Finite Group Databasebeta

← #29602002

Subgroups — #29602002

Order 15616.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D61C211yesmaximal, characteristic
2C2×#16839627C211yesmaximal, characteristic, complemented
3C2×#16839619C211yesmaximal, characteristic
4C2×C4×#11758198C211yesmaximal, characteristic
5#21444077C214yesmaximal, complemented
6#21444049C211yesmaximal, characteristic, complemented
7#21444048C211yesmaximal, characteristic, complemented
8#21444040C214yesmaximal
9C2×C61×#382C211yesmaximal, characteristic, complemented
10C23×C4×D61C2212yes—
11C23×C4×D61C2211yescharacteristic
12C22×C4⋊Dic61C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C2212yes—
15C2×C4⋊C4×D61—24no—
16C2×C4⋊C4×D61—24no—
17C2×C4⋊C4×D61—24no—
18C22×#11758198C2212yes—
19C22×#11758198C2212yes—
20C22×C4×Dic61C2211yescharacteristic
21#16839627—24no—
22#16839619—24no—
23C4×#11758198C418yescomplemented
24C2×#14519684C2214yes—
25C2×#14519684C2212yes—
26C2×#14519684C2214yes—
27C2×#14519684C2214yes—
28C2×#14519684C2212yes—
29C22×C4⋊C4×C61C2211yescharacteristic
30C22×C4⋊C4×C61C2211yescharacteristic
31C22×C42×C61C2211yescharacteristic
32C61×#382—24no—
33C2×#95×C61C2214yes—
34C24×D61C2311yescharacteristic
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38C23×Dic61C2311yescharacteristic
39C22×C4×D61—22no—
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—22no—
43C22×C4×D61—24no—
44C22×C4×D61—24no—
45C22×C4×D61—24no—
46C22×C4×D61—28no—
47C22×C4×D61—44no—
48C22×C4×D61C2×C414yes—
49C22×C4×D61D414yes—
50C22×C4×D61Q814yes—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—22no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61D414yes—
57C2×#11758196—24no—
58C2×#11758196—24no—
59C2×#11758196—24no—
60C4⋊C4×D61—232no—
61C2×#11758198—24no—
62C2×#11758198—24no—
63C2×#11758198—24no—
64C2×#11758198—28no—
65C2×#11758198C2×C418yes—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61C2×C414yescomplemented
69#14519684—24no—
70#14519684—28no—
71#14519684—44no—
72#14519684—44no—
73#14519684C2×C418yescomplemented
74C23×C4×C61C2312yes—
75C23×C4×C61C2312yes—
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61—22no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61D414yes—
84C2×C42×C61—22no—
85C2×C42×C61—22no—
86C2×C42×C61C2×C414yescomplemented
87#95×C61—28no—
88C24×C61C2411yescharacteristic
89C23×D61—21no—
90C23×D61—21no—
91C23×D61—21no—
92C23×D61—21no—
93C23×D61—22no—
94C23×D61—22no—
95C23×D61C22×C412yes—
96C23×D61C2×D412yes—
97C23×D61C2×Q812yes—
98C23×D61#4114yes—
99C23×D61#4114yes—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—24no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—24no—
108C22×C4×C61—22no—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—24no—
114C22×C4×C61—24no—
115C22×C4×C61C22×C414yes—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C2×D414yes—
118C22×C4×C61C2×D412yes—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2×D412yes—
121C22×C4×C61C2×Q812yes—
122C22×C4×C61#4112yes—
123C22×C4×C61#4112yes—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C22×Dic61—24no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—24no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—22no—
142C22×Dic61—24no—
143C22×Dic61C22×C414yes—
144C22×Dic61C22×C412yes—
145C22×Dic61C2×D414yes—
146C22×Dic61C2×D412yes—
147C22×Dic61C2×Q812yes—
148C2×C4×D61—28no—
149C2×C4×D61—28no—
150C2×C4×D61—28no—
151C2×C4×D61—28no—
152C2×C4×D61—28no—
153C2×C4×D61—28no—
154C2×C4×D61—216no—
155C2×C4×D61—416no—
156C2×C4×D61—416no—
157C2×C4×D61—48no—
158C2×C4×D61—48no—
159C2×C4×D61C4⋊C4116yes—
160C4⋊C4×C61—28no—
161C4⋊C4×C61—44no—
162C42×C61—28no—
163#11758198—216no—
164#11758198—48no—
165C4⋊Dic61—28no—
166C4⋊Dic61—44no—
167#11758196—216no—
168C4×Dic61—28no—
169C23×C61C23×C411yescharacteristic
170C23×C61C22×D411yescharacteristic
171C23×C61C22×D411yescharacteristic
172C23×C61C22×D411yescharacteristic
173C23×C61C22×Q811yescharacteristic
174C23×C61C2×#4112yes—
175C23×C61C2×#4112yes—
176C23×C61C2×#4111yescharacteristic
177C23×C61C2×#4111yescharacteristic
178C23×C61H(5,𝔽2)12yes—
179C23×C612−1+412yes—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—24no—
183C22×D61—22no—
184C22×D61—22no—
185C22×D61—22no—
186C22×D61—22no—
187C22×D61—22no—
188C22×D61—22no—
189C22×D61—24no—
190C22×D61—24no—
191C22×D61—24no—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—24no—
196C22×D61—24no—
197C22×D61—24no—
198C22×D61C2×C4⋊C414yes—
199C22×D61#11718yes—
200C22×D61#12414yes—
201C22×D61#12414yes—
202C22×D61#12518yes—
203C2×C4×C61—24no—
204C2×C4×C61—24no—
205C2×C4×C61—24no—
206C2×C4×C61—24no—
207C2×C4×C61—28no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—44no—
219C2×C4×C61—44no—
220C2×C4×C61C2×C4⋊C414yes—
221C2×C4×C61C4×D414yes—
222C2×C4×C61C4×D414yes—
223C2×C4×C61#12014yes—
224C2×C4×C61C22⋊Q814yes—
225C2×C4×C61C22⋊Q814yes—
226C2×C4×C61#12314yes—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—22no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—24no—
236C2×Dic61—24no—
237C2×Dic61—28no—
238C2×Dic61—42no—
239C2×Dic61—42no—
240C2×Dic61—44no—
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×C4⋊C414yes—
251C4×D61—232no—
252C4×D61—432no—
253#13881—611nomaximal, Sylow
254C22×C61C22×C4⋊C411yescharacteristic
255C22×C61C2×#11712yes—
256C22×C61C2×C4×D411yescharacteristic
257C22×C61C2×C4×D411yescharacteristic
258C22×C61#52012yes—
259C22×C61C2×#12011yescharacteristic
260C22×C61C2×C22⋊Q811yescharacteristic
261C22×C61C2×C22⋊Q811yescharacteristic
262C22×C61C2×#12311yescharacteristic
263C22×C61C2×#12411yescharacteristic
264C22×C61C2×#12411yescharacteristic
265C22×C61C2×#12512yes—
266C22×C61#53511yescharacteristic
267C22×C61#53611yescharacteristic
268C22×C61#54412yes—
269C22×C61#54612yes—
270C22×C61#54612yes—
271C22×C61#54812yes—
272C22×C61#54812yes—
273C22×C61#55212yes—
274C22×C61#55212yes—
275C22×C61#55312yes—
276C22×C61#55312yes—
277C2×D61—24no—
278C2×D61—28no—
279C2×D61—28no—
280C2×D61—24no—
281C2×D61—24no—
282C2×D61—28no—
283C2×D61—28no—
284C2×D61—24no—
285C2×D61—24no—
286C2×D61—24no—
287C2×D61#38218yescomplemented
288C4×C61—44no—
289C4×C61—44no—
290C4×C61—82no—
291C4×C61#38518yes—
292Dic61—24no—
293Dic61—44no—
294Dic61—44no—
295Dic61—82no—
296C23×C4⋊C4—611no—
297C2×#382—611no—
298C2×#382D6111yescharacteristic, complemented
299C2×#380—611no—
300C2×C4×C22⋊C4—611no—
301#1442—614no—
302C4⋊#95—611no—
303C4⋊#95—611no—
304C22⋊#95—614no—
305C2×C61C2×#38211yescharacteristic
306C2×C61C2×#38511yescharacteristic
307C2×C61#231711yescharacteristic
308C2×C61#235212yes—
309C2×C61#235212yes—
310C2×C61#243511yescharacteristic
311C2×C61#243611yescharacteristic
312C2×C61#244212yes—
313C2×C61#244212yes—
314C2×C61#250311yescharacteristic
315C2×C61#250411yescharacteristic
316D61—28no—
317C24×C4—611no—
318C24×C4—612no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—612no—
321C22×C4⋊C4—611no—
322C22×C4⋊C4—1224no—
323C22×C4⋊C4—1224no—
324C22×C4⋊C4—1224no—
325C22×C4⋊C4C2×D6111yescharacteristic
326C22×C4⋊C4C2×D6111yescharacteristic
327C22×C22⋊C4—612no—
328C22×C22⋊C4—612no—
329C22×C42—611no—
330C22×C42C2×D6111yescharacteristic
331#382—24no—
332#382—1224no—
333#380—1224no—
334C4×C22⋊C4—618no—
335C2×#95—612no—
336C2×#95—614no—
337C2×#95—612no—
338C2×#95—614no—
339C2×#95—614no—
340C2×#95C2×D6114yes—
341C61#1388111yescharacteristic, complemented, Sylow
342C25—611no—
343C23×C4—611no—
344C23×C4—614no—
345C23×C4—614no—
346C23×C4—614no—
347C23×C4—612no—
348C23×C4—612no—
349C23×C4—612no—
350C23×C4—1224no—
351C23×C4—1224no—
352C23×C4—1228no—
353C23×C4—1222no—
354C23×C4—1222no—
355C23×C4—1222no—
356C23×C4—1222no—
357C23×C4—1224no—
358C23×C4—2444no—
359C23×C4C22×D6112yes—
360C23×C4C22×D6112yes—
361C23×C4C22×D6112yes—
362C23×C4C22×D6111yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—614no—
369C2×C4⋊C4—12232no—
370C2×C4⋊C4—1224no—
371C2×C4⋊C4—1224no—
372C2×C4⋊C4—1222no—
373C2×C4⋊C4—1224no—
374C2×C4⋊C4—1222no—
375C2×C4⋊C4—1222no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4C22⋊D6114yes—
379C2×C22⋊C4—618no—
380C2×C22⋊C4—1224no—
381C2×C22⋊C4—1224no—
382C2×C22⋊C4—1224no—
383C2×C22⋊C4—1228no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—614no—
387C2×C42—1222no—
388C2×C42—1222no—
389C2×C42C4×D6114yescomplemented
390#95—28no—
391#95—618no—
392#95—1224no—
393#95—1228no—
394#95—2444no—
395#95—2444no—
396C24—612no—
397C24—612no—
398C24—612no—
399C24—614no—
400C24—614no—
401C24—1221no—
402C24—1221no—
403C24—1221no—
404C24—1221no—
405C24—1222no—
406C24—1222no—
407C24C23×D6111yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—612no—
424C22×C4—612no—
425C22×C4—612no—
426C22×C4—6116no—
427C22×C4—614no—
428C22×C4—614no—
429C22×C4—12216no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1221no—
433C22×C4—1221no—
434C22×C4—1228no—
435C22×C4—1228no—
436C22×C4—1228no—
437C22×C4—1228no—
438C22×C4—1228no—
439C22×C4—1228no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1224no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1224no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1224no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1224no—
455C22×C4—24416no—
456C22×C4—24416no—
457C22×C4—2448no—
458C22×C4—2448no—
459C22×C4C2×C4×D6114yes—
460C22×C4C2×C4×D6112yes—
461C22×C4#1175822212yes—
462C22×C4D4×D6112yes—
463C22×C4D4×D6112yes—
464C22×C4Q8×D6112yes—
465C22×C4#1175822612yes—
466C22×C4C2×C22⋊D6114yes—
467C22×C4C2×C22⋊D6112yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—12216no—
471C4⋊C4—1228no—
472C4⋊C4—2444no—
473C22⋊C4—12216no—
474C22⋊C4—2448no—
475C42—28no—
476C42—1228no—
477C23—614no—
478C23—618no—
479C23—614no—
480C23—614no—
481C23—618no—
482C23—1224no—
483C23—1224no—
484C23—1224no—
485C23—1224no—
486C23—1222no—
487C23—1222no—
488C23—1224no—
489C23—1222no—
490C23—1222no—
491C23—1222no—
492C23—1222no—
493C23—1222no—
494C23—1222no—
495C23—1224no—
496C23—1224no—
497C23—1224no—
498C23—1224no—
499C23—1224no—
500C23C22×C4×D6111yescharacteristic
501C23C2×D4×D6111yescharacteristic
502C23C2×D4×D6111yescharacteristic
503C23C22×C22⋊D6111yescharacteristic
504C23C2×Q8×D6111yescharacteristic
505C23#41×D6112yes—
506C23#41×D6112yes—
507C23C2×#1175822211yescharacteristic
508C23C2×#1175822611yescharacteristic
509C23#1451982312yes—
510C23#1451982412yes—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—614no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—12232no—
536C2×C4—1224no—
537C2×C4—1224no—
538C2×C4—1224no—
539C2×C4—1224no—
540C2×C4—1228no—
541C2×C4—24432no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4—2442no—
545C2×C4—2442no—
546C2×C4—2442no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2444no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4C4⋊C4×D6114yes—
555C2×C4C4×C22⋊D6114yes—
556C2×C4#1451973914yes—
557C2×C4#1451974514yes—
558C2×C4#1451975314yes—
559C2×C4#1451975514yes—
560C2×C4#1451976214yes—
561C22—618no—
562C22—1224no—
563C22—1224no—
564C22—1228no—
565C22—1228no—
566C22—1224no—
567C22—1228no—
568C22—1224no—
569C22—1228no—
570C22—1224no—
571C22—1224no—
572C22C2×C4⋊C4×D6111yescharacteristic
573C22#117×D6112yes—
574C22C2×C4×C22⋊D6111yescharacteristic
575C22C2×#1451973911yescharacteristic
576C22#1684024212yes—
577C22C2×#1451974511yescharacteristic
578C22C2×#1451975311yescharacteristic
579C22C2×#1451975511yescharacteristic
580C22C2×#1451976211yescharacteristic
581C22#124×D6111yescharacteristic
582C22#124×D6111yescharacteristic
583C22#125×D6112yes—
584C22#1684033011yescharacteristic
585C22#1684033211yescharacteristic
586C22#1684039212yes—
587C22#1684040012yes—
588C22#1684040512yes—
589C22#1684041912yes—
590C22#1684042012yes—
591C22#1684045012yes—
592C22#1684045112yes—
593C22#1684045612yes—
594C22#1684046512yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—1224no—
599C4—4882no—
600C4—2444no—
601C4—2444no—
602C4#1683963318yes—
603C2—1228no—
604C2#382×D6111yescharacteristic
605C2C2×#1683963311yescharacteristic
606C2#2144844411yescharacteristic
607C2#2144874512yes—
608C2#2144874912yes—
609C2#2144946711yescharacteristic
610C2#2144947411yescharacteristic
611C2#2144953912yes—
612C2#2144954612yes—
613C2#2145030111yescharacteristic
614C2#2145030811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#382C2
socle, cosocleC24×C61C24
prime radicalC24×D61C23
hypercenterC2×#382D61
hypocenterC61#13881
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
2. C2×#16839627C2semidirect
5. #21444077C2semidirect
6. #21444049C2semidirect
7. #21444048C2semidirect
9. C2×C61×#382C2semidirect
23. C4×#11758198C4semidirect
68. C2×C4×Dic61C2×C4semidirect
73. #14519684C2×C4semidirect
86. C2×C42×C61C2×C4semidirect
287. C2×D61#382semidirect
298. C2×#382D61semidirect
341. C61#13881semidirect
389. C2×C42C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D612
C2×#168396272
C2×#168396192
C2×C4×#117581982
#214440772
#214440492
#214440482
#214440402
C2×C61×#3822
#1388161