Finite Group Databasebeta

← #29602257

Subgroups — #29602257

Order 15616.

2574 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519739C211yesmaximal, characteristic
2C2×#16839664C211yesmaximal, characteristic
3C2×#16839663C211yesmaximal, characteristic
4C2×#16839631C211yesmaximal, characteristic, complemented
5C2×#16839621C214yesmaximal, complemented
6C2×C4×#11758196C211yesmaximal, characteristic, complemented
7#21444063C214yesmaximal, complemented
8#21444048C211yesmaximal, characteristic, complemented
9C61×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C23×C4×D61C2212yes—
12C2×#14519739—24no—
13C2×#14519739—24no—
14C2×#14519739C418yescomplemented
15C22×C4⋊Dic61C2211yescharacteristic
16C22×#11758196C2212yes—
17C22×#11758196C2212yes—
18C22×#11758196C2212yes—
19C22×#11758198C2214yes—
20C22×#11758198C2214yes—
21C22×#11758198C2212yes—
22C22×C4×Dic61C2211yescharacteristic
23C22×C4×Dic61C2211yescharacteristic
24#16839664—24no—
25#16839663—24no—
26#16839631—24no—
27#16839621—216no—
28C4×#11758196—24no—
29C2×#14519684C2214yes—
30C2×#14519684C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C2×#95×C61C2214yes—
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×Dic61C2311yescharacteristic
46C23×Dic61C2311yescharacteristic
47C22×C4×D61—24no—
48C22×C4×D61—24no—
49C22×C4×D61—28no—
50C22×C4×D61—28no—
51C22×C4×D61C2×C418yes—
52#14519739—232no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×#11758196—24no—
57C2×#11758196—24no—
58C2×#11758196—28no—
59C2×#11758196—24no—
60C2×#11758196—24no—
61C2×#11758196—44no—
62C2×#11758196D418yescomplemented
63C2×#11758198—24no—
64C2×#11758198—24no—
65C2×#11758198—28no—
66C2×#11758198—28no—
67C2×#11758198—28no—
68C2×#11758198—28no—
69C2×#11758198—44no—
70C2×#11758198—44no—
71C2×#11758198C2×C418yes—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—22no—
75C2×C4×Dic61C2×C414yes—
76C2×C4×Dic61D414yes—
77C2×C4×Dic61D414yes—
78#14519684—24no—
79#14519684—28no—
80C23×C4×C61C2312yes—
81C23×C4×C61C2312yes—
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—22no—
88C2×C4⋊C4×C61—22no—
89C2×C4⋊C4×C61C2×C414yes—
90C2×C42×C61—22no—
91C2×C42×C61—22no—
92C2×C42×C61—22no—
93#95×C61—28no—
94C24×C61C2411yescharacteristic
95C23×D61—22no—
96C23×D61—22no—
97C23×D61—24no—
98C23×D61—24no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—24no—
102C23×D61—24no—
103C23×D61C22×C414yes—
104C23×D61C2×D414yes—
105C23×D61C2×Q814yes—
106C22×C4×C61—24no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—24no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—24no—
121C22×C4×C61C22×C414yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C2×D414yes—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×Q812yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—24no—
139C22×Dic61—22no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61C22×C412yes—
144C22×Dic61C22×C412yes—
145C22×Dic61C2×D414yes—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×D412yes—
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150C22×Dic61C2×D414yes—
151C22×Dic61#4114yes—
152C2×D244—28no—
153C2×D244—28no—
154C2×D244—28no—
155C2×D244—28no—
156C2×D244—28no—
157C2×D244—28no—
158C2×D244C4⋊C4116yescomplemented
159C2×C4×D61—216no—
160C2×C4×D61—232no—
161C2×C4×D61—416no—
162C2×C4×D61—48no—
163C2×C4×D61—48no—
164C4⋊C4×C61—28no—
165C4⋊C4×C61—44no—
166C42×C61—44no—
167#11758198—216no—
168#11758198—416no—
169#11758198—416no—
170C4⋊Dic61—44no—
171#11758196—216no—
172#11758196—48no—
173#11758196—48no—
174C4×Dic61—44no—
175C4×Dic61C22⋊C4116yescomplemented
176C23×C61C23×C411yescharacteristic
177C23×C61C22×D411yescharacteristic
178C23×C61C22×D411yescharacteristic
179C23×C61C22×D411yescharacteristic
180C23×C61C22×D412yes—
181C23×C61C22×Q811yescharacteristic
182C23×C61C2×#4111yescharacteristic
183C23×C61C2×#4111yescharacteristic
184C23×C61C2×#4112yes—
185C23×C61H(5,𝔽2)12yes—
186C23×C612−1+412yes—
187C22×D61—24no—
188C22×D61—24no—
189C22×D61—28no—
190C22×D61—28no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—28no—
196C22×D61—28no—
197C22×D61—28no—
198C22×D61—28no—
199C22×D61—44no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—42no—
203C22×D61—42no—
204C22×D61—44no—
205C22×D61—44no—
206C22×D61—44no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61C2×C4⋊C418yes—
210C2×C4×C61—24no—
211C2×C4×C61—28no—
212C2×C4×C61—24no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—42no—
216C2×C4×C61—44no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
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×C22⋊C414yes—
228C2×C4×C61C2×C4⋊C414yes—
229C2×C4×C61#11714yes—
230C2×C4×C61#12114yes—
231C2×C4×C61#12114yes—
232C2×C4×C61C22⋊Q814yes—
233C2×C4×C61C22⋊Q814yes—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—24no—
238C2×Dic61—24no—
239C2×Dic61—24no—
240C2×Dic61—24no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—42no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249C2×Dic61—44no—
250C2×Dic61C2×C22⋊C414yes—
251C2×Dic61C2×C22⋊C414yes—
252C2×Dic61C4×D418yes—
253C2×Dic61#12118yes—
254C2×Dic61#12414yes—
255C2×Dic61#12714yes—
256D244—232no—
257C4×D61—432no—
258#13906—611nomaximal, Sylow
259C22×C61C22×C22⋊C411yescharacteristic
260C22×C61C22×C4⋊C411yescharacteristic
261C22×C61C2×#11711yescharacteristic
262C22×C61C2×C4×D412yes—
263C22×C61#52012yes—
264C22×C61C2×#12112yes—
265C22×C61C2×#12111yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×C22⋊Q811yescharacteristic
268C22×C61C2×C22⋊Q811yescharacteristic
269C22×C61C2×#12411yescharacteristic
270C22×C61C2×#12711yescharacteristic
271C22×C61#53511yescharacteristic
272C22×C61#53611yescharacteristic
273C22×C61#53712yes—
274C22×C61D4212yes—
275C22×C61#54712yes—
276C22×C61#54712yes—
277C22×C61D4×Q812yes—
278C22×C61#55012yes—
279C22×C61#55412yes—
280C22×C61#55512yes—
281C22×C61#55612yes—
282C2×D61—216no—
283C2×D61—28no—
284C2×D61—44no—
285C2×D61—44no—
286C2×D61—48no—
287C2×D61—48no—
288C2×D61—48no—
289C2×D61—44no—
290C2×D61—44no—
291C2×D61—44no—
292C2×D61—44no—
293C4×C61—44no—
294C4×C61—44no—
295C4×C61—82no—
296C4×C61#38018yescomplemented
297Dic61—24no—
298Dic61—44no—
299Dic61—82no—
300Dic61#39018yescomplemented
301C22×C4×D4—611no—
302C2×#390—611no—
303C2×#390—611no—
304C2×#384—611no—
305C2×#380—614no—
306C2×C4×C4⋊C4—611no—
307#1439—614no—
308C4⋊#95—611no—
309C4⋊#95D6111yescharacteristic, complemented
310C2×C61C2×#38011yescharacteristic
311C2×C61C2×#39011yescharacteristic
312C2×C61#231111yescharacteristic
313C2×C61D4×C4⋊C412yes—
314C2×C61#235412yes—
315C2×C61#253712yes—
316C2×C61#253812yes—
317C2×C61#254111yescharacteristic
318C2×C61#254211yescharacteristic
319C2×C61#254711yescharacteristic
320C2×C61#256311yescharacteristic
321D61—48no—
322C23×D4—611no—
323C24×C4—612no—
324C2×C4×D4—618no—
325C2×C4×D4—1224no—
326C2×C4×D4—1224no—
327C22×C4⋊C4—612no—
328C22×C4⋊C4—612no—
329C22×C4⋊C4—611no—
330C22×C4⋊C4—612no—
331C22×C4⋊C4C2×D6111yescharacteristic
332C22×C4⋊C4C2×D6111yescharacteristic
333C22×C22⋊C4—612no—
334C22×C22⋊C4—614no—
335C22×C22⋊C4—614no—
336C22×C42—611no—
337C22×C42—611no—
338C22×C42C2×D6111yescharacteristic
339#390—1224no—
340#390—1224no—
341#384—1224no—
342#380—12216no—
343C4×C4⋊C4—1224no—
344C2×#95—614no—
345C2×#95—612no—
346C2×#95C2×D6114yes—
347C61#1390611yescharacteristic, complemented, Sylow
348C25—612no—
349C22×D4—614no—
350C22×D4—614no—
351C22×D4—614no—
352C22×D4—1222no—
353C22×D4—1222no—
354C22×D4—1222no—
355C22×D4—1222no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—612no—
359C23×C4—618no—
360C23×C4—612no—
361C23×C4—1224no—
362C23×C4—1224no—
363C23×C4—1228no—
364C23×C4—1228no—
365C23×C4C22×D6112yes—
366C23×C4C22×D6112yes—
367C23×C4C22×D6112yes—
368C23×C4C22×D6111yescharacteristic
369C4×D4—12232no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—618no—
376C2×C4⋊C4—1224no—
377C2×C4⋊C4—1224no—
378C2×C4⋊C4—1228no—
379C2×C4⋊C4—1224no—
380C2×C4⋊C4—1224no—
381C2×C4⋊C4—1222no—
382C2×C4⋊C4—1222no—
383C2×C4⋊C4—1222no—
384C2×C4⋊C4—2444no—
385C2×C4⋊C4C4×D6114yes—
386C2×C22⋊C4—618no—
387C2×C22⋊C4—1224no—
388C2×C22⋊C4—1224no—
389C2×C22⋊C4—1228no—
390C2×C22⋊C4—1228no—
391C2×C22⋊C4—1228no—
392C2×C22⋊C4—1228no—
393C2×C22⋊C4—2444no—
394C2×C22⋊C4—2444no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—614no—
399C2×C42—614no—
400C2×C42—614no—
401C2×C42—1222no—
402C2×C42—1222no—
403C2×C42—1222no—
404#95—28no—
405#95—1224no—
406#95—1228no—
407C24—614no—
408C24—614no—
409C24—614no—
410C24—1224no—
411C24—1222no—
412C24—1222no—
413C24—1224no—
414C24—1222no—
415C24—1222no—
416C24—1224no—
417C24—1224no—
418C24C23×D6111yescharacteristic
419C2×D4—6116no—
420C2×D4—1228no—
421C2×D4—1228no—
422C2×D4—1228no—
423C2×D4—1228no—
424C2×D4—1228no—
425C2×D4—1228no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—612no—
444C22×C4—614no—
445C22×C4—614no—
446C22×C4—612no—
447C22×C4—612no—
448C22×C4—612no—
449C22×C4—614no—
450C22×C4—12216no—
451C22×C4—12232no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1224no—
461C22×C4—1222no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—1224no—
465C22×C4—24416no—
466C22×C4—2448no—
467C22×C4—2448no—
468C22×C4C2×C4×D6114yes—
469C22×C4C2×C4×D6112yes—
470C22×C4#1175822212yes—
471C22×C4#1175822212yes—
472C22×C4D4×D6114yes—
473C22×C4D4×D6112yes—
474C22×C4D4×D6112yes—
475C22×C4D4×D6112yes—
476C22×C4Q8×D6112yes—
477C4⋊C4—28no—
478C4⋊C4—44no—
479C4⋊C4—12216no—
480C4⋊C4—2448no—
481C4⋊C4—2448no—
482C4⋊C4—2444no—
483C22⋊C4—12216no—
484C22⋊C4—24416no—
485C22⋊C4—24416no—
486C42—44no—
487C42—6116no—
488C42—2444no—
489C23—618no—
490C23—1224no—
491C23—1228no—
492C23—1224no—
493C23—1228no—
494C23—1228no—
495C23—1224no—
496C23—1224no—
497C23—1224no—
498C23—1224no—
499C23—1228no—
500C23—1228no—
501C23—1228no—
502C23—2444no—
503C23—2444no—
504C23—2444no—
505C23—2442no—
506C23—2442no—
507C23—2442no—
508C23—2442no—
509C23—2444no—
510C23—2444no—
511C23—2444no—
512C23C22×C4×D6111yescharacteristic
513C23C2×D4×D6111yescharacteristic
514C23C2×D4×D6111yescharacteristic
515C23C2×D4×D6111yescharacteristic
516C23C2×D4×D6112yes—
517C23C2×Q8×D6111yescharacteristic
518C23#41×D6112yes—
519C23C2×#1175822211yescharacteristic
520C23C2×#1175822211yescharacteristic
521C23#1451982312yes—
522C23#1451982512yes—
523D4—12232no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—614no—
542C2×C4—618no—
543C2×C4—618no—
544C2×C4—614no—
545C2×C4—614no—
546C2×C4—614no—
547C2×C4—1224no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1224no—
552C2×C4—1224no—
553C2×C4—1224no—
554C2×C4—24432no—
555C2×C4—2442no—
556C2×C4—2442no—
557C2×C4—2442no—
558C2×C4—2444no—
559C2×C4—2444no—
560C2×C4—2442no—
561C2×C4—2442no—
562C2×C4—2442no—
563C2×C4—2442no—
564C2×C4C22⋊C4×D6114yes—
565C2×C4C4⋊C4×D6114yes—
566C2×C4#1451973414yes—
567C2×C4#1451975014yes—
568C2×C4#1451975014yes—
569C2×C4#1451975514yes—
570C2×C4#1451975514yes—
571C22—12216no—
572C22—1228no—
573C22—2444no—
574C22—2444no—
575C22—2448no—
576C22—2444no—
577C22—2444no—
578C22—2444no—
579C22—2448no—
580C22—2444no—
581C22—2448no—
582C22C2×C22⋊C4×D6111yescharacteristic
583C22C2×C4⋊C4×D6111yescharacteristic
584C22C2×#1451973411yescharacteristic
585C22C4×D4×D6112yes—
586C22#1684024412yes—
587C22#121×D6112yes—
588C22C2×#1451975011yescharacteristic
589C22C2×#1451975011yescharacteristic
590C22C2×#1451975511yescharacteristic
591C22C2×#1451975511yescharacteristic
592C22#124×D6111yescharacteristic
593C22#127×D6111yescharacteristic
594C22#1684033011yescharacteristic
595C22#1684033711yescharacteristic
596C22#1684034212yes—
597C22#1684039612yes—
598C22#1684041312yes—
599C22#1684041412yes—
600C22#1684042712yes—
601C22#1684043312yes—
602C22#1684046812yes—
603C22#1684047512yes—
604C22#1684048412yes—
605C4—44no—
606C4—44no—
607C4—82no—
608C4—618no—
609C4—1224no—
610C4—4882no—
611C4—2444no—
612C4#1683962118yescomplemented
613C2—2448no—
614C2C2×#1683962111yescharacteristic
615C2#390×D6111yescharacteristic
616C2#2144841011yescharacteristic
617C2#2144872012yes—
618C2#2144876412yes—
619C2#2145073412yes—
620C2#2145074012yes—
621C2#2145076411yescharacteristic
622C2#2145077111yescharacteristic
623C2#2145081611yescharacteristic
624C2#2145099111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC61×C4⋊#95C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenterC4⋊#95D61
hypocenterC61#13906
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#16839631C2semidirect
5. C2×#16839621C2semidirect
6. C2×C4×#11758196C2semidirect
7. #21444063C2semidirect
8. #21444048C2semidirect
9. C61×C4⋊#95C2semidirect
14. C2×#14519739C4semidirect
62. C2×#11758196D4semidirect
158. C2×D244C4⋊C4semidirect
175. C4×Dic61C22⋊C4semidirect
296. C4×C61#380semidirect
300. Dic61#390semidirect
309. C4⋊#95D61semidirect
347. C61#13906semidirect
612. C4#16839621semidirect

Maximal subgroups

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

subgroupindex
C22×#145197392
C2×#168396642
C2×#168396632
C2×#168396312
C2×#168396212
C2×C4×#117581962
#214440632
#214440482
C61×C4⋊#952
#1390661