Finite Group Databasebeta

← #29689069

Subgroups — #29689069

Order 15616.

1538 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C2×#16839724C211yesmaximal, characteristic
3C2×#16839619C211yesmaximal, characteristic
4#21446251C212yesmaximal, complemented
5#21446249C212yesmaximal, complemented
6#21446127C211yesmaximal, characteristic, complemented
7#21446126C211yesmaximal, characteristic, complemented
8#21446124C211yesmaximal, characteristic, complemented
9#21445904C212yesmaximal, complemented
10#21445752C212yesmaximal, complemented
11C61×#1872C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C22×#11758189C2212yes—
15C2×#14519757C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753C2211yescharacteristic
20C22⋊Q8×D61—24no—
21C22⋊Q8×D61C418yescomplemented
22C22×C4⋊Dic61C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61C2211yescharacteristic
25C22×#11758198—21no—
26C2×C22⋊C4×D61—21no—
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519700C2212yescomplemented
30#16839724—24no—
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2212yes—
33#16839619—21no—
34#16839619—21no—
35#16839619C2212yes—
36#16839619C2212yes—
37C2×#14519684—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C2×C22⋊Q8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yescomplemented
53C23×Dic61—21no—
54C23×Dic61D411yescharacteristic
55C22×C4×D61—21no—
56C22×C4×D61—21no—
57C22×C4×D61—42no—
58C22×C4×D61—42no—
59C22×C4×D61C2×C414yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D414yes—
63C22×C4×D61C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—24no—
66C2×#11758189—42no—
67C2×#11758189—42no—
68C22×#92418C2312yes—
69#14519757—22no—
70#14519757C2×C414yescomplemented
71#14519756—22no—
72#14519756C2×C414yes—
73#14519755—28no—
74#14519754—28no—
75#14519753—22no—
76#14519753C2×C414yescomplemented
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—42no—
80C2×C4⋊Dic61C2312yes—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C4⋊C4×D61—28no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61C2×C414yescomplemented
88C2×C22⋊Dic61—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—22no—
92C2×#11758198—22no—
93C2×#11758198—44no—
94C22⋊C4×D61—28no—
95#14519702—42no—
96#14519701—42no—
97#14519700—42no—
98C22⋊#92418—22no—
99#14519689—24no—
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×C61C2×C414yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—42no—
111C2×C4⋊C4×C61C2312yes—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114#103×C61—42no—
115C22⋊C8×C61—22no—
116C24×C61C2×D411yescharacteristic
117C23×D61—22no—
118C23×D61—22no—
119C23×D61—22no—
120C23×D61—22no—
121C23×D61—41no—
122C23×D61—41no—
123C23×D61C22⋊C414yes—
124C23×D61C2×D411yescharacteristic
125C23×D61C2×D411yescharacteristic
126C2×Q8×C61—21no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61C22×C412yes—
130C22×C4×C61—22no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61C22×C412yes—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D412yes—
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—22no—
145C22×Dic61—41no—
146C22×Dic61—41no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61C22⋊C414yes—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153Q8×D61—48no—
154Q8×D61—48no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—24no—
160C2×C4×D61—24no—
161C2×C4×D61—48no—
162C2×C4×D61—48no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61C22⋊C414yes—
172C2×C4×D61C22⋊C414yes—
173C2×C4×D61C22×C412yes—
174C2×C4×D61C22×C412yes—
175C2×C4×D61C2×D412yes—
176C2×C4×D61C2×D412yes—
177C2×Dic122—22no—
178C2×Dic122—22no—
179C2×Dic122—21no—
180C2×Dic122—24no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122C22×C412yes—
184C2×C8×C61—22no—
185C2×C8×C61—22no—
186C2×C8×C61—22no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—22no—
190C4⋊C4×C61—22no—
191C4⋊C4×C61—42no—
192C4⋊C4×C61C22×C412yes—
193C22⋊C4×C61—24no—
194C22⋊Dic61—24no—
195#11758198—24no—
196#11758198—24no—
197#11758198—44no—
198#11758198—44no—
199C4⋊Dic61—21no—
200C4⋊Dic61—24no—
201C4⋊Dic61—22no—
202C4⋊Dic61—22no—
203C4⋊Dic61—42no—
204C4⋊Dic61C22×C412yes—
205#11758196—24no—
206#11758196—24no—
207#11758196—24no—
208C2×#92418—22no—
209C2×#92418—22no—
210C2×#92418—22no—
211#11758189—48no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61C2×C22⋊C412yes—
217C23×C61#13712yes—
218C23×C61#13712yes—
219C23×C61C22×D411yescharacteristic
220C22×D61—21no—
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—28no—
231C22×D61—28no—
232C22×D61—44no—
233C22×D61—44no—
234C22×D61—44no—
235C22×D61—44no—
236C22×D61—44no—
237C22×D61C2×C22⋊C412yes—
238C22×D61C2×C22⋊C412yes—
239C22×D61AGL(1,ℤ/8)12yes—
240C22×D61#13712yes—
241Q8×C61—24no—
242Q8×C61—42no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—24no—
246C2×C4×C61—24no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—84no—
257C2×C4×C61C2×C22⋊C412yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61H(5,𝔽2)12yes—
261C2×C4×C61H(5,𝔽2)12yes—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—44no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—84no—
277C2×Dic61C2×C22⋊C412yes—
278C2×Dic61C2×C22⋊C412yes—
279C4×D61—24no—
280C4×D61—24no—
281C4×D61—24no—
282C4×D61—24no—
283C4×D61—44no—
284C4×D61—44no—
285C4×D61—48no—
286C4×D61—48no—
287Dic122—24no—
288Dic122—48no—
289Dic122—42no—
290C8×C61—42no—
291#92418—42no—
292#21765—611nomaximal, Sylow
293C22×C61—22no—
294C22×C61—22no—
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61#41914yes—
300C22×C61C22×C22⋊C411yescharacteristic
301C22×C61#53411yescharacteristic
302C22×C61#53411yescharacteristic
303C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
304C22×C61C2×#13711yescharacteristic
305C22×C61C2×#13711yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C2×D61—22no—
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—416no—
317C2×D61—41no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61#41714yes—
322C4×C61—22no—
323C4×C61—42no—
324C4×C61—42no—
325C4×C61—82no—
326C4×C61#51812yes—
327C4×C61#53412yes—
328Dic61—22no—
329Dic61—22no—
330Dic61—42no—
331Dic61—42no—
332Dic61—82no—
333C22×C22⋊Q8—611no—
334C2×#407—611no—
335C2×#380—611no—
336#1894—612no—
337#1894—612no—
338#1872—611no—
339#1872—611no—
340#1872—611no—
341#1872D6111yescharacteristic, complemented
342#1823—612no—
343#1787—612no—
344C2×C61—24no—
345C2×C61#230811yescharacteristic
346C2×C61C2×#41911yescharacteristic
347C2×C61C2×#41711yescharacteristic
348C2×C61#319311yescharacteristic
349C2×C61#319311yescharacteristic
350C2×C61#319511yescharacteristic
351C2×C61#319511yescharacteristic
352D61—24no—
353D61—42no—
354C23×Q8—611no—
355C24×C4—611no—
356C22×M16—612no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—618no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8—1221no—
362C2×C22⋊Q8—1221no—
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—
371C2×#103—612no—
372C2×#103—612no—
373C2×#103—612no—
374C2×#103C2×D6112yescomplemented
375#407—1224no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—612no—
378C2×C22⋊C8C2×D6111yescharacteristic
379#380—612no—
380#380—612no—
381#380—1221no—
382#380—1221no—
383C2×#95—1221no—
384#340—612no—
385#340—612no—
386#340—612no—
387#340C2×D6112yescomplemented
388C61#2176511yescharacteristic, complemented, Sylow
389C25—611no—
390C22×Q8—614no—
391C22×Q8—611no—
392C22×Q8—1221no—
393C22×Q8—1222no—
394C22×Q8—2442no—
395C22×Q8—2442no—
396C22×Q8C22×D6111yescharacteristic
397C23×C4—21no—
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C23×C4—614no—
402C23×C4—611no—
403C23×C4—611no—
404C23×C4—614no—
405C23×C4—1221no—
406C23×C4—1221no—
407C23×C4—1221no—
408C23×C4—2442no—
409C23×C4—2442no—
410C23×C4C22×D6111yescharacteristic
411C2×M16—1224no—
412C2×M16—2442no—
413C2×M16—2442no—
414C22×C8—612no—
415C22×C8C22×D6112yes—
416C22⋊Q8—22no—
417C22⋊Q8—614no—
418C22⋊Q8—614no—
419C22⋊Q8—614no—
420C22⋊Q8—1228no—
421C22⋊Q8—1228no—
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424C22⋊Q8—1222no—
425C22⋊Q8C4×D6114yes—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—42no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—614no—
431C2×C4⋊C4—612no—
432C2×C4⋊C4—1222no—
433C2×C4⋊C4—1228no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
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#103—42no—
451#103—2442no—
452#103—2442no—
453#103—2442no—
454C22⋊C8—22no—
455C22⋊C8—1224no—
456C22⋊C8—1222no—
457#95—1222no—
458#95—1222no—
459C24—611no—
460C24—611no—
461C24—614no—
462C24—1222no—
463C24—1222no—
464C24—1222no—
465C24—1222no—
466C24—2441no—
467C24—2441no—
468C24D4×D6111yescharacteristic
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—612no—
473C2×Q8—1224no—
474C2×Q8—1221no—
475C2×Q8—1222no—
476C2×Q8—1222no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×Q8—2448no—
480C2×Q8—2448no—
481C2×Q8C2×C4×D6112yes—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—614no—
493C22×C4—612no—
494C22×C4—612no—
495C22×C4—612no—
496C22×C4—612no—
497C22×C4—614no—
498C22×C4—614no—
499C22×C4—1224no—
500C22×C4—1224no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1222no—
504C22×C4—1222no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1222no—
508C22×C4—2448no—
509C22×C4—2448no—
510C22×C4—2441no—
511C22×C4—2441no—
512C22×C4—2441no—
513C22×C4—2441no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2441no—
519C22×C4—2441no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4C2×C4×D6112yes—
524C22×C4D4×D6112yes—
525C22×C4D4×D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527M16—2448no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—1222no—
532C2×C8—1222no—
533C2×C8—1222no—
534C4⋊C4—21no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—42no—
539C4⋊C4—612no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1224no—
543C4⋊C4—1221no—
544C4⋊C4—1224no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—2442no—
548C4⋊C4C2×C4×D6112yes—
549C22⋊C4—24no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C22⋊C4—2444no—
554C22⋊C4—2444no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—612no—
560C23—612no—
561C23—612no—
562C23—612no—
563C23—1228no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1228no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—2444no—
576C23—2444no—
577C23—2444no—
578C23—2444no—
579C23—2444no—
580C23C22⋊C4×D6112yes—
581C23#1451980612yes—
582C23#1451980912yes—
583C23C2×D4×D6111yescharacteristic
584Q8—24no—
585Q8—42no—
586Q8—1224no—
587Q8—2448no—
588Q8—2442no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—84no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1224no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1224no—
613C2×C4—1222no—
614C2×C4—1222no—
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×C4C22⋊C4×D6112yes—
629C2×C4C22×C4×D6111yescharacteristic
630C2×C4C2×D4×D6111yescharacteristic
631C2×C4#1451982212yes—
632C2×C4#1451982312yes—
633C8—42no—
634C8—2442no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—614no—
642C22—1222no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—1222no—
651C22—24416no—
652C22—2441no—
653C22—2441no—
654C22—2441no—
655C22—2441no—
656C22#1683976914yes—
657C22C2×C22⋊C4×D6111yescharacteristic
658C22#1684032211yescharacteristic
659C22#1684032311yescharacteristic
660C22AGL(1,ℤ/8)×D6111yescharacteristic
661C22#137×D6111yescharacteristic
662C22C2×#1451980611yescharacteristic
663C22C2×#1451980911yescharacteristic
664C4—22no—
665C4—42no—
666C4—42no—
667C4—82no—
668C4—1222no—
669C4—1222no—
670C4—4882no—
671C4—2442no—
672C4—2442no—
673C4#1684023412yes—
674C4#1684032012yes—
675C2—24no—
676C2—1224no—
677C2—2442no—
678C2#2144839311yescharacteristic
679C2C2×#1683976911yescharacteristic
680C2#417×D6111yescharacteristic
681C2#2145662211yescharacteristic
682C2#2145662411yescharacteristic
683C2#2145664111yescharacteristic
684C2#2145664211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1872C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1872D61
hypocenterC61#21765
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #21446251C2semidirect
5. #21446249C2semidirect
6. #21446127C2semidirect
7. #21446126C2semidirect
8. #21446124C2semidirect
9. #21445904C2semidirect
10. #21445752C2semidirect
11. C61×#1872C2semidirect
21. C22⋊Q8×D61C4semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
29. C2×#14519700C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
52. C2×Q8×D61C2×C4semidirect
70. #14519757C2×C4semidirect
76. #14519753C2×C4semidirect
87. C4⋊C4×D61C2×C4semidirect
341. #1872D61semidirect
374. C2×#103C2×D61semidirect
387. #340C2×D61semidirect
388. C61#21765semidirect

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×#168397242
C2×#168396192
#214462512
#214462492
#214461272
#214461262
#214461242
#214459042
#214457522
C61×#18722
#2176561