Finite Group Databasebeta

← #29602896

Subgroups — #29602896

Order 15616.

2558 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D244C211yesmaximal, characteristic
2C2×#16839663C211yesmaximal, characteristic
3C2×#16839663C211yesmaximal, characteristic
4C2×#16839633C214yesmaximal, complemented
5C2×#16839631C211yesmaximal, characteristic, complemented
6#21444063C214yesmaximal, complemented
7#21444048C211yesmaximal, characteristic, complemented
8C4×#14519684C211yesmaximal, characteristic, complemented
9C2×C61×#389C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×C4×D61C2212yes—
12C2×C4×D244—24no—
13C2×C4×D244—24no—
14C2×C4×D244—24no—
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2214yes—
17C22×#11758198C2212yes—
18C22×#11758198C2214yes—
19C22×#11758198C2214yes—
20C22×C4×Dic61C2211yescharacteristic
21C22×C4×Dic61C2211yescharacteristic
22#16839663—24no—
23#16839663C418yescomplemented
24#16839633—216no—
25#16839631—24no—
26C2×#14519684C2214yes—
27C2×#14519684C2212yes—
28C2×#14519684C2212yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C42×C61C2211yescharacteristic
33C61×#389—24no—
34C2×#95×C61C2212yes—
35C24×D61C2312yes—
36C22×D244—22no—
37C22×D244—22no—
38C22×D244—22no—
39C22×D244—22no—
40C22×D244C2×C414yes—
41C22×D244D414yes—
42C22×D244Q814yes—
43C23×Dic61C2312yes—
44C23×Dic61C2312yes—
45C23×Dic61C2312yes—
46C22×C4×D61—24no—
47C22×C4×D61—24no—
48C22×C4×D61—24no—
49C22×C4×D61—28no—
50C22×C4×D61—44no—
51C4×D244—232no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61D414yes—
55C2×#11758196—28no—
56C2×#11758196—28no—
57C2×#11758196—44no—
58C2×#11758198—24no—
59C2×#11758198—24no—
60C2×#11758198—24no—
61C2×#11758198—28no—
62C2×#11758198—28no—
63C2×#11758198—28no—
64C2×#11758198—28no—
65C2×#11758198—44no—
66C2×#11758198C2×C4116yescomplemented
67C2×C4×Dic61—22no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—22no—
72C2×C4×Dic61C2×C414yescomplemented
73#14519684—28no—
74#14519684—24no—
75#14519684D418yescomplemented
76C23×C4×C61C2312yes—
77C23×C4×C61C2312yes—
78C23×C4×C61C2311yescharacteristic
79C23×C4×C61C2311yescharacteristic
80C23×C4×C61C2311yescharacteristic
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61—22no—
84C2×C4⋊C4×C61—24no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—28no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61C2×C414yescomplemented
89C2×C42×C61—22no—
90C2×C42×C61—22no—
91C2×C42×C61D414yes—
92#95×C61—24no—
93C24×C61C2411yescharacteristic
94C23×D61—22no—
95C23×D61—22no—
96C23×D61—24no—
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—24no—
100C23×D61—24no—
101C23×D61—24no—
102C23×D61C22×C414yes—
103C23×D61C2×D414yes—
104C23×D61C2×Q814yes—
105C22×C4×C61—21no—
106C22×C4×C61—21no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—21no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—24no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—24no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61C22×C414yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D414yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—24no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61C22×C414yes—
147C22×Dic61C2×D414yes—
148C22×Dic61C2×D414yes—
149C22×Dic61C2×D414yes—
150C22×Dic61#4114yes—
151C2×D244—28no—
152C2×D244—28no—
153C2×D244—28no—
154C2×D244—28no—
155C2×D244—28no—
156C2×D244—28no—
157C2×D244C4⋊C4116yescomplemented
158C2×C4×D61—216no—
159C2×C4×D61—416no—
160C2×C4×D61—416no—
161C2×C4×D61—48no—
162C2×C4×D61—48no—
163C4⋊C4×C61—28no—
164C4⋊C4×C61—44no—
165C4⋊C4×C61—48no—
166C42×C61—28no—
167#11758198—216no—
168#11758198—232no—
169#11758198—416no—
170C4⋊Dic61—28no—
171#11758196—416no—
172C4×Dic61—28no—
173C4×Dic61—44no—
174C23×C61C23×C411yescharacteristic
175C23×C61C22×D411yescharacteristic
176C23×C61C22×D411yescharacteristic
177C23×C61C22×D411yescharacteristic
178C23×C61C22×D412yes—
179C23×C61C22×Q811yescharacteristic
180C23×C61C2×#4111yescharacteristic
181C23×C61C2×#4111yescharacteristic
182C23×C61C2×#4112yes—
183C23×C61H(5,𝔽2)12yes—
184C23×C612−1+412yes—
185C22×D61—28no—
186C22×D61—24no—
187C22×D61—24no—
188C22×D61—28no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—28no—
194C22×D61—28no—
195C22×D61—28no—
196C22×D61—28no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—44no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—44no—
204C22×D61—44no—
205C22×D61—44no—
206C22×D61—44no—
207C22×D61C2×C4⋊C418yes—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—28no—
213C2×C4×C61—24no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—44no—
220C2×C4×C61—42no—
221C2×C4×C61—42no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—44no—
226C2×C4×C61—44no—
227C2×C4×C61C2×C4⋊C414yes—
228C2×C4×C61C4×D414yes—
229C2×C4×C61C4×D414yes—
230C2×C4×C61#12014yes—
231C2×C4×C61C22⋊Q814yes—
232C2×C4×C61C22⋊Q814yes—
233C2×C4×C61#12314yes—
234C2×C4×C61#12714yes—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—28no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—44no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61C4×D418yescomplemented
254D244—232no—
255C4×D61—432no—
256#13977—611nomaximal, Sylow
257C22×C61C22×C4⋊C411yescharacteristic
258C22×C61C2×C4×D412yes—
259C22×C61C2×C4×D411yescharacteristic
260C22×C61C2×C4×D411yescharacteristic
261C22×C61#52012yes—
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×C22⋊Q811yescharacteristic
264C22×C61C2×C22⋊Q811yescharacteristic
265C22×C61C2×#12311yescharacteristic
266C22×C61C2×#12711yescharacteristic
267C22×C61#53211yescharacteristic
268C22×C61#53212yes—
269C22×C61#53511yescharacteristic
270C22×C61#53611yescharacteristic
271C22×C61#54212yes—
272C22×C61D4212yes—
273C22×C61D4212yes—
274C22×C61#54712yes—
275C22×C61#54712yes—
276C22×C61#54812yes—
277C22×C61D4×Q812yes—
278C22×C61#55312yes—
279C22×C61#55412yes—
280C2×D61—216no—
281C2×D61—28no—
282C2×D61—44no—
283C2×D61—44no—
284C2×D61—44no—
285C2×D61—48no—
286C2×D61—48no—
287C2×D61—44no—
288C2×D61—44no—
289C2×D61—48no—
290C2×D61—44no—
291C4×C61—24no—
292C4×C61—44no—
293C4×C61—42no—
294C4×C61—82no—
295C4×C61#38518yescomplemented
296Dic61—28no—
297Dic61—44no—
298Dic61—44no—
299C22×C4×D4—611no—
300C2×#390—611no—
301C2×#390—611no—
302C2×#389D6111yescharacteristic, complemented
303C2×#385—614no—
304C2×#384—611no—
305#1439—614no—
306C4⋊#95—611no—
307C4×#95—611no—
308C2×C61C2×#38511yescharacteristic
309C2×C61C4×#12711yescharacteristic
310C2×C61#232611yescharacteristic
311C2×C61D4×C4⋊C412yes—
312C2×C61#236112yes—
313C2×C61#242212yes—
314C2×C61#243211yescharacteristic
315C2×C61#243311yescharacteristic
316C2×C61#245212yes—
317C2×C61#254211yescharacteristic
318C2×C61#256611yescharacteristic
319D61—48no—
320C23×D4—611no—
321C24×C4—612no—
322C2×C4×D4—1224no—
323C2×C4×D4—1224no—
324C2×C4×D4—1224no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4—614no—
327C22×C4⋊C4C2×D6111yescharacteristic
328C22×C4⋊C4C2×D6112yes—
329C22×C4⋊C4C2×D6111yescharacteristic
330C22×C22⋊C4—614no—
331C22×C22⋊C4—612no—
332C22×C22⋊C4—614no—
333C22×C42—611no—
334C22×C42—611no—
335C22×C42C2×D6111yescharacteristic
336#390—618no—
337#390—1224no—
338#389—24no—
339#385—12216no—
340#384—1224no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—614no—
344C2×#95C2×D6112yes—
345C61#1397711yescharacteristic, complemented, Sylow
346C25—612no—
347C22×D4—614no—
348C22×D4—614no—
349C22×D4—614no—
350C22×D4—1222no—
351C22×D4—1222no—
352C22×D4—1222no—
353C22×D4—1222no—
354C23×C4—612no—
355C23×C4—612no—
356C23×C4—612no—
357C23×C4—1224no—
358C23×C4—1224no—
359C23×C4—1224no—
360C23×C4—1228no—
361C23×C4—2444no—
362C23×C4C22×D6112yes—
363C23×C4C22×D6112yes—
364C23×C4C22×D6111yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C4×D4—12232no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—24no—
375C2×C4⋊C4—614no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1228no—
379C2×C4⋊C4—1228no—
380C2×C4⋊C4—2444no—
381C2×C4⋊C4C4×D6114yescomplemented
382C2×C22⋊C4—6116no—
383C2×C22⋊C4—1224no—
384C2×C22⋊C4—1228no—
385C2×C22⋊C4—1228no—
386C2×C22⋊C4—1224no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1228no—
389C2×C22⋊C4—1228no—
390C2×C22⋊C4—2444no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—614no—
394C2×C42—1222no—
395C2×C42—1222no—
396C2×C42—1222no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42C22⋊D6114yes—
400#95—24no—
401#95—618no—
402#95—1228no—
403#95—1224no—
404C24—614no—
405C24—614no—
406C24—614no—
407C24—1224no—
408C24—1222no—
409C24—1222no—
410C24—1224no—
411C24—1222no—
412C24—1222no—
413C24—1224no—
414C24—1224no—
415C24C23×D6111yescharacteristic
416C2×D4—6116no—
417C2×D4—1228no—
418C2×D4—1228no—
419C2×D4—1228no—
420C2×D4—1228no—
421C2×D4—1228no—
422C2×D4—1228no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—614no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—12216no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—24416no—
460C22×C4—24416no—
461C22×C4—2448no—
462C22×C4—2448no—
463C22×C4C2×C4×D6114yes—
464C22×C4C2×C4×D6112yes—
465C22×C4#1175822212yes—
466C22×C4D4×D6114yes—
467C22×C4D4×D6112yes—
468C22×C4D4×D6112yes—
469C22×C4Q8×D6112yes—
470C22×C4#1175822612yes—
471C22×C4C2×C22⋊D6112yes—
472C22×C4C2×C22⋊D6112yes—
473C22×C4C2×C22⋊D6112yes—
474C4⋊C4—28no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—1228no—
478C4⋊C4—24416no—
479C22⋊C4—12232no—
480C22⋊C4—12216no—
481C22⋊C4—24416no—
482C42—28no—
483C42—1228no—
484C42—2444no—
485C23—618no—
486C23—1228no—
487C23—1228no—
488C23—1224no—
489C23—1224no—
490C23—1228no—
491C23—1224no—
492C23—1224no—
493C23—1224no—
494C23—1224no—
495C23—1228no—
496C23—1228no—
497C23—1228no—
498C23—2444no—
499C23—2442no—
500C23—2442no—
501C23—2444no—
502C23—2442no—
503C23—2442no—
504C23—2444no—
505C23—2444no—
506C23—2444no—
507C23—2444no—
508C23C22×C4×D6111yescharacteristic
509C23C2×D4×D6111yescharacteristic
510C23C2×D4×D6112yes—
511C23C2×D4×D6111yescharacteristic
512C23C22×C22⋊D6111yescharacteristic
513C23C2×Q8×D6111yescharacteristic
514C23#41×D6112yes—
515C23C2×#1175822211yescharacteristic
516C23C2×#1175822611yescharacteristic
517C23#1451982312yes—
518C23#1451982512yes—
519D4—12232no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—618no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1224no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—1224no—
548C2×C4—1224no—
549C2×C4—1228no—
550C2×C4—1224no—
551C2×C4—24432no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2444no—
558C2×C4—2442no—
559C2×C4C4⋊C4×D6114yes—
560C2×C4C4×C22⋊D6114yes—
561C2×C4#1451973914yes—
562C2×C4#1451974514yes—
563C2×C4#1451975314yes—
564C2×C4#1451975514yes—
565C2×C4#1451976214yes—
566C2×C4#1451977314yes—
567C22—12216no—
568C22—1228no—
569C22—2444no—
570C22—2444no—
571C22—2444no—
572C22—2448no—
573C22—2448no—
574C22—2444no—
575C22—2448no—
576C22—2444no—
577C22—2444no—
578C22C2×C4⋊C4×D6111yescharacteristic
579C22C4×D4×D6112yes—
580C22C2×C4×C22⋊D6111yescharacteristic
581C22C2×#1451973911yescharacteristic
582C22#1684024412yes—
583C22C2×#1451974511yescharacteristic
584C22C2×#1451975311yescharacteristic
585C22C2×#1451975511yescharacteristic
586C22C2×#1451976211yescharacteristic
587C22C2×#1451977311yescharacteristic
588C22#1684030811yescharacteristic
589C22#1684030912yes—
590C22#1684032611yescharacteristic
591C22#1684033611yescharacteristic
592C22#1684038412yes—
593C22#1684039612yes—
594C22#1684039612yes—
595C22#1684041412yes—
596C22#1684041512yes—
597C22#1684042212yes—
598C22#1684042712yes—
599C22#1684046512yes—
600C22#1684046812yes—
601C4—24no—
602C4—44no—
603C4—42no—
604C4—82no—
605C4—1228no—
606C4—2444no—
607C4—2444no—
608C4#1683963318yescomplemented
609C2—2448no—
610C2C2×#1683963311yescharacteristic
611C2C4×#1451977311yescharacteristic
612C2#2144852611yescharacteristic
613C2#2144872012yes—
614C2#2144883412yes—
615C2#2144933412yes—
616C2#2144944811yescharacteristic
617C2#2144945711yescharacteristic
618C2#2144967012yes—
619C2#2145077311yescharacteristic
620C2#2145102011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#389C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenterC2×#389D61
hypocenterC61#13977
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
4. C2×#16839633C2semidirect
5. C2×#16839631C2semidirect
6. #21444063C2semidirect
7. #21444048C2semidirect
8. C4×#14519684C2semidirect
9. C2×C61×#389C2semidirect
23. #16839663C4semidirect
66. C2×#11758198C2×C4semidirect
72. C2×C4×Dic61C2×C4semidirect
75. #14519684D4semidirect
88. C2×C4⋊C4×C61C2×C4semidirect
157. C2×D244C4⋊C4semidirect
253. C2×Dic61C4×D4semidirect
295. C4×C61#385semidirect
302. C2×#389D61semidirect
345. C61#13977semidirect
381. C2×C4⋊C4C4×D61semidirect
608. C4#16839633semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D2442
C2×#168396632
C2×#168396332
C2×#168396312
#214440632
#214440482
C4×#145196842
C2×C61×#3892
#1397761