Finite Group Databasebeta

← #29602891

Subgroups — #29602891

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
1C2×#16839665C211yesmaximal, characteristic
2C2×#16839665C211yesmaximal, characteristic
3C2×#16839660C211yesmaximal, characteristic, complemented
4C2×#16839635C214yesmaximal, complemented
5C2×#16839631C211yesmaximal, characteristic, complemented
6#21444062C214yesmaximal, complemented
7#21444048C211yesmaximal, characteristic, complemented
8C4×#14519684C211yesmaximal, characteristic, complemented
9C22×C4×D4×C61C211yesmaximal, characteristic
10C22×C4⋊Dic61C2211yescharacteristic
11C22×C4⋊Dic61C2211yescharacteristic
12C22×#11758196C2212yes—
13C22×#11758196C2214yes—
14C22×C22⋊Dic61C2214yes—
15C22×C22⋊Dic61C2214yes—
16C22×C4×Dic61C2211yescharacteristic
17C22×C4×Dic61C2211yescharacteristic
18#16839665—24no—
19#16839665C418yescomplemented
20#16839660—24no—
21#16839635—216no—
22#16839631—24no—
23C2×#14519684C2214yes—
24C2×#14519684C2212yes—
25C2×#14519684C2212yes—
26C2×#14519684C2212yes—
27C23×D4×C61C2211yescharacteristic
28C24×C4×C61C2212yes—
29C2×C4×D4×C61—24no—
30C2×C4×D4×C61—24no—
31C2×C4×D4×C61—24no—
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C22⋊C4×C61C2212yes—
34C22×C42×C61C2211yescharacteristic
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2312yes—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—22no—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61C2×C414yescomplemented
45C2×#11758196—28no—
46C2×#11758196—28no—
47C2×#11758196—24no—
48C2×#11758196—28no—
49C2×#11758196—44no—
50C2×C22⋊Dic61—28no—
51C2×C22⋊Dic61—28no—
52C2×C22⋊Dic61—28no—
53C2×C22⋊Dic61—28no—
54C2×C22⋊Dic61—44no—
55C2×C22⋊Dic61C2×C4116yescomplemented
56C2×C4×Dic61—22no—
57C2×C4×Dic61—22no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61C2×C414yescomplemented
62#14519684—24no—
63#14519684—28no—
64#14519684—24no—
65#14519684D418yescomplemented
66C25×C61C2312yes—
67C22×D4×C61—22no—
68C22×D4×C61—22no—
69C22×D4×C61—22no—
70C22×D4×C61—22no—
71C22×D4×C61C2×C414yes—
72C22×D4×C61D414yes—
73C22×D4×C61Q814yes—
74C23×C4×C61—24no—
75C23×C4×C61—24no—
76C23×C4×C61—24no—
77C23×C4×C61—28no—
78C23×C4×C61—44no—
79C23×C4×C61C2312yes—
80C23×C4×C61C2311yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C4×D4×C61—232no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61D414yes—
87C2×C22⋊C4×C61—24no—
88C2×C22⋊C4×C61—24no—
89C2×C22⋊C4×C61—24no—
90C2×C42×C61—22no—
91C2×C42×C61—22no—
92C2×C42×C61D414yes—
93C24×C61—22no—
94C24×C61—22no—
95C24×C61—24no—
96C24×C61—22no—
97C24×C61—22no—
98C24×C61—24no—
99C24×C61—24no—
100C24×C61—24no—
101C24×C61C22×C414yes—
102C24×C61C2×D414yes—
103C24×C61C2×Q814yes—
104C24×C61C2411yescharacteristic
105C2×D4×C61—28no—
106C2×D4×C61—28no—
107C2×D4×C61—28no—
108C2×D4×C61—28no—
109C2×D4×C61—28no—
110C2×D4×C61—28no—
111C2×D4×C61C4⋊C4116yescomplemented
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—22no—
115C22×C4×C61—216no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61—22no—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—416no—
125C22×C4×C61—416no—
126C22×C4×C61—48no—
127C22×C4×C61—48no—
128C22×C4×C61C22×C412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D414yes—
131C22×C4×C61C2×D414yes—
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×Q812yes—
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—22no—
143C22×Dic61—24no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—24no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—22no—
155C22×Dic61—24no—
156C22×Dic61—22no—
157C22×Dic61—24no—
158C22×Dic61C22×C414yes—
159C22×Dic61C22×C414yes—
160C22×Dic61C2×D414yes—
161C22×Dic61C2×D414yes—
162C22×Dic61#4114yes—
163C4⋊C4×C61—28no—
164C22⋊C4×C61—216no—
165C42×C61—28no—
166C22⋊Dic61—232no—
167C22⋊Dic61—416no—
168C4⋊Dic61—28no—
169C4⋊Dic61—44no—
170#11758196—416no—
171#11758196—48no—
172C4×Dic61—28no—
173C4×Dic61—44no—
174C23×C61—28no—
175C23×C61—24no—
176C23×C61—24no—
177C23×C61—28no—
178C23×C61—24no—
179C23×C61—24no—
180C23×C61—24no—
181C23×C61—24no—
182C23×C61—28no—
183C23×C61—28no—
184C23×C61—28no—
185C23×C61—28no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61—44no—
189C23×C61—42no—
190C23×C61—42no—
191C23×C61—44no—
192C23×C61—44no—
193C23×C61—44no—
194C23×C61—44no—
195C23×C61—44no—
196C23×C61C2×C4⋊C418yes—
197C23×C61C23×C411yescharacteristic
198C23×C61C22×D411yescharacteristic
199C23×C61C22×D411yescharacteristic
200C23×C61C22×D411yescharacteristic
201C23×C61C22×D412yes—
202C23×C61C22×Q811yescharacteristic
203C23×C61C2×#4111yescharacteristic
204C23×C61C2×#4111yescharacteristic
205C23×C61C2×#4112yes—
206C23×C61H(5,𝔽2)12yes—
207C23×C612−1+412yes—
208D4×C61—232no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—24no—
212C2×C4×C61—24no—
213C2×C4×C61—22no—
214C2×C4×C61—24no—
215C2×C4×C61—22no—
216C2×C4×C61—22no—
217C2×C4×C61—28no—
218C2×C4×C61—432no—
219C2×C4×C61—42no—
220C2×C4×C61—42no—
221C2×C4×C61—44no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61C2×C4⋊C414yes—
225C2×C4×C61C4×D414yes—
226C2×C4×C61C4×D414yes—
227C2×C4×C61#12014yes—
228C2×C4×C61C22⋊Q814yes—
229C2×C4×C61C22⋊Q814yes—
230C2×C4×C61#12314yes—
231C2×C4×C61#12714yes—
232C2×Dic61—24no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—28no—
240C2×Dic61—24no—
241C2×Dic61—24no—
242C2×Dic61—24no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—44no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61—42no—
252C2×Dic61—42no—
253C2×Dic61—44no—
254C2×Dic61—44no—
255C2×Dic61C4×D418yescomplemented
256#13977—611nomaximal, Sylow
257C22×C61—216no—
258C22×C61—28no—
259C22×C61—44no—
260C22×C61—44no—
261C22×C61—44no—
262C22×C61—48no—
263C22×C61—48no—
264C22×C61—44no—
265C22×C61—44no—
266C22×C61—48no—
267C22×C61—44no—
268C22×C61C22×C4⋊C411yescharacteristic
269C22×C61C2×C4×D412yes—
270C22×C61C2×C4×D411yescharacteristic
271C22×C61C2×C4×D411yescharacteristic
272C22×C61#52012yes—
273C22×C61C2×#12011yescharacteristic
274C22×C61C2×C22⋊Q811yescharacteristic
275C22×C61C2×C22⋊Q811yescharacteristic
276C22×C61C2×#12311yescharacteristic
277C22×C61C2×#12711yescharacteristic
278C22×C61#53211yescharacteristic
279C22×C61#53212yes—
280C22×C61#53511yescharacteristic
281C22×C61#53611yescharacteristic
282C22×C61#54212yes—
283C22×C61D4212yes—
284C22×C61D4212yes—
285C22×C61#54712yes—
286C22×C61#54712yes—
287C22×C61#54812yes—
288C22×C61D4×Q812yes—
289C22×C61#55312yes—
290C22×C61#55412yes—
291C4×C61—24no—
292C4×C61—42no—
293C4×C61—44no—
294C4×C61#38518yescomplemented
295Dic61—28no—
296Dic61—44no—
297Dic61—44no—
298Dic61—82no—
299C22×C4×D4D6111yescharacteristic
300C2×#390—611no—
301C2×#390—611no—
302C2×#389—611no—
303C2×#385—614no—
304C2×#384—611no—
305#1439—614no—
306C4⋊#95—611no—
307C4×#95—611no—
308C2×C61—48no—
309C2×C61C2×#38511yescharacteristic
310C2×C61C4×#12711yescharacteristic
311C2×C61#232611yescharacteristic
312C2×C61D4×C4⋊C412yes—
313C2×C61#236112yes—
314C2×C61#242212yes—
315C2×C61#243211yescharacteristic
316C2×C61#243311yescharacteristic
317C2×C61#245212yes—
318C2×C61#254211yescharacteristic
319C2×C61#256611yescharacteristic
320C23×D4C2×D6111yescharacteristic
321C24×C4C2×D6112yes—
322C2×C4×D4—24no—
323C2×C4×D4—24no—
324C2×C4×D4—24no—
325C22×C4⋊C4—611no—
326C22×C4⋊C4—612no—
327C22×C4⋊C4—611no—
328C22×C4⋊C4—614no—
329C22×C4⋊C4C2×D6111yescharacteristic
330C22×C22⋊C4—614no—
331C22×C22⋊C4—614no—
332C22×C22⋊C4C2×D6112yes—
333C22×C42—611no—
334C22×C42—611no—
335C22×C42C2×D6111yescharacteristic
336#390—618no—
337#390—1224no—
338#389—1224no—
339#385—12216no—
340#384—1224no—
341C2×#95—612no—
342C2×#95—612no—
343C2×#95—614no—
344C2×#95—612no—
345C61#1397711yescharacteristic, complemented, Sylow
346C25C22×D6112yes—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—22no—
351C22×D4Dic12214yes—
352C22×D4C4×D6114yes—
353C22×D4C22⋊D6114yes—
354C23×C4—24no—
355C23×C4—24no—
356C23×C4—24no—
357C23×C4—28no—
358C23×C4—44no—
359C23×C4—612no—
360C23×C4—612no—
361C23×C4—612no—
362C23×C4—612no—
363C23×C4C22×D6111yescharacteristic
364C23×C4C22×D6112yes—
365C23×C4C22×D6111yescharacteristic
366C23×C4C22×D6111yescharacteristic
367C4×D4—232no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—614no—
371C2×C4⋊C4—1222no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1228no—
374C2×C4⋊C4—1228no—
375C2×C4⋊C4—1228no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—2444no—
381C2×C4⋊C4C22⋊D6114yes—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—6116no—
386C2×C22⋊C4—1228no—
387C2×C22⋊C4—1228no—
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—618no—
401#95—1228no—
402#95—1224no—
403#95—1224no—
404C24—24no—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—22no—
410C24—24no—
411C24—24no—
412C24C2×Dic12214yes—
413C24C2×C4×D6114yes—
414C24C2×C22⋊D6114yes—
415C24C23×D6111yescharacteristic
416C2×D4—28no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—28no—
422C2×D4#11758196116yescomplemented
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—216no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—614no—
440C22×C4—614no—
441C22×C4—614no—
442C22×C4—614no—
443C22×C4—614no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1224no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1224no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1224no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—1222no—
462C22×C4—1224no—
463C22×C4C2×Dic12212yes—
464C22×C4C2×C4×D6112yes—
465C22×C4#1175822212yes—
466C22×C4#1175822212yes—
467C22×C4D4×D6114yes—
468C22×C4C2×C22⋊D6112yes—
469C22×C4C2×C22⋊D6114yes—
470C22×C4C2×C22⋊D6112yes—
471C22×C4C2×C22⋊D6112yes—
472C22×C4C2×C22⋊D6112yes—
473C22×C4C2×C22⋊D6112yes—
474C4⋊C4—28no—
475C4⋊C4—1228no—
476C4⋊C4—24416no—
477C4⋊C4—2444no—
478C4⋊C4—2448no—
479C22⋊C4—216no—
480C22⋊C4—12232no—
481C22⋊C4—24416no—
482C42—28no—
483C42—1228no—
484C42—2444no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23C2×#1175819618yes—
508C23C22×C4×D6111yescharacteristic
509C23C2×D4×D6112yes—
510C23C22×C22⋊D6111yescharacteristic
511C23C22×C22⋊D6111yescharacteristic
512C23C22×C22⋊D6111yescharacteristic
513C23C22×Dic12211yescharacteristic
514C23#41×D6112yes—
515C23C2×#1175822211yescharacteristic
516C23C2×#1175822211yescharacteristic
517C23#1451982212yes—
518C23#1451982412yes—
519D4—232no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—28no—
529C2×C4—432no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—618no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1224no—
541C2×C4—1224no—
542C2×C4—1228no—
543C2×C4—1224no—
544C2×C4—1224no—
545C2×C4—1224no—
546C2×C4—1224no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2444no—
551C2×C4—2444no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4—2444no—
557C2×C4—2442no—
558C2×C4—2442no—
559C2×C4C2×#1175819614yes—
560C2×C4C4×C22⋊D6114yes—
561C2×C4C4×C22⋊D6114yes—
562C2×C4#1451974614yes—
563C2×C4#1451975614yes—
564C2×C4#1451975614yes—
565C2×C4#1451976014yes—
566C2×C4#1451977314yes—
567C22—216no—
568C22—28no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—48no—
574C22—44no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22C22×#1175819611yescharacteristic
579C22C4×D4×D6112yes—
580C22C2×C4×C22⋊D6111yescharacteristic
581C22C2×C4×C22⋊D6111yescharacteristic
582C22#1684024112yes—
583C22C2×#1451974611yescharacteristic
584C22C2×#1451975611yescharacteristic
585C22C2×#1451975611yescharacteristic
586C22C2×#1451976011yescharacteristic
587C22C2×#1451977311yescharacteristic
588C22#1684030811yescharacteristic
589C22#1684030912yes—
590C22#1684032911yescharacteristic
591C22#1684033311yescharacteristic
592C22#1684037712yes—
593C22D4×C22⋊D6112yes—
594C22D4×C22⋊D6112yes—
595C22#1684041112yes—
596C22#1684041712yes—
597C22#1684041912yes—
598C22D4×Dic12212yes—
599C22#1684046212yes—
600C22#1684047112yes—
601C4—24no—
602C4—42no—
603C4—44no—
604C4—1228no—
605C4—4882no—
606C4—2444no—
607C4—2444no—
608C4#1683963518yescomplemented
609C2—48no—
610C2C2×#1683963511yescharacteristic
611C2C4×#1451977311yescharacteristic
612C2#2144853111yescharacteristic
613C2D4×#1175819612yes—
614C2#2144883112yes—
615C2#2144932712yes—
616C2#2144944311yescharacteristic
617C2#2144945211yescharacteristic
618C2#2144966312yes—
619C2#2145077211yescharacteristic
620C2#2145101511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×C4×D4×C61C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC22×C4×D4D61
hypocenterC61#13977
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
3. C2×#16839660C2semidirect
4. C2×#16839635C2semidirect
5. C2×#16839631C2semidirect
6. #21444062C2semidirect
7. #21444048C2semidirect
8. C4×#14519684C2semidirect
19. #16839665C4semidirect
44. C2×C4⋊Dic61C2×C4semidirect
55. C2×C22⋊Dic61C2×C4semidirect
61. C2×C4×Dic61C2×C4semidirect
65. #14519684D4semidirect
111. C2×D4×C61C4⋊C4semidirect
255. C2×Dic61C4×D4semidirect
294. C4×C61#385semidirect
345. C61#13977semidirect
422. C2×D4#11758196semidirect
608. C4#16839635semidirect

Maximal subgroups

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

subgroupindex
C2×#168396652
C2×#168396602
C2×#168396352
C2×#168396312
#214440622
#214440482
C4×#145196842
C22×C4×D4×C612
#1397761