Finite Group Databasebeta

← #16022749

Subgroups — #16022749

Order 3328.

3232 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719536C211yesmaximal, characteristic
2C2×#41×Dic13C211yesmaximal, characteristic, complemented
3C2×C4×Q8×D13C211yesmaximal, characteristic, complemented
4#13117906C213yesmaximal, complemented
5#13117861C213yesmaximal, complemented
6#13117857C213yesmaximal, complemented
7#13117758C213yesmaximal, complemented
8#13117754C211yesmaximal, characteristic, complemented
9Q8×#6103C211yesmaximal, characteristic, complemented
10#13117383C213yesmaximal, complemented
11#13117364C213yesmaximal, complemented
12#13117352C213yesmaximal, complemented
13C4×Q8×Dic13C211yesmaximal, characteristic, complemented
14C4×#88135C213yesmaximal, complemented
15C13×#2341C211yesmaximal, characteristic, complemented
16C22×C13×#41C2211yescharacteristic
17C22×Q8×D13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C2×C4×Q8×C13C2211yescharacteristic
20#11719536C4116yescomplemented
21#41×Dic13—28no—
22C2×Q8×Dic13C2211yescharacteristic
23C2×Q8×Dic13C2211yescharacteristic
24C2×Q8×Dic13C2211yescharacteristic
25C2×#88192C2213yes—
26C2×#88190C2213yes—
27C2×D4×Dic13C2213yes—
28C2×D4×Dic13C2213yes—
29C2×C4×C22⋊D13C2213yes—
30C2×#88180C2213yes—
31C2×#88178C2213yes—
32C22×C4×Dic13C2213yes—
33C4×Q8×D13—28no—
34C2×C4⋊C4×D13C2213yes—
35C2×#88141C2213yes—
36C2×#88135C2213yes—
37C2×#88135C2213yes—
38C2×C42×D13C2213yes—
39C2×C4×Dic26C2213yes—
40C13×#390C2213yes—
41C13×#390C2213yes—
42C13×#380C2213yes—
43C4×C13×C22⋊C4C2213yes—
44#11718648C2213yes—
45#11718638C2213yes—
46C4⋊C4×Dic13C2213yes—
47C4⋊C4×Dic13C2213yes—
48#11718635C2213yes—
49#11718634C2213yes—
50#11718617C2216yes—
51#11718607C2216yes—
52#11718606C2216yes—
53#11718605C2216yes—
54#11718605C2216yes—
55C22⋊C4×Dic13C2216yes—
56C22⋊C4×Dic13C2213yes—
57C22⋊C4×Dic13C2213yes—
58C4×#6103C2213yes—
59C4×C4⋊Dic13C2213yes—
60C4×#6101C2213yes—
61C42×Dic13C2213yes—
62#11718590C2213yes—
63#11718589C2213yes—
64#11718327C2213yes—
65#11718327C2211yescharacteristic
66#11718313C2216yes—
67#11718310C2213yes—
68#11718310C2213yes—
69C2×C13×#41—24no—
70C2×C13×#41—24no—
71C2×C13×#41C2×C418yescomplemented
72C22×Q8×C13C2311yescharacteristic
73C22×D4×C13C2313yes—
74C23×C4×C13C2313yes—
75C22×C22⋊D13C2313yes—
76C23×Dic13C2313yes—
77C23×Dic13C2313yes—
78C2×Q8×D13—24no—
79C2×Q8×D13—24no—
80C2×Q8×D13C2×C418yescomplemented
81C22×C4×D13C2313yes—
82C22×C4×D13C2313yes—
83C22×C4×D13C2311yescharacteristic
84C22×Dic26C2313yes—
85C4×Q8×C13—24no—
86C4×Q8×C13—24no—
87C2×C13×C4⋊C4C2313yes—
88C2×C13×C4⋊C4C2313yes—
89C2×C13×C22⋊C4C2313yes—
90C2×C13×C22⋊C4C2313yes—
91C2×C13×C22⋊C4C2316yes—
92C2×C42×C13C2313yes—
93C2×C42×C13C2313yes—
94C2×C22⋊Dic13C2316yes—
95C2×C22⋊Dic13C2316yes—
96C2×C22⋊Dic13C2313yes—
97Q8×Dic13—24no—
98Q8×Dic13C2×C418yescomplemented
99Q8×Dic13D418yescomplemented
100#88192C2×C4124yescomplemented
101#88190C2×C4124yescomplemented
102D4×Dic13—212no—
103D4×Dic13—212no—
104C4×C22⋊D13C2×C4124yescomplemented
105C2×#6103C2313yes—
106C2×#6103C2311yescharacteristic
107C2×#6103C2311yescharacteristic
108#88180—212no—
109C2×C4⋊Dic13C2313yes—
110C2×C4⋊Dic13C2313yes—
111C2×C4⋊Dic13C2313yes—
112#88178C2×C4124yescomplemented
113C2×#6101C2316yes—
114C2×#6101C2313yes—
115C2×#6101C2313yes—
116C2×#6101C2313yes—
117C2×C4×Dic13—212no—
118C2×C4×Dic13C2313yes—
119C2×C4×Dic13C2313yes—
120C2×C4×Dic13C2313yes—
121C2×C4×Dic13C2313yes—
122C2×C4×Dic13C2311yescharacteristic
123C2×C4×Dic13C2313yes—
124C2×C4×Dic13C2313yes—
125C2×C4×Dic13C2311yescharacteristic
126C2×C4×Dic13C2313yes—
127C2×C4×Dic13C2313yes—
128C2×C4×Dic13C2313yes—
129C4⋊C4×D13—212no—
130#88141—212no—
131#88135—212no—
132#88135—212no—
133C42×D13—212no—
134C4×Dic26—212no—
135C13×#95C2311yescharacteristic
136C13×#95C2313yes—
137#88071C2313yes—
138#88071C2313yes—
139#88071C2316yes—
140#88071C2316yes—
141#88071C2316yes—
142#88071C2311yescharacteristic
143#88071C2313yes—
144C4×#548—131nomaximal, Sylow
145C24×C13C2413yes—
146C23×D13C2411yescharacteristic
147C13×#41—232no—
148C2×Q8×C13C22×C414yes—
149C2×Q8×C13C2×D414yes—
150C2×Q8×C13#4114yes—
151C2×D4×C13—26no—
152C2×D4×C13—26no—
153C2×D4×C13C22×C4112yes—
154C22×C4×C13—26no—
155C22×C4×C13—26no—
156C22×C4×C13C22×C4112yes—
157C22×C4×C13C2413yes—
158C22×C4×C13C2413yes—
159C22×C4×C13C2411yescharacteristic
160C22×C4×C13C2411yescharacteristic
161C22×C4×C13C2411yescharacteristic
162C22×C4×C13C2413yes—
163C2×C22⋊D13—26no—
164C2×C22⋊D13—26no—
165C2×C22⋊D13C22×C4112yes—
166C22×Dic13—26no—
167C22×Dic13—26no—
168C22×Dic13—26no—
169C22×Dic13—26no—
170C22×Dic13—26no—
171C22×Dic13—43no—
172C22×Dic13C2413yes—
173C22×Dic13C2411yescharacteristic
174C22×Dic13C2413yes—
175C22×Dic13C2413yes—
176C22×Dic13C2413yes—
177C22×Dic13C2411yescharacteristic
178C22×Dic13C2411yescharacteristic
179Q8×D13—232no—
180C2×C4×D13—26no—
181C2×C4×D13—26no—
182C2×C4×D13—22no—
183C2×C4×D13—22no—
184C2×C4×D13—26no—
185C2×C4×D13—26no—
186C2×C4×D13—43no—
187C2×C4×D13C22×C4112yes—
188C2×C4×D13#4114yescomplemented
189C2×Dic26—26no—
190C2×Dic26—26no—
191C2×Dic26C22×C4112yes—
192C13×C4⋊C4—26no—
193C13×C4⋊C4—26no—
194C13×C22⋊C4—26no—
195C13×C22⋊C4—26no—
196C13×C22⋊C4—46no—
197C42×C13—26no—
198C42×C13—26no—
199C22⋊Dic13—212no—
200C22⋊Dic13C22×C4124yes—
201C22⋊Dic13C22×C4112yes—
202#6103—26no—
203#6103C22×C414yes—
204#6103#4114yescomplemented
205C4⋊Dic13—26no—
206C4⋊Dic13C22×C4112yes—
207C4⋊Dic13C2×D4112yes—
208#6101—26no—
209#6101—212no—
210#6101—26no—
211#6101C22×C4112yes—
212C4×Dic13—26no—
213C4×Dic13—26no—
214C4×Dic13—26no—
215C4×Dic13—26no—
216C4×Dic13—22no—
217C4×Dic13—26no—
218C4×Dic13—26no—
219C4×Dic13—22no—
220C4×Dic13—43no—
221C4×Dic13C22×C4112yes—
222C4×Dic13C2×D4112yes—
223C2×#548—131no—
224C2×C4×#41—131no—
225C22×C4×Q8—131no—
226#2367—133no—
227#2362—133no—
228#2361—133no—
229#2352—133no—
230#2351—131no—
231#2341D1311yescharacteristic, complemented
232Q8×C22⋊C4—131no—
233C4×#124—133no—
234C4×C22⋊Q8—133no—
235C4×#121—133no—
236C42×Q8—131no—
237C42×D4—133no—
238C23×C13—23no—
239C23×C13—23no—
240C23×C13—23no—
241C23×C13—23no—
242C23×C13—23no—
243C23×C13—23no—
244C23×C13C23×C416yes—
245C23×C13C2511yescharacteristic
246C22×D13—21no—
247C22×D13—21no—
248C22×D13—21no—
249C22×D13—21no—
250C22×D13C23×C412yes—
251C22×D13C2×#4112yes—
252C22×D13C2×#4112yes—
253Q8×C13C4×D4116yescomplemented
254D4×C13—224no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—21no—
260C2×C4×C13—21no—
261C2×C4×C13—224no—
262C2×C4×C13—23no—
263C2×C4×C13—23no—
264C2×C4×C13—23no—
265C2×C4×C13—23no—
266C2×C4×C13—23no—
267C2×C4×C13—23no—
268C2×C4×C13C23×C412yes—
269C2×C4×C13C23×C416yes—
270C2×C4×C13C22×D416yes—
271C2×C4×C13C2×#4112yes—
272C2×C4×C13C2×#4112yes—
273C2×C4×C13C2×#4116yes—
274C22⋊D13—224no—
275C2×Dic13—21no—
276C2×Dic13—23no—
277C2×Dic13—23no—
278C2×Dic13—23no—
279C2×Dic13—23no—
280C2×Dic13—23no—
281C2×Dic13—21no—
282C2×Dic13—21no—
283C2×Dic13—23no—
284C2×Dic13—21no—
285C2×Dic13—23no—
286C2×Dic13—23no—
287C2×Dic13—23no—
288C2×Dic13—412no—
289C2×Dic13—412no—
290C2×Dic13C23×C412yes—
291C2×Dic13C23×C416yes—
292C2×Dic13C23×C416yes—
293C2×Dic13C22×D416yes—
294C2×Dic13C22×D412yes—
295C2×Dic13C2×#4112yes—
296C2×Dic13C2×#4112yes—
297C2×Dic132−1+412yes—
298C4×D13—224no—
299C4×D13—28no—
300C4×D13—412no—
301Dic26—224no—
302C22×#41C2×D1311yescharacteristic
303C23×Q8—131no—
304#548—1316no—
305C2×#124—133no—
306C2×C22⋊Q8—133no—
307C2×#121—133no—
308C4×#41—268no—
309C2×C4×Q8—131no—
310C2×C4×Q8—131no—
311C2×C4×Q8—131no—
312C2×C4×Q8—133no—
313C2×C4×Q8—133no—
314C2×C4×Q8—268no—
315C2×C4×Q8C2×D1311yescharacteristic
316C2×C4×Q8C2×D1311yescharacteristic
317C2×C4×D4—133no—
318C2×C4×D4—133no—
319C2×C4×D4—133no—
320C2×C4×D4—133no—
321C2×C4×D4—133no—
322C2×#117—133no—
323C22×C4⋊C4—133no—
324C22×C42—133no—
325C22×C42—133no—
326#391—133no—
327#391—133no—
328#390C2×D1313yes—
329#390C2×D1313yes—
330#389—133no—
331#389—133no—
332#388—136no—
333#388—136no—
334#388—136no—
335#387—136no—
336#386—136no—
337#385—136no—
338#382—133no—
339#380—133no—
340#380C2×D1313yes—
341C4×C4⋊C4—133no—
342C4×C4⋊C4—133no—
343C4×C4⋊C4—133no—
344C4×C4⋊C4—133no—
345C4×C4⋊C4—133no—
346C4×C4⋊C4—133no—
347C4×C22⋊C4—133no—
348C4×C22⋊C4—133no—
349C4×C22⋊C4—136no—
350C4×C22⋊C4—133no—
351C4×C22⋊C4—133no—
352C4×C22⋊C4—131no—
353C4×C22⋊C4C2×D1313yes—
354C43—133no—
355C22×C13—26no—
356C22×C13—26no—
357C22×C13—26no—
358C22×C13—43no—
359C22×C13—43no—
360C22×C13—43no—
361C22×C13—43no—
362C22×C13C24×C411yescharacteristic
363C22×C13C23×D411yescharacteristic
364C22×C13C22×#4111yescharacteristic
365C22×C13C22×#4111yescharacteristic
366C22×C13C22×#4111yescharacteristic
367C22×C13C2×2−1+411yescharacteristic
368C22×C13#58511yescharacteristic
369C2×D13—22no—
370C2×D13—22no—
371C2×D13—22no—
372C2×D13—22no—
373C2×D13—22no—
374C2×D13—22no—
375C2×D13C4×#4114yescomplemented
376C4×C13—22no—
377C4×C13—22no—
378C4×C13—22no—
379C4×C13—43no—
380C4×C13—43no—
381C4×C13C2×C4×D4112yes—
382Dic13—26no—
383Dic13—26no—
384Dic13—22no—
385Dic13—26no—
386Dic13—22no—
387Dic13—43no—
388Dic13#54814yescomplemented
389C2×#41—24no—
390C2×#41—24no—
391C2×#41C4×D1318yescomplemented
392C22×Q8—138no—
393C22×Q8—133no—
394C22×Q8—264no—
395C22×Q8—264no—
396C22×Q8C22×D1311yescharacteristic
397C22×D4—133no—
398C22×D4C22×D1313yes—
399C23×C4—133no—
400C23×C4—131no—
401C23×C4—133no—
402C23×C4—133no—
403C23×C4—133no—
404C23×C4C22×D1313yes—
405#124—1324no—
406C22⋊Q8—1324no—
407#121—1324no—
408C4×Q8—24no—
409C4×Q8—24no—
410C4×Q8—138no—
411C4×Q8—138no—
412C4×Q8—2612no—
413C4×Q8—2612no—
414C4×Q8—264no—
415C4×D4—1324no—
416C4×D4—2612no—
417C4×D4—2612no—
418C4×D4—2612no—
419C4×D4—2612no—
420#117—2612no—
421C2×C4⋊C4—133no—
422C2×C4⋊C4—133no—
423C2×C4⋊C4—133no—
424C2×C4⋊C4—136no—
425C2×C4⋊C4—133no—
426C2×C4⋊C4—133no—
427C2×C4⋊C4—133no—
428C2×C4⋊C4—2612no—
429C2×C4⋊C4C22×D1313yes—
430C2×C4⋊C4C22×D1313yes—
431C2×C22⋊C4—133no—
432C2×C22⋊C4—136no—
433C2×C22⋊C4—131no—
434C2×C22⋊C4—131no—
435C2×C22⋊C4—136no—
436C2×C22⋊C4—133no—
437C2×C22⋊C4C22×D1313yes—
438C2×C22⋊C4C22×D1313yes—
439C2×C22⋊C4C22×D1316yes—
440C2×C42—133no—
441C2×C42—133no—
442C2×C42—133no—
443C2×C42—133no—
444C2×C42—133no—
445C2×C42—133no—
446C2×C42—133no—
447C2×C42—133no—
448C2×C42—131no—
449C2×C42—131no—
450C2×C42—133no—
451C2×C42—2612no—
452C2×C42—2612no—
453C2×C42C22×D1313yes—
454C2×C42C22×D1313yes—
455#95—131no—
456#95—133no—
457#95—133no—
458#95—133no—
459#95—136no—
460#95—136no—
461#95—136no—
462#95C22×D1311yescharacteristic
463#95C22×D1313yes—
464C2×C13—46no—
465C2×C13C22×C4×D411yescharacteristic
466C2×C13C2×C4×#4111yescharacteristic
467C2×C13C4×2−1+411yescharacteristic
468C2×C13C2×#54811yescharacteristic
469C2×C13D4×#4111yescharacteristic
470C2×C13#348011yescharacteristic
471C2×C13#348211yescharacteristic
472D13—24no—
473C24—131no—
474C24C23×D1313yes—
475#41—232no—
476C2×Q8—1312no—
477C2×Q8—2632no—
478C2×Q8—266no—
479C2×Q8—266no—
480C2×Q8C2×C4×D1314yes—
481C2×Q8D4×D1314yes—
482C2×Q8#612914yes—
483C2×D4—26no—
484C2×D4—26no—
485C2×D4—1312no—
486C2×D4—266no—
487C2×D4—266no—
488C2×D4C2×C4×D13112yes—
489C22×C4—26no—
490C22×C4—26no—
491C22×C4—131no—
492C22×C4—131no—
493C22×C4—131no—
494C22×C4—133no—
495C22×C4—133no—
496C22×C4—133no—
497C22×C4—1312no—
498C22×C4—134no—
499C22×C4—133no—
500C22×C4—262no—
501C22×C4—262no—
502C22×C4—266no—
503C22×C4—266no—
504C22×C4—266no—
505C22×C4—266no—
506C22×C4—266no—
507C22×C4—266no—
508C22×C4—266no—
509C22×C4—266no—
510C22×C4—266no—
511C22×C4—523no—
512C22×C4—523no—
513C22×C4C2×C4×D13112yes—
514C22×C4C23×D1313yes—
515C22×C4C23×D1313yes—
516C22×C4C23×D1311yescharacteristic
517C22×C4C23×D1311yescharacteristic
518C22×C4C23×D1311yescharacteristic
519C22×C4C23×D1313yes—
520C4⋊C4—26no—
521C4⋊C4—26no—
522C4⋊C4—1312no—
523C4⋊C4—1312no—
524C4⋊C4—1312no—
525C4⋊C4—2612no—
526C4⋊C4—266no—
527C4⋊C4—266no—
528C4⋊C4—266no—
529C22⋊C4—26no—
530C22⋊C4—26no—
531C22⋊C4—46no—
532C22⋊C4—1312no—
533C22⋊C4—1324no—
534C22⋊C4—134no—
535C22⋊C4—134no—
536C22⋊C4—266no—
537C22⋊C4—2612no—
538C42—26no—
539C42—26no—
540C42—1312no—
541C42—1312no—
542C42—262no—
543C42—266no—
544C42—266no—
545C42—266no—
546C42—266no—
547C42—262no—
548C42—266no—
549C42—266no—
550C42—523no—
551C13C4×#54811yescharacteristic, complemented, Sylow
552C23—23no—
553C23—23no—
554C23—23no—
555C23—23no—
556C23—23no—
557C23—23no—
558C23—132no—
559C23—132no—
560C23—132no—
561C23—261no—
562C23—261no—
563C23—261no—
564C23—261no—
565C23C22×C4×D1316yes—
566C23C24×D1311yescharacteristic
567Q8—2624no—
568Q8#88135116yescomplemented
569D4—224no—
570D4—2624no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—224no—
578C2×C4—23no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—23no—
584C2×C4—132no—
585C2×C4—132no—
586C2×C4—132no—
587C2×C4—132no—
588C2×C4—136no—
589C2×C4—136no—
590C2×C4—136no—
591C2×C4—132no—
592C2×C4—268no—
593C2×C4—261no—
594C2×C4—263no—
595C2×C4—263no—
596C2×C4—263no—
597C2×C4—263no—
598C2×C4—263no—
599C2×C4—261no—
600C2×C4—263no—
601C2×C4—261no—
602C2×C4—261no—
603C2×C4—2624no—
604C2×C4—263no—
605C2×C4—263no—
606C2×C4—263no—
607C2×C4—5212no—
608C2×C4—5212no—
609C2×C4—5212no—
610C2×C4C22×C4×D1312yes—
611C2×C4C22×C4×D1316yes—
612C2×C4C2×D4×D1316yes—
613C2×C4C2×#612916yes—
614C2×C4#41×D1312yes—
615C2×C4#41×D1312yes—
616C22—26no—
617C22—26no—
618C22—26no—
619C22—43no—
620C22—43no—
621C22—43no—
622C22—43no—
623C22—134no—
624C22—262no—
625C22—262no—
626C22—262no—
627C22—262no—
628C22—262no—
629C22—262no—
630C22C23×C4×D1311yescharacteristic
631C22C22×D4×D1311yescharacteristic
632C22C22×#612911yescharacteristic
633C22C2×#41×D1311yescharacteristic
634C22C2×#41×D1311yescharacteristic
635C22#1171965011yescharacteristic
636C22D13×2−1+411yescharacteristic
637C4—22no—
638C4—22no—
639C4—22no—
640C4—43no—
641C4—43no—
642C4—134no—
643C4—266no—
644C4—266no—
645C4—262no—
646C4—266no—
647C4—262no—
648C4—523no—
649C4C2×#88135112yes—
650C2—46no—
651C2—264no—
652C2C22×#8813511yescharacteristic
653C2C4×#41×D1311yescharacteristic
654C2#1312804711yescharacteristic
655C2D13×#54811yescharacteristic
656C2#1312832611yescharacteristic
657C2#1312843511yescharacteristic
658C2#1312845211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC13×#2341C2
socle, cosocleC23×C13C25
prime radicalC2×#11719536C2
hypercenter#2341D13
hypocenterC13C4×#548
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. C2×#41×Dic13C2semidirect
3. C2×C4×Q8×D13C2semidirect
4. #13117906C2semidirect
5. #13117861C2semidirect
6. #13117857C2semidirect
7. #13117758C2semidirect
8. #13117754C2semidirect
9. Q8×#6103C2semidirect
10. #13117383C2semidirect
11. #13117364C2semidirect
12. #13117352C2semidirect
13. C4×Q8×Dic13C2semidirect
14. C4×#88135C2semidirect
15. C13×#2341C2semidirect
20. #11719536C4semidirect
71. C2×C13×#41C2×C4semidirect
80. C2×Q8×D13C2×C4semidirect
98. Q8×Dic13C2×C4semidirect
99. Q8×Dic13D4semidirect
100. #88192C2×C4semidirect
101. #88190C2×C4semidirect
104. C4×C22⋊D13C2×C4semidirect
112. #88178C2×C4semidirect
188. C2×C4×D13#41semidirect
204. #6103#41semidirect
231. #2341D13semidirect
253. Q8×C13C4×D4semidirect
375. C2×D13C4×#41semidirect
388. Dic13#548semidirect
391. C2×#41C4×D13semidirect
551. C13C4×#548semidirect
568. Q8#88135semidirect

Maximal subgroups

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

subgroupindex
C2×#117195362
C2×#41×Dic132
C2×C4×Q8×D132
#131179062
#131178612
#131178572
#131177582
#131177542
Q8×#61032
#131173832
#131173642
#131173522
C4×Q8×Dic132
C4×#881352
C13×#23412
C4×#54813