Finite Group Databasebeta

← #29712651

Subgroups — #29712651

Order 15616.

1500 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C2×#16840092C211yesmaximal, characteristic, complemented
3C2×#16839619C211yesmaximal, characteristic
4#21447017C212yesmaximal, complemented
5#21447013C212yesmaximal, complemented
6#21446127C211yesmaximal, characteristic, complemented
7#21446124C211yesmaximal, characteristic, complemented
8#21446010C212yesmaximal, complemented
9#21445752C212yesmaximal, complemented
10#21445728C211yesmaximal, characteristic
11C2×C61×#483C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C2×#14519757C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519755—21no—
17C2×#14519754—21no—
18C2×#14519753C2211yescharacteristic
19C22⋊Q8×D61—24no—
20C22⋊Q8×D61—24no—
21C22×C4⋊Dic61C2211yescharacteristic
22C2×C4⋊C4×D61—21no—
23C2×C4⋊C4×D61C2211yescharacteristic
24C22×#11758198—21no—
25C2×C22⋊C4×D61—21no—
26#16840092—24no—
27C2×#14519708C2212yescomplemented
28C2×#14519702C2212yescomplemented
29C2×#14519701C2212yescomplemented
30C2×#14519700C2212yescomplemented
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2212yes—
33C2×#14519689C2212yes—
34#16839619—21no—
35#16839619—21no—
36#16839619C2212yes—
37#16839619C2212yes—
38C2×#14519684—21no—
39#16839525C2212yescomplemented
40#16839524C2212yescomplemented
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C61×#483—24no—
44C2×#106×C61C2212yescomplemented
45C2×#103×C61C2212yescomplemented
46C2×C22⋊C8×C61C2211yescharacteristic
47C24×D61D411yescharacteristic
48C22×Dic122—21no—
49C22×Dic122C2311yescharacteristic
50C2×Q8×D61—22no—
51C2×Q8×D61—22no—
52C2×Q8×D61—24no—
53C2×Q8×D61—42no—
54C23×Dic61—21no—
55C23×Dic61D411yescharacteristic
56C22×C4×D61—21no—
57C22×C4×D61—21no—
58C22×C4×D61—22no—
59C22×C4×D61—22no—
60C22×C4×D61—24no—
61C22×C4×D61—42no—
62C22×C4×D61D411yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C22×#92418C2312yes—
67#14519757—22no—
68#14519757—22no—
69#14519756—22no—
70#14519756D414yes—
71#14519755—28no—
72#14519754—28no—
73#14519753—22no—
74#14519753—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—42no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—28no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61—22no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—22no—
90C2×#11758198—22no—
91C2×#11758198—44no—
92C22⋊C4×D61—28no—
93#14519708—24no—
94#14519702—42no—
95#14519701—42no—
96#14519700—24no—
97C22⋊#92418—22no—
98#14519689—24no—
99#14519689—42no—
100#14519684—22no—
101#14519684—22no—
102C22×Q8×C61C2311yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61D414yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—22no—
110C2×C4⋊C4×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2312yes—
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C22⋊C4×C61—21no—
115#106×C61—24no—
116#103×C61—24no—
117C22⋊C8×C61—22no—
118C24×C61C2×D411yescharacteristic
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—22no—
122C23×D61—22no—
123C23×D61—41no—
124C23×D61—41no—
125C23×D61SD1614yes—
126C23×D61C2×D411yescharacteristic
127C23×D61C2×D411yescharacteristic
128C2×Q8×C61—21no—
129C2×Q8×C61—22no—
130C2×Q8×C61—22no—
131C2×Q8×C61C2×D412yes—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—22no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—42no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D412yes—
142C22×C4×C61C2×D412yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—22no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—41no—
150C22×Dic61—42no—
151C22×Dic61—42no—
152C22×Dic61—42no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155Q8×D61—48no—
156Q8×D61—48no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—48no—
171C2×C4×D61—48no—
172C2×C4×D61—41no—
173C2×C4×D61—41no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61#4114yes—
179C2×Dic122—22no—
180C2×Dic122—22no—
181C2×Dic122—24no—
182C2×Dic122—21no—
183C2×Dic122—42no—
184C2×Dic122—42no—
185C2×Dic122C2×D412yes—
186C2×C8×C61—22no—
187C2×C8×C61—22no—
188C2×C8×C61—22no—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—22no—
192C4⋊C4×C61—22no—
193C4⋊C4×C61—42no—
194C4⋊C4×C61C2×D412yes—
195C22⋊C4×C61—24no—
196C22⋊Dic61—24no—
197#11758198—24no—
198#11758198—24no—
199#11758198—44no—
200#11758198—44no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—22no—
204C4⋊Dic61—22no—
205C4⋊Dic61—42no—
206C4⋊Dic61C2×D412yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210C2×#92418—22no—
211C2×#92418—22no—
212C2×#92418—22no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61#12012yes—
218C23×C61C2×SD1612yes—
219C23×C61#13712yes—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—28no—
232C22×D61—28no—
233C22×D61—44no—
234C22×D61—44no—
235C22×D61—44no—
236C22×D61—44no—
237C22×D61—44no—
238C22×D61#12312yes—
239C22×D61C2×SD1612yes—
240C22×D61C2×SD1612yes—
241C22×D61#13712yes—
242Q8×C61—24no—
243Q8×C61—42no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—24no—
247C2×C4×C61—24no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—42no—
257C2×C4×C61—84no—
258C2×C4×C61#12012yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4112yes—
262C2×C4×C61H(5,𝔽2)12yes—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—44no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—42no—
276C2×Dic61—42no—
277C2×Dic61—42no—
278C2×Dic61—84no—
279C2×Dic61#12312yes—
280C4×D61—24no—
281C4×D61—24no—
282C4×D61—44no—
283C4×D61—44no—
284C4×D61—48no—
285C4×D61—48no—
286C4×D61—42no—
287C4×D61—42no—
288Dic122—24no—
289Dic122—48no—
290Dic122—42no—
291C8×C61—42no—
292#92418—42no—
293#24003—611nomaximal, Sylow
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61#44814yes—
301C22×C61C2×#12011yescharacteristic
302C22×C61C2×#12311yescharacteristic
303C22×C61#53411yescharacteristic
304C22×C61C22×SD1611yescharacteristic
305C22×C61C2×#13711yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—416no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61#48314yescomplemented
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—82no—
327C4×C61#54612yes—
328C4×C61#54612yes—
329Dic61—22no—
330Dic61—42no—
331Dic61—41no—
332Dic61—42no—
333Dic61—82no—
334C22×C22⋊Q8—611no—
335C2×#483—611no—
336C2×#483D6111yescharacteristic, complemented
337C2×#380—611no—
338#2035—612no—
339#2035—612no—
340#1872—611no—
341#1872—611no—
342#1853—612no—
343#1787—612no—
344#1779—611no—
345C2×C61—24no—
346C2×C61#240511yescharacteristic
347C2×C61C2×#44811yescharacteristic
348C2×C61#300711yescharacteristic
349C2×C61C2×#48311yescharacteristic
350C2×C61#309511yescharacteristic
351C2×C61#318711yescharacteristic
352C2×C61#319511yescharacteristic
353D61—24no—
354D61—42no—
355C23×Q8—611no—
356C24×C4—611no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C22⋊Q8—1221no—
362C2×C22⋊Q8—1224no—
363C2×C22⋊Q8—1224no—
364C2×C22⋊Q8C2×D6111yescharacteristic
365C22×C4⋊C4—611no—
366C22×C4⋊C4—611no—
367C22×C4⋊C4—1221no—
368C22×C4⋊C4C2×D6111yescharacteristic
369C22×C22⋊C4—1221no—
370C22×C22⋊C4—1221no—
371#483—24no—
372#483—1224no—
373C2×#106—612no—
374C2×#106C2×D6112yescomplemented
375C2×#103—612no—
376C2×#103—612no—
377C2×#103—612no—
378C2×#103C2×D6112yescomplemented
379C2×C22⋊C8—611no—
380C2×C22⋊C8—612no—
381C2×C22⋊C8—612no—
382C2×C22⋊C8C2×D6111yescharacteristic
383#380—612no—
384#380—612no—
385#380—1221no—
386#380—1221no—
387C2×#95—1221no—
388#340—612no—
389#340—612no—
390C61#2400311yescharacteristic, complemented, Sylow
391C25—611no—
392C22×Q8—611no—
393C22×Q8—1221no—
394C22×Q8—1224no—
395C22×Q8—1222no—
396C22×Q8—1222no—
397C22×Q8—2442no—
398C22×Q8C22×D6111yescharacteristic
399C23×C4—21no—
400C23×C4—611no—
401C23×C4—611no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—611no—
405C23×C4—1224no—
406C23×C4—1221no—
407C23×C4—1221no—
408C23×C4—1222no—
409C23×C4—1222no—
410C23×C4—1221no—
411C23×C4—2442no—
412C23×C4C22×D6111yescharacteristic
413C22×C8—612no—
414C22×C8C22×D6112yes—
415C22⋊Q8—22no—
416C22⋊Q8—614no—
417C22⋊Q8—1228no—
418C22⋊Q8—1222no—
419C22⋊Q8—1228no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8C22⋊D6114yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—612no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1228no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1222no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—2442no—
440C2×C4⋊C4C22×D6112yes—
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1228no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1222no—
448C2×C22⋊C4—1222no—
449C2×C22⋊C4—2444no—
450#106—24no—
451#106—1224no—
452#103—24no—
453#103—1224no—
454#103—2442no—
455#103—2442no—
456C22⋊C8—22no—
457C22⋊C8—1224no—
458C22⋊C8—1222no—
459C22⋊C8—2442no—
460#95—1222no—
461#95—1222no—
462C24—611no—
463C24—611no—
464C24—614no—
465C24—1222no—
466C24—1222no—
467C24—1222no—
468C24—1222no—
469C24—2441no—
470C24—2441no—
471C24D4×D6111yescharacteristic
472C2×Q8—21no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—612no—
476C2×Q8—1224no—
477C2×Q8—1221no—
478C2×Q8—1222no—
479C2×Q8—1222no—
480C2×Q8—2442no—
481C2×Q8—2442no—
482C2×Q8—2448no—
483C2×Q8—2448no—
484C2×Q8C2×C22⋊D6112yes—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—42no—
493C22×C4—611no—
494C22×C4—614no—
495C22×C4—611no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1224no—
499C22×C4—1224no—
500C22×C4—1222no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1222no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1222no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
512C22×C4—1222no—
513C22×C4—1222no—
514C22×C4—2441no—
515C22×C4—2441no—
516C22×C4—2448no—
517C22×C4—2448no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4—2441no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4D4×D6112yes—
527C22×C4D4×D6111yescharacteristic
528C22×C4C2×C22⋊D6112yes—
529C22×C4C23×D6111yescharacteristic
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—1222no—
534C2×C8—1222no—
535C2×C8—1222no—
536C4⋊C4—21no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4—612no—
542C4⋊C4—1224no—
543C4⋊C4—1224no—
544C4⋊C4—1224no—
545C4⋊C4—1221no—
546C4⋊C4—1224no—
547C4⋊C4—1222no—
548C4⋊C4—1222no—
549C4⋊C4—2442no—
550C4⋊C4C2×C22⋊D6112yes—
551C22⋊C4—24no—
552C22⋊C4—1224no—
553C22⋊C4—1224no—
554C22⋊C4—1224no—
555C22⋊C4—2444no—
556C22⋊C4—2444no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—612no—
562C23—612no—
563C23—612no—
564C23—612no—
565C23—1228no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1228no—
577C23—2444no—
578C23—2444no—
579C23—2444no—
580C23—2444no—
581C23—2444no—
582C23#1451974512yes—
583C23SD16×D6112yes—
584C23#1451980612yes—
585C23C2×D4×D6111yescharacteristic
586Q8—24no—
587Q8—42no—
588Q8—1224no—
589Q8—2448no—
590Q8—2442no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—84no—
605C2×C4—612no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1224no—
609C2×C4—1224no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—4884no—
616C2×C4—2444no—
617C2×C4—2444no—
618C2×C4—2448no—
619C2×C4—2448no—
620C2×C4—2444no—
621C2×C4—2442no—
622C2×C4—2442no—
623C2×C4—2442no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2442no—
629C2×C4—2442no—
630C2×C4#1451974512yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4C22×C22⋊D6111yescharacteristic
633C2×C4#41×D6112yes—
634C2×C4#1451982312yes—
635C8—42no—
636C8—2442no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—614no—
644C22—1222no—
645C22—1222no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22—24416no—
654C22—2441no—
655C22—2441no—
656C22—2441no—
657C22—2441no—
658C22#1683989914yes—
659C22C2×#1451974511yescharacteristic
660C22#123×D6111yescharacteristic
661C22#1684032311yescharacteristic
662C22C2×SD16×D6111yescharacteristic
663C22#137×D6111yescharacteristic
664C22C2×#1451980611yescharacteristic
665C22#1684056111yescharacteristic
666C4—22no—
667C4—42no—
668C4—42no—
669C4—82no—
670C4—1222no—
671C4—4882no—
672C4—2441no—
673C4—2442no—
674C4—2442no—
675C4#1684040012yes—
676C4#1684040412yes—
677C2—24no—
678C2—1224no—
679C2—2442no—
680C2#2144914311yescharacteristic
681C2C2×#1683989911yescharacteristic
682C2#2145507511yescharacteristic
683C2#483×D6111yescharacteristic
684C2#2145576211yescharacteristic
685C2#2145657511yescharacteristic
686C2#2145664211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#483C2
socleC23×C61C22×D4
prime radicalC24×D61D4
hypercenterC2×#483D61
hypocenterC61#24003
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
2. C2×#16840092C2semidirect
4. #21447017C2semidirect
5. #21447013C2semidirect
6. #21446127C2semidirect
7. #21446124C2semidirect
8. #21446010C2semidirect
9. #21445752C2semidirect
11. C2×C61×#483C2semidirect
27. C2×#14519708C22semidirect
28. C2×#14519702C22semidirect
29. C2×#14519701C22semidirect
30. C2×#14519700C22semidirect
39. #16839525C22semidirect
40. #16839524C22semidirect
44. C2×#106×C61C22semidirect
45. C2×#103×C61C22semidirect
322. C2×D61#483semidirect
336. C2×#483D61semidirect
374. C2×#106C2×D61semidirect
378. C2×#103C2×D61semidirect
390. C61#24003semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C2×#168400922
C2×#168396192
#214470172
#214470132
#214461272
#214461242
#214460102
#214457522
#214457282
C2×C61×#4832
#2400361