Finite Group Databasebeta

← #15881263

Subgroups — #15881263

Order 3328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718825C211yesmaximal, characteristic
2C2×SD16×Dic13C211yesmaximal, characteristic
3#13117754C211yesmaximal, characteristic, complemented
4D4×#6101C211yesmaximal, characteristic
5#13115574C211yesmaximal, characteristic, complemented
6#13115567C211yesmaximal, characteristic, complemented
7#13115102C211yesmaximal, characteristic, complemented
8#13115097C211yesmaximal, characteristic
9#13115090C211yesmaximal, characteristic, complemented
10#13115081C211yesmaximal, characteristic
11#13115003C211yesmaximal, characteristic, complemented
12#13114844C211yesmaximal, characteristic
13#13114839C211yesmaximal, characteristic, complemented
14#13114712C211yesmaximal, characteristic, complemented
15C2×C13×#440C211yesmaximal, characteristic
16C22×C13×SD16C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×D4×C13C2211yescharacteristic
19C2×Q8×Dic13C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×D4×Dic13C2211yescharacteristic
22C2×D4×Dic13C2211yescharacteristic
23C22×#6101—21no—
24C13×#440—24no—
25C2×C13×#107C2211yescharacteristic
26C2×C13×#103C2211yescharacteristic
27C2×C13×#102C2211yescharacteristic
28C2×C13×#97C2211yescharacteristic
29C2×#88075C2211yescharacteristic
30C2×#88075C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33#11718871—21no—
34#11718825C418yescomplemented
35SD16×Dic13—24no—
36C2×#88058C2211yescharacteristic
37C2×#88057C2211yescharacteristic
38C2×#88055C2211yescharacteristic
39C2×#88054C2211yescharacteristic
40C2×C8×Dic13C2211yescharacteristic
41#11718638—21no—
42#11718638C2211yescharacteristic
43#11718637C2211yescharacteristic
44#11718634—21no—
45#11718634C2211yescharacteristic
46#11718634C2211yescharacteristic
47C2×#88048C2211yescharacteristic
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×#6101—21no—
103C2×#6101—21no—
104C2×#6101—44no—
105C2×#6101D411yescharacteristic
106C2×#6101D411yescharacteristic
107C2×C4×Dic13—21no—
108C2×C4×Dic13—21no—
109C2×C4×Dic13C2311yescharacteristic
110C2×C4×Dic13C2311yescharacteristic
111C22×#1055C2311yescharacteristic
112C22×#1055C2311yescharacteristic
113C13×#107—22no—
114C13×#103—22no—
115C13×#102—22no—
116C13×#97—22no—
117#88075—22no—
118#88075C2×C414yescomplemented
119#88072—22no—
120#88072C2×C414yes—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071—21no—
127#88071D411yescharacteristic
128#88071D411yescharacteristic
129#88058C2×C414yescomplemented
130#88057—22no—
131#88055C2×C414yescomplemented
132#88054—22no—
133C8×Dic13—22no—
134#88048—41no—
135#88047—41no—
136#22677—131nomaximal, Sylow
137C24×C13—21no—
138C2×Q8×C13—21no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13C22×C412yes—
142C2×Q8×C13C2×D412yes—
143C2×Q8×C13C2×Q812yes—
144C2×D4×C13—21no—
145C2×D4×C13—21no—
146C2×D4×C13—21no—
147C2×D4×C13C22×C412yes—
148C2×D4×C13C2×D412yes—
149C2×D4×C13C2×Q812yes—
150C22×C4×C13—21no—
151C22×C4×C13—21no—
152C22×C4×C13—21no—
153C22×C4×C13—42no—
154C22×C4×C13—42no—
155C22×C4×C13—81no—
156C22×C4×C13C2×D411yescharacteristic
157C22×C4×C13C2×D411yescharacteristic
158C22×C4×C13C2411yescharacteristic
159C22×Dic13—21no—
160C22×Dic13—21no—
161C22×Dic13—24no—
162C22×Dic13—21no—
163C22×Dic13—21no—
164C22×Dic13—42no—
165C22×Dic13—42no—
166C22×Dic13—42no—
167C22×Dic13—42no—
168C22×Dic13—81no—
169C22×Dic13C2×D411yescharacteristic
170C22×Dic13C2×D411yescharacteristic
171C22×Dic13C2×D411yescharacteristic
172C22×Dic13C2×D411yescharacteristic
173C13×SD16C4⋊C4116yescomplemented
174C2×C8×C13—21no—
175C2×C8×C13—21no—
176C2×C8×C13—21no—
177C2×C8×C13C22×C412yes—
178C2×C8×C13C2×D412yes—
179C2×C8×C13C2×Q812yes—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—21no—
184C13×C4⋊C4—42no—
185C13×C22⋊C4—42no—
186C42×C13—21no—
187C42×C13—21no—
188C42×C13—42no—
189C22⋊Dic13—24no—
190C22⋊Dic13—42no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13—24no—
194C4⋊Dic13—21no—
195C4⋊Dic13—21no—
196C4⋊Dic13—21no—
197C4⋊Dic13—21no—
198C4⋊Dic13—42no—
199C4⋊Dic13C22×C412yes—
200C4⋊Dic13C22×C412yes—
201#6101—22no—
202#6101—41no—
203#6101—42no—
204#6101—42no—
205#6101—42no—
206#6101—42no—
207#6101—42no—
208#6101—81no—
209C4×Dic13—21no—
210C4×Dic13—21no—
211C4×Dic13—24no—
212C4×Dic13—42no—
213C4×Dic13C22×C412yes—
214C4×Dic13#4112yes—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055C22×C412yes—
221C2×#440—131no—
222C2×#440D1311yescharacteristic
223C2×C4×SD16—131no—
224#2351—131no—
225D4×C4⋊C4—131no—
226#1943—131no—
227#1942—131no—
228#1865—131no—
229#1864—131no—
230#1863—131no—
231#1862—131no—
232#1847—131no—
233#1804—131no—
234#1803—131no—
235#1771—131no—
236C23×C13—22no—
237C23×C13—22no—
238C23×C13—22no—
239C23×C13—41no—
240C23×C13—41no—
241C23×C13—41no—
242C23×C13—41no—
243C23×C13C22×D411yescharacteristic
244Q8×C13—22no—
245Q8×C13—22no—
246Q8×C13—22no—
247Q8×C13C2×C4⋊C414yescomplemented
248D4×C13—22no—
249D4×C13—22no—
250D4×C13—22no—
251D4×C13C2×C4⋊C414yes—
252C2×C4×C13—22no—
253C2×C4×C13—22no—
254C2×C4×C13—22no—
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—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13—84no—
268C2×C4×C13C23×C411yescharacteristic
269C2×C4×C13C22×D411yescharacteristic
270C2×C4×C13C22×Q811yescharacteristic
271C2×C4×C13C2×#4111yescharacteristic
272C2×C4×C13H(5,𝔽2)11yescharacteristic
273C2×C4×C132−1+411yescharacteristic
274C2×Dic13—21no—
275C2×Dic13—21no—
276C2×Dic13—21no—
277C2×Dic13—21no—
278C2×Dic13—21no—
279C2×Dic13—21no—
280C2×Dic13—22no—
281C2×Dic13—22no—
282C2×Dic13—21no—
283C2×Dic13—21no—
284C2×Dic13—21no—
285C2×Dic13—48no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288C2×Dic13—41no—
289C2×Dic13—41no—
290C2×Dic13—41no—
291C2×Dic13—41no—
292C2×Dic13—41no—
293C2×Dic13—41no—
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—84no—
297C2×Dic13C4×D412yes—
298C2×Dic13C4×D412yes—
299C2×Dic13AGL(1,ℤ/8)12yes—
300C8×C13—41no—
301C8×C13C2×C4⋊C414yescomplemented
302#1055—22no—
303#1055—41no—
304C22×SD16C2×D1311yescharacteristic
305C2×C4×Q8—131no—
306C2×C4×Q8—131no—
307C2×C4×Q8C2×D1311yescharacteristic
308C2×C4×D4—131no—
309C2×C4×D4—131no—
310C2×C4×D4C2×D1311yescharacteristic
311C22×C4⋊C4—261no—
312#440—24no—
313#440—138no—
314C4×SD16—264no—
315C2×#107—131no—
316C2×#107—131no—
317C2×#107C2×D1311yescharacteristic
318C2×#106—131no—
319C2×#106—131no—
320C2×C4⋊C8—131no—
321C2×#103—131no—
322C2×#103—131no—
323C2×#103C2×D1311yescharacteristic
324C2×#102—131no—
325C2×#102—131no—
326C2×#102C2×D1311yescharacteristic
327C2×#97—131no—
328C2×#97C2×D1311yescharacteristic
329C2×C4×C8—131no—
330#389—131no—
331#389—131no—
332#389—131no—
333#389—131no—
334#389—261no—
335#389—261no—
336#385—261no—
337#385—261no—
338#384—131no—
339#382—131no—
340#382—261no—
341#380—261no—
342C4×C4⋊C4—131no—
343C4×C4⋊C4—261no—
344#340—131no—
345#340—131no—
346#340—131no—
347#340—131no—
348#336—131no—
349#336—131no—
350C22×C13—24no—
351C22×C13—42no—
352C22×C13—42no—
353C22×C13—42no—
354C22×C13—42no—
355C22×C13—42no—
356C22×C13—42no—
357C22×C13C2×C4×D411yescharacteristic
358C22×C13D4211yescharacteristic
359C22×C13D4×Q811yescharacteristic
360C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
361C22×C13#57511yescharacteristic
362C22×C13#57711yescharacteristic
363C22×C13#57811yescharacteristic
364C4×C13—24no—
365C4×C13—41no—
366C4×C13—41no—
367C4×C13—81no—
368C4×C13—81no—
369C4×C13C22×C4⋊C411yescharacteristic
370C4×C13#52011yescharacteristic
371C4×C13#54711yescharacteristic
372C4×C13#55411yescharacteristic
373Dic13—22no—
374Dic13—22no—
375Dic13—22no—
376Dic13—42no—
377Dic13—42no—
378Dic13—41no—
379Dic13—81no—
380Dic13—81no—
381C22×Q8C22×D1311yescharacteristic
382C22×D4C22×D1311yescharacteristic
383C23×C4—21no—
384C23×C4—261no—
385C23×C4—261no—
386C2×SD16Dic2614yes—
387C2×SD16C4×D1314yes—
388C2×SD16C22⋊D1314yes—
389C22×C8—131no—
390C22×C8—131no—
391C22×C8C22×D1311yescharacteristic
392C22×C8C22×D1311yescharacteristic
393C4×Q8—22no—
394C4×Q8—41no—
395C4×Q8—134no—
396C4×Q8—262no—
397C4×Q8—262no—
398C4×Q8—521no—
399C4×D4—22no—
400C4×D4—41no—
401C4×D4—134no—
402C4×D4—262no—
403C4×D4—262no—
404C4×D4—521no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—131no—
407C2×C4⋊C4—131no—
408C2×C4⋊C4—131no—
409C2×C4⋊C4—131no—
410C2×C4⋊C4—131no—
411C2×C4⋊C4—131no—
412C2×C4⋊C4—261no—
413C2×C4⋊C4—261no—
414C2×C4⋊C4—261no—
415C2×C4⋊C4—261no—
416C2×C4⋊C4—261no—
417C2×C4⋊C4—261no—
418C2×C4⋊C4—261no—
419C2×C4⋊C4—261no—
420C2×C4⋊C4—524no—
421C2×C4⋊C4C22×D1311yescharacteristic
422C2×C4⋊C4C22×D1311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—261no—
425C2×C22⋊C4—261no—
426C2×C42—21no—
427C2×C42—131no—
428C2×C42—131no—
429C2×C42—261no—
430C2×C42—261no—
431C2×C42C22×D1311yescharacteristic
432#107—22no—
433#107—134no—
434#107—521no—
435#106—262no—
436#106—521no—
437C4⋊C8—262no—
438#103—22no—
439#103—134no—
440#103—262no—
441#102—22no—
442#102—134no—
443#102—262no—
444#97—22no—
445#97—134no—
446C4×C8—262no—
447#95—131no—
448#95—131no—
449#95—261no—
450#95—261no—
451#95—261no—
452#95—261no—
453#95—261no—
454#95—261no—
455C2×C13—44no—
456C2×C13D4×C4⋊C411yescharacteristic
457C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
458C2×C13#294711yescharacteristic
459C2×C13#326511yescharacteristic
460C2×C13#326911yescharacteristic
461C2×C13#338611yescharacteristic
462C2×C13#338811yescharacteristic
463C24—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8C2×Dic2612yes—
468C2×Q8C2×C4×D1312yes—
469C2×Q8C2×C22⋊D1312yes—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×Dic2612yes—
474C2×D4C2×C4×D1312yes—
475C2×D4C2×C22⋊D1312yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—81no—
482C22×C4—131no—
483C22×C4—131no—
484C22×C4—131no—
485C22×C4—131no—
486C22×C4—264no—
487C22×C4—261no—
488C22×C4—261no—
489C22×C4—261no—
490C22×C4—261no—
491C22×C4—522no—
492C22×C4—522no—
493C22×C4—522no—
494C22×C4—522no—
495C22×C4—1041no—
496C22×C4D4×D1311yescharacteristic
497C22×C4D4×D1311yescharacteristic
498C22×C4C23×D1311yescharacteristic
499SD16#6101116yescomplemented
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—132no—
504C2×C8—261no—
505C2×C8—261no—
506C2×C8—261no—
507C2×C8—261no—
508C2×C8—261no—
509C2×C8C2×Dic2612yes—
510C2×C8C2×C4×D1312yes—
511C2×C8C2×C22⋊D1312yes—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—42no—
517C4⋊C4—132no—
518C4⋊C4—132no—
519C4⋊C4—262no—
520C4⋊C4—264no—
521C4⋊C4—261no—
522C4⋊C4—261no—
523C4⋊C4—261no—
524C4⋊C4—261no—
525C4⋊C4—261no—
526C4⋊C4—261no—
527C4⋊C4—521no—
528C4⋊C4—522no—
529C4⋊C4—522no—
530C4⋊C4—522no—
531C4⋊C4—522no—
532C4⋊C4—522no—
533C4⋊C4—522no—
534C4⋊C4—1041no—
535C22⋊C4—42no—
536C22⋊C4—264no—
537C22⋊C4—522no—
538C42—21no—
539C42—21no—
540C42—42no—
541C42—132no—
542C42—132no—
543C42—264no—
544C42—261no—
545C42—261no—
546C42—522no—
547C13#2267711yescharacteristic, complemented, Sylow
548C23—22no—
549C23—22no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23C2×D4×D1311yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C2×#610114yescomplemented
560D4—22no—
561D4—22no—
562D4—22no—
563D4C2×#610114yes—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—84no—
580C2×C4—132no—
581C2×C4—132no—
582C2×C4—132no—
583C2×C4—261no—
584C2×C4—261no—
585C2×C4—261no—
586C2×C4—261no—
587C2×C4—261no—
588C2×C4—261no—
589C2×C4—261no—
590C2×C4—261no—
591C2×C4—262no—
592C2×C4—262no—
593C2×C4—261no—
594C2×C4—528no—
595C2×C4—521no—
596C2×C4—521no—
597C2×C4—521no—
598C2×C4—521no—
599C2×C4—521no—
600C2×C4—521no—
601C2×C4—521no—
602C2×C4—521no—
603C2×C4—521no—
604C2×C4—521no—
605C2×C4—1044no—
606C2×C4C22×Dic2611yescharacteristic
607C2×C4C22×C4×D1311yescharacteristic
608C2×C4#8825111yescharacteristic
609C2×C4#41×D1311yescharacteristic
610C2×C4#8825711yescharacteristic
611C2×C4C22×C22⋊D1311yescharacteristic
612C8—41no—
613C8—262no—
614C8—521no—
615C8C2×#610114yescomplemented
616C22—24no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22D4×Dic2611yescharacteristic
624C22C4×D4×D1311yescharacteristic
625C22D13×AGL(1,ℤ/8)11yescharacteristic
626C22#1171948011yescharacteristic
627C22#1171948211yescharacteristic
628C22#1171948411yescharacteristic
629C22D4×C22⋊D1311yescharacteristic
630C4—24no—
631C4—41no—
632C4—41no—
633C4—81no—
634C4—81no—
635C4—262no—
636C4—262no—
637C4—262no—
638C4—522no—
639C4—522no—
640C4—521no—
641C4—1041no—
642C4—1041no—
643C4#1171924411yescharacteristic
644C4#1171925111yescharacteristic
645C4C22×#610111yescharacteristic
646C4#1171952911yescharacteristic
647C2—44no—
648C2D4×#610111yescharacteristic
649C2#1312364111yescharacteristic
650C2#1312367611yescharacteristic
651C2#1312632511yescharacteristic
652C2#1312638511yescharacteristic
653C2#1312775811yescharacteristic
654C2#1312778211yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#440C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
cosocleC22×C4×C13C24
hypercenterC2×#440D13
hypocenterC13#22677
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
3. #13117754C2semidirect
5. #13115574C2semidirect
6. #13115567C2semidirect
7. #13115102C2semidirect
9. #13115090C2semidirect
11. #13115003C2semidirect
13. #13114839C2semidirect
14. #13114712C2semidirect
34. #11718825C4semidirect
87. Q8×Dic13C2×C4semidirect
118. #88075C2×C4semidirect
129. #88058C2×C4semidirect
131. #88055C2×C4semidirect
173. C13×SD16C4⋊C4semidirect
247. Q8×C13C2×C4⋊C4semidirect
301. C8×C13C2×C4⋊C4semidirect
499. SD16#6101semidirect
547. C13#22677semidirect
559. Q8C2×#6101semidirect
615. C8C2×#6101semidirect

Maximal subgroups

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

subgroupindex
C2×#117188252
C2×SD16×Dic132
#131177542
D4×#61012
#131155742
#131155672
#131151022
#131150972
#131150902
#131150812
#131150032
#131148442
#131148392
#131147122
C2×C13×#4402
#2267713