Finite Group Databasebeta

← #15920051

Subgroups — #15920051

Order 3328.

990 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
1C2×#11718681C211yesmaximal, characteristic, complemented
2C22×#88075C211yesmaximal, characteristic
3#13118538C211yesmaximal, characteristic
4#13116379C211yesmaximal, characteristic, complemented
5#13116217C211yesmaximal, characteristic, complemented
6#13116217C211yesmaximal, characteristic, complemented
7#13115977C211yesmaximal, characteristic, complemented
8#13115104C211yesmaximal, characteristic, complemented
9#13115104C211yesmaximal, characteristic, complemented
10#13115028C211yesmaximal, characteristic
11#13115016C211yesmaximal, characteristic
12#13114859C211yesmaximal, characteristic
13#13114841C211yesmaximal, characteristic, complemented
14#13114771C211yesmaximal, characteristic
15Q8×C13×C22⋊C4C211yesmaximal, characteristic
16C23×Q8×C13C2211yescharacteristic
17C2×C4×Q8×C13C2211yescharacteristic
18C2×C4×Q8×C13C2211yescharacteristic
19C22×C4⋊Dic13C2211yescharacteristic
20C23×#1055C2211yescharacteristic
21C2×#88132—21no—
22C2×#88123C2211yescharacteristic
23C13×#391—21no—
24C13×#391—21no—
25C13×#391C2211yescharacteristic
26C13×#391C2211yescharacteristic
27C13×#380—21no—
28C13×#380C2211yescharacteristic
29C4×C13×C22⋊C4—21no—
30C4×C13×C22⋊C4C2211yescharacteristic
31C2×#88075—24no—
32C2×#88075C2211yescharacteristic
33C2×#88075C2211yescharacteristic
34C2×#88075C2211yescharacteristic
35C2×#88075C2211yescharacteristic
36C2×#88075C2211yescharacteristic
37C2×#88075C2211yescharacteristic
38C2×#88075C2211yescharacteristic
39C2×#88075C2211yescharacteristic
40#11718873C2211yescharacteristic
41C2×C22⋊#1055C2211yescharacteristic
42#11718681—24no—
43C2×#88048C2211yescharacteristic
44C2×#88048C2211yescharacteristic
45C2×#88048C2211yescharacteristic
46C2×#88048C2211yescharacteristic
47#11718611—21no—
48#11718596C2211yescharacteristic
49#11718589C2211yescharacteristic
50#11718589C2211yescharacteristic
51C2×C4⋊#1055C2211yescharacteristic
52C2×C4⋊#1055C2211yescharacteristic
53#11718322—21no—
54#11718317—21no—
55#11718121C2211yescharacteristic
56#11718121C2211yescharacteristic
57#11718121C2211yescharacteristic
58#11718121C2211yescharacteristic
59#11718116C2211yescharacteristic
60#11718116C2211yescharacteristic
61C22×Q8×C13—21no—
62C22×Q8×C13—22no—
63C22×Q8×C13—22no—
64C22×Q8×C13—22no—
65C22×Q8×C13—41no—
66C22×Q8×C13—41no—
67C22×Q8×C13Q814yescomplemented
68C22×Q8×C13C2311yescharacteristic
69C22×Q8×C13C2311yescharacteristic
70C23×C4×C13—21no—
71C23×C4×C13C2311yescharacteristic
72C23×Dic13—21no—
73C22×Dic26—21no—
74C4×Q8×C13—22no—
75C4×Q8×C13—22no—
76C4×Q8×C13—41no—
77C4×Q8×C13—41no—
78C2×C13×C4⋊C4—21no—
79C2×C13×C4⋊C4—21no—
80C2×C13×C4⋊C4C2311yescharacteristic
81C2×C13×C4⋊C4C2311yescharacteristic
82C2×C13×C22⋊C4—21no—
83C2×C13×C22⋊C4D411yescharacteristic
84C2×C13×C22⋊C4D411yescharacteristic
85C2×C42×C13—21no—
86C2×C42×C13—21no—
87C2×C42×C13C2311yescharacteristic
88C2×C42×C13C2311yescharacteristic
89C2×C22⋊Dic13—21no—
90C2×C4⋊Dic13—22no—
91C2×C4⋊Dic13—21no—
92C2×C4⋊Dic13—21no—
93C2×C4⋊Dic13—21no—
94C2×C4⋊Dic13—21no—
95C2×C4⋊Dic13—41no—
96C2×C4⋊Dic13C2311yescharacteristic
97C2×C4⋊Dic13C2311yescharacteristic
98C2×C4⋊Dic13C2311yescharacteristic
99C2×C4⋊Dic13C2311yescharacteristic
100C2×#6101—21no—
101C2×#6101—21no—
102C22×#1055—22no—
103C22×#1055—22no—
104C22×#1055—41no—
105C22×#1055C2311yescharacteristic
106C22×#1055C2311yescharacteristic
107C22×#1055C2311yescharacteristic
108C22×#1055C2311yescharacteristic
109#88132—44no—
110#88123—22no—
111#88123—41no—
112C13×#95—21no—
113C13×#95—21no—
114C13×#95—21no—
115C13×#95—21no—
116C13×#95—21no—
117C13×#95D411yescharacteristic
118C13×#95D411yescharacteristic
119#88075—22no—
120#88075—22no—
121#88075—22no—
122#88075—22no—
123#88075—22no—
124#88075—41no—
125#88075—41no—
126#88075D414yes—
127#88071—21no—
128#88071—21no—
129#88071—21no—
130C22⋊#1055—22no—
131#88048—22no—
132#88048—22no—
133#88048—41no—
134#88048—41no—
135C4⋊#1055—22no—
136C4⋊#1055—41no—
137#25955—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—28no—
145C2×Q8×C13—21no—
146C2×Q8×C13—21no—
147C2×Q8×C13—21no—
148C2×Q8×C13—21no—
149C2×Q8×C13—42no—
150C2×Q8×C13—42no—
151C2×Q8×C13—44no—
152C2×Q8×C13—44no—
153C2×Q8×C13—44no—
154C2×Q8×C13C2×D412yes—
155C2×Q8×C13C2×Q812yes—
156C2×Q8×C13C2×Q812yes—
157C2×Q8×C13#4112yes—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13—21no—
161C22×C4×C13—21no—
162C22×C4×C13—21no—
163C22×C4×C13—24no—
164C22×C4×C13—21no—
165C22×C4×C13—21no—
166C22×C4×C13—21no—
167C22×C4×C13—42no—
168C22×C4×C13—42no—
169C22×C4×C13C2×D411yescharacteristic
170C22×C4×C13C2×D411yescharacteristic
171C22×C4×C13C2×D411yescharacteristic
172C22×C4×C13C2×D411yescharacteristic
173C22×C4×C13C2×D411yescharacteristic
174C22×C4×C13C2×Q812yes—
175C22×C4×C13#4112yes—
176C22×C4×C13C2411yescharacteristic
177C22×Dic13—21no—
178C22×Dic13—21no—
179C22×Dic13—21no—
180C22×Dic13—21no—
181C22×Dic13—42no—
182C22×Dic13—42no—
183C22×Dic13—81no—
184C2×Dic26—42no—
185C2×Dic26—42no—
186C2×Dic26—81no—
187C13×C4⋊C4—21no—
188C13×C4⋊C4—21no—
189C13×C4⋊C4—21no—
190C13×C4⋊C4—21no—
191C13×C4⋊C4—42no—
192C13×C4⋊C4—42no—
193C13×C22⋊C4—22no—
194C13×C22⋊C4—42no—
195C13×C22⋊C4SD1614yes—
196C42×C13—21no—
197C42×C13—21no—
198C42×C13—21no—
199C42×C13—21no—
200C42×C13—42no—
201C42×C13—42no—
202C22⋊Dic13—42no—
203C4⋊Dic13—21no—
204C4⋊Dic13—21no—
205C4⋊Dic13—21no—
206C4⋊Dic13—21no—
207C4⋊Dic13—21no—
208C4⋊Dic13—21no—
209C4⋊Dic13—21no—
210C4⋊Dic13—42no—
211C4⋊Dic13—42no—
212C4⋊Dic13—42no—
213C4⋊Dic13—81no—
214C4⋊Dic13C2×D412yes—
215#6101—42no—
216#6101—42no—
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×#1055—44no—
229C2×#1055C2×D412yes—
230C2×#475—131no—
231C22×#103—131no—
232#2440—131no—
233Q8×C22⋊C4D1311yescharacteristic
234#2096—131no—
235#2066—131no—
236#2066—131no—
237#2024—131no—
238#1865—131no—
239#1865—131no—
240#1853—131no—
241#1850—131no—
242#1807—131no—
243#1803—131no—
244#1787—131no—
245C23×C13—21no—
246C23×C13—21no—
247C23×C13—21no—
248C23×C13—21no—
249C23×C13C22⋊Q812yes—
250C23×C13C2×SD1612yes—
251C23×C13#13712yes—
252C23×C13C22×D411yescharacteristic
253Q8×C13—22no—
254Q8×C13—22no—
255Q8×C13—22no—
256Q8×C13—22no—
257Q8×C13—22no—
258Q8×C13—48no—
259Q8×C13—41no—
260Q8×C13—41no—
261Q8×C13C22⋊Q814yescomplemented
262C2×C4×C13—22no—
263C2×C4×C13—21no—
264C2×C4×C13—21no—
265C2×C4×C13—22no—
266C2×C4×C13—21no—
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—22no—
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—48no—
283C2×C4×C13—41no—
284C2×C4×C13—41no—
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×C13C22⋊Q812yes—
292C2×C4×C13C2×SD1612yes—
293C2×C4×C13C2×SD1612yes—
294C2×C4×C13C22×D411yescharacteristic
295C2×C4×C13C22×Q811yescharacteristic
296C2×C4×C13C2×#4111yescharacteristic
297C2×C4×C13C2×#4111yescharacteristic
298C2×C4×C13H(5,𝔽2)11yescharacteristic
299C2×C4×C13H(5,𝔽2)11yescharacteristic
300C2×Dic13—22no—
301C2×Dic13—41no—
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—84no—
313Dic26—84no—
314#1055—22no—
315#1055—41no—
316#1055—41no—
317#1055—41no—
318C23×Q8C2×D1311yescharacteristic
319C23×C8—131no—
320C2×C4⋊Q8—131no—
321C2×C22⋊Q8—261no—
322C2×C4×Q8C2×D1311yescharacteristic
323C2×C4×Q8C2×D1311yescharacteristic
324C22×C4⋊C4—131no—
325#475—264no—
326C2×#106—131no—
327C2×#106—131no—
328C2×#106—131no—
329C2×#106—131no—
330C2×C4⋊C8—131no—
331C2×C4⋊C8—131no—
332C2×#103—131no—
333C2×#103—131no—
334C2×#103—131no—
335C2×#103—131no—
336C2×#103—131no—
337C2×#103—131no—
338C2×#103—131no—
339C2×#103—131no—
340C2×#103—264no—
341C2×C22⋊C8—131no—
342#398—261no—
343#395—261no—
344#391—21no—
345#391—21no—
346#391C2×D1311yescharacteristic
347#391C2×D1311yescharacteristic
348#389—131no—
349#389—131no—
350#385—261no—
351#384—131no—
352#380—21no—
353#380—131no—
354#380C2×D1311yescharacteristic
355C4×C22⋊C4—21no—
356C4×C22⋊C4C2×D1311yescharacteristic
357#340—131no—
358#340—131no—
359#340—131no—
360#340—131no—
361#336—131no—
362#336—131no—
363C22×C13—22no—
364C22×C13—22no—
365C22×C13—22no—
366C22×C13—22no—
367C22×C13—22no—
368C22×C13—22no—
369C22×C13#47514yescomplemented
370C22×C13C2×C22⋊Q811yescharacteristic
371C22×C13D4211yescharacteristic
372C22×C13#54611yescharacteristic
373C22×C13C22×SD1611yescharacteristic
374C22×C13C2×#13711yescharacteristic
375C22×C13#57711yescharacteristic
376C22×C13#57811yescharacteristic
377C4×C13—22no—
378C4×C13—24no—
379C4×C13—22no—
380C4×C13—22no—
381C4×C13—42no—
382C4×C13—41no—
383C4×C13—41no—
384C4×C13—81no—
385C4×C13—81no—
386C4×C13C2×C22⋊Q811yescharacteristic
387C4×C13#54311yescharacteristic
388C4×C13#54611yescharacteristic
389C4×C13#55111yescharacteristic
390Dic13—42no—
391Dic13—81no—
392Dic13—81no—
393Dic13—81no—
394C22×Q8—22no—
395C22×Q8—21no—
396C22×Q8—22no—
397C22×Q8—22no—
398C22×Q8—41no—
399C22×Q8—41no—
400C22×Q8—261no—
401C22×Q8Dic2614yescomplemented
402C22×Q8C22×D1311yescharacteristic
403C22×Q8C22×D1311yescharacteristic
404C23×C4—21no—
405C23×C4—261no—
406C23×C4C22×D1311yescharacteristic
407C22×C8—131no—
408C22×C8—131no—
409C22×C8—131no—
410C22×C8—131no—
411C22×C8—262no—
412C22×C8—262no—
413C22×C8—521no—
414C4⋊Q8—262no—
415C4⋊Q8—521no—
416C22⋊Q8—524no—
417C4×Q8—22no—
418C4×Q8—22no—
419C4×Q8—41no—
420C4×Q8—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—131no—
424C2×C4⋊C4—131no—
425C2×C4⋊C4—131no—
426C2×C4⋊C4—131no—
427C2×C4⋊C4—262no—
428C2×C4⋊C4—261no—
429C2×C4⋊C4—261no—
430C2×C4⋊C4—261no—
431C2×C4⋊C4—261no—
432C2×C4⋊C4—261no—
433C2×C4⋊C4—261no—
434C2×C4⋊C4—521no—
435C2×C4⋊C4C22×D1311yescharacteristic
436C2×C4⋊C4C22×D1311yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—261no—
439C2×C22⋊C4C22⋊D1311yescharacteristic
440C2×C22⋊C4C22⋊D1311yescharacteristic
441C2×C42—21no—
442C2×C42—21no—
443C2×C42C22×D1311yescharacteristic
444C2×C42C22×D1311yescharacteristic
445#106—262no—
446#106—262no—
447#106—521no—
448#106—521no—
449C4⋊C8—262no—
450C4⋊C8—521no—
451#103—134no—
452#103—262no—
453#103—262no—
454#103—262no—
455#103—262no—
456#103—262no—
457#103—521no—
458#103—521no—
459C22⋊C8—262no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—261no—
466#95—261no—
467#95—261no—
468#95C22⋊D1311yescharacteristic
469#95C22⋊D1311yescharacteristic
470C2×C13—24no—
471C2×C13#243711yescharacteristic
472C2×C13C2×#47511yescharacteristic
473C2×C13#308211yescharacteristic
474C2×C13#327011yescharacteristic
475C2×C13D4×SD1611yescharacteristic
476C2×C13#329811yescharacteristic
477C2×C13#331611yescharacteristic
478C24C2×C22⋊D1311yescharacteristic
479C2×Q8—24no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—28no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—522no—
495C2×Q8—522no—
496C2×Q8—1041no—
497C2×Q8C2×Dic2612yes—
498C2×Q8C2×Dic2612yes—
499C2×Q8D4×D1312yes—
500C2×Q8#612912yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—24no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—261no—
513C22×C4—261no—
514C22×C4—261no—
515C22×C4—261no—
516C22×C4—522no—
517C22×C4—522no—
518C22×C4—1041no—
519C22×C4C2×Dic2612yes—
520C22×C4#612712yes—
521C22×C4C2×C22⋊D1311yescharacteristic
522C22×C4C2×C22⋊D1311yescharacteristic
523C22×C4C2×C22⋊D1311yescharacteristic
524C22×C4C2×C22⋊D1311yescharacteristic
525C22×C4C2×C22⋊D1311yescharacteristic
526C22×C4C23×D1311yescharacteristic
527C2×C8—132no—
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—
537C2×C8—261no—
538C2×C8—261no—
539C2×C8—524no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—132no—
547C4⋊C4—261no—
548C4⋊C4—261no—
549C4⋊C4—261no—
550C4⋊C4—261no—
551C4⋊C4—261no—
552C4⋊C4—261no—
553C4⋊C4—261no—
554C4⋊C4—522no—
555C4⋊C4—522no—
556C4⋊C4—522no—
557C4⋊C4—522no—
558C4⋊C4—522no—
559C4⋊C4—1041no—
560C22⋊C4—22no—
561C22⋊C4—42no—
562C22⋊C4—522no—
563C22⋊C4#610614yes—
564C42—21no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—42no—
569C42—42no—
570C13#2595511yescharacteristic, complemented, Sylow
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#8817812yes—
576C23C2×#610612yes—
577C23#8820512yes—
578C23C22×C22⋊D1311yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—48no—
585Q8—41no—
586Q8—41no—
587Q8—1044no—
588Q8#8813214yescomplemented
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—262no—
619C2×C4—521no—
620C2×C4—521no—
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—1044no—
631C2×C4#8817812yes—
632C2×C4C2×#610612yes—
633C2×C4C2×#610612yes—
634C2×C4C22×Dic2611yescharacteristic
635C2×C4C2×#612711yescharacteristic
636C2×C4C2×D4×D1311yescharacteristic
637C2×C4C2×#612911yescharacteristic
638C2×C4#8825111yescharacteristic
639C2×C4#8825111yescharacteristic
640C8—262no—
641C8—521no—
642C8—521no—
643C8—521no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1171868114yescomplemented
651C22C2×#8817811yescharacteristic
652C22D4×C22⋊D1311yescharacteristic
653C22#1171951111yescharacteristic
654C22C22×#610611yescharacteristic
655C22C2×#8820511yescharacteristic
656C22#1171954211yescharacteristic
657C22#1171954311yescharacteristic
658C4—22no—
659C4—24no—
660C4—22no—
661C4—22no—
662C4—41no—
663C4—42no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—522no—
668C4—1041no—
669C4—1041no—
670C4—1041no—
671C4C2×#8813211yescharacteristic
672C4#1171918811yescharacteristic
673C4#1171919511yescharacteristic
674C4#1171920011yescharacteristic
675C2—24no—
676C2#1311849711yescharacteristic
677C2C2×#1171868111yescharacteristic
678C2#1312467811yescharacteristic
679C2#1312640211yescharacteristic
680C2D4×#610611yescharacteristic
681C2#1312670111yescharacteristic
682C2#1312694711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D13
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingQ8×C13×C22⋊C4C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
hypercenterQ8×C22⋊C4D13
hypocenterC13#25955
LCSG ▷ C22×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#11718681C2semidirect
4. #13116379C2semidirect
5. #13116217C2semidirect
6. #13116217C2semidirect
7. #13115977C2semidirect
8. #13115104C2semidirect
9. #13115104C2semidirect
13. #13114841C2semidirect
67. C22×Q8×C13Q8semidirect
261. Q8×C13C22⋊Q8semidirect
369. C22×C13#475semidirect
401. C22×Q8Dic26semidirect
570. C13#25955semidirect
588. Q8#88132semidirect
650. C22#11718681semidirect

Maximal subgroups

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

subgroupindex
C2×#117186812
C22×#880752
#131185382
#131163792
#131162172
#131159772
#131151042
#131150282
#131150162
#131148592
#131148412
#131147712
Q8×C13×C22⋊C42
#2595513