Finite Group Databasebeta

← #16798729

Subgroups — #16798729

Order 3328.

1018 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×#11719129C211yesmaximal, characteristic
2C2×C4×#6122C211yesmaximal, characteristic
3#13130896C212yesmaximal, complemented
4#13130893C212yesmaximal, complemented
5#13130892C212yesmaximal, complemented
6#13130889C212yesmaximal, complemented
7#13130499C211yesmaximal, characteristic
8#13130499C211yesmaximal, characteristic
9#13130499C211yesmaximal, characteristic
10#13130499C211yesmaximal, characteristic
11C2×C4⋊C4×Dic13C211yesmaximal, characteristic
12C22×#6122C2211yescharacteristic
13C22×#6122C2211yescharacteristic
14C22×#6122C2211yescharacteristic
15C22×#6122C2211yescharacteristic
16C22×C13×C4⋊C4C411yescharacteristic
17C22×C4⋊Dic13C411yescharacteristic
18C22×C4×Dic13C2211yescharacteristic
19C22×C4×Dic13C2211yescharacteristic
20C22×C4×Dic13C2211yescharacteristic
21#11719129—24no—
22C2×#88112C2212yes—
23C4×#6122—24no—
24C2×#88110C2212yes—
25C2×C4⋊#1057C2212yes—
26C2×C4×#1057C2212yes—
27C2×#88071C411yescharacteristic
28C2×#88071C411yescharacteristic
29C4⋊C4×Dic13C411yescharacteristic
30C4⋊C4×Dic13C411yescharacteristic
31C4⋊C4×Dic13C411yescharacteristic
32C4⋊C4×Dic13C411yescharacteristic
33C4⋊C4×Dic13C2211yescharacteristic
34C4⋊C4×Dic13C2211yescharacteristic
35C4⋊C4×Dic13C2211yescharacteristic
36C4⋊C4×Dic13C2211yescharacteristic
37#11718290C2212yes—
38#11718290C2212yes—
39#11718290C2212yes—
40#11718290C2212yes—
41#11718290C2212yes—
42#11718290C2212yes—
43#11718290C2212yes—
44#11718290C2212yes—
45C23×C4×C13C2×C411yescharacteristic
46C23×C4×C13C2×C411yescharacteristic
47C23×C4×C13C2×C411yescharacteristic
48C23×Dic13C2×C411yescharacteristic
49C23×Dic13C2×C411yescharacteristic
50C23×Dic13C2×C411yescharacteristic
51C23×Dic13C2311yescharacteristic
52C2×#6122—22no—
53C2×#6122—22no—
54C2×#6122—22no—
55C2×#6122—22no—
56C2×#6122—22no—
57C2×#6122—22no—
58C2×#6122—22no—
59C2×#6122—22no—
60C2×#6122—22no—
61C2×#6122—22no—
62C2×#6122D414yes—
63C2×#6122Q814yes—
64C22×#1057C2312yes—
65C22×#1057C2312yes—
66C22×#1057C2312yes—
67C22×#1057C2312yes—
68C2×C13×C4⋊C4—24no—
69C2×C13×C4⋊C4C2×C411yescharacteristic
70C2×C13×C4⋊C4C2×C411yescharacteristic
71C2×C13×C4⋊C4C2×C411yescharacteristic
72C2×C13×C4⋊C4C2×C411yescharacteristic
73C2×C4⋊Dic13—24no—
74C2×C4⋊Dic13C2×C411yescharacteristic
75C2×C4⋊Dic13C2×C411yescharacteristic
76C2×C4⋊Dic13C2×C411yescharacteristic
77C2×C4⋊Dic13C2×C411yescharacteristic
78C2×C4×Dic13—22no—
79C2×C4×Dic13—22no—
80C2×C4×Dic13—22no—
81C2×C4×Dic13—22no—
82C2×C4×Dic13—22no—
83C2×C4×Dic13—22no—
84C2×C4×Dic13C2×C411yescharacteristic
85C2×C4×Dic13C2×C411yescharacteristic
86C2×C4×Dic13C2×C411yescharacteristic
87C2×C4×Dic13C2×C411yescharacteristic
88C2×C4×Dic13C2×C411yescharacteristic
89C2×C4×Dic13C2×C411yescharacteristic
90C2×C4×Dic13C2311yescharacteristic
91C2×C4×Dic13C2311yescharacteristic
92C2×C4×Dic13C2311yescharacteristic
93C2×C4×Dic13C2311yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C2×C4×Dic13C2311yescharacteristic
96#88112—24no—
97#88110—24no—
98C4⋊#1057—24no—
99C4×#1057—24no—
100#88071C2×C411yescharacteristic
101#88071C2×C411yescharacteristic
102#88071C2×C411yescharacteristic
103#88071C2×C411yescharacteristic
104#88071C2×C411yescharacteristic
105#88071C2×C411yescharacteristic
106#88071C2×C411yescharacteristic
107#88071C2×C411yescharacteristic
108#21023—131nomaximal, Sylow
109C24×C13C22×C411yescharacteristic
110C22×C4×C13—22no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—22no—
115C22×C4×C13—22no—
116C22×C4×C13—22no—
117C22×C4×C13—41no—
118C22×C4×C13—41no—
119C22×C4×C13C22×C411yescharacteristic
120C22×C4×C13C22×C411yescharacteristic
121C22×C4×C13C22×C411yescharacteristic
122C22×C4×C13C22×C411yescharacteristic
123C22×C4×C13C22×C411yescharacteristic
124C22×C4×C13C22×C411yescharacteristic
125C22×Dic13—22no—
126C22×Dic13—22no—
127C22×Dic13—22no—
128C22×Dic13—22no—
129C22×Dic13—22no—
130C22×Dic13—41no—
131C22×Dic13—41no—
132C22×Dic13—41no—
133C22×Dic13—41no—
134C22×Dic13C22×C411yescharacteristic
135C22×Dic13C22×C411yescharacteristic
136C22×Dic13C22×C411yescharacteristic
137C22×Dic13C22×C411yescharacteristic
138C22×Dic13C22×C411yescharacteristic
139C22×Dic13C22×C411yescharacteristic
140C22×Dic13C22×C411yescharacteristic
141C22×Dic13C2×D412yes—
142C22×Dic13C2×Q812yes—
143C22×Dic13#4112yes—
144C22×Dic13#4112yes—
145C22×Dic13#4112yes—
146C22×Dic13#4112yes—
147C22×Dic13C2411yescharacteristic
148#6122—28no—
149#6122—28no—
150#6122—44no—
151#6122—44no—
152C2×#1057—22no—
153C2×#1057—22no—
154C2×#1057—22no—
155C2×#1057—22no—
156C2×#1057—22no—
157C2×#1057—22no—
158C2×#1057—22no—
159C2×#1057—22no—
160C2×#1057—22no—
161C2×#1057—22no—
162C2×#1057C2×D414yes—
163C2×#1057C2×Q814yes—
164C13×C4⋊C4—22no—
165C13×C4⋊C4—22no—
166C13×C4⋊C4—22no—
167C13×C4⋊C4M1614yescomplemented
168C4⋊Dic13—22no—
169C4⋊Dic13—22no—
170C4⋊Dic13—22no—
171C4⋊Dic13M1614yescomplemented
172C4×Dic13—22no—
173C4×Dic13—22no—
174C4×Dic13—22no—
175C4×Dic13—22no—
176C4×Dic13—22no—
177C4×Dic13—22no—
178C4×Dic13—21no—
179C4×Dic13—21no—
180C4×Dic13—21no—
181C4×Dic13—21no—
182C4×Dic13—21no—
183C4×Dic13—21no—
184C4×Dic13—21no—
185C4×Dic13—21no—
186C4×Dic13—21no—
187C4×Dic13—21no—
188C4×Dic13—21no—
189C4×Dic13—21no—
190C2×#423—131no—
191C2×C4×M16—131no—
192C2×C4×C4⋊C4—131no—
193#1917—132no—
194#1917—132no—
195#1916—132no—
196#1916—132no—
197#1728—131no—
198#1728—131no—
199#1728—131no—
200#1728—131no—
201C23×C13—21no—
202C23×C13—21no—
203C23×C13—21no—
204C23×C13—21no—
205C23×C13#11712yes—
206C23×C13C4×D412yes—
207C23×C13C4×Q812yes—
208C23×C13C23×C411yescharacteristic
209C2×C4×C13—28no—
210C2×C4×C13—21no—
211C2×C4×C13—21no—
212C2×C4×C13—21no—
213C2×C4×C13—21no—
214C2×C4×C13—21no—
215C2×C4×C13—21no—
216C2×C4×C13—21no—
217C2×C4×C13—21no—
218C2×C4×C13—21no—
219C2×C4×C13—21no—
220C2×C4×C13—21no—
221C2×C4×C13—21no—
222C2×C4×C13—21no—
223C2×C4×C13—21no—
224C2×C4×C13—44no—
225C2×C4×C13—44no—
226C2×C4×C13C2×M1612yes—
227C2×C4×C13C2×M1612yes—
228C2×C4×C13C2×M1612yes—
229C2×C4×C13C2×M1612yes—
230C2×Dic13—21no—
231C2×Dic13—21no—
232C2×Dic13—21no—
233C2×Dic13—21no—
234C2×Dic13—21no—
235C2×Dic13—21no—
236C2×Dic13—21no—
237C2×Dic13—21no—
238C2×Dic13—21no—
239C2×Dic13—21no—
240C2×Dic13—21no—
241C2×Dic13—21no—
242C2×Dic13—21no—
243C2×Dic13—21no—
244C2×Dic13—21no—
245C2×Dic13—21no—
246C2×Dic13—44no—
247C2×Dic13—44no—
248C2×Dic13—44no—
249C2×Dic13#11712yes—
250C2×Dic13C4×D412yes—
251C2×Dic13C4×Q812yes—
252C2×Dic13C22⋊Q814yes—
253C2×Dic13#12314yes—
254C2×Dic13#12514yes—
255C2×Dic13#12614yes—
256C2×Dic13C2×M1612yes—
257C2×Dic13C2×M1612yes—
258C2×Dic13C22×D411yescharacteristic
259C2×Dic13C22×Q811yescharacteristic
260C2×Dic13C2×#4111yescharacteristic
261C2×Dic13C2×#4111yescharacteristic
262C2×Dic13C2×#4111yescharacteristic
263C2×Dic13C2×#4111yescharacteristic
264#1057—24no—
265#1057—24no—
266#1057—42no—
267#1057—42no—
268C22×M16—131no—
269C22×M16—131no—
270C22×M16—131no—
271C22×M16—131no—
272C22×C4⋊C4—131no—
273C22×C4⋊C4#26111yescharacteristic
274C22×C42—131no—
275C22×C42—131no—
276C22×C42—131no—
277#423—264no—
278C2×C4⋊C8—132no—
279C2×C4⋊C8—132no—
280C4×M16—264no—
281C2×#97—132no—
282C2×C4×C8—132no—
283C4×C4⋊C4—131no—
284C4×C4⋊C4—131no—
285C4×C4⋊C4—131no—
286C4×C4⋊C4—131no—
287C4×C4⋊C4—131no—
288C4×C4⋊C4—131no—
289C4×C4⋊C4—131no—
290C4×C4⋊C4—131no—
291C2×#95—131no—
292C2×#95—131no—
293#336—132no—
294#336—132no—
295#336—132no—
296#336—132no—
297#336—132no—
298#336—132no—
299#336—132no—
300#336—132no—
301C22×C13—22no—
302C22×C13—22no—
303C22×C13—22no—
304C22×C13—22no—
305C22×C13—22no—
306C22×C13—22no—
307C22×C13#38714yes—
308C22×C13C2×#11711yescharacteristic
309C22×C13C2×C4×D411yescharacteristic
310C22×C13C2×C4×Q811yescharacteristic
311C22×C13C22×M1611yescharacteristic
312C22×C13C22×M1611yescharacteristic
313C22×C13#56811yescharacteristic
314C22×C13#56811yescharacteristic
315C4×C13—22no—
316C4×C13—22no—
317C4×C13—22no—
318C4×C13—22no—
319C4×C13—41no—
320C4×C13—41no—
321Dic13—22no—
322Dic13—22no—
323Dic13—41no—
324Dic13—41no—
325Dic13—41no—
326Dic13—41no—
327Dic13#38714yes—
328Dic13C2×C22⋊Q811yescharacteristic
329Dic13C2×#12311yescharacteristic
330Dic13C2×#12511yescharacteristic
331Dic13C2×#12611yescharacteristic
332C23×C4—131no—
333C23×C4—131no—
334C23×C4—131no—
335C23×C4—131no—
336C23×C4C2×#26111yescharacteristic
337C23×C4C2×#26111yescharacteristic
338C23×C4C2×#26111yescharacteristic
339C2×M16—134no—
340C2×M16—134no—
341C2×M16—262no—
342C2×M16—262no—
343C2×M16—262no—
344C2×M16—262no—
345C2×M16—262no—
346C2×M16—262no—
347C2×M16—262no—
348C2×M16—262no—
349C2×M16—262no—
350C2×M16—262no—
351C22×C8—132no—
352C22×C8—132no—
353C22×C8—132no—
354C22×C8—132no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—131no—
357C2×C4⋊C4—131no—
358C2×C4⋊C4—131no—
359C2×C4⋊C4—131no—
360C2×C4⋊C4—264no—
361C2×C4⋊C4C2×#26111yescharacteristic
362C2×C4⋊C4C2×#26111yescharacteristic
363C2×C4⋊C4C2×#26111yescharacteristic
364C2×C4⋊C4C2×#26111yescharacteristic
365C2×C42—131no—
366C2×C42—131no—
367C2×C42—131no—
368C2×C42—131no—
369C2×C42—131no—
370C2×C42—131no—
371C2×C42—131no—
372C2×C42—131no—
373C2×C42—131no—
374C2×C42—131no—
375C2×C42—131no—
376C2×C42—131no—
377C2×C42—262no—
378C2×C42—262no—
379C2×C42—262no—
380C2×C42—262no—
381C2×C42—262no—
382C2×C42—262no—
383C4⋊C8—264no—
384C4⋊C8—264no—
385#97—264no—
386C4×C8—264no—
387#95—131no—
388#95—131no—
389#95—131no—
390#95—131no—
391#95—131no—
392#95—131no—
393#95—131no—
394#95—131no—
395C2×C13—24no—
396C2×C13C2×#38711yescharacteristic
397C2×C13#292011yescharacteristic
398C2×C13#292111yescharacteristic
399C2×C13D4×M1611yescharacteristic
400C2×C13D4×M1611yescharacteristic
401C2×C13Q8×M1611yescharacteristic
402C2×C13Q8×M1611yescharacteristic
403C24C22×#26111yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—132no—
414C22×C4—132no—
415C22×C4—132no—
416C22×C4—132no—
417C22×C4—132no—
418C22×C4—132no—
419C22×C4—131no—
420C22×C4—131no—
421C22×C4—131no—
422C22×C4—131no—
423C22×C4—131no—
424C22×C4—131no—
425C22×C4—131no—
426C22×C4—131no—
427C22×C4—262no—
428C22×C4—262no—
429C22×C4—262no—
430C22×C4—262no—
431C22×C4—262no—
432C22×C4—521no—
433C22×C4—521no—
434C22×C4—521no—
435C22×C4—521no—
436C22×C4C22×#26111yescharacteristic
437C22×C4C22×#26111yescharacteristic
438C22×C4C22×#26111yescharacteristic
439C22×C4C22×#26111yescharacteristic
440C22×C4C22×#26111yescharacteristic
441C22×C4C22×#26111yescharacteristic
442M16—268no—
443M16—268no—
444M16—524no—
445M16—524no—
446C2×C8—134no—
447C2×C8—134no—
448C2×C8—262no—
449C2×C8—262no—
450C2×C8—262no—
451C2×C8—262no—
452C2×C8—262no—
453C2×C8—262no—
454C2×C8—262no—
455C2×C8—262no—
456C2×C8—262no—
457C2×C8—262no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—134no—
462C4⋊C4—262no—
463C4⋊C4—262no—
464C4⋊C4—262no—
465C4⋊C4#612214yescomplemented
466C42—261no—
467C42—261no—
468C42—261no—
469C42—261no—
470C42—261no—
471C42—261no—
472C42—261no—
473C42—261no—
474C42—262no—
475C42—262no—
476C42—261no—
477C42—261no—
478C42—262no—
479C42—262no—
480C42—261no—
481C42—261no—
482C42—262no—
483C42—262no—
484C13#2102311yescharacteristic, complemented, Sylow
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23#8823712yes—
490C23D4×#26112yes—
491C23Q8×#26112yes—
492C23C23×#26111yescharacteristic
493C2×C4—28no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—134no—
511C2×C4—134no—
512C2×C4—134no—
513C2×C4—134no—
514C2×C4—131no—
515C2×C4—131no—
516C2×C4—132no—
517C2×C4—131no—
518C2×C4—131no—
519C2×C4—132no—
520C2×C4—131no—
521C2×C4—131no—
522C2×C4—132no—
523C2×C4—132no—
524C2×C4—132no—
525C2×C4—261no—
526C2×C4—261no—
527C2×C4—261no—
528C2×C4—261no—
529C2×C4—261no—
530C2×C4—261no—
531C2×C4—261no—
532C2×C4—261no—
533C2×C4—261no—
534C2×C4—261no—
535C2×C4—261no—
536C2×C4—261no—
537C2×C4—261no—
538C2×C4—261no—
539C2×C4—261no—
540C2×C4—261no—
541C2×C4—524no—
542C2×C4—524no—
543C2×C4—524no—
544C2×C4D13⋊M1612yes—
545C2×C4C2×#612212yes—
546C2×C4C2×#612212yes—
547C2×C4C2×#612212yes—
548C8—264no—
549C8—264no—
550C8—522no—
551C8—522no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22#1171908214yes—
559C22C2×D13⋊M1611yescharacteristic
560C22C2×#8823711yescharacteristic
561C22#1171962111yescharacteristic
562C22C2×D4×#26111yescharacteristic
563C22#1171962511yescharacteristic
564C22C2×Q8×#26111yescharacteristic
565C22C22×#612211yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—131no—
573C4—131no—
574C4—131no—
575C4—131no—
576C4—134no—
577C4—262no—
578C4—262no—
579C4—521no—
580C4—521no—
581C4—521no—
582C4—521no—
583C2—24no—
584C2C2×#1171908211yescharacteristic
585C2#1313151211yescharacteristic
586C2#1313151811yescharacteristic
587C2D4×#612211yescharacteristic
588C2#1313157611yescharacteristic
589C2Q8×#612211yescharacteristic
590C2#1313163711yescharacteristic
subgroupG/H
center, FrattiniC23C23×#261
commutator, socleC23×C13C23×C4
FittingC22×C13×C4⋊C4C4
prime radicalC24×C13C22×C4
cosocleC22×Dic13C24
hypercenterC22×C4⋊C4#261
hypocenterC13#21023
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #13130896C2semidirect
4. #13130893C2semidirect
5. #13130892C2semidirect
6. #13130889C2semidirect
167. C13×C4⋊C4M16semidirect
171. C4⋊Dic13M16semidirect
465. C4⋊C4#6122semidirect
484. C13#21023semidirect

Maximal subgroups

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

subgroupindex
C2×#117191292
C2×C4×#61222
#131308962
#131308932
#131308922
#131308892
#131304992
C2×C4⋊C4×Dic132
#2102313