Finite Group Databasebeta

← #29686577

Subgroups — #29686577

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840002C211yesmaximal, characteristic
2C2×#16839867C211yesmaximal, characteristic, complemented
3C2×#97×D61C211yesmaximal, characteristic
4C4×#14519757C211yesmaximal, characteristic
5C4×#14519753C211yesmaximal, characteristic
6#21446687C211yesmaximal, characteristic, complemented
7#21446556C211yesmaximal, characteristic, complemented
8#21446511C211yesmaximal, characteristic, complemented
9#21446507C211yesmaximal, characteristic, complemented
10#21446241C211yesmaximal, characteristic
11#21446239C211yesmaximal, characteristic
12#21445843C211yesmaximal, characteristic
13#21445717C211yesmaximal, characteristic, complemented
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1763C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519753C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16840002C418yescomplemented
24#16839867—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35#97×D61—24no—
36#16839669C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839647—21no—
41#16839643—21no—
42#16839636—21no—
43#16839633—21no—
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46C4×#11758196—21no—
47C4×C4⋊Dic61—21no—
48C4×C4⋊Dic61C2211yescharacteristic
49C4×#11758198—21no—
50C4×#11758198—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758202—22no—
67C2×#11758202—22no—
68C2×#11758202C2×C414yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C2×C8×D61—22no—
72C2×C8×D61—22no—
73C2×C8×D61—22no—
74C2×C8×D61—22no—
75C2×C8×D61—22no—
76C2×C8×D61C2×C414yes—
77#14519757—22no—
78#14519757C2×C414yes—
79#14519753—22no—
80#14519753C2×C414yes—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61D411yescharacteristic
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—22no—
101C42×D61—22no—
102#14519710C2×C414yescomplemented
103#14519708—22no—
104#14519708—22no—
105#14519702—22no—
106#14519702C2×C414yescomplemented
107#14519701—22no—
108#14519700—22no—
109#14519688D414yes—
110#14519687—22no—
111#14519687—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121C22×Q8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×Q8×C61—22no—
125C4×Q8×C61—22no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#103×C61—22no—
134#103×C61C2×C414yescomplemented
135#97×C61—22no—
136#95×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C4×D61C22×C412yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C22×C412yes—
181C2×C8×C61#4112yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—42no—
188C4⋊C4×C61C22×C412yes—
189C42×C61—42no—
190C42×C61—42no—
191C42×C61C2×D412yes—
192C42×C61#4112yes—
193#11758202—28no—
194#11758198—24no—
195#11758198—24no—
196#11758198—42no—
197#11758198—42no—
198C4⋊Dic61—21no—
199C4⋊Dic61—24no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61C22×C412yes—
206#11758196—24no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C22×C412yes—
218C2×#92418C2×D412yes—
219C2×#92418C2×D412yes—
220C2×#92418#4112yes—
221C8×D61—28no—
222C8×D61—28no—
223C23×C61C22×D411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61C4×D412yes—
229C22×D61#13712yes—
230C22×D61#13712yes—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—41no—
234Q8×C61—41no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61#12112yes—
252C2×C4×C61#12112yes—
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C4×D412yes—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—44no—
285C4×D61—44no—
286Dic122—22no—
287Dic122—22no—
288Dic122—41no—
289Dic122—41no—
290C8×C61—22no—
291C8×C61—22no—
292#92418—22no—
293#92418C4×D414yes—
294#21527—611nomaximal, Sylow
295C22×C61C2×C4×D411yescharacteristic
296C22×C61C2×#12111yescharacteristic
297C22×C61#53211yescharacteristic
298C22×C61C2×#13711yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57511yescharacteristic
302C2×D61—22no—
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61#44114yescomplemented
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61C2×C4×D411yescharacteristic
316C4×C61C4×#4111yescharacteristic
317C4×C61#52511yescharacteristic
318C4×C61#52911yescharacteristic
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—41no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326C2×#467—611no—
327C2×#441—611no—
328C22×#97—611no—
329C4×C22⋊Q8—611no—
330C4×C22⋊Q8—611no—
331#1971—611no—
332#1943—611no—
333#1935—611no—
334#1935—611no—
335#1892—611no—
336#1892—611no—
337#1807—611no—
338#1776—611no—
339#1763—611no—
340#1763D6111yescharacteristic, complemented
341C2×C61C4×#12111yescharacteristic
342C2×C61C2×#44111yescharacteristic
343C2×C61#294311yescharacteristic
344C2×C61#305611yescharacteristic
345C2×C61#305611yescharacteristic
346C2×C61#318111yescharacteristic
347C2×C61#318111yescharacteristic
348D61—24no—
349C22×Q16—611no—
350C23×C8—611no—
351C23×C8—611no—
352C2×C22⋊Q8—611no—
353C2×C22⋊Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×Q8C2×D6111yescharacteristic
356C22×C42—611no—
357#467—618no—
358#441—1224no—
359C2×#107—611no—
360C2×#106—611no—
361C2×#106—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103C2×D6111yescharacteristic
367C2×#103C2×D6111yescharacteristic
368C2×#97—611no—
369C2×#97—611no—
370C2×#97—611no—
371C2×#97—1224no—
372C2×#97C2×D6111yescharacteristic
373#391—611no—
374#391—611no—
375#389—611no—
376#389—611no—
377#387—1221no—
378#387—1221no—
379#385—1221no—
380#385—1221no—
381#380—611no—
382#380—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—1221no—
385C4×C4⋊C4—1221no—
386C4×C4⋊C4C2×D6111yescharacteristic
387C4×C22⋊C4—1221no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340C2×D6111yescharacteristic
394#340C2×D6111yescharacteristic
395C61#2152711yescharacteristic, complemented, Sylow
396C22×Q8—611no—
397C22×Q8C22×D6111yescharacteristic
398C23×C4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C2×Q16—614no—
402C2×Q16—1222no—
403C2×Q16—1222no—
404C22×C8—611no—
405C22×C8—611no—
406C22×C8—614no—
407C22×C8—1222no—
408C22×C8—1222no—
409C22×C8—1222no—
410C22×C8—1222no—
411C22×C8—1222no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22⋊Q8—614no—
415C22⋊Q8—614no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C42—21no—
439C2×C42—21no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1222no—
444C2×C42—1222no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42C22×D6111yescharacteristic
448#107—614no—
449#106—1222no—
450#106—1222no—
451#103—22no—
452#103—614no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103C4×D6114yescomplemented
457#97—22no—
458#97—614no—
459#97—1222no—
460#97—1222no—
461#95—21no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—611no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—612no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8C2×C4×D6112yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—611no—
489C22×C4—612no—
490C22×C4—612no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1222no—
502C22×C4—1222no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—2441no—
506C22×C4—2441no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510Q16—1228no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—612no—
516C2×C8—612no—
517C2×C8—612no—
518C2×C8—612no—
519C2×C8—1228no—
520C2×C8—1228no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8C2×C4×D6112yes—
524C2×C8#1175822212yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—612no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4C2×C4×D6112yes—
543C22⋊C4—1224no—
544C22⋊C4—1224no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C42—42no—
548C42—42no—
549C42—1221no—
550C42—1221no—
551C42—1222no—
552C42—1222no—
553C42—2442no—
554C42—2442no—
555C42#1175822212yes—
556C42D4×D6112yes—
557C23—612no—
558C23—612no—
559C23—612no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23C2×D4×D6111yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—1222no—
570Q8—1222no—
571Q8—2441no—
572Q8—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—612no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4#1451975012yes—
617C2×C4#1451975012yes—
618C2×C4C22×C4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4C2×#1175822211yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C8—22no—
625C8—22no—
626C8—614no—
627C8—1222no—
628C22—614no—
629C22—1222no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22C4×D4×D6111yescharacteristic
636C22C2×#1451975011yescharacteristic
637C22#1684030911yescharacteristic
638C22#137×D6111yescharacteristic
639C22#137×D6111yescharacteristic
640C22#1684055811yescharacteristic
641C22#1684056111yescharacteristic
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—1222no—
649C4—4881no—
650C4—4881no—
651C4—2442no—
652C4—2441no—
653C4—2441no—
654C4—2442no—
655C4C4×D4×D6111yescharacteristic
656C4C4×#1175822211yescharacteristic
657C4#1684027111yescharacteristic
658C4#1684029011yescharacteristic
659C2—1224no—
660C2C4×#1451975011yescharacteristic
661C2#441×D6111yescharacteristic
662C2#2145461411yescharacteristic
663C2#2145544711yescharacteristic
664C2#2145545211yescharacteristic
665C2#2145652411yescharacteristic
666C2#2145652611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1763C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1763D61
hypocenterC61#21527
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
2. C2×#16839867C2semidirect
6. #21446687C2semidirect
7. #21446556C2semidirect
8. #21446511C2semidirect
9. #21446507C2semidirect
13. #21445717C2semidirect
14. #21445650C2semidirect
15. C61×#1763C2semidirect
23. #16840002C4semidirect
68. C2×#11758202C2×C4semidirect
102. #14519710C2×C4semidirect
106. #14519702C2×C4semidirect
134. #103×C61C2×C4semidirect
308. C2×D61#441semidirect
340. #1763D61semidirect
395. C61#21527semidirect
456. #103C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168400022
C2×#168398672
C2×#97×D612
C4×#145197572
C4×#145197532
#214466872
#214465562
#214465112
#214465072
#214462412
#214462392
#214458432
#214457172
#214456502
C61×#17632
#2152761