22×C4—131no—
415C22×C4—134no—
416C22×C4—262no—
417C22×C4—262no—
418C22×C4—264no—
419C22×C4—262no—
420C22×C4—261no—
421C22×C4—261no—
422C22×C4—521no—
423C22×C4—521no—
424C22×C4—522no—
425C22×C4—522no—
426C22×C4C2×Dic2612yes—
427C22×C4C2×C4×D1312yes—
428C22×C4#612712yes—
429C22×C4D4×D1311yescharacteristic
430C22×C4D4×D1312yes—
431C22×C4#612912yes—
432C22×C4Q8×D1312yes—
433C22×C4C23×D1311yescharacteristic
434M16—44no—
435M16—524no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—262no—
440C2×C8—262no—
441C2×C8—262no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—132no—
447C4⋊C4—132no—
448C4⋊C4—134no—
449C4⋊C4—264no—
450C4⋊C4—264no—
451C4⋊C4—264no—
452C4⋊C4—264no—
453C4⋊C4—264no—
454C4⋊C4—264no—
455C4⋊C4—264no—
456C4⋊C4—262no—
457C4⋊C4—261no—
458C4⋊C4—261no—
459C4⋊C4C2×Dic2614yes—
460C4⋊C4C2×C4×D1312yes—
461C4⋊C4C2×C4×D1312yes—
462C22⋊C4—24no—
463C22⋊C4—264no—
464C42—24no—
465C42—134no—
466C42—134no—
467C42—132no—
468C42—132no—
469C42—132no—
470C42—132no—
471C42—264no—
472C42—524no—
473C42—524no—
474C13#2131211yescharacteristic, complemented, Sylow
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23#8813212yes—
480C23C22⋊C4×D1312yes—
481C23#8815012yes—
482C23C2×D4×D1311yescharacteristic
483Q8—528no—
484Q8—528no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—132no—
496C2×C4—132no—
497C2×C4—132no—
498C2×C4—132no—
499C2×C4—132no—
500C2×C4—132no—
501C2×C4—268no—
502C2×C4—261no—
503C2×C4—261no—
504C2×C4—261no—
505C2×C4—261no—
506C2×C4—261no—
507C2×C4—261no—
508C2×C4—262no—
509C2×C4—262no—
510C2×C4—264no—
511C2×C4—524no—
512C2×C4—528no—
513C2×C4—522no—
514C2×C4—521no—
515C2×C4—521no—
516C2×C4—521no—
517C2×C4—522no—
518C2×C4—521no—
519C2×C4C4×Dic2614yes—
520C2×C4#8813212yes—
521C2×C4C22⋊C4×D1312yes—
522C2×C4#8813914yes—
523C2×C4#8814114yes—
524C2×C4#8814214yes—
525C2×C4#8815012yes—
526C2×C4C22×Dic2611yescharacteristic
527C2×C4C22×C4×D1311yescharacteristic
528C2×C4C2×#612711yescharacteristic
529C2×C4C2×D4×D1311yescharacteristic
530C2×C4C2×#612911yescharacteristic
531C2×C4C2×Q8×D1311yescharacteristic
532C8—42no—
533C8—522no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22#1171830714yes—
541C22C2×#8813211yescharacteristic
542C22C2×C22⋊C4×D1311yescharacteristic
543C22C2×#8815011yescharacteristic
544C22D13×AGL(1,ℤ/8)11yescharacteristic
545C22#1171947811yescharacteristic
546C22D13×#13711yescharacteristic
547C22#1171948211yescharacteristic
548C4—42no—
549C4—44no—
550C4—42no—
551C4—134no—
552C4—262no—
553C4—262no—
554C4—262no—
555C4—522no—
556C4—522no—
557C4—524no—
558C4#1171830714yes—
559C4C2×C4×Dic2611yescharacteristic
560C4C2×#8813911yescharacteristic
561C4C2×#8814111yescharacteristic
562C4C2×#8814211yescharacteristic
563C2—24no—
564C2C2×#1171830711yescharacteristic
565C2D13×#41711yescharacteristic
566C2#1312335211yescharacteristic
567C2#1312463011yescharacteristic
568C2#1312463211yescharacteristic
569C2#1312463811yescharacteristic
570C2#1312464011yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1736C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenter#1736D13
hypocenterC13#21312
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #13115792C2semidirect
5. #13115790C2semidirect
6. #13115786C2semidirect
7. #13115782C2semidirect
21. #11719222C4semidirect
61. C2×C13×C4⋊C4Q8semidirect
74. C2×C4⋊Dic13Q8semidirect
98. #88144C2×C4semidirect
99. #88144C2×C4semidirect
102. #88142C2×C4semidirect
103. #88142C2×C4semidirect
368. C2×C4⋊C4Dic26semidirect
474. C13#21312semidirect

Maximal subgroups

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

subgroupindex
C2×#117192222
C2×#117187792
C2×C4⋊C4×Dic132
#131157922
#131157902
#131157862
#131157822
#131145572
#131145562
#131145552
C13×#17362
#2131213