Finite Group Databasebeta

← #16023867

Subgroups — #16023867

Order 3328.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11719279C211yesmaximal, characteristic
2C2×#11719279C211yesmaximal, characteristic
3C2×C4×#6131C211yesmaximal, characteristic
4#13117757C211yesmaximal, characteristic, complemented
5#13117754C211yesmaximal, characteristic, complemented
6Q8×#6101C211yesmaximal, characteristic, complemented
7#13117747C213yesmaximal, complemented
8#13117745C213yesmaximal, complemented
9#13117737C213yesmaximal, complemented
10#13117736C213yesmaximal, complemented
11#13117457C213yesmaximal, complemented
12#13117451C213yesmaximal, complemented
13#13117316C213yesmaximal, complemented
14D13×#389C213yesmaximal, complemented
15Q8×C13×C4⋊C4C211yesmaximal, characteristic, complemented
16C22×#6131C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C2×C4×Q8×C13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×Q8×Dic13C2211yescharacteristic
22C2×Q8×Dic13C2211yescharacteristic
23#11719279—28no—
24#11719279C4116yescomplemented
25C4×#6131—28no—
26C2×#88147C2213yes—
27C2×#88147C2213yes—
28C2×#88147C2213yes—
29C2×#88146C2213yes—
30C2×C4⋊C4×D13C2213yes—
31C2×C4⋊C4×D13C2216yes—
32C2×C4⋊C4×D13C2213yes—
33C2×C4×D52C2213yes—
34C2×C4×D52C2213yes—
35C2×C4×D52C2213yes—
36C2×#88126C2213yes—
37C2×#88126C2213yes—
38C2×C42×D13C2213yes—
39C13×#389C2213yes—
40C13×#389C2213yes—
41C13×#384C2213yes—
42C4×C13×C4⋊C4C2213yes—
43#11718650C2216yes—
44#11718650C2216yes—
45#11718650C2213yes—
46#11718650C2213yes—
47#11718648C2216yes—
48#11718646C2213yes—
49#11718646C2213yes—
50#11718638C2213yes—
51#11718637C2213yes—
52#11718634C2213yes—
53#11718634C2213yes—
54#11718633C2216yes—
55#11718633C2213yes—
56#11718633C2211yescharacteristic
57#11718597C2213yes—
58C4×C4⋊Dic13C2213yes—
59C4×#6101C2213yes—
60C4×#6101C2213yes—
61#11718589C2213yes—
62#11718589C2213yes—
63#11718329C2213yes—
64#11718329C2213yes—
65#11718328C2216yes—
66D13×#95C2216yes—
67#11718308C2213yes—
68#11718308C2213yes—
69C22×Q8×C13C2311yescharacteristic
70C2×#6131C2×C418yes—
71C2×#6131D418yes—
72C2×#6131Q818yes—
73C22×D52C2313yes—
74C22×C4×D13C2313yes—
75C22×C4×D13C2313yes—
76C22×C4×D13C2316yes—
77C22×C4×D13C2316yes—
78C22×C4×D13C2313yes—
79C4×Q8×C13—24no—
80C4×Q8×C13—24no—
81C4×Q8×C13C2×C418yescomplemented
82C2×C13×C4⋊C4C2311yescharacteristic
83C2×C13×C4⋊C4C2313yes—
84C2×C13×C4⋊C4C2313yes—
85C2×C13×C4⋊C4C2313yes—
86C2×C13×C4⋊C4C2316yes—
87C2×C13×C4⋊C4C2313yes—
88C2×C42×C13C2313yes—
89C2×C42×C13C2313yes—
90C2×C42×C13C2313yes—
91Q8×Dic13—24no—
92Q8×Dic13—24no—
93Q8×Dic13C2×C418yescomplemented
94C2×#6103C2313yes—
95C2×#6103C2313yes—
96C2×#6103C2316yes—
97C2×#6103C2313yes—
98C2×#6103C2313yes—
99C2×C4⋊Dic13C2313yes—
100C2×C4⋊Dic13C2313yes—
101C2×C4⋊Dic13C2313yes—
102C2×C4⋊Dic13C2313yes—
103C2×C4⋊Dic13C2313yes—
104C2×#6101C2311yescharacteristic
105C2×#6101C2313yes—
106C2×#6101C2313yes—
107C2×#6101C2311yescharacteristic
108C2×#6101C2316yes—
109C2×#6101C2313yes—
110C2×#6101C2313yes—
111C2×#6101C2311yescharacteristic
112C2×#6101C2313yes—
113C2×C4×Dic13C2311yescharacteristic
114C2×C4×Dic13C2313yes—
115C2×C4×Dic13C2313yes—
116C2×C4×Dic13C2313yes—
117C2×C4×Dic13C2313yes—
118#88147—212no—
119#88147—212no—
120#88147C2×C4124yescomplemented
121#88146—212no—
122C4⋊C4×D13—212no—
123C4⋊C4×D13—412no—
124C4⋊C4×D13C2×C4124yescomplemented
125C4×D52—212no—
126C4×D52—212no—
127C4×D52C2×C4124yescomplemented
128#88126—212no—
129#88126C2×C4124yescomplemented
130C42×D13—212no—
131C13×#95C2313yes—
132C13×#95C2313yes—
133#88071C2313yes—
134#88071C2313yes—
135#88071C2316yes—
136#88071C2316yes—
137#88071C2311yescharacteristic
138#88071C2313yes—
139#88071C2311yescharacteristic
140#88071C2313yes—
141#34250—131nomaximal, Sylow
142C23×D13C2413yes—
143C2×Q8×C13C22×C414yes—
144C2×Q8×C13C2×D414yes—
145C2×Q8×C13C2×Q814yes—
146C22×C4×C13C2413yes—
147C22×C4×C13C2411yescharacteristic
148C22×C4×C13C2413yes—
149C22×C4×C13C2411yescharacteristic
150C22×C4×C13C2413yes—
151C22×C4×C13C2411yescharacteristic
152C22×C4×C13C2413yes—
153C22×Dic13C2413yes—
154C22×Dic13C2411yescharacteristic
155C22×Dic13C2413yes—
156C22×Dic13C2411yescharacteristic
157C22×Dic13C2413yes—
158C22×Dic13C2411yescharacteristic
159C22×Dic13C2411yescharacteristic
160#6131C4⋊C4164yescomplemented
161C2×D52C22×C4112yes—
162C2×D52C2×D4112yes—
163C2×D52C2×Q8112yes—
164C2×C4×D13—26no—
165C2×C4×D13—26no—
166C2×C4×D13—26no—
167C2×C4×D13—26no—
168C2×C4×D13—212no—
169C2×C4×D13—46no—
170C2×C4×D13—46no—
171C2×C4×D13—43no—
172C2×C4×D13—43no—
173C2×C4×D13—46no—
174C2×C4×D13—46no—
175C2×C4×D13C22×C4124yes—
176C2×C4×D13C22×C4112yes—
177C2×C4×D13C2×D4112yes—
178C2×C4×D13C2×Q8112yes—
179C13×C4⋊C4—26no—
180C13×C4⋊C4—26no—
181C13×C4⋊C4—22no—
182C13×C4⋊C4—46no—
183C13×C4⋊C4—43no—
184C13×C4⋊C4C22×C4112yes—
185C42×C13—26no—
186C42×C13—26no—
187C42×C13C22×C4112yes—
188#6103—26no—
189#6103—26no—
190#6103—212no—
191#6103C22×C4112yes—
192#6103C22×C4112yes—
193C4⋊Dic13—26no—
194C4⋊Dic13—26no—
195C4⋊Dic13—26no—
196C4⋊Dic13—43no—
197C4⋊Dic13C22×C4112yes—
198#6101—22no—
199#6101—22no—
200#6101—26no—
201#6101—26no—
202#6101—46no—
203#6101—43no—
204#6101—43no—
205#6101C22×C414yes—
206#6101C22×C4112yes—
207C4×Dic13—26no—
208C4×Dic13—26no—
209C4×Dic13—22no—
210C4×Dic13—26no—
211C4×Dic13C22×C4112yes—
212C2×#520—131no—
213C2×#520—131no—
214C2×C4×#41—131no—
215#2351—131no—
216#2351—131no—
217Q8×C4⋊C4—131no—
218Q8×C4⋊C4D1311yescharacteristic, complemented
219#2349—133no—
220#2349—133no—
221D4×C4⋊C4—133no—
222D4×C4⋊C4—133no—
223#2317—133no—
224#2316—133no—
225#2296—133no—
226C2×#389—133no—
227C23×C13C2511yescharacteristic
228C22×D13—23no—
229C22×D13—23no—
230C22×D13—23no—
231C22×D13—23no—
232C22×D13C23×C416yes—
233C22×D13C22×D416yes—
234C22×D13C22×Q816yes—
235Q8×C13C2×C4⋊C4116yescomplemented
236C2×C4×C13—21no—
237C2×C4×C13—23no—
238C2×C4×C13—21no—
239C2×C4×C13—23no—
240C2×C4×C13—21no—
241C2×C4×C13—21no—
242C2×C4×C13—23no—
243C2×C4×C13—21no—
244C2×C4×C13—23no—
245C2×C4×C13—23no—
246C2×C4×C13—23no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—23no—
250C2×C4×C13—23no—
251C2×C4×C13C23×C412yes—
252C2×C4×C13C23×C416yes—
253C2×C4×C13C23×C416yes—
254C2×C4×C13C22×D416yes—
255C2×C4×C13C22×Q816yes—
256C2×C4×C132−1+412yes—
257C2×Dic13—21no—
258C2×Dic13—23no—
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—23no—
262C2×Dic13—21no—
263C2×Dic13—21no—
264C2×Dic13—23no—
265C2×Dic13—23no—
266C2×Dic13—23no—
267C2×Dic13—21no—
268C2×Dic13—23no—
269C2×Dic13—23no—
270C2×Dic13—23no—
271C2×Dic13—21no—
272C2×Dic13C23×C412yes—
273C2×Dic13C23×C416yes—
274C2×Dic13C23×C412yes—
275C2×Dic13C22×D412yes—
276C2×Dic13C22×Q812yes—
277C2×Dic13H(5,𝔽2)12yes—
278D52C2×C4⋊C4148yescomplemented
279C4×D13—424no—
280C4×D13—412no—
281C4×D13—412no—
282C4×D13—812no—
283C4×D13C2×C4⋊C4148yescomplemented
284C22×#41—131no—
285#520—1316no—
286#520—268no—
287C4×#41—268no—
288C2×C4×Q8—131no—
289C2×C4×Q8—131no—
290C2×C4×Q8—131no—
291C2×C4×Q8C2×D1311yescharacteristic
292C2×C4×Q8C2×D1311yescharacteristic
293C2×C4×Q8C2×D1311yescharacteristic
294C2×C4×D4—133no—
295C2×C4×D4—133no—
296C2×C4×D4—133no—
297C2×C4×D4—133no—
298C2×C4×D4—133no—
299C2×C4×D4—133no—
300C2×#117—133no—
301C2×#117—133no—
302C2×#117—133no—
303C22×C4⋊C4—133no—
304C22×C4⋊C4—133no—
305C22×C4⋊C4—136no—
306C22×C42—133no—
307#389—136no—
308#389—133no—
309#389—133no—
310#389—133no—
311#389—133no—
312#389—133no—
313#389—131no—
314#389—133no—
315#389—133no—
316#389—133no—
317#389—133no—
318#389C2×D1313yes—
319#389C2×D1313yes—
320#385—136no—
321#385—136no—
322#385—133no—
323#385—133no—
324#385—133no—
325#385—133no—
326#384—133no—
327#384C2×D1313yes—
328#382—133no—
329#382—133no—
330#380—136no—
331#380—136no—
332C4×C4⋊C4—133no—
333C4×C4⋊C4—133no—
334C4×C4⋊C4—133no—
335C4×C4⋊C4C2×D1313yes—
336C2×#95—136no—
337C22×C13C24×C411yescharacteristic
338C22×C13C23×D411yescharacteristic
339C22×C13C23×Q811yescharacteristic
340C22×C13C2×H(5,𝔽2)11yescharacteristic
341C22×C13C2×2−1+411yescharacteristic
342C22×C13#58511yescharacteristic
343C22×C13#58511yescharacteristic
344C2×D13—26no—
345C2×D13—26no—
346C2×D13—26no—
347C2×D13—26no—
348C2×D13—26no—
349C2×D13—26no—
350C2×D13C22×C4⋊C4112yes—
351C4×C13—26no—
352C4×C13—22no—
353C4×C13—22no—
354C4×C13—43no—
355C4×C13—43no—
356C4×C13—41no—
357C4×C13C22×C4⋊C4112yes—
358Dic13—26no—
359Dic13—22no—
360Dic13—22no—
361Dic13—43no—
362Dic13—43no—
363Dic13—41no—
364Dic13C22×C4⋊C414yes—
365C2×#41—138no—
366C2×#41—138no—
367C2×#41—138no—
368C22×Q8C22×D1311yescharacteristic
369C22×D4—133no—
370C23×C4—133no—
371C23×C4—136no—
372C23×C4—133no—
373C23×C4—133no—
374C23×C4—136no—
375C4×Q8—24no—
376C4×Q8—24no—
377C4×Q8—138no—
378C4×Q8—264no—
379C4×Q8—264no—
380C4×Q8C4×D1318yescomplemented
381C4×D4—1324no—
382C4×D4—1324no—
383C4×D4—2612no—
384C4×D4—2612no—
385C4×D4—2612no—
386C4×D4—2612no—
387#117—1324no—
388#117—2612no—
389#117—2612no—
390C2×C4⋊C4—133no—
391C2×C4⋊C4—133no—
392C2×C4⋊C4—131no—
393C2×C4⋊C4—1324no—
394C2×C4⋊C4—133no—
395C2×C4⋊C4—133no—
396C2×C4⋊C4—133no—
397C2×C4⋊C4—133no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—133no—
400C2×C4⋊C4—131no—
401C2×C4⋊C4—136no—
402C2×C4⋊C4—133no—
403C2×C4⋊C4—133no—
404C2×C4⋊C4—133no—
405C2×C4⋊C4—2612no—
406C2×C4⋊C4—5212no—
407C2×C4⋊C4C22×D1313yes—
408C2×C4⋊C4C22×D1313yes—
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C4⋊C4C22×D1316yes—
411C2×C4⋊C4C22×D1313yes—
412C2×C4⋊C4C22×D1313yes—
413C2×C22⋊C4—133no—
414C2×C22⋊C4—133no—
415C2×C22⋊C4—136no—
416C2×C22⋊C4—133no—
417C2×C22⋊C4—133no—
418C2×C42—133no—
419C2×C42—133no—
420C2×C42—133no—
421C2×C42—133no—
422C2×C42—131no—
423C2×C42—2612no—
424C2×C42C22×D1313yes—
425C2×C42C22×D1313yes—
426C2×C42C22×D1313yes—
427#95—133no—
428#95—136no—
429#95—136no—
430#95—131no—
431#95—131no—
432#95—133no—
433#95—133no—
434#95—133no—
435#95C22×D1313yes—
436#95C22×D1313yes—
437C2×C13C23×C4⋊C411yescharacteristic
438C2×C13C4×H(5,𝔽2)11yescharacteristic
439C2×C13C4×2−1+411yescharacteristic
440C2×C13#348811yescharacteristic
441C2×C13#348911yescharacteristic
442C2×C13#350311yescharacteristic
443C2×C13#350411yescharacteristic
444D13—212no—
445C24—133no—
446#41—1364no—
447C2×Q8C2×C4×D1314yes—
448C2×Q8D4×D1314yes—
449C2×Q8Q8×D1314yes—
450C2×D4—1312no—
451C2×D4—1312no—
452C2×D4—1312no—
453C22×C4—131no—
454C22×C4—1312no—
455C22×C4—1312no—
456C22×C4—1312no—
457C22×C4—131no—
458C22×C4—133no—
459C22×C4—131no—
460C22×C4—1324no—
461C22×C4—133no—
462C22×C4—131no—
463C22×C4—133no—
464C22×C4—266no—
465C22×C4—266no—
466C22×C4—266no—
467C22×C4—2612no—
468C22×C4—266no—
469C22×C4—526no—
470C22×C4—526no—
471C22×C4—523no—
472C22×C4—523no—
473C22×C4—526no—
474C22×C4—526no—
475C22×C4C23×D1313yes—
476C22×C4C23×D1311yescharacteristic
477C22×C4C23×D1313yes—
478C22×C4C23×D1311yescharacteristic
479C22×C4C23×D1313yes—
480C22×C4C23×D1311yescharacteristic
481C22×C4C23×D1313yes—
482C4⋊C4—26no—
483C4⋊C4—26no—
484C4⋊C4—22no—
485C4⋊C4—46no—
486C4⋊C4—43no—
487C4⋊C4—1312no—
488C4⋊C4—1312no—
489C4⋊C4—134no—
490C4⋊C4—266no—
491C4⋊C4—266no—
492C4⋊C4—266no—
493C4⋊C4—262no—
494C4⋊C4—266no—
495C4⋊C4—266no—
496C4⋊C4—262no—
497C4⋊C4—526no—
498C4⋊C4—523no—
499C4⋊C4—523no—
500C4⋊C4—523no—
501C4⋊C4C2×C4×D13112yes—
502C22⋊C4—1312no—
503C22⋊C4—1312no—
504C22⋊C4—266no—
505C22⋊C4—2612no—
506C22⋊C4—266no—
507C42—26no—
508C42—26no—
509C42—1312no—
510C42—262no—
511C42—266no—
512C42—266no—
513C42—266no—
514C42C2×C4×D13112yes—
515C13#3425011yescharacteristic, complemented, Sylow
516C23—136no—
517C23—136no—
518C23—136no—
519C23—263no—
520C23—263no—
521C23—263no—
522C23—263no—
523C23C24×D1311yescharacteristic
524Q8C4⋊C4×D13116yescomplemented
525D4—1348no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—23no—
530C2×C4—23no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—23no—
534C2×C4—23no—
535C2×C4—21no—
536C2×C4—23no—
537C2×C4—23no—
538C2×C4—23no—
539C2×C4—23no—
540C2×C4—21no—
541C2×C4—132no—
542C2×C4—132no—
543C2×C4—132no—
544C2×C4—1348no—
545C2×C4—132no—
546C2×C4—136no—
547C2×C4—132no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—261no—
551C2×C4—261no—
552C2×C4—263no—
553C2×C4—263no—
554C2×C4—261no—
555C2×C4—263no—
556C2×C4—263no—
557C2×C4—261no—
558C2×C4—263no—
559C2×C4—263no—
560C2×C4—263no—
561C2×C4—261no—
562C2×C4—263no—
563C2×C4—5224no—
564C2×C4—5212no—
565C2×C4—5212no—
566C2×C4—10412no—
567C2×C4C22×C4×D1312yes—
568C2×C4C22×C4×D1316yes—
569C2×C4C22×C4×D1316yes—
570C2×C4C2×D4×D1316yes—
571C2×C4C2×Q8×D1316yes—
572C2×C4#8825412yes—
573C22—1312no—
574C22—266no—
575C22—266no—
576C22—266no—
577C22—266no—
578C22—266no—
579C22—266no—
580C22C23×C4×D1311yescharacteristic
581C22C22×D4×D1311yescharacteristic
582C22C22×Q8×D1311yescharacteristic
583C22C2×#8825411yescharacteristic
584C22#1171964811yescharacteristic
585C22D13×H(5,𝔽2)11yescharacteristic
586C22#1171965211yescharacteristic
587C4—22no—
588C4—26no—
589C4—22no—
590C4—43no—
591C4—43no—
592C4—41no—
593C4—134no—
594C4—262no—
595C4—266no—
596C4—262no—
597C4—523no—
598C4—521no—
599C4—523no—
600C4C2×C4⋊C4×D13112yes—
601C2—2612no—
602C2C22×C4⋊C4×D1311yescharacteristic
603C2#1312804211yescharacteristic
604C2C4×#8825411yescharacteristic
605C2#1312850411yescharacteristic
606C2#1312851311yescharacteristic
607C2#1312865411yescharacteristic
608C2#1312866111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingQ8×C13×C4⋊C4C2
socle, cosocleC23×C13C25
prime radicalC22×#6131C22
hypercenterQ8×C4⋊C4D13
hypocenterC13#34250
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #13117757C2semidirect
5. #13117754C2semidirect
6. Q8×#6101C2semidirect
7. #13117747C2semidirect
8. #13117745C2semidirect
9. #13117737C2semidirect
10. #13117736C2semidirect
11. #13117457C2semidirect
12. #13117451C2semidirect
13. #13117316C2semidirect
14. D13×#389C2semidirect
15. Q8×C13×C4⋊C4C2semidirect
24. #11719279C4semidirect
81. C4×Q8×C13C2×C4semidirect
93. Q8×Dic13C2×C4semidirect
120. #88147C2×C4semidirect
124. C4⋊C4×D13C2×C4semidirect
127. C4×D52C2×C4semidirect
129. #88126C2×C4semidirect
160. #6131C4⋊C4semidirect
218. Q8×C4⋊C4D13semidirect
235. Q8×C13C2×C4⋊C4semidirect
278. D52C2×C4⋊C4semidirect
283. C4×D13C2×C4⋊C4semidirect
380. C4×Q8C4×D13semidirect
515. C13#34250semidirect
524. Q8C4⋊C4×D13semidirect

Maximal subgroups

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

subgroupindex
C2×#117192792
C2×C4×#61312
#131177572
#131177542
Q8×#61012
#131177472
#131177452
#131177372
#131177362
#131174572
#131174512
#131173162
D13×#3892
Q8×C13×C4⋊C42
#3425013