Finite Group Databasebeta

← #29601989

Subgroups — #29601989

Order 15616.

2006 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519736C211yesmaximal, characteristic
2C2×#16839632C211yesmaximal, characteristic, complemented
3C2×#16839628C211yesmaximal, characteristic, complemented
4C2×#16839622C211yesmaximal, characteristic
5#21444064C214yesmaximal
6#21444053C214yesmaximal, complemented
7#21444049C211yesmaximal, characteristic, complemented
8#21444049C211yesmaximal, characteristic, complemented
9C2×C4×C22⋊C4×C61C211yesmaximal, characteristic
10C2×#14519736—24no—
11C2×#14519736—24no—
12C2×#14519736—24no—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2214yes—
16C22×C4⋊Dic61C2211yescharacteristic
17C22×C4⋊Dic61C2211yescharacteristic
18C22×C22⋊Dic61C2212yes—
19C22×C22⋊Dic61C2212yes—
20C22×C4×Dic61C2212yes—
21#16839632—24no—
22#16839628—24no—
23#16839622—24no—
24C2×#14519684C2212yes—
25C2×#14519684C2214yes—
26C2×#14519684C2214yes—
27C2×#14519684C2214yes—
28C24×C4×C61C2211yescharacteristic
29C22×C22⋊C4×C61C2212yes—
30C22×C42×C61C2211yescharacteristic
31C22×C42×C61C2211yescharacteristic
32C4×C22⋊C4×C61C418yescomplemented
33C2×#95×C61C2212yes—
34C23×Dic61C2312yes—
35C23×Dic61C2312yes—
36C23×Dic61C2312yes—
37C23×Dic61C2312yes—
38#14519736—232no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—22no—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—22no—
47C2×C4⋊Dic61—28no—
48C2×C4⋊Dic61—22no—
49C2×C4⋊Dic61—22no—
50C2×C4⋊Dic61—28no—
51C2×C4⋊Dic61—44no—
52C2×C4⋊Dic61D414yes—
53C2×C4⋊Dic61D414yes—
54C2×C22⋊Dic61—24no—
55C2×C22⋊Dic61—24no—
56C2×C22⋊Dic61—24no—
57C2×C22⋊Dic61—24no—
58C2×C22⋊Dic61—44no—
59C2×C4×Dic61—24no—
60C2×C4×Dic61—24no—
61C2×C4×Dic61—24no—
62#14519684—24no—
63#14519684—28no—
64#14519684—28no—
65#14519684—44no—
66C25×C61C2311yescharacteristic
67C23×C4×C61—22no—
68C23×C4×C61—22no—
69C23×C4×C61—22no—
70C23×C4×C61—22no—
71C23×C4×C61C2×C414yes—
72C23×C4×C61D414yes—
73C23×C4×C61Q814yes—
74C23×C4×C61C2312yes—
75C23×C4×C61C2311yescharacteristic
76C23×C4×C61C2312yes—
77C23×C4×C61C2311yescharacteristic
78C2×C22⋊C4×C61—24no—
79C2×C22⋊C4×C61—24no—
80C2×C22⋊C4×C61C2×C418yes—
81C2×C42×C61—22no—
82C2×C42×C61—22no—
83C2×C42×C61—22no—
84C2×C42×C61—22no—
85C2×C42×C61C2×C414yes—
86C2×C42×C61C2×C414yes—
87#95×C61C2×C418yes—
88C24×C61—21no—
89C24×C61—21no—
90C24×C61—22no—
91C24×C61—22no—
92C24×C61—21no—
93C24×C61—21no—
94C24×C61—22no—
95C24×C61—22no—
96C24×C61C22×C412yes—
97C24×C61C2×D412yes—
98C24×C61C2×Q812yes—
99C24×C61C2411yescharacteristic
100C22×C4×C61—22no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—28no—
109C22×C4×C61—28no—
110C22×C4×C61—28no—
111C22×C4×C61—28no—
112C22×C4×C61—28no—
113C22×C4×C61—28no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61C4⋊C4116yescomplemented
121C22×C4×C61C22×C414yes—
122C22×C4×C61C22×C412yes—
123C22×C4×C61C22×C414yes—
124C22×C4×C61C22×C412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×Q812yes—
130C22×C4×C61C2×Q812yes—
131C22×C4×C61#4114yes—
132C22×C4×C61#4114yes—
133C22×C4×C61#4112yes—
134C22×C4×C61#4112yes—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—24no—
140C22×Dic61—24no—
141C22×Dic61—24no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—24no—
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—24no—
152C22×Dic61—24no—
153C22×Dic61C2×D414yes—
154C22×Dic61C2×D414yes—
155C22⋊C4×C61—216no—
156C42×C61—28no—
157C42×C61—28no—
158C22⋊Dic61—216no—
159C22⋊Dic61—48no—
160C4⋊Dic61—28no—
161C4⋊Dic61—28no—
162C4⋊Dic61—416no—
163C4⋊Dic61—44no—
164C4⋊Dic61—44no—
165C4×Dic61—216no—
166C23×C61—24no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—24no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—24no—
180C23×C61—42no—
181C23×C61—42no—
182C23×C61—42no—
183C23×C61—42no—
184C23×C61—42no—
185C23×C61—42no—
186C23×C61—42no—
187C23×C61—42no—
188C23×C61C2×C4⋊C414yes—
189C23×C61C23×C411yescharacteristic
190C23×C61C22×D411yescharacteristic
191C23×C61C22×D411yescharacteristic
192C23×C61C22×D411yescharacteristic
193C23×C61C22×Q811yescharacteristic
194C23×C61C2×#4112yes—
195C23×C61C2×#4112yes—
196C23×C61C2×#4111yescharacteristic
197C23×C61C2×#4111yescharacteristic
198C23×C61H(5,𝔽2)12yes—
199C23×C612−1+412yes—
200C2×C4×C61—24no—
201C2×C4×C61—22no—
202C2×C4×C61—22no—
203C2×C4×C61—24no—
204C2×C4×C61—22no—
205C2×C4×C61—22no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—24no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—24no—
212C2×C4×C61—232no—
213C2×C4×C61—24no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61C2×C4⋊C414yes—
221C2×C4×C61C2×C4⋊C414yes—
222C2×C4×C61#11718yescomplemented
223C2×C4×C61C4×D414yes—
224C2×C4×C61C4×D414yes—
225C2×C4×C61#12014yes—
226C2×C4×C61C22⋊Q814yes—
227C2×C4×C61C22⋊Q814yes—
228C2×C4×C61#12314yes—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—28no—
232C2×Dic61—28no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—42no—
236C2×Dic61—42no—
237C2×Dic61—44no—
238C2×Dic61—44no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61—44no—
246C2×Dic61—44no—
247C2×Dic61—44no—
248C2×Dic61—44no—
249#13879—611nomaximal, Sylow
250C22×C61—24no—
251C22×C61—24no—
252C22×C61—24no—
253C22×C61—44no—
254C22×C61—44no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—44no—
258C22×C61—44no—
259C22×C61—42no—
260C22×C61—42no—
261C22×C61C22×C4⋊C411yescharacteristic
262C22×C61C2×#11712yes—
263C22×C61C2×C4×D411yescharacteristic
264C22×C61C2×C4×D411yescharacteristic
265C22×C61#52012yes—
266C22×C61C2×#12011yescharacteristic
267C22×C61C2×C22⋊Q811yescharacteristic
268C22×C61C2×C22⋊Q811yescharacteristic
269C22×C61C2×#12311yescharacteristic
270C22×C61#53211yescharacteristic
271C22×C61#53211yescharacteristic
272C22×C61#53312yes—
273C22×C61#53511yescharacteristic
274C22×C61#53611yescharacteristic
275C22×C61#54412yes—
276C22×C61#54612yes—
277C22×C61#54712yes—
278C22×C61#54812yes—
279C22×C61#55012yes—
280C22×C61#55212yes—
281C22×C61#55312yes—
282C22×C61#55512yes—
283C22×C61#55612yes—
284C4×C61—28no—
285C4×C61—24no—
286C4×C61—44no—
287C4×C61#38518yes—
288Dic61—44no—
289Dic61—44no—
290Dic61—82no—
291Dic61—82no—
292C22×#117—611no—
293C2×#384—611no—
294C2×#382—611no—
295C2×#380—611no—
296C2×C4×C22⋊C4D6111yescharacteristic
297#1439—614no—
298#1436—614no—
299C4⋊#95—611no—
300C4⋊#95—611no—
301C2×C61—44no—
302C2×C61#228511yescharacteristic
303C2×C61C2×#38511yescharacteristic
304C2×C61#231711yescharacteristic
305C2×C61#235212yes—
306C2×C61#235412yes—
307C2×C61#243111yescharacteristic
308C2×C61#243211yescharacteristic
309C2×C61#243912yes—
310C2×C61#244112yes—
311C2×C61#249911yescharacteristic
312C2×C61#250011yescharacteristic
313C24×C4C2×D6111yescharacteristic
314C2×#117—1224no—
315C2×#117—1224no—
316C2×#117—1224no—
317C22×C4⋊C4—611no—
318C22×C4⋊C4—611no—
319C22×C4⋊C4—611no—
320C22×C4⋊C4—611no—
321C22×C4⋊C4—614no—
322C22×C22⋊C4—612no—
323C22×C22⋊C4—612no—
324C22×C22⋊C4C2×D6112yes—
325C22×C42—612no—
326C22×C42C2×D6111yescharacteristic
327C22×C42C2×D6111yescharacteristic
328#384—1224no—
329#382—1224no—
330#380—1224no—
331C4×C22⋊C4Dic6118yescomplemented
332C2×#95—614no—
333C2×#95—612no—
334C2×#95—614no—
335C2×#95—614no—
336C2×#95C2×D6112yes—
337C61#1387911yescharacteristic, complemented, Sylow
338C25C22×D6111yescharacteristic
339C23×C4—22no—
340C23×C4—22no—
341C23×C4—22no—
342C23×C4—22no—
343C23×C4—612no—
344C23×C4—612no—
345C23×C4—612no—
346C23×C4—612no—
347C23×C4Dic12214yes—
348C23×C4D24414yes—
349C23×C4C2×Dic6114yes—
350C23×C4C22×D6112yes—
351C23×C4C22×D6111yescharacteristic
352C23×C4C22×D6112yes—
353C23×C4C22×D6111yescharacteristic
354#117—12232no—
355C2×C4⋊C4—614no—
356C2×C4⋊C4—614no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1222no—
360C2×C4⋊C4—1222no—
361C2×C4⋊C4—1228no—
362C2×C4⋊C4—1222no—
363C2×C4⋊C4—1222no—
364C2×C4⋊C4—1228no—
365C2×C4⋊C4—1222no—
366C2×C4⋊C4—1222no—
367C2×C4⋊C4—1222no—
368C2×C4⋊C4—1222no—
369C2×C4⋊C4—2444no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—1224no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1224no—
376C2×C22⋊C4—2444no—
377C2×C22⋊C4C2×Dic6118yes—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—1224no—
383C2×C42—1224no—
384C2×C42—1224no—
385C2×C42C2×Dic6114yes—
386C2×C42C2×Dic6114yes—
387#95—1228no—
388#95—1224no—
389#95—1228no—
390#95—2444no—
391#95C2×Dic6118yes—
392C24—21no—
393C24—21no—
394C24—21no—
395C24—21no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24C2×Dic12212yes—
401C24C2×D24412yes—
402C24C22×Dic6112yes—
403C24C23×D6111yescharacteristic
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—614no—
425C22×C4—614no—
426C22×C4—1222no—
427C22×C4—1222no—
428C22×C4—1222no—
429C22×C4—1222no—
430C22×C4—1224no—
431C22×C4—1224no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4C4⋊Dic61116yescomplemented
445C22×C4C2×Dic12212yes—
446C22×C4C2×Dic12212yes—
447C22×C4C2×D24412yes—
448C22×C4C2×D24412yes—
449C22×C4#1175822214yes—
450C22×C4#1175822214yes—
451C22×C4D4×D6112yes—
452C22×C4D4×D6112yes—
453C22×C4#1175822412yes—
454C22×C4#1175822412yes—
455C22×C4C22×Dic6114yes—
456C22×C4C22×Dic6112yes—
457C22×C4C22×Dic6114yes—
458C22×C4C22×Dic6112yes—
459C4⋊C4—1228no—
460C4⋊C4—1228no—
461C4⋊C4—24416no—
462C4⋊C4—2444no—
463C4⋊C4—2444no—
464C22⋊C4—216no—
465C22⋊C4—12216no—
466C22⋊C4—2448no—
467C42—28no—
468C42—28no—
469C42—12216no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23C2×C4⋊Dic6114yes—
493C23C23×Dic6111yescharacteristic
494C23C2×D4×D6111yescharacteristic
495C23C2×D4×D6111yescharacteristic
496C23C22×D24411yescharacteristic
497C23C22×Dic12211yescharacteristic
498C23C2×#1175822411yescharacteristic
499C23C2×#1175822411yescharacteristic
500C23C2×#1175822212yes—
501C23C2×#1175822212yes—
502C23#1451982312yes—
503C23#1451982412yes—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—232no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—1224no—
525C2×C4—1224no—
526C2×C4—1228no—
527C2×C4—1228no—
528C2×C4—2444no—
529C2×C4—2444no—
530C2×C4—2442no—
531C2×C4—2442no—
532C2×C4—2442no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4—2444no—
538C2×C4—2442no—
539C2×C4—2442no—
540C2×C4—2442no—
541C2×C4—2442no—
542C2×C4—2444no—
543C2×C4—2444no—
544C2×C4C2×C4⋊Dic6114yes—
545C2×C4C2×C4⋊Dic6114yes—
546C2×C4#1451973618yescomplemented
547C2×C4D4×Dic6114yes—
548C2×C4D4×Dic6114yes—
549C2×C4#1451974714yes—
550C2×C4#1451975414yes—
551C2×C4#1451975414yes—
552C2×C4#1451976114yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22C22×C4⋊Dic6111yescharacteristic
565C22C2×#1451973612yes—
566C22C2×D4×Dic6111yescharacteristic
567C22C2×D4×Dic6111yescharacteristic
568C22#1684024512yes—
569C22C2×#1451974711yescharacteristic
570C22C2×#1451975411yescharacteristic
571C22C2×#1451975411yescharacteristic
572C22C2×#1451976111yescharacteristic
573C22#1684031211yescharacteristic
574C22#1684031211yescharacteristic
575C22#1684031912yes—
576C22#1684033111yescharacteristic
577C22#1684033811yescharacteristic
578C22#1684039012yes—
579C22#1684040812yes—
580C22#1684041712yes—
581C22#1684042112yes—
582C22#1684043312yes—
583C22#1684045212yes—
584C22#1684045812yes—
585C22#1684047712yes—
586C22#1684048212yes—
587C4—24no—
588C4—28no—
589C4—44no—
590C4—4882no—
591C4—4882no—
592C4—2444no—
593C4—2444no—
594C4#1683963718yes—
595C2—44no—
596C2#2144822611yescharacteristic
597C2C2×#1683963711yescharacteristic
598C2#2144844211yescharacteristic
599C2#2144874812yes—
600C2#2144876812yes—
601C2#2144943811yescharacteristic
602C2#2144944711yescharacteristic
603C2#2144951012yes—
604C2#2144953212yes—
605C2#2145027211yescharacteristic
606C2#2145028111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C4×C22⋊C4×C61C2
socle, cosocleC24×C61C24
prime radicalC25×C61C23
hypercenterC2×C4×C22⋊C4D61
hypocenterC61#13879
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839632C2semidirect
3. C2×#16839628C2semidirect
6. #21444053C2semidirect
7. #21444049C2semidirect
8. #21444049C2semidirect
32. C4×C22⋊C4×C61C4semidirect
120. C22×C4×C61C4⋊C4semidirect
222. C2×C4×C61#117semidirect
331. C4×C22⋊C4Dic61semidirect
337. C61#13879semidirect
444. C22×C4C4⋊Dic61semidirect
546. C2×C4#14519736semidirect

Maximal subgroups

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

subgroupindex
C22×#145197362
C2×#168396322
C2×#168396282
C2×#168396222
#214440642
#214440532
#214440492
C2×C4×C22⋊C4×C612
#1387961