Finite Group Databasebeta

← #15784898

Subgroups — #15784898

Order 3328.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#88146C211yesmaximal, characteristic
2C2×#11718637C211yesmaximal, characteristic, complemented
3C2×#11718648C211yesmaximal, characteristic
4C2×C4×#6103C211yesmaximal, characteristic
5#13113080C214yesmaximal
6#13113067C214yesmaximal, complemented
7#13113066C211yesmaximal, characteristic, complemented
8#13113065C211yesmaximal, characteristic, complemented
9C2×C13×#382C211yesmaximal, characteristic, complemented
10C23×C4×D13C2211yescharacteristic
11C22×C13×C4⋊C4C2211yescharacteristic
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C42×C13C2211yescharacteristic
14C22×#6103C2212yes—
15C22×#6103C2212yes—
16C22×#6103C2212yes—
17C22×C4⋊Dic13C2211yescharacteristic
18C22×C4⋊Dic13C2211yescharacteristic
19C22×#6101C2214yes—
20C22×C4×Dic13C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C2×#88146—24no—
23C2×#88146—24no—
24C2×#88146—24no—
25C13×#382—24no—
26C2×C13×#95C2214yes—
27C2×#88071C2212yes—
28C2×#88071C2214yes—
29C2×#88071C2214yes—
30C2×#88071C2212yes—
31#11718648—24no—
32#11718637—24no—
33C4×#6103C418yescomplemented
34C24×D13C2311yescharacteristic
35C23×C4×C13C2312yes—
36C23×C4×C13C2312yes—
37C23×C4×C13C2312yes—
38C23×C4×C13C2311yescharacteristic
39C23×Dic13C2312yes—
40C23×Dic13C2312yes—
41C23×Dic13C2311yescharacteristic
42C23×Dic13C2312yes—
43C22×C4×D13—22no—
44C22×C4×D13—22no—
45C22×C4×D13—22no—
46C22×C4×D13—22no—
47C22×C4×D13C2×C414yes—
48C22×C4×D13D414yes—
49C22×C4×D13Q814yes—
50C2×C13×C4⋊C4—22no—
51C2×C13×C4⋊C4—22no—
52C2×C13×C4⋊C4—22no—
53C2×C13×C4⋊C4—22no—
54C2×C13×C4⋊C4—22no—
55C2×C13×C4⋊C4D414yes—
56C2×C42×C13—22no—
57C2×C42×C13—22no—
58C2×C42×C13C2×C414yes—
59C2×#6103—24no—
60C2×#6103—24no—
61C2×#6103—24no—
62C2×#6103—24no—
63C2×#6103—24no—
64C2×#6103—24no—
65C2×#6103—44no—
66C2×#6103C2×C418yes—
67C2×C4⋊Dic13—22no—
68C2×C4⋊Dic13—22no—
69C2×C4⋊Dic13—22no—
70C2×C4⋊Dic13—22no—
71C2×C4⋊Dic13—22no—
72C2×C4⋊Dic13D414yes—
73C2×#6101—28no—
74C2×#6101—28no—
75C2×#6101—44no—
76C2×C4×Dic13—24no—
77C2×C4×Dic13—22no—
78C2×C4×Dic13—22no—
79C2×C4×Dic13—24no—
80C2×C4×Dic13—24no—
81C2×C4×Dic13C2×C414yes—
82#88146—232no—
83C13×#95—28no—
84#88071—24no—
85#88071—28no—
86#88071—44no—
87#88071C2×C418yes—
88#13879—131nomaximal, Sylow
89C24×C13C2411yescharacteristic
90C23×D13—21no—
91C23×D13—21no—
92C23×D13—22no—
93C23×D13—22no—
94C23×D13—21no—
95C23×D13—21no—
96C23×D13—22no—
97C23×D13—22no—
98C23×D13C22×C412yes—
99C23×D13C2×D412yes—
100C23×D13C2×Q812yes—
101C22×C4×C13—22no—
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—24no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—22no—
108C22×C4×C13—22no—
109C22×C4×C13—22no—
110C22×C4×C13—24no—
111C22×C4×C13—22no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—24no—
115C22×C4×C13—24no—
116C22×C4×C13C22×C414yes—
117C22×C4×C13C22×C412yes—
118C22×C4×C13C2×D412yes—
119C22×C4×C13C2×D412yes—
120C22×C4×C13C2×D414yes—
121C22×C4×C13C2×D412yes—
122C22×C4×C13C2×Q812yes—
123C22×C4×C13#4112yes—
124C22×C4×C13#4112yes—
125C22×Dic13—21no—
126C22×Dic13—21no—
127C22×Dic13—22no—
128C22×Dic13—22no—
129C22×Dic13—21no—
130C22×Dic13—21no—
131C22×Dic13—22no—
132C22×Dic13—22no—
133C22×Dic13—22no—
134C22×Dic13—22no—
135C22×Dic13—24no—
136C22×Dic13—24no—
137C22×Dic13—22no—
138C22×Dic13—24no—
139C22×Dic13—22no—
140C22×Dic13—22no—
141C22×Dic13—24no—
142C22×Dic13C22×C412yes—
143C22×Dic13C22×C414yes—
144C22×Dic13C2×D414yes—
145C22×Dic13C2×D412yes—
146C22×Dic13C2×Q812yes—
147C22×Dic13#4114yes—
148C22×Dic13#4114yes—
149C2×C4×D13—28no—
150C2×C4×D13—28no—
151C2×C4×D13—28no—
152C2×C4×D13—28no—
153C2×C4×D13—28no—
154C2×C4×D13—28no—
155C2×C4×D13C4⋊C4116yescomplemented
156C13×C4⋊C4—28no—
157C13×C4⋊C4—44no—
158C42×C13—28no—
159#6103—216no—
160#6103—216no—
161#6103—48no—
162C4⋊Dic13—28no—
163C4⋊Dic13—44no—
164#6101—416no—
165C4×Dic13—216no—
166C4×Dic13—28no—
167C22×#117—131no—
168C2×#384—131no—
169C2×#382D1311yescharacteristic, complemented
170C2×#380—131no—
171C2×C4×C22⋊C4—131no—
172#1439—134no—
173#1436—134no—
174C4⋊#95—131no—
175C4⋊#95—131no—
176C23×C13C23×C411yescharacteristic
177C23×C13C22×D411yescharacteristic
178C23×C13C22×D411yescharacteristic
179C23×C13C22×D411yescharacteristic
180C23×C13C22×Q811yescharacteristic
181C23×C13C2×#4112yes—
182C23×C13C2×#4112yes—
183C23×C13C2×#4111yescharacteristic
184C23×C13C2×#4111yescharacteristic
185C23×C13H(5,𝔽2)12yes—
186C23×C132−1+412yes—
187C22×D13—24no—
188C22×D13—22no—
189C22×D13—22no—
190C22×D13—24no—
191C22×D13—22no—
192C22×D13—22no—
193C22×D13—22no—
194C22×D13—22no—
195C22×D13—22no—
196C22×D13—22no—
197C22×D13—24no—
198C22×D13—22no—
199C22×D13—22no—
200C22×D13—24no—
201C22×D13—42no—
202C22×D13—42no—
203C22×D13—42no—
204C22×D13—42no—
205C22×D13—42no—
206C22×D13—42no—
207C22×D13—42no—
208C22×D13—42no—
209C22×D13C2×C4⋊C414yes—
210C2×C4×C13—24no—
211C2×C4×C13—24no—
212C2×C4×C13—24no—
213C2×C4×C13—24no—
214C2×C4×C13—28no—
215C2×C4×C13—42no—
216C2×C4×C13—42no—
217C2×C4×C13—42no—
218C2×C4×C13—44no—
219C2×C4×C13—42no—
220C2×C4×C13—42no—
221C2×C4×C13—42no—
222C2×C4×C13—42no—
223C2×C4×C13—42no—
224C2×C4×C13—42no—
225C2×C4×C13—44no—
226C2×C4×C13—44no—
227C2×C4×C13C2×C4⋊C414yes—
228C2×C4×C13C4×D414yes—
229C2×C4×C13C4×D414yes—
230C2×C4×C13#12014yes—
231C2×C4×C13C22⋊Q814yes—
232C2×C4×C13C22⋊Q814yes—
233C2×C4×C13#12314yes—
234C2×Dic13—24no—
235C2×Dic13—22no—
236C2×Dic13—22no—
237C2×Dic13—24no—
238C2×Dic13—22no—
239C2×Dic13—22no—
240C2×Dic13—22no—
241C2×Dic13—22no—
242C2×Dic13—24no—
243C2×Dic13—24no—
244C2×Dic13—24no—
245C2×Dic13—24no—
246C2×Dic13—24no—
247C2×Dic13—28no—
248C2×Dic13—42no—
249C2×Dic13—42no—
250C2×Dic13—44no—
251C2×Dic13—42no—
252C2×Dic13—42no—
253C2×Dic13—42no—
254C2×Dic13—44no—
255C2×Dic13—44no—
256C2×Dic13C2×C4⋊C414yes—
257C2×Dic13#11718yescomplemented
258C4×D13—232no—
259C24×C4—131no—
260C2×#117—264no—
261C2×#117—264no—
262C2×#117—264no—
263C22×C4⋊C4—131no—
264C22×C4⋊C4—131no—
265C22×C4⋊C4—134no—
266C22×C4⋊C4C2×D1311yescharacteristic
267C22×C4⋊C4C2×D1311yescharacteristic
268C22×C22⋊C4—132no—
269C22×C22⋊C4—132no—
270C22×C22⋊C4—132no—
271C22×C42—131no—
272C22×C42—132no—
273C22×C42C2×D1311yescharacteristic
274#384—264no—
275#382—24no—
276#380—264no—
277C4×C22⋊C4—138no—
278C2×#95—132no—
279C2×#95—134no—
280C2×#95—132no—
281C2×#95—134no—
282C2×#95C2×D1314yes—
283C22×C13C22×C4⋊C411yescharacteristic
284C22×C13C2×#11712yes—
285C22×C13C2×C4×D411yescharacteristic
286C22×C13C2×C4×D411yescharacteristic
287C22×C13#52012yes—
288C22×C13C2×#12011yescharacteristic
289C22×C13C2×C22⋊Q811yescharacteristic
290C22×C13C2×C22⋊Q811yescharacteristic
291C22×C13C2×#12311yescharacteristic
292C22×C13#53211yescharacteristic
293C22×C13#53211yescharacteristic
294C22×C13#53312yes—
295C22×C13#53511yescharacteristic
296C22×C13#53611yescharacteristic
297C22×C13#54412yes—
298C22×C13#54612yes—
299C22×C13#54712yes—
300C22×C13#54812yes—
301C22×C13#55012yes—
302C22×C13#55212yes—
303C22×C13#55312yes—
304C22×C13#55512yes—
305C22×C13#55612yes—
306C2×D13—24no—
307C2×D13—24no—
308C2×D13—24no—
309C2×D13—44no—
310C2×D13—44no—
311C2×D13—42no—
312C2×D13—42no—
313C2×D13—44no—
314C2×D13—44no—
315C2×D13—42no—
316C2×D13—42no—
317C4×C13—44no—
318C4×C13—44no—
319C4×C13—82no—
320C4×C13#38518yes—
321Dic13—28no—
322Dic13—24no—
323Dic13—44no—
324Dic13—82no—
325C25—131no—
326C23×C4—132no—
327C23×C4—131no—
328C23×C4—134no—
329C23×C4—134no—
330C23×C4—134no—
331C23×C4—132no—
332C23×C4—132no—
333C23×C4—262no—
334C23×C4—262no—
335C23×C4—262no—
336C23×C4—262no—
337C23×C4C22×D1312yes—
338C23×C4C22×D1312yes—
339C23×C4C22×D1312yes—
340C23×C4C22×D1311yescharacteristic
341#117—2632no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—134no—
348C2×C4⋊C4—262no—
349C2×C4⋊C4—262no—
350C2×C4⋊C4—268no—
351C2×C4⋊C4—268no—
352C2×C4⋊C4—262no—
353C2×C4⋊C4—262no—
354C2×C4⋊C4—262no—
355C2×C4⋊C4—524no—
356C2×C4⋊C4C22⋊D1314yes—
357C2×C22⋊C4—138no—
358C2×C22⋊C4—264no—
359C2×C22⋊C4—264no—
360C2×C22⋊C4—264no—
361C2×C22⋊C4—264no—
362C2×C22⋊C4—264no—
363C2×C22⋊C4—264no—
364C2×C22⋊C4—524no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—134no—
368C2×C42—264no—
369C2×C42—264no—
370C2×C42—262no—
371C2×C42—262no—
372C2×C42—264no—
373C2×C42C4×D1314yes—
374#95—28no—
375#95—138no—
376#95—268no—
377#95—264no—
378#95—524no—
379C2×C13#228511yescharacteristic
380C2×C13C2×#38511yescharacteristic
381C2×C13#231711yescharacteristic
382C2×C13#235212yes—
383C2×C13#235412yes—
384C2×C13#243111yescharacteristic
385C2×C13#243211yescharacteristic
386C2×C13#243912yes—
387C2×C13#244112yes—
388C2×C13#249911yescharacteristic
389C2×C13#250011yescharacteristic
390D13—44no—
391C24—132no—
392C24—132no—
393C24—132no—
394C24—261no—
395C24—261no—
396C24—261no—
397C24—262no—
398C24—262no—
399C24—262no—
400C24—262no—
401C24—261no—
402C24C23×D1311yescharacteristic
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—132no—
419C22×C4—134no—
420C22×C4—132no—
421C22×C4—132no—
422C22×C4—134no—
423C22×C4—134no—
424C22×C4—1316no—
425C22×C4—134no—
426C22×C4—261no—
427C22×C4—261no—
428C22×C4—262no—
429C22×C4—262no—
430C22×C4—261no—
431C22×C4—262no—
432C22×C4—262no—
433C22×C4—261no—
434C22×C4—268no—
435C22×C4—268no—
436C22×C4—268no—
437C22×C4—268no—
438C22×C4—268no—
439C22×C4—268no—
440C22×C4—262no—
441C22×C4—264no—
442C22×C4—264no—
443C22×C4—262no—
444C22×C4—262no—
445C22×C4—262no—
446C22×C4—262no—
447C22×C4—264no—
448C22×C4—264no—
449C22×C4C2×C4×D1314yes—
450C22×C4C2×C4×D1312yes—
451C22×C4#612712yes—
452C22×C4D4×D1312yes—
453C22×C4D4×D1312yes—
454C22×C4Q8×D1312yes—
455C22×C4#613112yes—
456C22×C4C2×C22⋊D1314yes—
457C22×C4C2×C22⋊D1312yes—
458C4⋊C4—28no—
459C4⋊C4—44no—
460C4⋊C4—268no—
461C4⋊C4—5216no—
462C4⋊C4—524no—
463C22⋊C4—2616no—
464C22⋊C4—2616no—
465C22⋊C4—528no—
466C42—28no—
467C42—2616no—
468C42—268no—
469C13#1387911yescharacteristic, complemented, Sylow
470C23—134no—
471C23—262no—
472C23—262no—
473C23—262no—
474C23—264no—
475C23—264no—
476C23—262no—
477C23—262no—
478C23—262no—
479C23—262no—
480C23—262no—
481C23—262no—
482C23—262no—
483C23—264no—
484C23—264no—
485C23—522no—
486C23—522no—
487C23—522no—
488C23—522no—
489C23—522no—
490C23—522no—
491C23—522no—
492C23—522no—
493C23C22×C4×D1311yescharacteristic
494C23C2×#612711yescharacteristic
495C23C2×D4×D1311yescharacteristic
496C23C2×D4×D1311yescharacteristic
497C23C2×Q8×D1311yescharacteristic
498C23C2×#613111yescharacteristic
499C23#41×D1312yes—
500C23#41×D1312yes—
501C23#8825612yes—
502C23#8825712yes—
503C23C22×C22⋊D1311yescharacteristic
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—134no—
522C2×C4—138no—
523C2×C4—264no—
524C2×C4—262no—
525C2×C4—262no—
526C2×C4—262no—
527C2×C4—264no—
528C2×C4—262no—
529C2×C4—262no—
530C2×C4—262no—
531C2×C4—264no—
532C2×C4—264no—
533C2×C4—264no—
534C2×C4—264no—
535C2×C4—2632no—
536C2×C4—264no—
537C2×C4—268no—
538C2×C4—524no—
539C2×C4—522no—
540C2×C4—522no—
541C2×C4—522no—
542C2×C4—524no—
543C2×C4—522no—
544C2×C4—522no—
545C2×C4—524no—
546C2×C4C4⋊C4×D1314yes—
547C2×C4#8814714yes—
548C2×C4#8814814yes—
549C2×C4#8815014yes—
550C2×C4C4×C22⋊D1314yes—
551C2×C4#8819114yes—
552C2×C4#8820014yes—
553C22—264no—
554C22—264no—
555C22—264no—
556C22—522no—
557C22—522no—
558C22—524no—
559C22—524no—
560C22—522no—
561C22—524no—
562C22—524no—
563C22—522no—
564C22C2×C4⋊C4×D1311yescharacteristic
565C22C2×#8814711yescharacteristic
566C22C2×#8814811yescharacteristic
567C22C2×#8815011yescharacteristic
568C22D13×#11712yes—
569C22#1171922812yes—
570C22#1171923712yes—
571C22#1171923812yes—
572C22#1171937311yescharacteristic
573C22#1171937311yescharacteristic
574C22#1171937911yescharacteristic
575C22#1171938011yescharacteristic
576C22#1171938112yes—
577C22#1171938212yes—
578C22#1171939312yes—
579C22#1171939412yes—
580C22#1171939912yes—
581C22#1171940012yes—
582C22C2×C4×C22⋊D1311yescharacteristic
583C22C2×#8819111yescharacteristic
584C22C2×#8820011yescharacteristic
585C22#1171952912yes—
586C22#1171953912yes—
587C4—44no—
588C4—44no—
589C4—82no—
590C4—264no—
591C4—268no—
592C4—524no—
593C4—1042no—
594C4#1171865018yes—
595C2—524no—
596C2#1311724211yescharacteristic
597C2C2×#1171865011yescharacteristic
598C2#1311746111yescharacteristic
599C2#1311776212yes—
600C2#1311778612yes—
601C2#1311845811yescharacteristic
602C2#1311846711yescharacteristic
603C2#1311853012yes—
604C2#1311854512yes—
605C2#1311929211yescharacteristic
606C2#1311930111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D13
commutatorC23×C13C23×C4
FittingC2×C13×#382C2
socle, cosocleC24×C13C24
prime radicalC24×D13C23
hypercenterC2×#382D13
hypocenterC13#13879
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#11718637C2semidirect
6. #13113067C2semidirect
7. #13113066C2semidirect
8. #13113065C2semidirect
9. C2×C13×#382C2semidirect
33. C4×#6103C4semidirect
155. C2×C4×D13C4⋊C4semidirect
169. C2×#382D13semidirect
257. C2×Dic13#117semidirect
469. C13#13879semidirect

Maximal subgroups

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

subgroupindex
C22×#881462
C2×#117186372
C2×#117186482
C2×C4×#61032
#131130802
#131130672
#131130662
#131130652
C2×C13×#3822
#1387913