Finite Group Databasebeta

← #15866306

Subgroups — #15866306

Order 3328.

1058 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718542C211yesmaximal, characteristic
2C2×#11718522C211yesmaximal, characteristic
3C2×#11718368C211yesmaximal, characteristic
4#13117317C211yesmaximal, characteristic
5D13×#389C211yesmaximal, characteristic
6#13115579C211yesmaximal, characteristic, complemented
7#13115576C211yesmaximal, characteristic, complemented
8#13115571C211yesmaximal, characteristic, complemented
9#13115570C211yesmaximal, characteristic, complemented
10#13115568C211yesmaximal, characteristic, complemented
11#13115563C211yesmaximal, characteristic, complemented
12#13115562C211yesmaximal, characteristic, complemented
13#13114554C211yesmaximal, characteristic
14#13114554C211yesmaximal, characteristic
15C13×#1943C211yesmaximal, characteristic, complemented
16C22×#6094C2211yescharacteristic
17C22×#6094C2211yescharacteristic
18C2×#88146C2211yescharacteristic
19C2×#88146C2211yescharacteristic
20C2×C4⋊C4×D13—21no—
21C2×C4⋊C4×D13C2211yescharacteristic
22C2×C4⋊C4×D13C2211yescharacteristic
23C2×C42×D13C2211yescharacteristic
24C2×C13×#107C2211yescharacteristic
25C2×C13×#106C2211yescharacteristic
26C2×C13×#97C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C13×#389C2211yescharacteristic
29C2×#88058C2211yescharacteristic
30C2×#88057C2211yescharacteristic
31C2×C8×Dic13C2211yescharacteristic
32C2×C8×Dic13C2211yescharacteristic
33#11718650—21no—
34#11718646C2211yescharacteristic
35#11718646C2211yescharacteristic
36C4⋊C4×Dic13—21no—
37#11718634C2211yescharacteristic
38#11718634C2211yescharacteristic
39#11718633—21no—
40C2×#88048C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44#11718589C2211yescharacteristic
45#11718589C2211yescharacteristic
46C2×#88043C2211yescharacteristic
47#11718542C418yescomplemented
48#11718522—24no—
49#11718368—24no—
50#11718329—21no—
51D13×#95—21no—
52#11718310—21no—
53#11718308—21no—
54C13×#340C2211yescharacteristic
55C13×#340C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718121C2211yescharacteristic
62C22×C4×D13—21no—
63C22×C4×D13—21no—
64C22×C4×D13D411yescharacteristic
65C22×C4×D13D411yescharacteristic
66C22×C4×D13C2311yescharacteristic
67C22×C8×C13C2311yescharacteristic
68C22×C8×C13C2311yescharacteristic
69C2×C13×C4⋊C4—21no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4C2311yescharacteristic
72C2×C13×C4⋊C4C2311yescharacteristic
73C2×C13×C4⋊C4C2311yescharacteristic
74C2×C13×C4⋊C4C2311yescharacteristic
75C2×C42×C13C2311yescharacteristic
76C2×#6103—21no—
77C2×#6103—21no—
78C2×C4⋊Dic13—21no—
79C2×C4⋊Dic13—21no—
80C2×C4⋊Dic13C2311yescharacteristic
81C2×C4⋊Dic13C2311yescharacteristic
82C2×C4⋊Dic13C2311yescharacteristic
83C2×C4⋊Dic13C2311yescharacteristic
84C2×#6101—21no—
85C2×#6101—21no—
86C2×#6101—21no—
87C2×#6101—21no—
88C2×#6101—21no—
89C2×#6101—21no—
90C2×C4×Dic13—21no—
91C2×C4×Dic13—21no—
92C2×C4×Dic13D411yescharacteristic
93C2×C4×Dic13D411yescharacteristic
94C2×C4×Dic13C2311yescharacteristic
95C22×#1055C2311yescharacteristic
96C22×#1055C2311yescharacteristic
97C2×#6094—22no—
98C2×#6094—22no—
99C2×#6094C2×C414yes—
100C2×#6094D414yes—
101C2×#6094Q814yes—
102C2×#6094Q814yes—
103#88146—22no—
104#88146—22no—
105#88146—22no—
106#88146C2×C414yes—
107C4⋊C4×D13—22no—
108C4⋊C4×D13—22no—
109C4⋊C4×D13—22no—
110C4⋊C4×D13—44no—
111C4⋊C4×D13C2×C414yes—
112C42×D13—22no—
113C42×D13—22no—
114C13×#107C2×C414yescomplemented
115C13×#106—22no—
116C13×#97—22no—
117C13×#95—21no—
118C13×#95—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88058—22no—
126#88057C2×C414yescomplemented
127C8×Dic13—22no—
128C8×Dic13—22no—
129#88048—22no—
130#88048C2×C414yescomplemented
131#88047—22no—
132#88047C2×C414yescomplemented
133#88043—22no—
134#21159—131nomaximal, Sylow
135C23×D13C2×D411yescharacteristic
136C22×C4×C13—21no—
137C22×C4×C13—21no—
138C22×C4×C13—21no—
139C22×C4×C13—21no—
140C22×C4×C13C2×D411yescharacteristic
141C22×C4×C13C2×D411yescharacteristic
142C22×C4×C13C2411yescharacteristic
143C22×Dic13—21no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—21no—
147C22×Dic13C2×D411yescharacteristic
148C22×Dic13C2×D411yescharacteristic
149C22×Dic13C2×D411yescharacteristic
150C2×C4×D13—24no—
151C2×C4×D13—22no—
152C2×C4×D13—22no—
153C2×C4×D13—22no—
154C2×C4×D13—22no—
155C2×C4×D13—41no—
156C2×C4×D13—41no—
157C2×C4×D13—42no—
158C2×C4×D13—42no—
159C2×C4×D13—42no—
160C2×C4×D13—42no—
161C2×C4×D13—42no—
162C2×C4×D13C22×C412yes—
163C2×C4×D13C2×D412yes—
164C2×C4×D13C2×Q812yes—
165C2×C4×D13C2×Q812yes—
166C2×C4×D13#4112yes—
167C2×C4×D13#4112yes—
168C2×C8×C13—21no—
169C2×C8×C13—21no—
170C2×C8×C13C22×C412yes—
171C2×C8×C13C2×D412yes—
172C2×C8×C13C2×Q812yes—
173C2×C8×C13C2×Q812yes—
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4—21no—
180C13×C4⋊C4—42no—
181C13×C4⋊C4—42no—
182C13×C4⋊C4C22×C412yes—
183C13×C4⋊C4C22×C412yes—
184C42×C13—21no—
185C42×C13—21no—
186#6103—24no—
187#6103—42no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—21no—
194C4⋊Dic13—42no—
195C4⋊Dic13—42no—
196C4⋊Dic13C22×C412yes—
197C4⋊Dic13C22×C412yes—
198#6101—24no—
199#6101—42no—
200#6101—42no—
201#6101—42no—
202#6101—42no—
203#6101—42no—
204C4×Dic13—21no—
205C4×Dic13—21no—
206C4×Dic13—24no—
207C4×Dic13—22no—
208C4×Dic13—22no—
209C4×Dic13—42no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055C22×C412yes—
213C2×#1055C2×D412yes—
214C2×#1055C2×Q812yes—
215C2×#1055C2×Q812yes—
216#6094—28no—
217#6094C4⋊C4116yescomplemented
218C2×#428—131no—
219C2×#428—131no—
220C2×C4×M16—131no—
221#2296—131no—
222C2×#389—131no—
223#1944—131no—
224#1944—131no—
225#1943—131no—
226#1943D1311yescharacteristic, complemented
227#1942—131no—
228#1942—131no—
229#1941—131no—
230#1941—131no—
231#1736—131no—
232#1736—131no—
233C23×C13C22×D411yescharacteristic
234C22×D13—21no—
235C22×D13—21no—
236C22×D13—21no—
237C22×D13—21no—
238C22×D13C4×D412yes—
239C22×D13#12112yes—
240C22×D13C4⋊Q812yes—
241C2×C4×C13—22no—
242C2×C4×C13—22no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—41no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13—41no—
254C2×C4×C13—41no—
255C2×C4×C13—41no—
256C2×C4×C13—41no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13C23×C411yescharacteristic
260C2×C4×C13C22×D411yescharacteristic
261C2×C4×C13C22×Q811yescharacteristic
262C2×C4×C13C22×Q811yescharacteristic
263C2×C4×C13C2×#4111yescharacteristic
264C2×C4×C13C2×#4111yescharacteristic
265C2×Dic13—21no—
266C2×Dic13—21no—
267C2×Dic13—21no—
268C2×Dic13—21no—
269C2×Dic13—22no—
270C2×Dic13—22no—
271C2×Dic13—41no—
272C2×Dic13—41no—
273C2×Dic13—41no—
274C2×Dic13—41no—
275C2×Dic13—41no—
276C2×Dic13—41no—
277C2×Dic13—41no—
278C2×Dic13—41no—
279C2×Dic13—41no—
280C2×Dic13—41no—
281C2×Dic13C4×D412yes—
282C2×Dic13#12112yes—
283C2×Dic13C4⋊Q812yes—
284C2×Dic13AGL(1,ℤ/8)12yes—
285C2×Dic13#13712yes—
286C4×D13—44no—
287C4×D13—44no—
288C4×D13—48no—
289C4×D13—84no—
290C4×D13C2×C4⋊C414yes—
291C4×D13C4×Q814yes—
292C4×D13C22⋊Q814yes—
293C4×D13#12514yes—
294C8×C13—22no—
295C8×C13C2×C4⋊C414yescomplemented
296#1055—22no—
297#1055C2×C4⋊C414yescomplemented
298C22×M16—131no—
299C22×M16—131no—
300C2×#117—131no—
301C2×#117—131no—
302C22×C4⋊C4—131no—
303C22×C4⋊C4—131no—
304C22×C4⋊C4—261no—
305C22×C42—131no—
306#428—138no—
307#428—264no—
308C2×#107—131no—
309C2×#107—131no—
310C2×#107—131no—
311C2×#107C2×D1311yescharacteristic
312C2×#106—131no—
313C2×#106—131no—
314C2×#106—131no—
315C2×#106C2×D1311yescharacteristic
316C4×M16—264no—
317C2×#97—131no—
318C2×#97C2×D1311yescharacteristic
319C2×C4×C8—131no—
320C2×C4×C8—131no—
321#389—131no—
322#389—131no—
323#389—131no—
324#389—131no—
325#389—131no—
326#389—131no—
327#389—261no—
328#389—261no—
329#389C2×D1311yescharacteristic
330#389C2×D1311yescharacteristic
331#385—261no—
332#385—261no—
333C4×C4⋊C4—261no—
334C4×C4⋊C4—261no—
335C2×#95—261no—
336#340—131no—
337#340—131no—
338#340—131no—
339#340—131no—
340#340—131no—
341#340—131no—
342#340C2×D1311yescharacteristic
343#340C2×D1311yescharacteristic
344C22×C13C2×C4×D411yescharacteristic
345C22×C13C2×#12111yescharacteristic
346C22×C13C2×C4⋊Q811yescharacteristic
347C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C13C2×#13711yescharacteristic
349C22×C13#57511yescharacteristic
350C22×C13#57511yescharacteristic
351C2×D13—22no—
352C2×D13—22no—
353C2×D13—22no—
354C2×D13—22no—
355C2×D13—22no—
356C2×D13—22no—
357C2×D13#38914yes—
358C4×C13—42no—
359C4×C13—42no—
360C4×C13—41no—
361C4×C13—41no—
362C4×C13—81no—
363C4×C13—81no—
364C4×C13C22×C4⋊C411yescharacteristic
365C4×C13C2×C4×Q811yescharacteristic
366C4×C13C2×C22⋊Q811yescharacteristic
367C4×C13C2×#12511yescharacteristic
368Dic13—22no—
369Dic13—22no—
370Dic13—42no—
371Dic13—42no—
372Dic13—81no—
373Dic13—81no—
374Dic13#38914yes—
375C23×C4—131no—
376C23×C4—131no—
377C23×C4—131no—
378C23×C4—261no—
379C23×C4—261no—
380C2×M16—134no—
381C2×M16—134no—
382C2×M16—134no—
383C2×M16—134no—
384C2×M16—262no—
385C2×M16—262no—
386C22×C8—131no—
387C22×C8—131no—
388C22×C8C22×D1311yescharacteristic
389C22×C8C22×D1311yescharacteristic
390#117—134no—
391#117—262no—
392#117—262no—
393#117—262no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—131no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—134no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—262no—
402C2×C4⋊C4—262no—
403C2×C4⋊C4—262no—
404C2×C4⋊C4—261no—
405C2×C4⋊C4—261no—
406C2×C4⋊C4—261no—
407C2×C4⋊C4—261no—
408C2×C4⋊C4—261no—
409C2×C4⋊C4—261no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4—524no—
413C2×C4⋊C4C22×D1311yescharacteristic
414C2×C4⋊C4C22×D1311yescharacteristic
415C2×C4⋊C4C22×D1311yescharacteristic
416C2×C4⋊C4C22×D1311yescharacteristic
417C2×C22⋊C4—261no—
418C2×C22⋊C4—261no—
419C2×C42—131no—
420C2×C42—131no—
421C2×C42—131no—
422C2×C42—262no—
423C2×C42—262no—
424C2×C42—261no—
425C2×C42—261no—
426C2×C42C22×D1311yescharacteristic
427#107—134no—
428#107—262no—
429#107—262no—
430#107C4×D1314yescomplemented
431#106—22no—
432#106—134no—
433#106—134no—
434#106—262no—
435#97—22no—
436#97—262no—
437C4×C8—262no—
438C4×C8—262no—
439#95—21no—
440#95—21no—
441#95—261no—
442#95—261no—
443#95—261no—
444#95—261no—
445#95—261no—
446#95—261no—
447C2×C13C2×#38911yescharacteristic
448C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
449C2×C13C4×#13711yescharacteristic
450C2×C13#305511yescharacteristic
451C2×C13#305611yescharacteristic
452C2×C13#316511yescharacteristic
453C2×C13#316611yescharacteristic
454D13—24no—
455C24—131no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—131no—
461C22×C4—132no—
462C22×C4—132no—
463C22×C4—132no—
464C22×C4—132no—
465C22×C4—132no—
466C22×C4—132no—
467C22×C4—131no—
468C22×C4—131no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4—264no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4—261no—
477C22×C4—261no—
478C22×C4—521no—
479C22×C4—521no—
480C22×C4—522no—
481C22×C4—522no—
482C22×C4—522no—
483C22×C4—522no—
484C22×C4—522no—
485C22×C4D4×D1311yescharacteristic
486C22×C4D4×D1311yescharacteristic
487C22×C4C23×D1311yescharacteristic
488M16—1316no—
489M16—268no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—132no—
493C2×C8—132no—
494C2×C8—132no—
495C2×C8—132no—
496C2×C8—261no—
497C2×C8—261no—
498C2×C8C2×C4×D1312yes—
499C2×C8D4×D1312yes—
500C2×C8Q8×D1312yes—
501C2×C8Q8×D1312yes—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—132no—
511C4⋊C4—132no—
512C4⋊C4—264no—
513C4⋊C4—261no—
514C4⋊C4—261no—
515C4⋊C4—261no—
516C4⋊C4—261no—
517C4⋊C4—261no—
518C4⋊C4—261no—
519C4⋊C4—522no—
520C4⋊C4—522no—
521C4⋊C4—522no—
522C4⋊C4—522no—
523C4⋊C4—522no—
524C4⋊C4—522no—
525C4⋊C4—522no—
526C4⋊C4C2×C4×D1312yes—
527C4⋊C4C2×C4×D1312yes—
528C22⋊C4—264no—
529C22⋊C4—522no—
530C42—21no—
531C42—21no—
532C42—262no—
533C42—262no—
534C42—264no—
535C42—261no—
536C42—261no—
537C42—522no—
538C13#2115911yescharacteristic, complemented, Sylow
539C23—132no—
540C23—132no—
541C23—132no—
542C23—261no—
543C23—261no—
544C23—261no—
545C23—261no—
546C23C2×D4×D1311yescharacteristic
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—132no—
566C2×C4—132no—
567C2×C4—132no—
568C2×C4—134no—
569C2×C4—134no—
570C2×C4—134no—
571C2×C4—134no—
572C2×C4—132no—
573C2×C4—132no—
574C2×C4—261no—
575C2×C4—261no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—262no—
579C2×C4—262no—
580C2×C4—524no—
581C2×C4—524no—
582C2×C4—528no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4—521no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—521no—
591C2×C4—521no—
592C2×C4—521no—
593C2×C4—1044no—
594C2×C4C22×C4×D1311yescharacteristic
595C2×C4C2×D4×D1311yescharacteristic
596C2×C4C2×Q8×D1311yescharacteristic
597C2×C4C2×Q8×D1311yescharacteristic
598C2×C4#41×D1311yescharacteristic
599C2×C4#41×D1311yescharacteristic
600C8—22no—
601C8—134no—
602C8—262no—
603C8C4⋊C4×D1314yescomplemented
604C22—134no—
605C22—262no—
606C22—262no—
607C22—262no—
608C22—262no—
609C22—262no—
610C22—262no—
611C22C4×D4×D1311yescharacteristic
612C22D13×#12111yescharacteristic
613C22D13×C4⋊Q811yescharacteristic
614C22D13×AGL(1,ℤ/8)11yescharacteristic
615C22#1171947811yescharacteristic
616C22D13×#13711yescharacteristic
617C22#1171948211yescharacteristic
618C4—42no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—81no—
623C4—81no—
624C4—134no—
625C4—262no—
626C4—262no—
627C4—522no—
628C4—522no—
629C4—1041no—
630C4—1041no—
631C4C2×C4⋊C4×D1311yescharacteristic
632C4C4×Q8×D1311yescharacteristic
633C4D13×C22⋊Q811yescharacteristic
634C4D13×#12511yescharacteristic
635C2—264no—
636C2D13×#38911yescharacteristic
637C2#1312364711yescharacteristic
638C2#1312365211yescharacteristic
639C2#1312445011yescharacteristic
640C2#1312446111yescharacteristic
641C2#1312538111yescharacteristic
642C2#1312538711yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#1943C2
socleC23×C13C22×D4
prime radicalC23×D13C2×D4
cosocleC22×C4×C13C24
hypercenter#1943D13
hypocenterC13#21159
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1943

Complements

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

NH ≅ G/Ntype
6. #13115579C2semidirect
7. #13115576C2semidirect
8. #13115571C2semidirect
9. #13115570C2semidirect
10. #13115568C2semidirect
11. #13115563C2semidirect
12. #13115562C2semidirect
15. C13×#1943C2semidirect
47. #11718542C4semidirect
114. C13×#107C2×C4semidirect
126. #88057C2×C4semidirect
130. #88048C2×C4semidirect
132. #88047C2×C4semidirect
217. #6094C4⋊C4semidirect
226. #1943D13semidirect
295. C8×C13C2×C4⋊C4semidirect
297. #1055C2×C4⋊C4semidirect
430. #107C4×D13semidirect
538. C13#21159semidirect
603. C8C4⋊C4×D13semidirect

Maximal subgroups

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

subgroupindex
C2×#117185422
C2×#117185222
C2×#117183682
#131173172
D13×#3892
#131155792
#131155762
#131155712
#131155702
#131155682
#131155632
#131155622
#131145542
C13×#19432
#2115913