Finite Group Databasebeta

← #15865616

Subgroups — #15865616

Order 3328.

1122 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×#11718627C211yesmaximal, characteristic
2C2×#11718774C211yesmaximal, characteristic
3C2×#11718762C211yesmaximal, characteristic, complemented
4#13117454C211yesmaximal, characteristic, complemented
5#13117429C211yesmaximal, characteristic
6#13115534C212yesmaximal, complemented
7#13115527C212yesmaximal, complemented
8#13115520C212yesmaximal, complemented
9#13115513C212yesmaximal, complemented
10#13114555C211yesmaximal, characteristic, complemented
11C2×C13×#340C211yesmaximal, characteristic, complemented
12C22×#6127C2211yescharacteristic
13C23×C8×C13C2211yescharacteristic
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C13×C4⋊C4C2211yescharacteristic
16C2×#88180C2211yescharacteristic
17C2×#88180C2211yescharacteristic
18C22×#6099C2211yescharacteristic
19#11718874—22no—
20#11718873—22no—
21#11718774—24no—
22C2×#88061C2211yescharacteristic
23C2×#88061C2211yescharacteristic
24C2×#88056C2211yescharacteristic
25C2×#88056C2211yescharacteristic
26#11718762—24no—
27C2×#88055C2212yes—
28C2×C8×Dic13C2212yes—
29#11718649C2212yes—
30#11718648—21no—
31#11718648—21no—
32#11718646—21no—
33#11718646—21no—
34#11718646C2212yes—
35#11718646C2212yes—
36#11718635C2212yes—
37C2×#88050C2212yes—
38#11718627C418yescomplemented
39C2×#88049C2212yes—
40C13×#340C2212yes—
41C13×#340C2212yes—
42#11718121C2212yes—
43#11718121C2212yes—
44C23×C4×C13—21no—
45C23×C4×C13—21no—
46C23×C4×C13C2311yescharacteristic
47C22×C22⋊D13—21no—
48C2×#6127—22no—
49C2×#6127—22no—
50C2×#6127—22no—
51C2×#6127—22no—
52C2×#6127—22no—
53C2×#6127C2×C414yes—
54C22×D52C2311yescharacteristic
55C22×C4×D13—21no—
56C22×Dic26C2311yescharacteristic
57C22×C8×C13—22no—
58C22×C8×C13—22no—
59C22×C8×C13D414yes—
60C22×C8×C13C2311yescharacteristic
61C22×C8×C13C2311yescharacteristic
62C2×C13×C4⋊C4—21no—
63C2×C13×C4⋊C4—22no—
64C2×C13×C4⋊C4—22no—
65C2×C13×C4⋊C4—22no—
66C2×C13×C4⋊C4—21no—
67C2×C13×C4⋊C4C2×C414yes—
68C2×C13×C4⋊C4C2312yes—
69C2×C13×C4⋊C4C2312yes—
70C2×C22⋊Dic13—21no—
71C2×C22⋊Dic13D412yes—
72C2×#6103—22no—
73C2×#6103—22no—
74#88180—22no—
75#88180—22no—
76#88180—22no—
77#88180—22no—
78C2×C4⋊Dic13—22no—
79C2×C4⋊Dic13—22no—
80C2×C4⋊Dic13D411yescharacteristic
81C2×C4⋊Dic13D411yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×C4×Dic13—21no—
85C2×C4×Dic13C2312yes—
86C2×#6099—22no—
87C2×#6099—22no—
88C2×#6099C2×C414yes—
89C22×#1055C2312yes—
90#88071—22no—
91#88071—22no—
92#88071—22no—
93#88071—22no—
94#88061—22no—
95#88061—22no—
96#88056—22no—
97#88056—22no—
98#88055—24no—
99C8×Dic13—24no—
100#88050C2×C418yescomplemented
101#88049C2×C418yescomplemented
102#21092—131nomaximal, Sylow
103C24×C13C2×D411yescharacteristic
104C23×D13—21no—
105C22×C4×C13—22no—
106C22×C4×C13—24no—
107C22×C4×C13—22no—
108C22×C4×C13—42no—
109C22×C4×C13—81no—
110C22×C4×C13—81no—
111C22×C4×C13—81no—
112C22×C4×C13—81no—
113C22×C4×C13C22×C412yes—
114C22×C4×C13C2×D412yes—
115C22×C4×C13C2×D411yescharacteristic
116C22×C4×C13#4112yes—
117C22×C4×C13#4112yes—
118C22×C4×C13#4112yes—
119C22×C4×C13#4112yes—
120C22×C4×C13C2411yescharacteristic
121C2×C22⋊D13—24no—
122C2×C22⋊D13—42no—
123C2×C22⋊D13—42no—
124C22×Dic13—21no—
125C22×Dic13—21no—
126C22×Dic13—21no—
127C22×Dic13C2×D412yes—
128C22×Dic13C2×D412yes—
129#6127—28no—
130#6127—28no—
131#6127—44no—
132#6127—44no—
133C2×D52—21no—
134C2×D52—21no—
135C2×D52—21no—
136C2×D52—21no—
137C2×D52—21no—
138C2×D52C22×C412yes—
139C2×C4×D13—24no—
140C2×C4×D13—42no—
141C2×C4×D13—42no—
142C2×Dic26—21no—
143C2×Dic26—21no—
144C2×Dic26—21no—
145C2×Dic26—21no—
146C2×Dic26—21no—
147C2×Dic26C22×C412yes—
148C2×C8×C13—28no—
149C2×C8×C13—21no—
150C2×C8×C13—21no—
151C2×C8×C13—21no—
152C2×C8×C13—21no—
153C2×C8×C13C2×D412yes—
154C2×C8×C13C2×D412yes—
155C13×C4⋊C4—24no—
156C13×C4⋊C4—22no—
157C13×C4⋊C4—22no—
158C13×C4⋊C4—22no—
159C13×C4⋊C4—42no—
160C13×C4⋊C4C22×C414yes—
161C22⋊Dic13—24no—
162C22⋊Dic13—42no—
163#6103—82no—
164#6103—82no—
165C4⋊Dic13—22no—
166C4⋊Dic13—22no—
167C4⋊Dic13—21no—
168C4⋊Dic13—21no—
169C4⋊Dic13—21no—
170C4⋊Dic13—21no—
171C4⋊Dic13—82no—
172C4⋊Dic13—82no—
173C4×Dic13—22no—
174C4×Dic13—22no—
175C4×Dic13—42no—
176#6099—28no—
177C2×#1055—22no—
178C2×#1055—22no—
179C2×#1055C22×C414yes—
180C2×#417—131no—
181C2×#416—131no—
182C2×#405—131no—
183#2316—131no—
184#2311—131no—
185#1936—132no—
186#1935—132no—
187#1934—132no—
188#1933—132no—
189#1736—131no—
190C2×#340D1311yescharacteristic, complemented
191C23×C13—21no—
192C23×C13—21no—
193C23×C13—21no—
194C23×C13—21no—
195C23×C13C4×D412yes—
196C23×C13#12112yes—
197C23×C13#12412yes—
198C23×C13C22×D411yescharacteristic
199C22×D13—22no—
200C22×D13—41no—
201C22×D13—41no—
202C22×D13—41no—
203C22×D13—41no—
204C22×D13—41no—
205C22×D13—41no—
206C2×C4×C13—24no—
207C2×C4×C13—42no—
208C2×C4×C13—42no—
209C2×C4×C13—42no—
210C2×C4×C13—42no—
211C2×C4×C13—42no—
212C2×C4×C13—84no—
213C2×C4×C13—84no—
214C2×C4×C13#11714yes—
215C2×C4×C13C4×D414yes—
216C2×C4×C13C4×D412yes—
217C2×C4×C13#12112yes—
218C2×C4×C13#12314yes—
219C2×C4×C13#12412yes—
220C2×C4×C13#12614yes—
221C2×C4×C13C23×C411yescharacteristic
222C2×C4×C13C22×D411yescharacteristic
223C2×C4×C13C2×#4111yescharacteristic
224C2×C4×C13C2×#4111yescharacteristic
225C2×C4×C13C2×#4111yescharacteristic
226C2×C4×C13C2×#4111yescharacteristic
227C22⋊D13—48no—
228C2×Dic13—22no—
229C2×Dic13—22no—
230C2×Dic13—22no—
231C2×Dic13—22no—
232C2×Dic13—22no—
233C2×Dic13—22no—
234C2×Dic13—22no—
235C2×Dic13—41no—
236C2×Dic13—41no—
237C2×Dic13—41no—
238C2×Dic13—41no—
239C2×Dic13—41no—
240C2×Dic13—41no—
241C2×Dic13—41no—
242C2×Dic13—41no—
243D52—22no—
244D52—22no—
245D52—41no—
246D52—41no—
247C4×D13—48no—
248Dic26—22no—
249Dic26—22no—
250Dic26—41no—
251Dic26—41no—
252C8×C13—22no—
253C8×C13—22no—
254#1055—24no—
255C22×#41—131no—
256C22×M16—131no—
257C23×C8C2×D1311yescharacteristic
258C2×#117—131no—
259C2×#117—131no—
260C22×C4⋊C4C2×D1311yescharacteristic
261C22×C4⋊C4C2×D1311yescharacteristic
262#417—138no—
263#416—264no—
264C2×#103—132no—
265C2×#103—131no—
266C2×#103—131no—
267C2×#102—132no—
268C2×#102—131no—
269C2×#102—131no—
270#405—264no—
271C2×#97—132no—
272C2×C4×C8—132no—
273#391—132no—
274#390—132no—
275#389—132no—
276#389—132no—
277#389—261no—
278#389—261no—
279#386—262no—
280#380—262no—
281#380—261no—
282#380—261no—
283#340—132no—
284#340—132no—
285#340C2×D1312yes—
286#340C2×D1312yes—
287C22×C13—22no—
288C22×C13—22no—
289C22×C13—22no—
290C22×C13—22no—
291C22×C13—22no—
292C22×C13—22no—
293C22×C13#38814yes—
294C22×C13C2×C4×D411yescharacteristic
295C22×C13C2×#12111yescharacteristic
296C22×C13C2×#12411yescharacteristic
297C22×C13#57611yescharacteristic
298C22×C13#57711yescharacteristic
299C22×C13#57711yescharacteristic
300C22×C13#57811yescharacteristic
301C2×D13—42no—
302C2×D13—42no—
303C2×D13—42no—
304C2×D13—81no—
305C2×D13—81no—
306C2×D13—81no—
307C2×D13—81no—
308C4×C13—44no—
309C4×C13—82no—
310C4×C13#38814yes—
311C4×C13C2×#11711yescharacteristic
312C4×C13C2×C4×D411yescharacteristic
313C4×C13C2×#12311yescharacteristic
314C4×C13C2×#12611yescharacteristic
315Dic13—42no—
316Dic13—42no—
317Dic13—42no—
318Dic13—81no—
319Dic13—81no—
320C2×#41—134no—
321C2×#41—262no—
322C2×#41—262no—
323C2×#41—262no—
324C2×#41—262no—
325C2×#41—262no—
326C22×Q8—131no—
327C22×D4—131no—
328C22×D4—261no—
329C23×C4—21no—
330C23×C4—21no—
331C23×C4—261no—
332C23×C4C22×D1311yescharacteristic
333C2×M16—134no—
334C2×M16—262no—
335C2×M16—262no—
336C22×C8—22no—
337C22×C8—22no—
338C22×C8—132no—
339C22×C8D5214yes—
340C22×C8C22×D1311yescharacteristic
341C22×C8C22×D1311yescharacteristic
342#117—262no—
343#117—262no—
344#117—262no—
345#117—262no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—131no—
352C2×C4⋊C4—131no—
353C2×C4⋊C4—131no—
354C2×C4⋊C4—131no—
355C2×C4⋊C4—262no—
356C2×C4⋊C4—262no—
357C2×C4⋊C4C4×D1314yes—
358C2×C4⋊C4C22×D1312yes—
359C2×C4⋊C4C22×D1312yes—
360C2×C22⋊C4—132no—
361C2×C22⋊C4—261no—
362C2×C22⋊C4—262no—
363C2×C22⋊C4—262no—
364C2×C42—132no—
365C2×C42—261no—
366#103—138no—
367#103—262no—
368#103—262no—
369#102—138no—
370#102—262no—
371#102—262no—
372#97—264no—
373C4×C8—264no—
374#95—262no—
375#95—262no—
376#95—262no—
377#95—262no—
378C2×C13—24no—
379C2×C13C2×#38811yescharacteristic
380C2×C13#294611yescharacteristic
381C2×C13#294711yescharacteristic
382C2×C13#306011yescharacteristic
383C2×C13#306111yescharacteristic
384C2×C13#313611yescharacteristic
385C2×C13#313711yescharacteristic
386D13—82no—
387C24—261no—
388C24D4×D1311yescharacteristic
389#41—268no—
390#41—268no—
391#41—524no—
392#41—524no—
393C2×Q8—132no—
394C2×Q8—261no—
395C2×Q8—261no—
396C2×Q8—261no—
397C2×Q8—261no—
398C2×Q8—261no—
399C2×D4—132no—
400C2×D4—264no—
401C2×D4—261no—
402C2×D4—261no—
403C2×D4—261no—
404C2×D4—261no—
405C2×D4—261no—
406C2×D4—522no—
407C2×D4—522no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—132no—
417C22×C4—132no—
418C22×C4—261no—
419C22×C4—264no—
420C22×C4—261no—
421C22×C4—261no—
422C22×C4—522no—
423C22×C4—522no—
424C22×C4C2×C4×D1312yes—
425C22×C4C2×D5212yes—
426C22×C4#612712yes—
427C22×C4D4×D1311yescharacteristic
428C22×C4#612912yes—
429C22×C4#612912yes—
430C22×C4#613112yes—
431C22×C4C23×D1311yescharacteristic
432M16—268no—
433C2×C8—28no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—134no—
439C2×C8—262no—
440C2×C8—262no—
441C2×C8C2×D5212yes—
442C2×C8C2×D5212yes—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—42no—
448C4⋊C4—262no—
449C4⋊C4—262no—
450C4⋊C4—261no—
451C4⋊C4—261no—
452C4⋊C4—261no—
453C4⋊C4—261no—
454C4⋊C4—1042no—
455C4⋊C4—1042no—
456C4⋊C4C2×C4×D1314yes—
457C22⋊C4—264no—
458C22⋊C4—522no—
459C22⋊C4—1042no—
460C22⋊C4—1042no—
461C42—262no—
462C42—262no—
463C42—522no—
464C13#2109211yescharacteristic, complemented, Sylow
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—262no—
470C23—521no—
471C23—521no—
472C23—521no—
473C23—521no—
474C23—521no—
475C23—521no—
476C23#8813512yes—
477C23#8813912yes—
478C23#8814912yes—
479C23C2×D4×D1311yescharacteristic
480Q8—262no—
481Q8—262no—
482Q8—521no—
483Q8—521no—
484D4—262no—
485D4—262no—
486D4—528no—
487D4—521no—
488D4—521no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—262no—
498C2×C4—262no—
499C2×C4—262no—
500C2×C4—262no—
501C2×C4—262no—
502C2×C4—262no—
503C2×C4—262no—
504C2×C4—521no—
505C2×C4—521no—
506C2×C4—528no—
507C2×C4—521no—
508C2×C4—521no—
509C2×C4—521no—
510C2×C4—521no—
511C2×C4—521no—
512C2×C4—521no—
513C2×C4C4×D5214yes—
514C2×C4#8813512yes—
515C2×C4#8813912yes—
516C2×C4#8814014yes—
517C2×C4#8814614yes—
518C2×C4#8814912yes—
519C2×C4#8815214yes—
520C2×C4C22×C4×D1311yescharacteristic
521C2×C4C22×D5211yescharacteristic
522C2×C4C2×#612711yescharacteristic
523C2×C4C2×#612911yescharacteristic
524C2×C4C2×#612911yescharacteristic
525C2×C4C2×#613111yescharacteristic
526C8—22no—
527C8—22no—
528C8—264no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—522no—
536C22—522no—
537C22—522no—
538C22—1041no—
539C22—1041no—
540C22—1041no—
541C22—1041no—
542C22#1171833114yes—
543C22C2×#8813511yescharacteristic
544C22C2×#8813911yescharacteristic
545C22C2×#8814911yescharacteristic
546C22#1171946211yescharacteristic
547C22#1171946711yescharacteristic
548C22#1171946711yescharacteristic
549C22#1171947111yescharacteristic
550C4—44no—
551C4—82no—
552C4—522no—
553C4—522no—
554C4—522no—
555C4—1041no—
556C4—1041no—
557C4#1171833114yes—
558C4C2×C4×D5211yescharacteristic
559C4C2×#8814011yescharacteristic
560C4C2×#8814611yescharacteristic
561C4C2×#8815211yescharacteristic
562C2—24no—
563C2—1042no—
564C2C2×#1171833111yescharacteristic
565C2#1312366211yescharacteristic
566C2#1312367311yescharacteristic
567C2#1312449911yescharacteristic
568C2#1312450811yescharacteristic
569C2#1312518811yescharacteristic
570C2#1312519911yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#340C2
socleC23×C13C22×D4
prime radicalC22×#6127C22
cosocleC22×C4×C13C24
hypercenterC2×#340D13
hypocenterC13#21092
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C2×#11718762C2semidirect
4. #13117454C2semidirect
6. #13115534C2semidirect
7. #13115527C2semidirect
8. #13115520C2semidirect
9. #13115513C2semidirect
10. #13114555C2semidirect
11. C2×C13×#340C2semidirect
38. #11718627C4semidirect
100. #88050C2×C4semidirect
101. #88049C2×C4semidirect
190. C2×#340D13semidirect
464. C13#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#117186272
C2×#117187742
C2×#117187622
#131174542
#131174292
#131155342
#131155272
#131155202
#131155132
#131145552
C2×C13×#3402
#2109213