Finite Group Databasebeta

← #15926002

Subgroups — #15926002

Order 3328.

856 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718752C211yesmaximal, characteristic, complemented
2C2×C4×#6105C211yesmaximal, characteristic
3#13117756C211yesmaximal, characteristic, complemented
4#13116357C211yesmaximal, characteristic
5#13116317C211yesmaximal, characteristic, complemented
6#13116224C211yesmaximal, characteristic
7#13116166C211yesmaximal, characteristic, complemented
8#13115566C211yesmaximal, characteristic, complemented
9#13115395C211yesmaximal, characteristic, complemented
10#13115194C211yesmaximal, characteristic
11#13115194C211yesmaximal, characteristic
12#13115103C211yesmaximal, characteristic, complemented
13#13115089C211yesmaximal, characteristic, complemented
14#13115083C211yesmaximal, characteristic
15C13×#2452C211yesmaximal, characteristic
16C2×C13×C4⋊Q8C2211yescharacteristic
17C2×C13×C22⋊Q8—21no—
18C2×C4×D4×C13C2211yescharacteristic
19C22×#6105C2211yescharacteristic
20C2×C4×Dic26C2211yescharacteristic
21C2×C4×Dic26C2211yescharacteristic
22C2×C4×Dic26C2211yescharacteristic
23C13×#396—21no—
24C13×#393—21no—
25C13×#390C2211yescharacteristic
26C13×#390C2211yescharacteristic
27C13×#389C2211yescharacteristic
28C13×#389C2211yescharacteristic
29C13×#385—21no—
30C4×C13×C4⋊C4C2211yescharacteristic
31C2×#88072C2211yescharacteristic
32C2×#88072C2211yescharacteristic
33C2×#88072C2211yescharacteristic
34#11718752—24no—
35C4×#6105—24no—
36#11718633—21no—
37#11718633—21no—
38C2×#88050C2211yescharacteristic
39C2×#88050C2211yescharacteristic
40C2×#88050C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88048C2211yescharacteristic
43C2×#88047C2211yescharacteristic
44C2×#88047C2211yescharacteristic
45C2×#88047C2211yescharacteristic
46#11718597C2211yescharacteristic
47#11718589C2211yescharacteristic
48#11718589C2211yescharacteristic
49C2×C4⋊#1055C2211yescharacteristic
50C2×C4×#1055C2211yescharacteristic
51#11718311—21no—
52#11718310—21no—
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718116C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13—21no—
62C22×D4×C13C2311yescharacteristic
63C23×C4×C13—21no—
64C22×Dic26C2311yescharacteristic
65C13×C4⋊Q8—22no—
66C13×C4⋊Q8—41no—
67C13×C22⋊Q8—44no—
68C4×D4×C13—22no—
69C4×D4×C13—41no—
70C2×C13×C4⋊C4—21no—
71C2×C13×C4⋊C4—21no—
72C2×C13×C4⋊C4—21no—
73C2×C13×C4⋊C4D411yescharacteristic
74C2×C13×C4⋊C4D411yescharacteristic
75C2×C13×C4⋊C4C2311yescharacteristic
76C2×C13×C4⋊C4C2311yescharacteristic
77C2×C13×C4⋊C4C2311yescharacteristic
78C2×C13×C22⋊C4—21no—
79C2×C13×C22⋊C4—21no—
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C13×C22⋊C4—21no—
83C2×C42×C13C2311yescharacteristic
84C2×C42×C13C2311yescharacteristic
85C2×C42×C13C2311yescharacteristic
86C2×#6105—22no—
87C2×#6105—22no—
88C2×#6105Q814yescomplemented
89C2×C4⋊Dic13C2311yescharacteristic
90C2×C4⋊Dic13C2311yescharacteristic
91C2×C4⋊Dic13C2311yescharacteristic
92C2×#6101—21no—
93C2×#6101—21no—
94C2×#6101—21no—
95C2×#6101—21no—
96C2×#6101—21no—
97C2×C4×Dic13—21no—
98C2×C4×Dic13—21no—
99C2×C4×Dic13—21no—
100C22×#1055C2311yescharacteristic
101C22×#1055C2311yescharacteristic
102C22×#1055C2311yescharacteristic
103C22×#1055C2311yescharacteristic
104C4×Dic26—22no—
105C4×Dic26—22no—
106C4×Dic26—22no—
107C4×Dic26—22no—
108C4×Dic26—41no—
109C4×Dic26—41no—
110C13×#95—21no—
111C13×#95—21no—
112C13×#95D411yescharacteristic
113C13×#95D411yescharacteristic
114#88072—22no—
115#88072—22no—
116#88072—41no—
117#88071—21no—
118#88071—21no—
119#88071—21no—
120#88071—21no—
121#88050—22no—
122#88050—22no—
123#88050D414yes—
124#88048—22no—
125#88048—41no—
126#88047—22no—
127#88047—22no—
128#88047—22no—
129C4⋊#1055—22no—
130C4×#1055—22no—
131#26377—131nomaximal, Sylow
132C24×C13—21no—
133C2×Q8×C13—42no—
134C2×Q8×C13—42no—
135C2×Q8×C13—81no—
136C2×D4×C13—21no—
137C2×D4×C13—21no—
138C2×D4×C13—21no—
139C2×D4×C13—21no—
140C2×D4×C13—21no—
141C2×D4×C13C2×Q812yes—
142C22×C4×C13—21no—
143C22×C4×C13—21no—
144C22×C4×C13—21no—
145C22×C4×C13—42no—
146C22×C4×C13—42no—
147C22×C4×C13—81no—
148C22×C4×C13C2×D411yescharacteristic
149C22×C4×C13C2×D411yescharacteristic
150C22×C4×C13C2×D411yescharacteristic
151C22×C4×C13C2×D411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13C2×D411yescharacteristic
154C22×C4×C13C2411yescharacteristic
155C22×Dic13—21no—
156C22×Dic13—21no—
157C22×Dic13—21no—
158C22×Dic13—21no—
159C2×Dic26—21no—
160C2×Dic26—21no—
161C2×Dic26—21no—
162C2×Dic26C2×D412yes—
163C2×Dic26C2×Q812yes—
164C2×Dic26#4112yes—
165C13×C4⋊C4—22no—
166C13×C4⋊C4—22no—
167C13×C4⋊C4—21no—
168C13×C4⋊C4—21no—
169C13×C4⋊C4—21no—
170C13×C4⋊C4—21no—
171C13×C4⋊C4—21no—
172C13×C4⋊C4—42no—
173C13×C4⋊C4—42no—
174C13×C4⋊C4—42no—
175C13×C4⋊C4C2×D412yes—
176C13×C22⋊C4—42no—
177C13×C22⋊C4—42no—
178C13×C22⋊C4—42no—
179C13×C22⋊C4—42no—
180C13×C22⋊C4—81no—
181C42×C13—21no—
182C42×C13—21no—
183C42×C13—21no—
184C42×C13—21no—
185C42×C13—21no—
186C42×C13—21no—
187#6105—28no—
188C4⋊Dic13—21no—
189C4⋊Dic13—21no—
190C4⋊Dic13—21no—
191C4⋊Dic13—21no—
192C4⋊Dic13—21no—
193C4⋊Dic13#4112yes—
194#6101—24no—
195#6101—42no—
196#6101—42no—
197#6101—42no—
198#6101—42no—
199C4×Dic13—24no—
200C4×Dic13—42no—
201C4×Dic13—42no—
202C2×#1055—21no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—21no—
210C2×#1055—21no—
211C2×#1055—21no—
212C2×#1055C2×D412yes—
213C2×#1055C2×Q812yes—
214C2×#477—131no—
215C2×C4×SD16—131no—
216#2452D1311yescharacteristic
217#2351—131no—
218#2092—131no—
219#2085—131no—
220#2067—131no—
221#2058—131no—
222#1942—131no—
223#1916—131no—
224#1880—131no—
225#1880—131no—
226#1865—131no—
227#1863—131no—
228#1862—131no—
229C23×C13—22no—
230C23×C13—41no—
231C23×C13—41no—
232C23×C13—41no—
233C23×C13—41no—
234C23×C13—41no—
235C23×C13—41no—
236C23×C13C22×D411yescharacteristic
237Q8×C13—84no—
238D4×C13—22no—
239D4×C13—22no—
240D4×C13—41no—
241D4×C13—41no—
242C2×C4×C13—21no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—22no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—21no—
258C2×C4×C13—21no—
259C2×C4×C13—41no—
260C2×C4×C13—41no—
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×C13#13512yes—
269C2×C4×C13AGL(1,ℤ/8)12yes—
270C2×C4×C13C22×D411yescharacteristic
271C2×C4×C13C22×Q811yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C2×C4×C13C2×#4111yescharacteristic
274C2×C4×C13H(5,𝔽2)11yescharacteristic
275C2×C4×C132−1+411yescharacteristic
276C2×Dic13—22no—
277C2×Dic13—22no—
278C2×Dic13—22no—
279C2×Dic13—22no—
280C2×Dic13—41no—
281C2×Dic13—41no—
282C2×Dic13—41no—
283C2×Dic13—41no—
284C2×Dic13—41no—
285C2×Dic13—41no—
286C2×Dic13—41no—
287C2×Dic13—41no—
288Dic26—22no—
289Dic26—22no—
290Dic26—22no—
291Dic26C22⋊Q814yescomplemented
292#1055—22no—
293#1055—22no—
294#1055—41no—
295#1055—41no—
296C22×SD16—131no—
297C2×C4⋊Q8C2×D1311yescharacteristic
298C2×C22⋊Q8—21no—
299C2×C4×Q8—131no—
300C2×C4×Q8—131no—
301C2×C4×Q8—131no—
302C2×C4×D4C2×D1311yescharacteristic
303#477—264no—
304C4×SD16—264no—
305C2×#107—131no—
306C2×#107—131no—
307C2×#107—131no—
308C2×#106—131no—
309C2×#106—131no—
310C2×C4⋊C8—131no—
311C2×#103—131no—
312C2×#103—131no—
313C2×#103—131no—
314C2×#102—131no—
315C2×#102—131no—
316C2×#102—131no—
317C2×C4×C8—131no—
318#396—21no—
319#393—21no—
320#390C2×D1311yescharacteristic
321#390C2×D1311yescharacteristic
322#389—131no—
323#389—131no—
324#389—261no—
325#389—261no—
326#389C2×D1311yescharacteristic
327#389C2×D1311yescharacteristic
328#385—21no—
329#382—131no—
330#382—261no—
331C4×C4⋊C4—261no—
332C4×C4⋊C4C2×D1311yescharacteristic
333#340—131no—
334#340—131no—
335#340—131no—
336#340—131no—
337#336—131no—
338#336—131no—
339#336—131no—
340#336—131no—
341C22×C13—42no—
342C22×C13—42no—
343C22×C13—42no—
344C22×C13—81no—
345C22×C13—81no—
346C22×C13—81no—
347C22×C13—81no—
348C22×C13#52511yescharacteristic
349C22×C13D4211yescharacteristic
350C22×C13D4×Q811yescharacteristic
351C22×C13C2×#13511yescharacteristic
352C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
353C22×C13#57711yescharacteristic
354C22×C13#57811yescharacteristic
355C4×C13—22no—
356C4×C13—22no—
357C4×C13—41no—
358C4×C13—41no—
359C4×C13—41no—
360C4×C13—42no—
361C4×C13—41no—
362C4×C13—81no—
363C4×C13—81no—
364C4×C13C2×C22⋊Q811yescharacteristic
365C4×C13#53911yescharacteristic
366C4×C13#54711yescharacteristic
367C4×C13#55411yescharacteristic
368Dic13—24no—
369Dic13—42no—
370Dic13—81no—
371Dic13—81no—
372C22×Q8—21no—
373C22×Q8—131no—
374C22×D4C22×D1311yescharacteristic
375C23×C4—21no—
376C2×SD16—134no—
377C2×SD16—262no—
378C2×SD16—262no—
379C22×C8—131no—
380C22×C8—131no—
381C22×C8—131no—
382C22×C8—131no—
383C4⋊Q8—22no—
384C4⋊Q8—41no—
385C22⋊Q8—44no—
386C4×Q8—262no—
387C4×Q8—262no—
388C4×Q8—262no—
389C4×Q8—262no—
390C4×Q8—521no—
391C4×Q8—521no—
392C4×D4—22no—
393C4×D4—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—131no—
398C2×C4⋊C4—131no—
399C2×C4⋊C4—131no—
400C2×C4⋊C4—261no—
401C2×C4⋊C4—261no—
402C2×C4⋊C4—261no—
403C2×C4⋊C4—261no—
404C2×C4⋊C4—261no—
405C2×C4⋊C4C22⋊D1311yescharacteristic
406C2×C4⋊C4C22⋊D1311yescharacteristic
407C2×C4⋊C4C22×D1311yescharacteristic
408C2×C4⋊C4C22×D1311yescharacteristic
409C2×C4⋊C4C22×D1311yescharacteristic
410C2×C22⋊C4—21no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C42—261no—
416C2×C42—261no—
417C2×C42—261no—
418C2×C42C22×D1311yescharacteristic
419C2×C42C22×D1311yescharacteristic
420C2×C42C22×D1311yescharacteristic
421#107—262no—
422#107—262no—
423#107—262no—
424#106—262no—
425#106—521no—
426C4⋊C8—262no—
427#103—134no—
428#103—262no—
429#103—262no—
430#102—262no—
431#102—262no—
432#102—521no—
433C4×C8—262no—
434#95—21no—
435#95—21no—
436#95—261no—
437#95—261no—
438#95—261no—
439#95—261no—
440#95C22⋊D1311yescharacteristic
441#95C22⋊D1311yescharacteristic
442C2×C13—82no—
443C2×C13#244911yescharacteristic
444C2×C13#310611yescharacteristic
445C2×C13#312111yescharacteristic
446C2×C13#326511yescharacteristic
447C2×C13#328511yescharacteristic
448C2×C13#338111yescharacteristic
449C2×C13#338611yescharacteristic
450C24—21no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—81no—
454C2×Q8—132no—
455C2×Q8—132no—
456C2×Q8—132no—
457C2×Q8—261no—
458C2×Q8—261no—
459C2×Q8—261no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4Q8×D1312yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—261no—
473C22×C4—261no—
474C22×C4—261no—
475C22×C4—261no—
476C22×C4C2×C22⋊D1311yescharacteristic
477C22×C4C2×C22⋊D1311yescharacteristic
478C22×C4C2×C22⋊D1311yescharacteristic
479C22×C4C2×C22⋊D1311yescharacteristic
480C22×C4C2×C22⋊D1311yescharacteristic
481C22×C4C2×C22⋊D1311yescharacteristic
482C22×C4C23×D1311yescharacteristic
483SD16—268no—
484C2×C8—132no—
485C2×C8—132no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—261no—
492C2×C8—261no—
493C2×C8—261no—
494C2×C8—261no—
495C2×C8—261no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—132no—
507C4⋊C4—264no—
508C4⋊C4—261no—
509C4⋊C4—261no—
510C4⋊C4—261no—
511C4⋊C4—261no—
512C4⋊C4—261no—
513C4⋊C4—522no—
514C4⋊C4—522no—
515C4⋊C4—522no—
516C4⋊C4—522no—
517C4⋊C4D4×D1312yes—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—81no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—264no—
530C42—522no—
531C42—522no—
532C13#2637711yescharacteristic, complemented, Sylow
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×C22⋊D1311yescharacteristic
541Q8—84no—
542Q8—134no—
543Q8—262no—
544Q8—262no—
545Q8—262no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—41no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—262no—
577C2×C4—262no—
578C2×C4—262no—
579C2×C4—262no—
580C2×C4—521no—
581C2×C4—521no—
582C2×C4—521no—
583C2×C4—521no—
584C2×C4—521no—
585C2×C4—521no—
586C2×C4—521no—
587C2×C4—521no—
588C2×C4#8818612yes—
589C2×C4#8820412yes—
590C2×C4C2×D4×D1311yescharacteristic
591C2×C4#8825111yescharacteristic
592C2×C4C2×Q8×D1311yescharacteristic
593C2×C4#8825411yescharacteristic
594C2×C4#41×D1311yescharacteristic
595C2×C4#41×D1311yescharacteristic
596C8—262no—
597C8—262no—
598C8—521no—
599C8—521no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22C2×#8818611yescharacteristic
608C22D4×C22⋊D1311yescharacteristic
609C22Q8×C22⋊D1311yescharacteristic
610C22C2×#8820411yescharacteristic
611C22#1171953311yescharacteristic
612C22#1171954111yescharacteristic
613C22#1171954311yescharacteristic
614C4—22no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—264no—
624C4—522no—
625C4—1041no—
626C4—1041no—
627C4D13×C22⋊Q811yescharacteristic
628C4#1171933611yescharacteristic
629C4#1171934111yescharacteristic
630C4#1171934311yescharacteristic
631C2—82no—
632C2#1311864211yescharacteristic
633C2#1312489011yescharacteristic
634C2#1312505911yescharacteristic
635C2#1312632711yescharacteristic
636C2#1312657311yescharacteristic
637C2#1312771111yescharacteristic
638C2#1312775711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#2452C2
socleC23×C13C22×D4
prime radicalC22×D4×C13C23
hypercenter#2452D13
hypocenterC13#26377
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#11718752C2semidirect
3. #13117756C2semidirect
5. #13116317C2semidirect
7. #13116166C2semidirect
8. #13115566C2semidirect
9. #13115395C2semidirect
12. #13115103C2semidirect
13. #13115089C2semidirect
88. C2×#6105Q8semidirect
291. Dic26C22⋊Q8semidirect
532. C13#26377semidirect

Maximal subgroups

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

subgroupindex
C2×#117187522
C2×C4×#61052
#131177562
#131163572
#131163172
#131162242
#131161662
#131155662
#131153952
#131151942
#131151032
#131150892
#131150832
C13×#24522
#2637713