Finite Group Databasebeta

← #15874234

Subgroups — #15874234

Order 3328.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117520C211yesmaximal, characteristic
2#13117422C211yesmaximal, characteristic
3#13115708C211yesmaximal, characteristic, complemented
4#13115698C211yesmaximal, characteristic, complemented
5#13115528C211yesmaximal, characteristic, complemented
6#13115514C211yesmaximal, characteristic, complemented
7#13115479C211yesmaximal, characteristic, complemented
8#13115472C211yesmaximal, characteristic, complemented
9#13115179C211yesmaximal, characteristic, complemented
10#13115179C211yesmaximal, characteristic, complemented
11#13115149C211yesmaximal, characteristic
12#13115037C211yesmaximal, characteristic
13#13114771C211yesmaximal, characteristic
14#13114604C211yesmaximal, characteristic
15C2×C13×#451C211yesmaximal, characteristic
16C23×Q8×C13C2211yescharacteristic
17C22×C13×Q16C2211yescharacteristic
18C22×C13×Q16C2211yescharacteristic
19C2×C13×C22⋊Q8C2211yescharacteristic
20C2×Q8×Dic13C2211yescharacteristic
21C2×#88180C2211yescharacteristic
22C2×#88180C2211yescharacteristic
23C13×#451C418yescomplemented
24C2×C13×#103C2211yescharacteristic
25C2×C13×#103C2211yescharacteristic
26C2×C13×C22⋊C8C2211yescharacteristic
27#11718891—21no—
28#11718891—21no—
29#11718891C2211yescharacteristic
30#11718891C2211yescharacteristic
31#11718891C2211yescharacteristic
32#11718891C2211yescharacteristic
33#11718891C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35C2×#88075C2211yescharacteristic
36C2×#88075C2211yescharacteristic
37C2×#88075C2211yescharacteristic
38C2×#88075C2211yescharacteristic
39C2×#88075C2211yescharacteristic
40#11718873C2211yescharacteristic
41#11718872—21no—
42#11718872—21no—
43C2×C22⋊#1055C2211yescharacteristic
44C2×#88058C2211yescharacteristic
45C2×#88057C2211yescharacteristic
46C2×#88055C2211yescharacteristic
47C2×C8×Dic13C2211yescharacteristic
48#11718646—21no—
49#11718646C2211yescharacteristic
50#11718642—21no—
51C4⋊C4×Dic13C2211yescharacteristic
52#11718612—21no—
53#11718612—21no—
54#11718143C2211yescharacteristic
55#11718143C2211yescharacteristic
56#11718142C2211yescharacteristic
57#11718142C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718121C2211yescharacteristic
60C22×Q8×C13—21no—
61C22×Q8×C13—22no—
62C22×Q8×C13—41no—
63C22×Q8×C13—41no—
64C22×Q8×C13—41no—
65C22×Q8×C13—41no—
66C22×Q8×C13C2×C414yes—
67C22×Q8×C13C2311yescharacteristic
68C22×Q8×C13C2311yescharacteristic
69C22×Q8×C13C2311yescharacteristic
70C23×C4×C13—21no—
71C23×C4×C13C2311yescharacteristic
72C2×C13×Q16—41no—
73C2×C13×Q16—41no—
74C2×C13×Q16—41no—
75C2×C13×Q16—41no—
76C2×C13×Q16C2×C414yes—
77C2×C13×Q16C2×C414yes—
78C22×C8×C13C2311yescharacteristic
79C22×C8×C13C2311yescharacteristic
80C13×C22⋊Q8—22no—
81C13×C22⋊Q8C2×C414yes—
82C2×C13×C4⋊C4—21no—
83C2×C13×C4⋊C4C2311yescharacteristic
84C2×C13×C22⋊C4—21no—
85C2×C22⋊Dic13—21no—
86C2×C22⋊Dic13—21no—
87C2×C22⋊Dic13D411yescharacteristic
88C2×C22⋊Dic13D411yescharacteristic
89Q8×Dic13—41no—
90Q8×Dic13—41no—
91#88180—41no—
92#88180—41no—
93#88180—41no—
94#88180—41no—
95C2×C4⋊Dic13—21no—
96C2×C4⋊Dic13—21no—
97C2×C4⋊Dic13D411yescharacteristic
98C2×C4⋊Dic13D411yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×C4⋊Dic13C2311yescharacteristic
101C2×C4⋊Dic13C2311yescharacteristic
102C2×C4⋊Dic13C2311yescharacteristic
103C2×C4×Dic13—21no—
104C2×C4×Dic13—21no—
105C2×C4×Dic13—21no—
106C2×C4×Dic13C2311yescharacteristic
107C2×C4×Dic13C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109C22×#1055C2311yescharacteristic
110C13×#103C2×C414yes—
111C13×#103C2×C414yes—
112C13×C22⋊C8C2×C414yes—
113#88075—41no—
114#88075—41no—
115#88075—41no—
116#88075—41no—
117#88075—41no—
118#88075—41no—
119#88071—21no—
120#88071—21no—
121#88071—21no—
122#88071—21no—
123#88071—21no—
124#88071—21no—
125#88071—21no—
126#88071—21no—
127#88071—21no—
128#88071—21no—
129#88071—21no—
130#88071—21no—
131#88071—21no—
132C22⋊#1055—22no—
133#88058—41no—
134#88057—41no—
135#88055—41no—
136C8×Dic13—41no—
137#22054—131nomaximal, Sylow
138C24×C13C2×D411yescharacteristic
139C2×Q8×C13—24no—
140C2×Q8×C13—21no—
141C2×Q8×C13—21no—
142C2×Q8×C13—21no—
143C2×Q8×C13—21no—
144C2×Q8×C13—21no—
145C2×Q8×C13—21no—
146C2×Q8×C13—21no—
147C2×Q8×C13—21no—
148C2×Q8×C13—21no—
149C2×Q8×C13—21no—
150C2×Q8×C13—21no—
151C2×Q8×C13—21no—
152C2×Q8×C13—21no—
153C2×Q8×C13—21no—
154C2×Q8×C13—21no—
155C2×Q8×C13—44no—
156C2×Q8×C13—44no—
157C2×Q8×C13—42no—
158C2×Q8×C13—42no—
159C2×Q8×C13—44no—
160C2×Q8×C13—44no—
161C2×Q8×C13C22×C412yes—
162C2×Q8×C13C22×C412yes—
163C2×Q8×C13C22×C412yes—
164C22×C4×C13—21no—
165C22×C4×C13—21no—
166C22×C4×C13—21no—
167C22×C4×C13—21no—
168C22×C4×C13—24no—
169C22×C4×C13—21no—
170C22×C4×C13—21no—
171C22×C4×C13—21no—
172C22×C4×C13—21no—
173C22×C4×C13—42no—
174C22×C4×C13—42no—
175C22×C4×C13C22×C412yes—
176C22×C4×C13C2×D411yescharacteristic
177C22×C4×C13#4112yes—
178C22×C4×C13C2411yescharacteristic
179C22×Dic13—21no—
180C22×Dic13—21no—
181C22×Dic13—21no—
182C22×Dic13—21no—
183C22×Dic13C2×D411yescharacteristic
184C22×Dic13C2×D411yescharacteristic
185C22×Dic13C2×D411yescharacteristic
186C22×Dic13C2×D411yescharacteristic
187C13×Q16—44no—
188C13×Q16—44no—
189C2×C8×C13—21no—
190C2×C8×C13—21no—
191C2×C8×C13—21no—
192C2×C8×C13—21no—
193C2×C8×C13C22×C412yes—
194C2×C8×C13C22×C412yes—
195C13×C4⋊C4—24no—
196C13×C4⋊C4—21no—
197C13×C4⋊C4C22×C412yes—
198C13×C22⋊C4—24no—
199C22⋊Dic13—41no—
200C22⋊Dic13—41no—
201C22⋊Dic13—42no—
202C22⋊Dic13—81no—
203C4⋊Dic13—21no—
204C4⋊Dic13—21no—
205C4⋊Dic13—21no—
206C4⋊Dic13—21no—
207C4⋊Dic13—21no—
208C4⋊Dic13—21no—
209C4⋊Dic13—21no—
210C4⋊Dic13—21no—
211C4⋊Dic13—41no—
212C4⋊Dic13—41no—
213C4⋊Dic13—42no—
214C4⋊Dic13—81no—
215C4×Dic13—21no—
216C4×Dic13—21no—
217C4×Dic13—21no—
218C4×Dic13—21no—
219C4×Dic13—42no—
220C4×Dic13—42no—
221C4×Dic13—81no—
222C2×#1055—21no—
223C2×#1055—21no—
224C2×#1055—21no—
225C2×#1055—21no—
226C2×#1055—21no—
227C2×#1055—21no—
228C2×#451D1311yescharacteristic
229#2323—131no—
230#2310—131no—
231#1971—131no—
232#1969—131no—
233#1935—131no—
234#1933—131no—
235#1928—131no—
236#1927—131no—
237#1878—131no—
238#1878—131no—
239#1873—131no—
240#1854—131no—
241#1787—131no—
242#1750—131no—
243C23×C13—21no—
244C23×C13—21no—
245C23×C13—21no—
246C23×C13—21no—
247C23×C13—21no—
248C23×C13—21no—
249C23×C13C4×D412yes—
250C23×C13C22×D411yescharacteristic
251Q8×C13—22no—
252Q8×C13—22no—
253Q8×C13—22no—
254Q8×C13—22no—
255Q8×C13—22no—
256Q8×C13—22no—
257Q8×C13—48no—
258Q8×C13—41no—
259Q8×C13—41no—
260Q8×C13—41no—
261Q8×C13—41no—
262Q8×C13—41no—
263Q8×C13—41no—
264C2×C4×C13—21no—
265C2×C4×C13—21no—
266C2×C4×C13—22no—
267C2×C4×C13—22no—
268C2×C4×C13—22no—
269C2×C4×C13—22no—
270C2×C4×C13—22no—
271C2×C4×C13—22no—
272C2×C4×C13—22no—
273C2×C4×C13—22no—
274C2×C4×C13—48no—
275C2×C4×C13—41no—
276C2×C4×C13—41no—
277C2×C4×C13—41no—
278C2×C4×C13—41no—
279C2×C4×C13—41no—
280C2×C4×C13—41no—
281C2×C4×C13—41no—
282C2×C4×C13—41no—
283C2×C4×C13C4×D412yes—
284C2×C4×C13C23×C411yescharacteristic
285C2×C4×C13C2×#4111yescharacteristic
286C2×C4×C13H(5,𝔽2)11yescharacteristic
287C2×C4×C13H(5,𝔽2)11yescharacteristic
288C2×C4×C13H(5,𝔽2)11yescharacteristic
289C2×C4×C13H(5,𝔽2)11yescharacteristic
290C2×Dic13—21no—
291C2×Dic13—21no—
292C2×Dic13—21no—
293C2×Dic13—21no—
294C2×Dic13—21no—
295C2×Dic13—21no—
296C2×Dic13—21no—
297C2×Dic13—21no—
298C2×Dic13—21no—
299C2×Dic13—21no—
300C2×Dic13—21no—
301C2×Dic13—21no—
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—
312C2×Dic13—41no—
313C2×Dic13—41no—
314C8×C13—22no—
315C8×C13—22no—
316#1055—41no—
317#1055—41no—
318C23×Q8C2×D1311yescharacteristic
319C22×Q16C2×D1311yescharacteristic
320C22×Q16C2×D1311yescharacteristic
321C2×C22⋊Q8C2×D1311yescharacteristic
322C2×C4×Q8—131no—
323C2×#117—131no—
324C2×#117—131no—
325#451Dic1318yescomplemented
326C2×#107—131no—
327C2×#106—131no—
328C2×#103—131no—
329C2×#103—131no—
330C2×#103—131no—
331C2×#103—131no—
332C2×#103—131no—
333C2×#103—131no—
334C2×#103C2×D1311yescharacteristic
335C2×#103C2×D1311yescharacteristic
336C2×C22⋊C8—131no—
337C2×C22⋊C8C2×D1311yescharacteristic
338C2×#97—131no—
339C2×C4×C8—131no—
340#391—131no—
341#391—131no—
342#391—131no—
343#391—131no—
344#391—131no—
345#391—261no—
346#391—261no—
347#389—131no—
348#389—261no—
349#388—261no—
350#388—261no—
351#387—261no—
352#381—261no—
353#381—261no—
354#380—131no—
355C4×C4⋊C4—131no—
356#340—131no—
357#340—131no—
358#340—131no—
359#340—131no—
360#336—131no—
361#336—131no—
362C22×C13—22no—
363C22×C13—22no—
364C22×C13—22no—
365C22×C13—41no—
366C22×C13—41no—
367C22×C13—41no—
368C22×C13—41no—
369C22×C13C2×C4×D411yescharacteristic
370C22×C13#53411yescharacteristic
371C22×C13#53511yescharacteristic
372C22×C13#57711yescharacteristic
373C22×C13#57811yescharacteristic
374C22×C13#57811yescharacteristic
375C22×C13#57811yescharacteristic
376C4×C13—22no—
377C4×C13—42no—
378C4×C13—42no—
379C4×C13—42no—
380C4×C13—42no—
381C4×C13#51811yescharacteristic
382C4×C13#51811yescharacteristic
383C4×C13#53811yescharacteristic
384C4×C13#53811yescharacteristic
385Dic13—41no—
386Dic13—41no—
387Dic13—41no—
388Dic13—41no—
389Dic13—81no—
390Dic13—81no—
391Dic13—81no—
392Dic13—81no—
393C22×Q8—21no—
394C22×Q8—22no—
395C22×Q8—41no—
396C22×Q8—41no—
397C22×Q8—41no—
398C22×Q8—41no—
399C22×Q8C2×Dic1314yes—
400C22×Q8C22×D1311yescharacteristic
401C22×Q8C22×D1311yescharacteristic
402C22×Q8C22×D1311yescharacteristic
403C23×C4—21no—
404C23×C4C22×D1311yescharacteristic
405C2×Q16—41no—
406C2×Q16—41no—
407C2×Q16—41no—
408C2×Q16—41no—
409C2×Q16C2×Dic1314yes—
410C2×Q16C2×Dic1314yes—
411C22×C8—131no—
412C22×C8—131no—
413C22×C8C22×D1311yescharacteristic
414C22×C8C22×D1311yescharacteristic
415C22⋊Q8—22no—
416C22⋊Q8C2×Dic1314yes—
417C4×Q8—521no—
418C4×Q8—521no—
419#117—521no—
420#117—521no—
421#117—521no—
422#117—521no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—131no—
425C2×C4⋊C4—131no—
426C2×C4⋊C4—131no—
427C2×C4⋊C4—131no—
428C2×C4⋊C4—131no—
429C2×C4⋊C4—131no—
430C2×C4⋊C4—261no—
431C2×C4⋊C4—261no—
432C2×C4⋊C4C22×D1311yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—131no—
435C2×C22⋊C4—131no—
436C2×C22⋊C4—261no—
437C2×C22⋊C4—261no—
438C2×C42—131no—
439C2×C42—131no—
440C2×C42—261no—
441C2×C42—261no—
442C2×C42—261no—
443#107—521no—
444#106—521no—
445#103—521no—
446#103—521no—
447#103—521no—
448#103—521no—
449#103—521no—
450#103—521no—
451#103C2×Dic1314yes—
452#103C2×Dic1314yes—
453C22⋊C8—262no—
454C22⋊C8C2×Dic1314yes—
455#97—521no—
456C4×C8—521no—
457#95—261no—
458#95—261no—
459#95—261no—
460#95—261no—
461#95—261no—
462#95—261no—
463#95—261no—
464#95—261no—
465#95—261no—
466#95—261no—
467#95—261no—
468#95—261no—
469#95—261no—
470C2×C13—42no—
471C2×C13#232111yescharacteristic
472C2×C13#294711yescharacteristic
473C2×C13#295011yescharacteristic
474C2×C13#320111yescharacteristic
475C2×C13#320211yescharacteristic
476C2×C13#322311yescharacteristic
477C2×C13#322511yescharacteristic
478C24D4×D1311yescharacteristic
479C2×Q8—24no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—42no—
498C2×Q8—42no—
499C2×Q8—44no—
500C2×Q8—44no—
501C2×Q8C22×Dic1312yes—
502C2×Q8C22×Dic1312yes—
503C2×Q8C22×Dic1312yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—42no—
514C22×C4—42no—
515C22×C4—131no—
516C22×C4—131no—
517C22×C4—131no—
518C22×C4—131no—
519C22×C4—261no—
520C22×C4—261no—
521C22×C4—261no—
522C22×C4—261no—
523C22×C4D4×D1311yescharacteristic
524C22×C4#612912yes—
525C22×C4C22×Dic1312yes—
526C22×C4C23×D1311yescharacteristic
527Q16—44no—
528Q16—44no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—261no—
534C2×C8—261no—
535C2×C8—261no—
536C2×C8—261no—
537C2×C8—261no—
538C2×C8—261no—
539C2×C8C22×Dic1312yes—
540C2×C8C22×Dic1312yes—
541C4⋊C4—24no—
542C4⋊C4—21no—
543C4⋊C4—261no—
544C4⋊C4—261no—
545C4⋊C4—261no—
546C4⋊C4—261no—
547C4⋊C4—261no—
548C4⋊C4—261no—
549C4⋊C4—261no—
550C4⋊C4—261no—
551C4⋊C4—521no—
552C4⋊C4—521no—
553C4⋊C4—522no—
554C4⋊C4—1041no—
555C4⋊C4C22×Dic1312yes—
556C22⋊C4—24no—
557C22⋊C4—521no—
558C22⋊C4—521no—
559C22⋊C4—522no—
560C22⋊C4—1041no—
561C42—261no—
562C42—261no—
563C42—261no—
564C42—261no—
565C42—522no—
566C42—522no—
567C42—1041no—
568C13#2205411yescharacteristic, complemented, Sylow
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23D4×Dic1312yes—
576C23C2×D4×D1311yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—48no—
584Q8—41no—
585Q8—41no—
586Q8—41no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—261no—
610C2×C4—261no—
611C2×C4—261no—
612C2×C4—261no—
613C2×C4—261no—
614C2×C4—261no—
615C2×C4—261no—
616C2×C4—261no—
617C2×C4—261no—
618C2×C4—261no—
619C2×C4—261no—
620C2×C4—261no—
621C2×C4—521no—
622C2×C4—521no—
623C2×C4—521no—
624C2×C4—521no—
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×C4D4×Dic1312yes—
634C2×C4C2×#612911yescharacteristic
635C2×C4#8825111yescharacteristic
636C2×C4#8825111yescharacteristic
637C2×C4#8825111yescharacteristic
638C2×C4#8825111yescharacteristic
639C2×C4C23×Dic1311yescharacteristic
640C8—22no—
641C8—22no—
642C8—521no—
643C8—521no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22#1171929411yescharacteristic
652C22#1171929711yescharacteristic
653C22#1171947111yescharacteristic
654C22#1171947111yescharacteristic
655C22#1171948311yescharacteristic
656C22#1171948411yescharacteristic
657C22C2×D4×Dic1311yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—521no—
664C4—521no—
665C4—521no—
666C4—521no—
667C4—1041no—
668C4—1041no—
669C4—1041no—
670C4—1041no—
671C4#1171929811yescharacteristic
672C4#1171929811yescharacteristic
673C4#1171950311yescharacteristic
674C4#1171950311yescharacteristic
675C2—42no—
676C2#1311749711yescharacteristic
677C2#1312368011yescharacteristic
678C2#1312370411yescharacteristic
679C2#1312572211yescharacteristic
680C2#1312572911yescharacteristic
681C2#1312591511yescharacteristic
682C2#1312595111yescharacteristic
subgroupG/H
centerC23C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC2×C13×#451C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenterC2×#451D13
hypocenterC13#22054
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #13115708C2semidirect
4. #13115698C2semidirect
5. #13115528C2semidirect
6. #13115514C2semidirect
7. #13115479C2semidirect
8. #13115472C2semidirect
9. #13115179C2semidirect
10. #13115179C2semidirect
23. C13×#451C4semidirect
325. #451Dic13semidirect
568. C13#22054semidirect

Maximal subgroups

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

subgroupindex
#131175202
#131174222
#131157082
#131156982
#131155282
#131155142
#131154792
#131154722
#131151792
#131151492
#131150372
#131147712
#131146042
C2×C13×#4512
#2205413