Finite Group Databasebeta

← #15881335

Subgroups — #15881335

Order 3328.

986 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×Dic13C211yesmaximal, characteristic
2C2×SD16×Dic13C211yesmaximal, characteristic
3#13117754C211yesmaximal, characteristic, complemented
4#13117741C211yesmaximal, characteristic
5#13115564C211yesmaximal, characteristic, complemented
6#13115562C211yesmaximal, characteristic, complemented
7#13115090C211yesmaximal, characteristic, complemented
8#13115088C211yesmaximal, characteristic, complemented
9#13115083C211yesmaximal, characteristic
10#13115081C211yesmaximal, characteristic
11#13114997C211yesmaximal, characteristic, complemented
12#13114849C211yesmaximal, characteristic, complemented
13#13114844C211yesmaximal, characteristic
14C8×#6101C211yesmaximal, characteristic, complemented
15C2×C4×C13×SD16C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×D4×Dic13C2211yescharacteristic
22C2×D4×Dic13C2211yescharacteristic
23C4×C13×SD16—24no—
24C2×C13×#106C2211yescharacteristic
25C2×C13×#103C2211yescharacteristic
26C2×C13×#102C2211yescharacteristic
27C2×C4×C8×C13C2211yescharacteristic
28C2×#88075C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88072C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32#11718871—21no—
33C2×#88071—21no—
34SD16×Dic13—24no—
35SD16×Dic13C418yescomplemented
36C2×#88057C2211yescharacteristic
37C2×#88057C2211yescharacteristic
38C2×#88054C2211yescharacteristic
39C2×C8×Dic13C2211yescharacteristic
40C2×C8×Dic13C2211yescharacteristic
41#11718638—21no—
42#11718638C2211yescharacteristic
43#11718638C2211yescharacteristic
44#11718634—21no—
45#11718634C2211yescharacteristic
46#11718634C2211yescharacteristic
47C2×#88047C2211yescharacteristic
48C2×#88047C2211yescharacteristic
49#11718607—21no—
50#11718603—21no—
51C4×#6101—21no—
52C4×#6101C2211yescharacteristic
53#11718589—21no—
54#11718589C2211yescharacteristic
55#11718589C2211yescharacteristic
56#11718143C2211yescharacteristic
57#11718143C2211yescharacteristic
58#11718142C2211yescharacteristic
59#11718121C2211yescharacteristic
60#11718121C2211yescharacteristic
61#11718116C2211yescharacteristic
62C22×Q8×C13C2311yescharacteristic
63C22×D4×C13C2311yescharacteristic
64C23×C4×C13—21no—
65C23×Dic13—21no—
66C23×Dic13—21no—
67C2×C13×SD16C2×C414yes—
68C2×C13×SD16D414yes—
69C2×C13×SD16Q814yes—
70C22×C8×C13C2311yescharacteristic
71C22×C8×C13C2311yescharacteristic
72C4×Q8×C13—22no—
73C4×Q8×C13—41no—
74C4×D4×C13—22no—
75C4×D4×C13—41no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C22⋊C4—21no—
80C2×C42×C13—21no—
81C2×C42×C13C2311yescharacteristic
82C2×C22⋊Dic13—21no—
83C2×C22⋊Dic13—21no—
84Q8×Dic13—22no—
85Q8×Dic13—22no—
86Q8×Dic13—41no—
87Q8×Dic13C2×C414yescomplemented
88D4×Dic13—22no—
89D4×Dic13—22no—
90D4×Dic13—41no—
91D4×Dic13C2×C414yes—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13C2311yescharacteristic
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×#6101—21no—
99C2×#6101—21no—
100C2×#6101—21no—
101C2×#6101—21no—
102C2×#6101D411yescharacteristic
103C2×#6101D411yescharacteristic
104C2×C4×Dic13—21no—
105C2×C4×Dic13—21no—
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C13×#106—22no—
111C13×#103—22no—
112C13×#102—22no—
113C4×C8×C13—22no—
114#88075—22no—
115#88075C2×C414yescomplemented
116#88072—22no—
117#88072C2×C414yes—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071D411yescharacteristic
127#88071D411yescharacteristic
128#88057—22no—
129#88057C2×C414yescomplemented
130#88054—22no—
131C8×Dic13—22no—
132C8×Dic13C2×C414yescomplemented
133#88047—41no—
134#88047—41no—
135#22683—131nomaximal, Sylow
136C24×C13—21no—
137C2×Q8×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13C22×C412yes—
141C2×Q8×C13C2×D412yes—
142C2×Q8×C13C2×Q812yes—
143C2×D4×C13—21no—
144C2×D4×C13—21no—
145C2×D4×C13—21no—
146C2×D4×C13C22×C412yes—
147C2×D4×C13C2×D412yes—
148C2×D4×C13C2×Q812yes—
149C22×C4×C13—21no—
150C22×C4×C13—21no—
151C22×C4×C13—21no—
152C22×C4×C13—42no—
153C22×C4×C13—42no—
154C22×C4×C13—81no—
155C22×C4×C13C2×D411yescharacteristic
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2411yescharacteristic
158C22×Dic13—21no—
159C22×Dic13—21no—
160C22×Dic13—24no—
161C22×Dic13—21no—
162C22×Dic13—21no—
163C22×Dic13—42no—
164C22×Dic13—42no—
165C22×Dic13—81no—
166C22×Dic13—81no—
167C22×Dic13—81no—
168C22×Dic13C2×D411yescharacteristic
169C22×Dic13C2×D411yescharacteristic
170C22×Dic13C2×D411yescharacteristic
171C22×Dic13C2×D411yescharacteristic
172C13×SD16C4⋊C4116yescomplemented
173C2×C8×C13—21no—
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13C22×C412yes—
177C2×C8×C13C2×D412yes—
178C2×C8×C13C2×Q812yes—
179C13×C4⋊C4—21no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—42no—
184C13×C22⋊C4—42no—
185C42×C13—21no—
186C42×C13—21no—
187C42×C13—42no—
188C22⋊Dic13—24no—
189C22⋊Dic13—42no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—24no—
193C4⋊Dic13—21no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—42no—
198C4⋊Dic13C22×C412yes—
199C4⋊Dic13C22×C412yes—
200#6101—22no—
201#6101—22no—
202#6101—42no—
203#6101—42no—
204#6101—81no—
205#6101—81no—
206C4×Dic13—21no—
207C4×Dic13—21no—
208C4×Dic13—24no—
209C4×Dic13—42no—
210C4×Dic13C22×C412yes—
211C4×Dic13#4112yes—
212C2×#1055—21no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055C22×C412yes—
218C2×C4×SD16—131no—
219C2×C4×SD16—131no—
220C2×C4×SD16D1311yescharacteristic
221#2351—131no—
222#2349—131no—
223#1941—131no—
224#1941—131no—
225#1863—131no—
226#1863—131no—
227#1862—131no—
228#1862—131no—
229#1846—131no—
230#1805—131no—
231#1804—131no—
232C8×C4⋊C4—131no—
233C23×C13—22no—
234C23×C13—22no—
235C23×C13—22no—
236C23×C13—41no—
237C23×C13—41no—
238C23×C13—41no—
239C23×C13—41no—
240C23×C13C22×D411yescharacteristic
241Q8×C13—22no—
242Q8×C13—22no—
243Q8×C13—22no—
244Q8×C13C2×C4⋊C414yes—
245D4×C13—22no—
246D4×C13—22no—
247D4×C13—22no—
248D4×C13C2×C4⋊C414yes—
249C2×C4×C13—22no—
250C2×C4×C13—22no—
251C2×C4×C13—21no—
252C2×C4×C13—22no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—41no—
258C2×C4×C13—41no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—84no—
264C2×C4×C13#13512yes—
265C2×C4×C13C23×C411yescharacteristic
266C2×C4×C13C22×D411yescharacteristic
267C2×C4×C13C22×Q811yescharacteristic
268C2×C4×C13C2×#4111yescharacteristic
269C2×C4×C13H(5,𝔽2)11yescharacteristic
270C2×C4×C132−1+411yescharacteristic
271C2×Dic13—21no—
272C2×Dic13—21no—
273C2×Dic13—21no—
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—22no—
278C2×Dic13—22no—
279C2×Dic13—21no—
280C2×Dic13—21no—
281C2×Dic13—21no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—84no—
293C2×Dic13—84no—
294C2×Dic13C4×D412yes—
295C2×Dic13C4×D412yes—
296C2×Dic13#13512yes—
297C8×C13—41no—
298C8×C13C2×C4⋊C414yes—
299#1055—22no—
300#1055—41no—
301C22×SD16C2×D1311yescharacteristic
302C2×C4×Q8—131no—
303C2×C4×Q8—131no—
304C2×C4×Q8C2×D1311yescharacteristic
305C2×C4×D4—131no—
306C2×C4×D4—131no—
307C2×C4×D4C2×D1311yescharacteristic
308C4×SD16—24no—
309C4×SD16—138no—
310C4×SD16—264no—
311C2×#107—131no—
312C2×#107—131no—
313C2×#106—131no—
314C2×#106—131no—
315C2×#106C2×D1311yescharacteristic
316C2×C4⋊C8—131no—
317C2×#103—131no—
318C2×#103—131no—
319C2×#103C2×D1311yescharacteristic
320C2×#102—131no—
321C2×#102—131no—
322C2×#102C2×D1311yescharacteristic
323C2×C4×C8—131no—
324C2×C4×C8—131no—
325C2×C4×C8C2×D1311yescharacteristic
326#389—131no—
327#389—131no—
328#389—131no—
329#389—131no—
330#389—261no—
331#389—261no—
332#385—261no—
333#385—261no—
334#382—131no—
335#382—131no—
336#382—261no—
337#380—261no—
338C4×C4⋊C4—131no—
339C4×C4⋊C4—261no—
340C2×#95—261no—
341#340—131no—
342#340—131no—
343#340—131no—
344#340—131no—
345#336—131no—
346#336—131no—
347C22×C13—24no—
348C22×C13—42no—
349C22×C13—42no—
350C22×C13—42no—
351C22×C13—42no—
352C22×C13—42no—
353C22×C13—42no—
354C22×C13C2×C4×D411yescharacteristic
355C22×C13D4211yescharacteristic
356C22×C13D4×Q811yescharacteristic
357C22×C13C2×#13511yescharacteristic
358C22×C13C2×#13511yescharacteristic
359C22×C13#57711yescharacteristic
360C22×C13#57711yescharacteristic
361C4×C13—24no—
362C4×C13—22no—
363C4×C13—41no—
364C4×C13—81no—
365C4×C13—81no—
366C4×C13C22×C4⋊C411yescharacteristic
367C4×C13#52011yescharacteristic
368C4×C13#54711yescharacteristic
369C4×C13#55411yescharacteristic
370Dic13—22no—
371Dic13—22no—
372Dic13—22no—
373Dic13—42no—
374Dic13—42no—
375Dic13—41no—
376Dic13—81no—
377Dic13—81no—
378C22×Q8C22×D1311yescharacteristic
379C22×D4C22×D1311yescharacteristic
380C23×C4—21no—
381C23×C4—261no—
382C23×C4—261no—
383C2×SD16Dic2614yes—
384C2×SD16C4×D1314yes—
385C2×SD16C22⋊D1314yes—
386C22×C8—131no—
387C22×C8—131no—
388C22×C8C22×D1311yescharacteristic
389C22×C8C22×D1311yescharacteristic
390C4×Q8—22no—
391C4×Q8—41no—
392C4×Q8—134no—
393C4×Q8—262no—
394C4×Q8—262no—
395C4×Q8—521no—
396C4×D4—22no—
397C4×D4—41no—
398C4×D4—134no—
399C4×D4—262no—
400C4×D4—262no—
401C4×D4—521no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—131no—
404C2×C4⋊C4—131no—
405C2×C4⋊C4—131no—
406C2×C4⋊C4—131no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—261no—
410C2×C4⋊C4—261no—
411C2×C4⋊C4—261no—
412C2×C4⋊C4—261no—
413C2×C4⋊C4—261no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4C22×D1311yescharacteristic
416C2×C4⋊C4C22×D1311yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—261no—
419C2×C22⋊C4—261no—
420C2×C42—21no—
421C2×C42—131no—
422C2×C42—131no—
423C2×C42—261no—
424C2×C42—261no—
425C2×C42C22×D1311yescharacteristic
426#107—521no—
427#107—521no—
428#106—22no—
429#106—134no—
430#106—262no—
431C4⋊C8—262no—
432#103—22no—
433#103—134no—
434#103—262no—
435#102—22no—
436#102—134no—
437#102—262no—
438C4×C8—22no—
439C4×C8—134no—
440C4×C8—262no—
441#95—131no—
442#95—131no—
443#95—261no—
444#95—261no—
445#95—261no—
446#95—261no—
447#95—261no—
448#95—261no—
449#95—261no—
450#95—261no—
451C2×C13—44no—
452C2×C13D4×C4⋊C411yescharacteristic
453C2×C13C4×#13511yescharacteristic
454C2×C13#294911yescharacteristic
455C2×C13#328211yescharacteristic
456C2×C13#328211yescharacteristic
457C2×C13#338111yescharacteristic
458C2×C13#338111yescharacteristic
459C24—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8C2×Dic2612yes—
464C2×Q8C2×C4×D1312yes—
465C2×Q8C2×C22⋊D1312yes—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4C2×Dic2612yes—
470C2×D4C2×C4×D1312yes—
471C2×D4C2×C22⋊D1312yes—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—81no—
478C22×C4—131no—
479C22×C4—131no—
480C22×C4—131no—
481C22×C4—131no—
482C22×C4—264no—
483C22×C4—261no—
484C22×C4—261no—
485C22×C4—261no—
486C22×C4—261no—
487C22×C4—522no—
488C22×C4—522no—
489C22×C4—1041no—
490C22×C4—1041no—
491C22×C4—1041no—
492C22×C4D4×D1311yescharacteristic
493C22×C4D4×D1311yescharacteristic
494C22×C4C23×D1311yescharacteristic
495SD16#6101116yescomplemented
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—132no—
500C2×C8—261no—
501C2×C8—261no—
502C2×C8—261no—
503C2×C8—261no—
504C2×C8—261no—
505C2×C8C2×Dic2612yes—
506C2×C8C2×C4×D1312yes—
507C2×C8C2×C22⋊D1312yes—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—42no—
513C4⋊C4—132no—
514C4⋊C4—132no—
515C4⋊C4—262no—
516C4⋊C4—264no—
517C4⋊C4—262no—
518C4⋊C4—261no—
519C4⋊C4—261no—
520C4⋊C4—261no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—522no—
525C4⋊C4—522no—
526C4⋊C4—522no—
527C4⋊C4—1041no—
528C4⋊C4—1041no—
529C22⋊C4—42no—
530C22⋊C4—264no—
531C22⋊C4—522no—
532C42—21no—
533C42—21no—
534C42—42no—
535C42—132no—
536C42—132no—
537C42—264no—
538C42—261no—
539C42—261no—
540C42—522no—
541C13#2268311yescharacteristic, complemented, Sylow
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C2×D4×D1311yescharacteristic
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8C2×#610114yes—
554D4—22no—
555D4—22no—
556D4—22no—
557D4C2×#610114yes—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—132no—
574C2×C4—132no—
575C2×C4—132no—
576C2×C4—261no—
577C2×C4—261no—
578C2×C4—261no—
579C2×C4—261no—
580C2×C4—261no—
581C2×C4—261no—
582C2×C4—261no—
583C2×C4—261no—
584C2×C4—262no—
585C2×C4—262no—
586C2×C4—261no—
587C2×C4—521no—
588C2×C4—521no—
589C2×C4—521no—
590C2×C4—521no—
591C2×C4—521no—
592C2×C4—521no—
593C2×C4—521no—
594C2×C4—521no—
595C2×C4—521no—
596C2×C4—521no—
597C2×C4—1044no—
598C2×C4—1044no—
599C2×C4#8817012yes—
600C2×C4C22×Dic2611yescharacteristic
601C2×C4C22×C4×D1311yescharacteristic
602C2×C4#8825111yescharacteristic
603C2×C4#41×D1311yescharacteristic
604C2×C4#8825711yescharacteristic
605C2×C4C22×C22⋊D1311yescharacteristic
606C8—41no—
607C8—262no—
608C8—521no—
609C8C2×#610114yes—
610C22—24no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22D4×Dic2611yescharacteristic
618C22C4×D4×D1311yescharacteristic
619C22C2×#8817011yescharacteristic
620C22#1171946711yescharacteristic
621C22D13×#13511yescharacteristic
622C22#1171947511yescharacteristic
623C22D4×C22⋊D1311yescharacteristic
624C4—24no—
625C4—22no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—262no—
630C4—262no—
631C4—262no—
632C4—522no—
633C4—522no—
634C4—521no—
635C4—1041no—
636C4—1041no—
637C4#1171924411yescharacteristic
638C4#1171925111yescharacteristic
639C4C22×#610111yescharacteristic
640C4#1171952911yescharacteristic
641C2—44no—
642C2D4×#610111yescharacteristic
643C2C4×#8817011yescharacteristic
644C2#1312369011yescharacteristic
645C2#1312653711yescharacteristic
646C2#1312654311yescharacteristic
647C2#1312770911yescharacteristic
648C2#1312771211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C4×C13×SD16C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×C4×SD16D13
hypocenterC13#22683
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
3. #13117754C2semidirect
5. #13115564C2semidirect
6. #13115562C2semidirect
7. #13115090C2semidirect
8. #13115088C2semidirect
11. #13114997C2semidirect
12. #13114849C2semidirect
14. C8×#6101C2semidirect
35. SD16×Dic13C4semidirect
87. Q8×Dic13C2×C4semidirect
115. #88075C2×C4semidirect
129. #88057C2×C4semidirect
132. C8×Dic13C2×C4semidirect
172. C13×SD16C4⋊C4semidirect
495. SD16#6101semidirect
541. C13#22683semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×Dic132
#131177542
#131177412
#131155642
#131155622
#131150902
#131150882
#131150832
#131150812
#131149972
#131148492
#131148442
C8×#61012
C2×C4×C13×SD162
#2268313