Finite Group Databasebeta

← #15784229

Subgroups — #15784229

Order 3328.

1862 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic26C211yesmaximal, characteristic
2C2×#11718307C212yesmaximal
3C2×#11718310C211yesmaximal, characteristic
4C2×#11718309C211yesmaximal, characteristic
5#13113070C212yesmaximal
6#13113067C212yesmaximal
7#13113064C212yesmaximal
8#13113063C211yesmaximal, characteristic
9#13113060C211yesmaximal, characteristic
10#13113060C211yesmaximal, characteristic
11C13×#1434C211yesmaximal, characteristic
12C23×Dic26C2211yescharacteristic
13C22×C42×C13C2211yescharacteristic
14C22×C4⋊Dic13C2211yescharacteristic
15C22×#6101C2211yescharacteristic
16C22×#6101C2211yescharacteristic
17C22×#6101C2212yes—
18C22×C4×Dic13C2211yescharacteristic
19C22×C4×Dic13C2212yes—
20C22×C4×Dic13C2212yes—
21C22×C4×Dic13C2211yescharacteristic
22C2×C4×Dic26—24no—
23C2×C4×Dic26—24no—
24C2×C4×Dic26C418yescomplemented
25C2×C13×#95C2211yescharacteristic
26C2×C13×#95C2211yescharacteristic
27C2×C13×#95C2212yes—
28C2×C13×#95C2212yes—
29C2×#88071C2211yescharacteristic
30C2×#88071C2211yescharacteristic
31C2×#88071C2212yes—
32C2×#88071C2212yes—
33C2×#88071C2212yes—
34C2×#88071C2212yes—
35C2×#88071C2212yes—
36C2×#88071C2212yes—
37C2×#88071C2212yes—
38#11718310—24no—
39#11718309—24no—
40#11718307C4116yescomplemented
41C23×C4×C13C2312yes—
42C23×C4×C13C2312yes—
43C23×C4×C13C2311yescharacteristic
44C23×C4×C13C2311yescharacteristic
45C23×C4×C13C2311yescharacteristic
46C23×Dic13C2311yescharacteristic
47C23×Dic13C2312yes—
48C23×Dic13C2312yes—
49C23×Dic13C2311yescharacteristic
50C23×Dic13C2311yescharacteristic
51C23×Dic13C2311yescharacteristic
52C22×Dic26—22no—
53C22×Dic26—22no—
54C22×Dic26—24no—
55C22×Dic26C2×C414yes—
56C22×Dic26C2×C418yes—
57C2×C42×C13—22no—
58C2×C42×C13—22no—
59C2×C42×C13C2×C414yes—
60C2×C4⋊Dic13—22no—
61C2×C4⋊Dic13—22no—
62C2×C4⋊Dic13C2×C414yes—
63C2×#6101—22no—
64C2×#6101—22no—
65C2×#6101—22no—
66C2×#6101—22no—
67C2×#6101—24no—
68C2×#6101—24no—
69C2×#6101C2×C414yes—
70C2×#6101C2×C414yes—
71C2×#6101C2×C418yes—
72C2×C4×Dic13—22no—
73C2×C4×Dic13—22no—
74C2×C4×Dic13—24no—
75C2×C4×Dic13—24no—
76C2×C4×Dic13—24no—
77C2×C4×Dic13—42no—
78C2×C4×Dic13—42no—
79C2×C4×Dic13C2×C414yes—
80C2×C4×Dic13C2×C418yes—
81C2×C4×Dic13C2×C414yes—
82C2×C4×Dic13D414yes—
83C2×C4×Dic13D414yes—
84C4×Dic26—232no—
85C13×#95—22no—
86C13×#95—22no—
87C13×#95—42no—
88C13×#95C2×C418yes—
89#88071—22no—
90#88071—22no—
91#88071—24no—
92#88071—42no—
93#88071—42no—
94#88071—42no—
95#88071C2×C418yes—
96#88071C2×C418yes—
97#88071C2×C418yes—
98#13786—131nomaximal, Sylow
99C24×C13C2411yescharacteristic
100C22×C4×C13—21no—
101C22×C4×C13—21no—
102C22×C4×C13—22no—
103C22×C4×C13—22no—
104C22×C4×C13—22no—
105C22×C4×C13—22no—
106C22×C4×C13—22no—
107C22×C4×C13—21no—
108C22×C4×C13—21no—
109C22×C4×C13—22no—
110C22×C4×C13—22no—
111C22×C4×C13—21no—
112C22×C4×C13—22no—
113C22×C4×C13—22no—
114C22×C4×C13—21no—
115C22×C4×C13—22no—
116C22×C4×C13—21no—
117C22×C4×C13—22no—
118C22×C4×C13—22no—
119C22×C4×C13—21no—
120C22×C4×C13—22no—
121C22×C4×C13—22no—
122C22×C4×C13—22no—
123C22×C4×C13C22×C414yes—
124C22×C4×C13C22×C414yes—
125C22×C4×C13C22×C412yes—
126C22×C4×C13C22×C414yes—
127C22×C4×C13C22×C412yes—
128C22×C4×C13C22×C412yes—
129C22×C4×C13C2×D412yes—
130C22×C4×C13C2×D412yes—
131C22×Dic13—21no—
132C22×Dic13—21no—
133C22×Dic13—22no—
134C22×Dic13—21no—
135C22×Dic13—21no—
136C22×Dic13—21no—
137C22×Dic13—21no—
138C22×Dic13—22no—
139C22×Dic13—22no—
140C22×Dic13—21no—
141C22×Dic13—21no—
142C22×Dic13—22no—
143C22×Dic13—22no—
144C22×Dic13—21no—
145C22×Dic13—21no—
146C22×Dic13—22no—
147C22×Dic13—22no—
148C22×Dic13—22no—
149C22×Dic13—22no—
150C22×Dic13—22no—
151C22×Dic13—21no—
152C22×Dic13—21no—
153C22×Dic13—22no—
154C22×Dic13—22no—
155C22×Dic13—22no—
156C22×Dic13C22×C414yes—
157C22×Dic13C22×C414yes—
158C22×Dic13C22×C412yes—
159C22×Dic13C22×C414yes—
160C22×Dic13C22×C412yes—
161C22×Dic13C22×C412yes—
162C22×Dic13C22×C412yes—
163C22×Dic13C22×C414yes—
164C22×Dic13C2×D412yes—
165C22×Dic13C2×D412yes—
166C22×Dic13C2×D412yes—
167C22×Dic13C2×D412yes—
168C22×Dic13#4114yes—
169C2×Dic26—28no—
170C2×Dic26—28no—
171C2×Dic26—216no—
172C2×Dic26—216no—
173C2×Dic26C42116yescomplemented
174C42×C13—28no—
175C4⋊Dic13—28no—
176#6101—28no—
177#6101—28no—
178#6101—216no—
179C4×Dic13—28no—
180C4×Dic13—216no—
181C4×Dic13—48no—
182C4×Dic13C22⋊C4116yes—
183C22×C4×Q8—131no—
184C2×#391—132no—
185C2×C4×C4⋊C4—131no—
186C2×#376—131no—
187#1436—132no—
188#1436—132no—
189C4⋊#95—132no—
190#1434—131no—
191#1434—131no—
192#1434—131no—
193#1434D1311yescharacteristic
194C23×C13C23×C411yescharacteristic
195C23×C13C23×C412yes—
196C23×C13C22×D411yescharacteristic
197C23×C13C22×D411yescharacteristic
198C23×C13C2×#4112yes—
199C23×C13H(5,𝔽2)11yescharacteristic
200C23×C13H(5,𝔽2)11yescharacteristic
201C23×C132−1+412yes—
202C23×C132−1+412yes—
203C23×C132−1+411yescharacteristic
204C23×C132−1+411yescharacteristic
205C2×C4×C13—22no—
206C2×C4×C13—22no—
207C2×C4×C13—24no—
208C2×C4×C13—24no—
209C2×C4×C13—22no—
210C2×C4×C13—22no—
211C2×C4×C13—24no—
212C2×C4×C13—24no—
213C2×C4×C13—22no—
214C2×C4×C13—24no—
215C2×C4×C13—24no—
216C2×C4×C13—24no—
217C2×C4×C13—22no—
218C2×C4×C13—22no—
219C2×C4×C13—24no—
220C2×C4×C13—24no—
221C2×C4×C13—24no—
222C2×C4×C13—42no—
223C2×C4×C13—41no—
224C2×C4×C13—41no—
225C2×C4×C13—42no—
226C2×C4×C13—42no—
227C2×C4×C13—41no—
228C2×C4×C13—41no—
229C2×C4×C13—42no—
230C2×C4×C13—42no—
231C2×C4×C13—42no—
232C2×C4×C13—42no—
233C2×C4×C13—42no—
234C2×C4×C13C2×C4214yes—
235C2×C4×C13C2×C22⋊C414yes—
236C2×Dic13—22no—
237C2×Dic13—22no—
238C2×Dic13—24no—
239C2×Dic13—24no—
240C2×Dic13—22no—
241C2×Dic13—22no—
242C2×Dic13—22no—
243C2×Dic13—22no—
244C2×Dic13—22no—
245C2×Dic13—22no—
246C2×Dic13—22no—
247C2×Dic13—24no—
248C2×Dic13—24no—
249C2×Dic13—24no—
250C2×Dic13—22no—
251C2×Dic13—22no—
252C2×Dic13—24no—
253C2×Dic13—24no—
254C2×Dic13—24no—
255C2×Dic13—42no—
256C2×Dic13—41no—
257C2×Dic13—41no—
258C2×Dic13—42no—
259C2×Dic13—42no—
260C2×Dic13—41no—
261C2×Dic13—41no—
262C2×Dic13—42no—
263C2×Dic13—42no—
264C2×Dic13—42no—
265C2×Dic13—42no—
266C2×Dic13—42no—
267C2×Dic13C2×C4214yes—
268C2×Dic13C2×C4214yes—
269C2×Dic13C2×C22⋊C414yes—
270C2×Dic13C2×C22⋊C414yes—
271C2×Dic13#11714yes—
272C2×Dic13C4×D418yes—
273C2×Dic13C4×D418yes—
274Dic26—232no—
275C23×Q8—131no—
276C2×C4×Q8—138no—
277C2×C4×Q8—264no—
278C2×C4×Q8—264no—
279C22×C4⋊C4—131no—
280C22×C4⋊C4—131no—
281C22×C4⋊C4—132no—
282C22×C4⋊C4—131no—
283C22×C42—131no—
284C22×C42—131no—
285C22×C42—132no—
286C22×C42—132no—
287C22×C42C2×D1311yescharacteristic
288#391—1316no—
289C4×C4⋊C4—264no—
290#376—264no—
291C2×#95—132no—
292C2×#95—132no—
293C2×#95—132no—
294C2×#95—131no—
295C2×#95—131no—
296C2×#95—132no—
297C2×#95—132no—
298C2×#95—132no—
299C2×#95—132no—
300C2×#95C2×D1311yescharacteristic
301C2×#95C2×D1311yescharacteristic
302C2×#95C2×D1312yes—
303C2×#95C2×D1312yes—
304C22×C13C22×C4211yescharacteristic
305C22×C13C22×C22⋊C411yescharacteristic
306C22×C13C2×#11711yescharacteristic
307C22×C13C2×C4×D412yes—
308C22×C13C2×C4×D412yes—
309C22×C13#51911yescharacteristic
310C22×C13#51911yescharacteristic
311C22×C13#51912yes—
312C22×C13#51912yes—
313C22×C13#52011yescharacteristic
314C22×C13#52011yescharacteristic
315C22×C13#52012yes—
316C22×C13#52012yes—
317C22×C13#53511yescharacteristic
318C22×C13#53611yescharacteristic
319C22×C13#53612yes—
320C22×C13#53612yes—
321C22×C13#53711yescharacteristic
322C22×C13#53711yescharacteristic
323C22×C13#54112yes—
324C22×C13#54112yes—
325C22×C13#54212yes—
326C22×C13#54212yes—
327C4×C13—24no—
328C4×C13—24no—
329C4×C13—42no—
330C4×C13—44no—
331C4×C13—44no—
332Dic13—24no—
333Dic13—24no—
334Dic13—42no—
335Dic13—44no—
336Dic13—44no—
337Dic13C4×C22⋊C418yes—
338C22×Q8—134no—
339C22×Q8—138no—
340C22×Q8—262no—
341C22×Q8—262no—
342C22×Q8—264no—
343C23×C4—131no—
344C23×C4—131no—
345C23×C4—131no—
346C23×C4—131no—
347C23×C4—132no—
348C23×C4—132no—
349C23×C4C22×D1311yescharacteristic
350C23×C4C22×D1312yes—
351C23×C4C22×D1312yes—
352C23×C4C22×D1311yescharacteristic
353C23×C4C22×D1311yescharacteristic
354C4×Q8—2632no—
355C2×C4⋊C4—134no—
356C2×C4⋊C4—134no—
357C2×C4⋊C4—134no—
358C2×C4⋊C4—138no—
359C2×C4⋊C4—262no—
360C2×C4⋊C4—262no—
361C2×C4⋊C4—262no—
362C2×C4⋊C4—262no—
363C2×C4⋊C4—262no—
364C2×C4⋊C4—262no—
365C2×C4⋊C4—264no—
366C2×C4⋊C4—264no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—134no—
370C2×C42—134no—
371C2×C42—134no—
372C2×C42—134no—
373C2×C42—138no—
374C2×C42—264no—
375C2×C42—264no—
376C2×C42—262no—
377C2×C42—262no—
378C2×C42—264no—
379C2×C42—522no—
380C2×C42—522no—
381C2×C42C4×D1314yes—
382#95—22no—
383#95—22no—
384#95—42no—
385#95—138no—
386#95—138no—
387#95—138no—
388#95—264no—
389#95—262no—
390#95—262no—
391#95—522no—
392#95—522no—
393#95—522no—
394#95C4×D1318yes—
395C2×C13C2×C4×C22⋊C411yescharacteristic
396C2×C13#227611yescharacteristic
397C2×C13#227611yescharacteristic
398C2×C13#231011yescharacteristic
399C2×C13#231111yescharacteristic
400C2×C13#232011yescharacteristic
401C2×C13#232011yescharacteristic
402C2×C13#232312yes—
403C2×C13#232312yes—
404C2×C13#232612yes—
405C2×C13#232712yes—
406C24C23×D1311yescharacteristic
407C2×Q8—1316no—
408C2×Q8—268no—
409C2×Q8—268no—
410C2×Q8—2616no—
411C2×Q8—2616no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—132no—
436C22×C4—134no—
437C22×C4—132no—
438C22×C4—132no—
439C22×C4—134no—
440C22×C4—134no—
441C22×C4—132no—
442C22×C4—132no—
443C22×C4—132no—
444C22×C4—132no—
445C22×C4—134no—
446C22×C4—134no—
447C22×C4—132no—
448C22×C4—261no—
449C22×C4—261no—
450C22×C4—261no—
451C22×C4—261no—
452C22×C4—262no—
453C22×C4—261no—
454C22×C4—261no—
455C22×C4—262no—
456C22×C4—262no—
457C22×C4—261no—
458C22×C4—261no—
459C22×C4—261no—
460C22×C4—262no—
461C22×C4—261no—
462C22×C4—262no—
463C22×C4—262no—
464C22×C4—262no—
465C22×C4—261no—
466C22×C4—262no—
467C22×C4—262no—
468C22×C4—261no—
469C22×C4—262no—
470C22×C4—262no—
471C22×C4—262no—
472C22×C4—262no—
473C22×C4C2×C4×D1314yes—
474C22×C4C2×C4×D1314yes—
475C22×C4C2×C4×D1312yes—
476C22×C4C2×C4×D1314yes—
477C22×C4C2×C4×D1312yes—
478C22×C4C2×C4×D1312yes—
479C22×C4D4×D1312yes—
480C22×C4D4×D1312yes—
481C4⋊C4—268no—
482C4⋊C4—268no—
483C4⋊C4—268no—
484C4⋊C4—2616no—
485C42—28no—
486C42—1316no—
487C42—2616no—
488C42—268no—
489C42—528no—
490C13#1378611yescharacteristic, complemented, Sylow
491C23C22×C4×D1311yescharacteristic
492C23C22×C4×D1312yes—
493C23C2×D4×D1311yescharacteristic
494C23C2×D4×D1311yescharacteristic
495C23#8825111yescharacteristic
496C23#8825411yescharacteristic
497C23#41×D1312yes—
498C23#8825611yescharacteristic
499C23#8825711yescharacteristic
500C23#8825712yes—
501C23#8825712yes—
502Q8—2632no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—134no—
533C2×C4—134no—
534C2×C4—134no—
535C2×C4—138no—
536C2×C4—138no—
537C2×C4—134no—
538C2×C4—134no—
539C2×C4—262no—
540C2×C4—262no—
541C2×C4—262no—
542C2×C4—262no—
543C2×C4—264no—
544C2×C4—264no—
545C2×C4—262no—
546C2×C4—262no—
547C2×C4—262no—
548C2×C4—262no—
549C2×C4—262no—
550C2×C4—262no—
551C2×C4—264no—
552C2×C4—264no—
553C2×C4—262no—
554C2×C4—264no—
555C2×C4—264no—
556C2×C4—264no—
557C2×C4—264no—
558C2×C4—522no—
559C2×C4—521no—
560C2×C4—521no—
561C2×C4—521no—
562C2×C4—522no—
563C2×C4—521no—
564C2×C4—522no—
565C2×C4—522no—
566C2×C4—522no—
567C2×C4—522no—
568C2×C4—522no—
569C2×C4—522no—
570C2×C4C42×D1314yes—
571C2×C4C22⋊C4×D1314yes—
572C22C2×C42×D1311yescharacteristic
573C22C2×C22⋊C4×D1311yescharacteristic
574C22#1171920911yescharacteristic
575C22#1171922111yescharacteristic
576C22#1171922112yes—
577C22#1171922111yescharacteristic
578C22D13×#11711yescharacteristic
579C22#1171922811yescharacteristic
580C22#1171922812yes—
581C22C4×D4×D1312yes—
582C22C4×D4×D1312yes—
583C22#1171925112yes—
584C22#1171927712yes—
585C22#1171931211yescharacteristic
586C22#1171931311yescharacteristic
587C22#1171933311yescharacteristic
588C22#1171933411yescharacteristic
589C22#1171935312yes—
590C22#1171935312yes—
591C22#1171937412yes—
592C22#1171940112yes—
593C22#1171940612yes—
594C22#1171941512yes—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—44no—
599C4—42no—
600C4—138no—
601C4—264no—
602C4—264no—
603C4—524no—
604C4—524no—
605C4—522no—
606C2C4×C22⋊C4×D1311yescharacteristic
607C2#1311719911yescharacteristic
608C2#1311720011yescharacteristic
609C2#1311742111yescharacteristic
610C2#1311742511yescharacteristic
611C2#1311748411yescharacteristic
612C2#1311748611yescharacteristic
613C2#1311751512yes—
614C2#1311751712yes—
615C2#1311754512yes—
616C2#1311755112yes—
subgroupG/H
center, FrattiniC24C23×D13
commutatorC22×C13C22×C42
FittingC13×#1434C2
socle, prime radical, cosocleC24×C13C24
hypercenter#1434D13
hypocenterC13#13786
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
24. C2×C4×Dic26C4semidirect
40. #11718307C4semidirect
173. C2×Dic26C42semidirect
490. C13#13786semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic262
C2×#117183072
C2×#117183102
C2×#117183092
#131130702
#131130672
#131130642
#131130632
#131130602
C13×#14342
#1378613