Finite Group Databasebeta

← #15880219

Subgroups — #15880219

Order 3328.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718893C211yesmaximal, characteristic
2#13117703C211yesmaximal, characteristic, complemented
3#13115536C211yesmaximal, characteristic, complemented
4#13115522C211yesmaximal, characteristic, complemented
5#13115460C211yesmaximal, characteristic, complemented
6#13115453C211yesmaximal, characteristic, complemented
7#13115104C211yesmaximal, characteristic, complemented
8#13115090C211yesmaximal, characteristic, complemented
9#13114924C211yesmaximal, characteristic, complemented
10#13114867C211yesmaximal, characteristic, complemented
11#13114776C211yesmaximal, characteristic
12#13114666C211yesmaximal, characteristic, complemented
13C4×#88075C211yesmaximal, characteristic, complemented
14#13114620C211yesmaximal, characteristic, complemented
15C13×#2341C211yesmaximal, characteristic
16C22×C13×#41C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C22×#6099C2211yescharacteristic
21C2×#88131—21no—
22C13×#390—21no—
23C13×#390—21no—
24C13×#390C2211yescharacteristic
25C13×#390C2211yescharacteristic
26C13×#380—21no—
27C13×#380C2211yescharacteristic
28C4×C13×C22⋊C4—21no—
29C4×C13×C22⋊C4C2211yescharacteristic
30#11718893C418yescomplemented
31C2×#88075C2211yescharacteristic
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35C2×#88072C2211yescharacteristic
36C2×#88072C2211yescharacteristic
37C2×#88072C2211yescharacteristic
38C2×#88072C2211yescharacteristic
39#11718873C2211yescharacteristic
40C2×C22⋊#1055C2211yescharacteristic
41C2×#88048C2211yescharacteristic
42C2×#88047C2211yescharacteristic
43#11718604—21no—
44C4×C4⋊Dic13C2211yescharacteristic
45C4×C4⋊Dic13C2211yescharacteristic
46#11718589C2211yescharacteristic
47#11718589C2211yescharacteristic
48C2×#88043C2211yescharacteristic
49C2×C4×#1055C2211yescharacteristic
50#11718314—21no—
51#11718314—21no—
52#11718121C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718116C2211yescharacteristic
59#11718116C2211yescharacteristic
60C2×C13×#41—22no—
61C2×C13×#41—22no—
62C2×C13×#41—22no—
63C2×C13×#41—41no—
64C2×C13×#41—41no—
65C2×C13×#41C2×C414yes—
66C22×Q8×C13C2311yescharacteristic
67C22×D4×C13—21no—
68C22×D4×C13C2311yescharacteristic
69C23×C4×C13—21no—
70C23×C4×C13C2311yescharacteristic
71C23×Dic13—21no—
72C4×Q8×C13—22no—
73C4×Q8×C13—22no—
74C4×Q8×C13—41no—
75C4×Q8×C13—41no—
76C2×C13×C4⋊C4—21no—
77C2×C13×C4⋊C4—21no—
78C2×C13×C4⋊C4C2311yescharacteristic
79C2×C13×C4⋊C4C2311yescharacteristic
80C2×C13×C22⋊C4—21no—
81C2×C13×C22⋊C4—21no—
82C2×C13×C22⋊C4—21no—
83C2×C13×C22⋊C4—21no—
84C2×C13×C22⋊C4—21no—
85C2×C13×C22⋊C4D411yescharacteristic
86C2×C13×C22⋊C4D411yescharacteristic
87C2×C42×C13—21no—
88C2×C42×C13—21no—
89C2×C42×C13C2311yescharacteristic
90C2×C42×C13C2311yescharacteristic
91C2×C22⋊Dic13—21no—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—22no—
94C2×C4⋊Dic13C2×C414yes—
95C2×C4⋊Dic13C2311yescharacteristic
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4⋊Dic13C2311yescharacteristic
99C2×#6101—21no—
100C2×#6101—21no—
101C2×C4×Dic13—21no—
102C2×C4×Dic13—21no—
103C2×#6099—41no—
104C2×#6099—41no—
105C2×#6099C2×C414yes—
106C22×#1055C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110#88131—44no—
111C13×#95—21no—
112C13×#95D411yescharacteristic
113C13×#95D411yescharacteristic
114#88075—22no—
115#88075—22no—
116#88075C2×C414yes—
117#88075C2×C414yes—
118#88072—41no—
119#88072—41no—
120#88072C2×C414yes—
121#88072C2×C414yes—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071—21no—
127C22⋊#1055—22no—
128#88048—41no—
129#88047—41no—
130#88043—41no—
131C4×#1055—41no—
132#22583—131nomaximal, Sylow
133C24×C13—21no—
134C24×C13C2×D411yescharacteristic
135C13×#41—28no—
136C13×#41—44no—
137C13×#41—44no—
138C13×#41—44no—
139C2×Q8×C13—21no—
140C2×Q8×C13—21no—
141C2×Q8×C13—21no—
142C2×Q8×C13C22×C412yes—
143C2×Q8×C13#4112yes—
144C2×Q8×C13#4112yes—
145C2×D4×C13—24no—
146C2×D4×C13—21no—
147C2×D4×C13—21no—
148C2×D4×C13—21no—
149C2×D4×C13—21no—
150C2×D4×C13—21no—
151C2×D4×C13—42no—
152C2×D4×C13—42no—
153C2×D4×C13C22×C412yes—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—24no—
160C22×C4×C13—21no—
161C22×C4×C13—21no—
162C22×C4×C13—42no—
163C22×C4×C13—42no—
164C22×C4×C13C22×C412yes—
165C22×C4×C13C2×D411yescharacteristic
166C22×C4×C13C2×D411yescharacteristic
167C22×C4×C13C2×D411yescharacteristic
168C22×C4×C13C2×D411yescharacteristic
169C22×C4×C13C2×D411yescharacteristic
170C22×C4×C13#4112yes—
171C22×C4×C13C2411yescharacteristic
172C22×Dic13—24no—
173C22×Dic13—21no—
174C22×Dic13—21no—
175C22×Dic13—21no—
176C22×Dic13—21no—
177C22×Dic13—42no—
178C22×Dic13—42no—
179C13×C4⋊C4—21no—
180C13×C4⋊C4—21no—
181C13×C4⋊C4—21no—
182C13×C4⋊C4—21no—
183C13×C4⋊C4—42no—
184C13×C4⋊C4—42no—
185C13×C22⋊C4—22no—
186C13×C22⋊C4—41no—
187C13×C22⋊C4—42no—
188C13×C22⋊C4—42no—
189C13×C22⋊C4—81no—
190C13×C22⋊C4—81no—
191C13×C22⋊C4—81no—
192C42×C13—21no—
193C42×C13—21no—
194C42×C13—21no—
195C42×C13—21no—
196C42×C13—42no—
197C42×C13—42no—
198C22⋊Dic13—42no—
199C4⋊Dic13—24no—
200C4⋊Dic13—21no—
201C4⋊Dic13—21no—
202C4⋊Dic13—21no—
203C4⋊Dic13—21no—
204C4⋊Dic13—21no—
205C4⋊Dic13—21no—
206C4⋊Dic13C22×C412yes—
207C4⋊Dic13C22×C412yes—
208#6101—42no—
209#6101—42no—
210C4×Dic13—42no—
211C4×Dic13—42no—
212#6099—44no—
213C2×#1055—21no—
214C2×#1055—21no—
215C2×#1055—21no—
216C2×#1055—21no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—21no—
222C2×#1055—21no—
223C2×#1055C22×C412yes—
224C2×#1055C22×C412yes—
225C2×#417—131no—
226#2344—131no—
227#2341D1311yescharacteristic
228#1936—131no—
229#1934—131no—
230#1926—131no—
231#1925—131no—
232#1865—131no—
233#1863—131no—
234#1824—131no—
235#1809—131no—
236#1788—131no—
237#1763—131no—
238C4×#103—131no—
239#1753—131no—
240C23×C13—21no—
241C23×C13—21no—
242C23×C13—21no—
243C23×C13—21no—
244C23×C13—21no—
245C23×C13—21no—
246C23×C13—22no—
247C23×C13—41no—
248C23×C13—41no—
249C23×C13—41no—
250C23×C13—41no—
251C23×C13—41no—
252C23×C13—41no—
253C23×C13C4×D412yes—
254C23×C13C22×D411yescharacteristic
255Q8×C13—22no—
256Q8×C13—22no—
257Q8×C13—22no—
258Q8×C13#11714yescomplemented
259D4×C13—22no—
260D4×C13—22no—
261D4×C13—48no—
262D4×C13—41no—
263D4×C13—41no—
264C2×C4×C13—22no—
265C2×C4×C13—21no—
266C2×C4×C13—22no—
267C2×C4×C13—22no—
268C2×C4×C13—21no—
269C2×C4×C13—21no—
270C2×C4×C13—21no—
271C2×C4×C13—22no—
272C2×C4×C13—22no—
273C2×C4×C13—22no—
274C2×C4×C13—22no—
275C2×C4×C13—21no—
276C2×C4×C13—21no—
277C2×C4×C13—21no—
278C2×C4×C13—21no—
279C2×C4×C13—21no—
280C2×C4×C13—21no—
281C2×C4×C13—21no—
282C2×C4×C13—21no—
283C2×C4×C13—21no—
284C2×C4×C13—48no—
285C2×C4×C13—41no—
286C2×C4×C13—41no—
287C2×C4×C13—41no—
288C2×C4×C13—41no—
289C2×C4×C13—41no—
290C2×C4×C13—41no—
291C2×C4×C13C4×D412yes—
292C2×C4×C13AGL(1,ℤ/8)12yes—
293C2×C4×C13C23×C411yescharacteristic
294C2×C4×C13C2×#4111yescharacteristic
295C2×C4×C13C2×#4111yescharacteristic
296C2×C4×C13C2×#4111yescharacteristic
297C2×C4×C13H(5,𝔽2)11yescharacteristic
298C2×C4×C13H(5,𝔽2)11yescharacteristic
299C2×Dic13—22no—
300C2×Dic13—22no—
301C2×Dic13—48no—
302C2×Dic13—41no—
303C2×Dic13—41no—
304C2×Dic13—41no—
305C2×Dic13—41no—
306C2×Dic13—41no—
307C2×Dic13—41no—
308C2×Dic13—41no—
309C2×Dic13—41no—
310C2×Dic13—41no—
311C2×Dic13—41no—
312#1055—22no—
313#1055—22no—
314#1055—41no—
315#1055—41no—
316C22×#41C2×D1311yescharacteristic
317C22×M16—131no—
318C2×C4×Q8C2×D1311yescharacteristic
319C2×C4×Q8C2×D1311yescharacteristic
320C2×#117—261no—
321C22×C4⋊C4—131no—
322C2×#107—131no—
323C2×#106—131no—
324#417—138no—
325C2×#103—131no—
326C2×#103—131no—
327C2×#103—131no—
328C2×#103—131no—
329C2×#102—131no—
330C2×#102—131no—
331C2×#102—131no—
332C2×#102—131no—
333C2×C22⋊C8—131no—
334C2×#97—131no—
335C2×C4×C8—131no—
336#390—21no—
337#390—21no—
338#390C2×D1311yescharacteristic
339#390C2×D1311yescharacteristic
340#389—131no—
341#389—131no—
342#387—261no—
343#387—261no—
344#381—261no—
345#380—21no—
346#380—131no—
347#380C2×D1311yescharacteristic
348C4×C4⋊C4—131no—
349C4×C4⋊C4—131no—
350C4×C22⋊C4—21no—
351C4×C22⋊C4C2×D1311yescharacteristic
352#340—131no—
353#340—131no—
354#340—131no—
355#340—131no—
356#340—131no—
357#340—131no—
358#336—131no—
359#336—131no—
360C22×C13—22no—
361C22×C13—22no—
362C22×C13—22no—
363C22×C13—41no—
364C22×C13—41no—
365C22×C13—41no—
366C22×C13—41no—
367C22×C13—42no—
368C22×C13—42no—
369C22×C13—42no—
370C22×C13—81no—
371C22×C13—81no—
372C22×C13—81no—
373C22×C13—81no—
374C22×C13C2×C4×D411yescharacteristic
375C22×C13#54611yescharacteristic
376C22×C13#54611yescharacteristic
377C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
378C22×C13#57511yescharacteristic
379C22×C13#57711yescharacteristic
380C22×C13#57811yescharacteristic
381C4×C13—22no—
382C4×C13—24no—
383C4×C13—22no—
384C4×C13—41no—
385C4×C13—41no—
386C4×C13—41no—
387C4×C13—81no—
388C4×C13—81no—
389C4×C13C2×#11711yescharacteristic
390C4×C13#51811yescharacteristic
391C4×C13#55111yescharacteristic
392C4×C13#55111yescharacteristic
393Dic13—42no—
394Dic13—42no—
395Dic13—81no—
396Dic13—81no—
397C2×#41—22no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—41no—
401C2×#41—41no—
402C2×#41C4×D1314yes—
403C22×Q8C22×D1311yescharacteristic
404C22×D4—21no—
405C22×D4C22×D1311yescharacteristic
406C23×C4—21no—
407C23×C4—261no—
408C23×C4C22×D1311yescharacteristic
409C2×M16—134no—
410C2×M16—521no—
411C2×M16—521no—
412C22×C8—131no—
413C22×C8—131no—
414C22×C8—131no—
415C22×C8—131no—
416C4×Q8—22no—
417C4×Q8—22no—
418C4×Q8—41no—
419C4×Q8—41no—
420#117—524no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—134no—
424C2×C4⋊C4—131no—
425C2×C4⋊C4—131no—
426C2×C4⋊C4—131no—
427C2×C4⋊C4—131no—
428C2×C4⋊C4—262no—
429C2×C4⋊C4—261no—
430C2×C4⋊C4—261no—
431C2×C4⋊C4—261no—
432C2×C4⋊C4C22×D1311yescharacteristic
433C2×C4⋊C4C22×D1311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—261no—
440C2×C22⋊C4C22⋊D1311yescharacteristic
441C2×C22⋊C4C22⋊D1311yescharacteristic
442C2×C42—21no—
443C2×C42—21no—
444C2×C42—261no—
445C2×C42—261no—
446C2×C42C22×D1311yescharacteristic
447C2×C42C22×D1311yescharacteristic
448#107—521no—
449#106—521no—
450#103—134no—
451#103—134no—
452#103—262no—
453#103—262no—
454#102—134no—
455#102—134no—
456#102—521no—
457#102—521no—
458C22⋊C8—262no—
459#97—521no—
460C4×C8—521no—
461#95—21no—
462#95—261no—
463#95—261no—
464#95—261no—
465#95—261no—
466#95—261no—
467#95C22⋊D1311yescharacteristic
468#95C22⋊D1311yescharacteristic
469C2×C13—42no—
470C2×C13—82no—
471C2×C13#234211yescharacteristic
472C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
473C2×C13#294711yescharacteristic
474C2×C13#330811yescharacteristic
475C2×C13#331711yescharacteristic
476C2×C13#332711yescharacteristic
477C2×C13#333211yescharacteristic
478C24—21no—
479C24C2×C22⋊D1311yescharacteristic
480#41—28no—
481#41—44no—
482#41—44no—
483#41—44no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8C2×C4×D1312yes—
488C2×Q8#612912yes—
489C2×Q8#612912yes—
490C2×D4—24no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—42no—
497C2×D4—42no—
498C2×D4C2×C4×D1312yes—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—24no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—264no—
510C22×C4—261no—
511C22×C4—261no—
512C22×C4—261no—
513C22×C4—261no—
514C22×C4—522no—
515C22×C4—522no—
516C22×C4C2×C4×D1312yes—
517C22×C4#612712yes—
518C22×C4C2×C22⋊D1311yescharacteristic
519C22×C4C2×C22⋊D1311yescharacteristic
520C22×C4C2×C22⋊D1311yescharacteristic
521C22×C4C2×C22⋊D1311yescharacteristic
522C22×C4C2×C22⋊D1311yescharacteristic
523C22×C4C23×D1311yescharacteristic
524M16—524no—
525C2×C8—132no—
526C2×C8—132no—
527C2×C8—261no—
528C2×C8—261no—
529C2×C8—261no—
530C2×C8—261no—
531C2×C8—261no—
532C2×C8—261no—
533C2×C8—261no—
534C2×C8—261no—
535C2×C8—261no—
536C2×C8—261no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—132no—
544C4⋊C4—132no—
545C4⋊C4—264no—
546C4⋊C4—261no—
547C4⋊C4—261no—
548C4⋊C4—261no—
549C4⋊C4—261no—
550C4⋊C4—261no—
551C4⋊C4—261no—
552C4⋊C4—522no—
553C4⋊C4—522no—
554C22⋊C4—22no—
555C22⋊C4—41no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—81no—
559C22⋊C4—81no—
560C22⋊C4—81no—
561C22⋊C4—522no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—42no—
568C42—522no—
569C42—522no—
570C13#2258311yescharacteristic, complemented, Sylow
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C4×C22⋊D1312yes—
585C23C22×C22⋊D1311yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8#8813114yescomplemented
590D4—22no—
591D4—22no—
592D4—48no—
593D4—41no—
594D4—41no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—48no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—262no—
623C2×C4—262no—
624C2×C4—528no—
625C2×C4—521no—
626C2×C4—521no—
627C2×C4—521no—
628C2×C4—521no—
629C2×C4—521no—
630C2×C4—521no—
631C2×C4—521no—
632C2×C4—521no—
633C2×C4—521no—
634C2×C4—521no—
635C2×C4C4×C22⋊D1312yes—
636C2×C4#8820312yes—
637C2×C4C22×C4×D1311yescharacteristic
638C2×C4C2×#612711yescharacteristic
639C2×C4C2×#612911yescharacteristic
640C2×C4C2×#612911yescharacteristic
641C2×C4#8825111yescharacteristic
642C2×C4#8825111yescharacteristic
643C8—262no—
644C8—262no—
645C8—521no—
646C8—521no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×C4×C22⋊D1311yescharacteristic
662C22#1171951111yescharacteristic
663C22#1171951111yescharacteristic
664C22C2×#8820311yescharacteristic
665C22#1171952711yescharacteristic
666C22#1171954111yescharacteristic
667C22#1171954311yescharacteristic
668C4—22no—
669C4—22no—
670C4—24no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—522no—
677C4—522no—
678C4—1041no—
679C4—1041no—
680C4C2×#8813111yescharacteristic
681C4#1171919111yescharacteristic
682C4#1171920011yescharacteristic
683C4#1171920011yescharacteristic
684C2—42no—
685C2—82no—
686C2#1311768111yescharacteristic
687C2C4×#8820311yescharacteristic
688C2#1312367511yescharacteristic
689C2#1312682711yescharacteristic
690C2#1312696211yescharacteristic
691C2#1312710411yescharacteristic
692C2#1312716311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC13×#2341C2
socleC23×C13C22×D4
prime radicalC22×C13×#41C22
cosocleC22×C4×C13C24
hypercenter#2341D13
hypocenterC13#22583
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
2. #13117703C2semidirect
3. #13115536C2semidirect
4. #13115522C2semidirect
5. #13115460C2semidirect
6. #13115453C2semidirect
7. #13115104C2semidirect
8. #13115090C2semidirect
9. #13114924C2semidirect
10. #13114867C2semidirect
12. #13114666C2semidirect
13. C4×#88075C2semidirect
14. #13114620C2semidirect
30. #11718893C4semidirect
258. Q8×C13#117semidirect
570. C13#22583semidirect
589. Q8#88131semidirect

Maximal subgroups

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

subgroupindex
C2×#117188932
#131177032
#131155362
#131155222
#131154602
#131154532
#131151042
#131150902
#131149242
#131148672
#131147762
#131146662
C4×#880752
#131146202
C13×#23412
#2258313