Finite Group Databasebeta

← #15899372

Subgroups — #15899372

Order 3328.

1042 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11718727C211yesmaximal, characteristic
2C2×C4×#6106C211yesmaximal, characteristic
3#13120194C211yesmaximal, characteristic
4#13116171C211yesmaximal, characteristic, complemented
5#13116166C211yesmaximal, characteristic, complemented
6#13115979C211yesmaximal, characteristic
7#13115969C211yesmaximal, characteristic
8#13115802C211yesmaximal, characteristic, complemented
9#13115773C211yesmaximal, characteristic, complemented
10#13115191C211yesmaximal, characteristic
11#13115191C211yesmaximal, characteristic
12#13115104C211yesmaximal, characteristic, complemented
13#13115090C211yesmaximal, characteristic, complemented
14#13114851C211yesmaximal, characteristic, complemented
15C2×C13×#557C211yesmaximal, characteristic, complemented
16C13×#557—24no—
17C13×#557—24no—
18C2×C13×C4⋊Q8—21no—
19C2×C13×C4⋊Q8C2211yescharacteristic
20C2×C13×#125—21no—
21C2×C13×#125—21no—
22C2×C13×#125C2211yescharacteristic
23C2×C13×#125C2211yescharacteristic
24C2×C4×Q8×C13—21no—
25C2×C4×Q8×C13C2211yescharacteristic
26C2×C4×Q8×C13C2211yescharacteristic
27C2×C4×Q8×C13C2211yescharacteristic
28C2×C4×Q8×C13C2211yescharacteristic
29C22×#6106C2211yescharacteristic
30C2×C4×D52C2211yescharacteristic
31#11718892—21no—
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35#11718727—24no—
36C4×#6106—24no—
37#11718653—21no—
38#11718650—21no—
39#11718650—21no—
40C2×#88049C2211yescharacteristic
41C2×#88049C2211yescharacteristic
42C2×#88049C2211yescharacteristic
43C2×#88048C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88047C2211yescharacteristic
46#11718600C2211yescharacteristic
47#11718600C2211yescharacteristic
48#11718597C2211yescharacteristic
49#11718589C2211yescharacteristic
50#11718589C2211yescharacteristic
51C2×C4⋊#1055C2211yescharacteristic
52C2×C4×#1055C2211yescharacteristic
53#11718121C2211yescharacteristic
54#11718121C2211yescharacteristic
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61#11718116C2211yescharacteristic
62#11718116C2211yescharacteristic
63C22×Q8×C13—21no—
64C22×Q8×C13D411yescharacteristic
65C22×Q8×C13C2311yescharacteristic
66C22×D52C2311yescharacteristic
67C22×C4×D13—21no—
68C13×C4⋊Q8—28no—
69C13×C4⋊Q8—22no—
70C13×C4⋊Q8—22no—
71C13×#125—28no—
72C13×#125—28no—
73C13×#125—22no—
74C13×#125—22no—
75C13×#125—22no—
76C13×#125D414yes—
77C4×Q8×C13—28no—
78C4×Q8×C13—22no—
79C4×Q8×C13—22no—
80C4×Q8×C13—22no—
81C4×Q8×C13—22no—
82C4×Q8×C13—22no—
83C4×Q8×C13—22no—
84C4×Q8×C13—22no—
85C4×Q8×C13D414yes—
86C2×C13×C4⋊C4—21no—
87C2×C13×C4⋊C4—21no—
88C2×C13×C4⋊C4—21no—
89C2×C13×C4⋊C4—21no—
90C2×C13×C4⋊C4—21no—
91C2×C13×C4⋊C4—21no—
92C2×C13×C4⋊C4—21no—
93C2×C13×C4⋊C4—21no—
94C2×C13×C4⋊C4D411yescharacteristic
95C2×C13×C4⋊C4D411yescharacteristic
96C2×C13×C4⋊C4D411yescharacteristic
97C2×C13×C4⋊C4C2311yescharacteristic
98C2×C13×C4⋊C4C2311yescharacteristic
99C2×C13×C4⋊C4C2311yescharacteristic
100C2×C42×C13—21no—
101C2×C42×C13—21no—
102C2×C42×C13—21no—
103C2×C42×C13C2311yescharacteristic
104C2×C42×C13C2311yescharacteristic
105C2×C42×C13C2311yescharacteristic
106C2×#6106—22no—
107C2×#6106—22no—
108C2×#6106Q814yescomplemented
109C2×#6103—21no—
110C2×#6103—21no—
111C2×#6103—21no—
112C2×#6103—21no—
113C2×#6103—21no—
114C2×C4⋊Dic13C2311yescharacteristic
115C2×C4⋊Dic13C2311yescharacteristic
116C2×C4⋊Dic13C2311yescharacteristic
117C2×#6101—21no—
118C2×#6101—21no—
119C22×#1055C2311yescharacteristic
120C22×#1055C2311yescharacteristic
121C22×#1055C2311yescharacteristic
122C22×#1055C2311yescharacteristic
123C4×D52—22no—
124C4×D52—22no—
125#88075—22no—
126#88075—22no—
127#88075—22no—
128#88071—21no—
129#88071—21no—
130#88071—21no—
131#88071—21no—
132#88049—22no—
133#88049—22no—
134#88049—22no—
135#88048—22no—
136#88048—22no—
137#88047—22no—
138C4⋊#1055—22no—
139C4×#1055—22no—
140#24334—131nomaximal, Sylow
141C23×D13—21no—
142C2×Q8×C13—24no—
143C2×Q8×C13—22no—
144C2×Q8×C13—22no—
145C2×Q8×C13—21no—
146C2×Q8×C13—21no—
147C2×Q8×C13—21no—
148C2×Q8×C13—41no—
149C2×Q8×C13—42no—
150C2×Q8×C13—42no—
151C2×Q8×C13C2×D412yes—
152C2×Q8×C13C2×Q812yes—
153C2×Q8×C13#4112yes—
154C22×C4×C13—21no—
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13C2×D411yescharacteristic
159C22×C4×C13C2×D411yescharacteristic
160C22×C4×C13C2×D411yescharacteristic
161C22×C4×C13C2×D411yescharacteristic
162C22×C4×C13C2×D411yescharacteristic
163C22×C4×C13C2×D411yescharacteristic
164C22×C4×C13C2411yescharacteristic
165C22×Dic13—21no—
166C22×Dic13—21no—
167C22×Dic13—21no—
168C2×D52—21no—
169C2×D52—21no—
170C2×D52—21no—
171C2×D52—21no—
172C2×D52—21no—
173C2×D52C2×Q812yes—
174C2×C4×D13—42no—
175C2×C4×D13—81no—
176C2×C4×D13—81no—
177C13×C4⋊C4—24no—
178C13×C4⋊C4—24no—
179C13×C4⋊C4—24no—
180C13×C4⋊C4—24no—
181C13×C4⋊C4—24no—
182C13×C4⋊C4—24no—
183C13×C4⋊C4—24no—
184C13×C4⋊C4—24no—
185C13×C4⋊C4—22no—
186C13×C4⋊C4—22no—
187C13×C4⋊C4—22no—
188C13×C4⋊C4—21no—
189C13×C4⋊C4—21no—
190C13×C4⋊C4—21no—
191C13×C4⋊C4C2×D412yes—
192C13×C4⋊C4C2×D412yes—
193C13×C4⋊C4C2×D412yes—
194C42×C13—24no—
195C42×C13—24no—
196C42×C13—24no—
197C42×C13—21no—
198C42×C13—21no—
199C42×C13—21no—
200C42×C13—21no—
201C42×C13C2×D412yes—
202C42×C13C2×D412yes—
203#6106—28no—
204#6103—42no—
205#6103—42no—
206#6103—42no—
207#6103—81no—
208#6103—81no—
209C4⋊Dic13—21no—
210C4⋊Dic13—21no—
211C4⋊Dic13—21no—
212C4⋊Dic13—21no—
213C4⋊Dic13—21no—
214C4⋊Dic13—21no—
215#6101—81no—
216#6101—81no—
217C2×#1055—21no—
218C2×#1055—21no—
219C2×#1055—21no—
220C2×#1055—21no—
221C2×#1055—21no—
222C2×#1055—21no—
223C2×#1055—21no—
224C2×#1055—21no—
225C2×#1055—21no—
226C2×#1055—21no—
227C2×#1055—21no—
228C2×#1055C2×Q812yes—
229C2×#557D1311yescharacteristic, complemented
230C2×#478—131no—
231C2×C4×SD16—131no—
232#2580—131no—
233#2059—131no—
234#2058—131no—
235#2025—131no—
236#2023—131no—
237#1991—131no—
238#1984—131no—
239#1880—131no—
240#1880—131no—
241#1865—131no—
242#1863—131no—
243#1805—131no—
244C23×C13C22×D411yescharacteristic
245C22×D13—22no—
246C22×D13—41no—
247C22×D13—41no—
248C22×D13—41no—
249C22×D13—41no—
250C22×D13—41no—
251C22×D13—41no—
252Q8×C13—22no—
253Q8×C13—22no—
254Q8×C13—22no—
255Q8×C13—44no—
256Q8×C13—48no—
257Q8×C13C22⋊Q814yes—
258C2×C4×C13—22no—
259C2×C4×C13—21no—
260C2×C4×C13—22no—
261C2×C4×C13—22no—
262C2×C4×C13—21no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—21no—
267C2×C4×C13—21no—
268C2×C4×C13—21no—
269C2×C4×C13—22no—
270C2×C4×C13—21no—
271C2×C4×C13—22no—
272C2×C4×C13—22no—
273C2×C4×C13—21no—
274C2×C4×C13—21no—
275C2×C4×C13—21no—
276C2×C4×C13—21no—
277C2×C4×C13—21no—
278C2×C4×C13—21no—
279C2×C4×C13—41no—
280C2×C4×C13—41no—
281C2×C4×C13—41no—
282C2×C4×C13—41no—
283C2×C4×C13—41no—
284C2×C4×C13—41no—
285C2×C4×C13#12012yes—
286C2×C4×C13#12012yes—
287C2×C4×C13#13512yes—
288C2×C4×C13C22×D411yescharacteristic
289C2×C4×C13C22×D411yescharacteristic
290C2×C4×C13C22×Q811yescharacteristic
291C2×C4×C13C2×#4111yescharacteristic
292C2×C4×C132−1+411yescharacteristic
293C2×C4×C132−1+411yescharacteristic
294C2×Dic13—41no—
295C2×Dic13—41no—
296C2×Dic13—41no—
297C2×Dic13—41no—
298C2×Dic13—41no—
299C2×Dic13—41no—
300C2×Dic13—41no—
301C2×Dic13—41no—
302C2×Dic13—41no—
303D52—22no—
304D52—22no—
305D52—41no—
306D52—41no—
307C4×D13—84no—
308#1055—22no—
309#1055—41no—
310#1055—41no—
311#1055—41no—
312C22×SD16—131no—
313#557—24no—
314#557—24no—
315C2×C4⋊Q8—21no—
316C2×C4⋊Q8C2×D1311yescharacteristic
317C2×#125—21no—
318C2×#125—21no—
319C2×#125C2×D1311yescharacteristic
320C2×#125C2×D1311yescharacteristic
321C2×C4×Q8—21no—
322C2×C4×Q8C2×D1311yescharacteristic
323C2×C4×Q8C2×D1311yescharacteristic
324C2×C4×Q8C2×D1311yescharacteristic
325C2×C4×Q8C2×D1311yescharacteristic
326C2×C4×D4—131no—
327#478—264no—
328C4×SD16—264no—
329C2×#107—131no—
330C2×#106—131no—
331C2×#106—131no—
332C2×C4⋊C8—131no—
333C2×#103—131no—
334C2×#103—131no—
335C2×#103—131no—
336C2×#102—131no—
337C2×#102—131no—
338C2×#102—131no—
339C2×C4×C8—131no—
340#399—261no—
341#393—261no—
342#390—131no—
343#390—131no—
344#389—131no—
345#389—131no—
346#385—261no—
347#385—261no—
348#382—131no—
349#340—131no—
350#340—131no—
351#340—131no—
352#340—131no—
353#340—131no—
354#340—131no—
355#336—131no—
356#336—131no—
357#336—131no—
358#336—131no—
359C22×C13C2×#12011yescharacteristic
360C22×C13#54811yescharacteristic
361C22×C13D4×Q811yescharacteristic
362C22×C13C2×#13511yescharacteristic
363C22×C13#57511yescharacteristic
364C22×C13#57711yescharacteristic
365C22×C13#57711yescharacteristic
366C2×D13—42no—
367C2×D13—42no—
368C2×D13—42no—
369C2×D13—81no—
370C2×D13—81no—
371C2×D13—81no—
372C2×D13—81no—
373C4×C13—24no—
374C4×C13—22no—
375C4×C13—22no—
376C4×C13—22no—
377C4×C13—41no—
378C4×C13—42no—
379C4×C13—42no—
380C4×C13—41no—
381C4×C13—41no—
382C4×C13—42no—
383C4×C13C2×C22⋊Q811yescharacteristic
384C4×C13#53611yescharacteristic
385C4×C13#54811yescharacteristic
386C4×C13D4×Q811yescharacteristic
387Dic13—81no—
388Dic13—81no—
389Dic13—81no—
390C22×Q8—21no—
391C22×Q8C22⋊D1311yescharacteristic
392C22×Q8C22×D1311yescharacteristic
393C22×D4—131no—
394C23×C4—261no—
395C2×SD16—134no—
396C2×SD16—262no—
397C2×SD16—262no—
398C22×C8—131no—
399C22×C8—131no—
400C22×C8—131no—
401C22×C8—131no—
402C4⋊Q8—28no—
403C4⋊Q8—22no—
404C4⋊Q8—22no—
405#125—28no—
406#125—28no—
407#125—22no—
408#125—22no—
409#125—22no—
410#125C22⋊D1314yes—
411C4×Q8—28no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×Q8—22no—
417C4×Q8—22no—
418C4×Q8—22no—
419C4×Q8C22⋊D1314yes—
420C4×D4—262no—
421C4×D4—262no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—131no—
431C2×C4⋊C4—131no—
432C2×C4⋊C4—131no—
433C2×C4⋊C4—261no—
434C2×C4⋊C4—261no—
435C2×C4⋊C4C22⋊D1311yescharacteristic
436C2×C4⋊C4C22⋊D1311yescharacteristic
437C2×C4⋊C4C22⋊D1311yescharacteristic
438C2×C4⋊C4C22×D1311yescharacteristic
439C2×C4⋊C4C22×D1311yescharacteristic
440C2×C4⋊C4C22×D1311yescharacteristic
441C2×C22⋊C4—261no—
442C2×C22⋊C4—261no—
443C2×C22⋊C4—261no—
444C2×C22⋊C4—261no—
445C2×C22⋊C4—261no—
446C2×C42—21no—
447C2×C42—21no—
448C2×C42—21no—
449C2×C42C22×D1311yescharacteristic
450C2×C42C22×D1311yescharacteristic
451C2×C42C22×D1311yescharacteristic
452#107—262no—
453#106—262no—
454#106—262no—
455C4⋊C8—262no—
456#103—262no—
457#103—262no—
458#103—262no—
459#102—262no—
460#102—262no—
461#102—262no—
462C4×C8—262no—
463#95—261no—
464#95—261no—
465#95—261no—
466#95—261no—
467C2×C13#240911yescharacteristic
468C2×C13#300211yescharacteristic
469C2×C13#301011yescharacteristic
470C2×C13#336511yescharacteristic
471C2×C13#337111yescharacteristic
472C2×C13#338111yescharacteristic
473C2×C13#338811yescharacteristic
474D13—82no—
475C24—261no—
476C2×Q8—24no—
477C2×Q8—22no—
478C2×Q8—22no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—41no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8Q8×D1312yes—
486C2×Q8#613112yes—
487C2×Q8C2×C22⋊D1312yes—
488C2×D4—132no—
489C2×D4—261no—
490C2×D4—261no—
491C2×D4—261no—
492C2×D4—261no—
493C2×D4—261no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—261no—
499C22×C4—261no—
500C22×C4—261no—
501C22×C4—522no—
502C22×C4—1041no—
503C22×C4—1041no—
504C22×C4C2×C22⋊D1311yescharacteristic
505C22×C4C2×C22⋊D1311yescharacteristic
506C22×C4C2×C22⋊D1311yescharacteristic
507C22×C4C2×C22⋊D1311yescharacteristic
508C22×C4C2×C22⋊D1311yescharacteristic
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C23×D1311yescharacteristic
511SD16—268no—
512C2×C8—132no—
513C2×C8—261no—
514C2×C8—261no—
515C2×C8—261no—
516C2×C8—261no—
517C2×C8—261no—
518C2×C8—261no—
519C2×C8—261no—
520C2×C8—261no—
521C2×C8—261no—
522C2×C8—261no—
523C2×C8—261no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—261no—
539C4⋊C4—261no—
540C4⋊C4—261no—
541C4⋊C4—261no—
542C4⋊C4—261no—
543C4⋊C4—261no—
544C4⋊C4—1041no—
545C4⋊C4—1041no—
546C4⋊C4C2×C22⋊D1312yes—
547C4⋊C4C2×C22⋊D1312yes—
548C4⋊C4C2×C22⋊D1312yes—
549C22⋊C4—522no—
550C22⋊C4—522no—
551C22⋊C4—522no—
552C22⋊C4—1041no—
553C22⋊C4—1041no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—21no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42C2×C22⋊D1312yes—
562C42C2×C22⋊D1312yes—
563C13#2433411yescharacteristic, complemented, Sylow
564C23—262no—
565C23—521no—
566C23—521no—
567C23—521no—
568C23—521no—
569C23—521no—
570C23—521no—
571C23C22×C22⋊D1311yescharacteristic
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—44no—
576Q8—48no—
577Q8#8820014yes—
578D4—262no—
579D4—262no—
580D4—521no—
581D4—521no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—521no—
610C2×C4—521no—
611C2×C4—521no—
612C2×C4—521no—
613C2×C4—521no—
614C2×C4—521no—
615C2×C4—521no—
616C2×C4—521no—
617C2×C4—521no—
618C2×C4—1044no—
619C2×C4#8820412yes—
620C2×C4#8820712yes—
621C2×C4#8820712yes—
622C2×C4C2×Q8×D1311yescharacteristic
623C2×C4C2×#613111yescharacteristic
624C2×C4#8825411yescharacteristic
625C2×C4#8825411yescharacteristic
626C2×C4C22×C22⋊D1311yescharacteristic
627C2×C4C22×C22⋊D1311yescharacteristic
628C8—262no—
629C8—521no—
630C8—521no—
631C8—521no—
632C22—522no—
633C22—522no—
634C22—522no—
635C22—1041no—
636C22—1041no—
637C22—1041no—
638C22—1041no—
639C22Q8×C22⋊D1311yescharacteristic
640C22#1171952211yescharacteristic
641C22C2×#8820411yescharacteristic
642C22#1171952711yescharacteristic
643C22#1171954211yescharacteristic
644C22#1171954211yescharacteristic
645C22C2×#8820711yescharacteristic
646C4—22no—
647C4—22no—
648C4—22no—
649C4—24no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—42no—
655C4—42no—
656C4—1041no—
657C4—1041no—
658C4—1041no—
659C4C2×#8820011yescharacteristic
660C4Q8×C22⋊D1311yescharacteristic
661C4#1171952111yescharacteristic
662C4#1171952211yescharacteristic
663C2—1042no—
664C2#1311819511yescharacteristic
665C2#1312405411yescharacteristic
666C2#1312412211yescharacteristic
667C2#1312753211yescharacteristic
668C2#1312760411yescharacteristic
669C2#1312770711yescharacteristic
670C2#1312777511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC2×C13×#557C2
socleC23×C13C22×D4
prime radicalC22×D52C23
hypercenterC2×#557D13
hypocenterC13#24334
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #13116171C2semidirect
5. #13116166C2semidirect
8. #13115802C2semidirect
9. #13115773C2semidirect
12. #13115104C2semidirect
13. #13115090C2semidirect
14. #13114851C2semidirect
15. C2×C13×#557C2semidirect
108. C2×#6106Q8semidirect
229. C2×#557D13semidirect
563. C13#24334semidirect

Maximal subgroups

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

subgroupindex
C2×#117187272
C2×C4×#61062
#131201942
#131161712
#131161662
#131159792
#131159692
#131158022
#131157732
#131151912
#131151042
#131150902
#131148512
C2×C13×#5572
#2433413