Finite Group Databasebeta

← #29708074

Subgroups — #29708074

Order 15616.

1202 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×#16839803C211yesmaximal, characteristic
3C2×#16839801C211yesmaximal, characteristic
4#389×D61C211yesmaximal, characteristic
5#21447158C211yesmaximal, characteristic, complemented
6#21447157C211yesmaximal, characteristic, complemented
7#21447154C211yesmaximal, characteristic, complemented
8#21447151C211yesmaximal, characteristic, complemented
9#21447150C211yesmaximal, characteristic, complemented
10#21447146C211yesmaximal, characteristic, complemented
11#21447145C211yesmaximal, characteristic, complemented
12#21445537C211yesmaximal, characteristic
13#21445537C211yesmaximal, characteristic
14#21445511C211yesmaximal, characteristic
15C61×#2059C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18#16840318—24no—
19#16840318—24no—
20C2×#14519776C2211yescharacteristic
21C2×#14519776C2211yescharacteristic
22C2×#14519769C2211yescharacteristic
23C2×#14519755—21no—
24C2×#14519755—21no—
25C2×#14519742—21no—
26C2×#14519742—21no—
27C2×#14519735C2211yescharacteristic
28C2×#14519735C2211yescharacteristic
29C2×C4⋊C4×D61—21no—
30C2×C4⋊C4×D61C2211yescharacteristic
31C2×C4⋊C4×D61C2211yescharacteristic
32C2×C42×D61C2211yescharacteristic
33#16839803—24no—
34#16839801—24no—
35C2×#14519711C2211yescharacteristic
36C2×#14519710C2211yescharacteristic
37C2×#14519710C2211yescharacteristic
38C2×#14519709C2211yescharacteristic
39C2×#14519708C2211yescharacteristic
40C2×#14519708C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839659—21no—
44#16839658C2211yescharacteristic
45#16839657—21no—
46#95×D61—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×#125×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61D411yescharacteristic
68C22×C4×D61D411yescharacteristic
69C22×C4×D61C2311yescharacteristic
70C2×#11758189—22no—
71C2×#11758189—22no—
72C2×#11758189—22no—
73C2×#11758189—22no—
74C2×#11758189Q814yescomplemented
75C2×#11758189Q814yescomplemented
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78#14519776—22no—
79#14519776—22no—
80#14519776—22no—
81#14519776—22no—
82#14519769—22no—
83#14519769—22no—
84#14519755—28no—
85#14519755—28no—
86#14519742—28no—
87#14519742—28no—
88#14519735—22no—
89#14519735—22no—
90#14519735D414yes—
91#14519735D414yes—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—22no—
107C4⋊C4×D61—22no—
108C4⋊C4×D61—44no—
109C4⋊C4×D61Q814yescomplemented
110C2×#11758198—21no—
111C2×#11758198—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61—21no—
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61C2311yescharacteristic
117C42×D61—22no—
118C42×D61—22no—
119#14519711—22no—
120#14519710—22no—
121#14519710—22no—
122#14519709—22no—
123#14519708—22no—
124#14519708—22no—
125#14519684—21no—
126#14519684—21no—
127#14519684—21no—
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130#125×C61—22no—
131#125×C61—22no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61—21no—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C4⋊C4×C61C2311yescharacteristic
139C2×C42×C61C2311yescharacteristic
140#107×C61—22no—
141#106×C61—22no—
142#95×C61—21no—
143C23×D61C2×D411yescharacteristic
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—22no—
159C2×C4×D61—22no—
160C2×C4×D61—24no—
161C2×C4×D61—22no—
162C2×C4×D61—22no—
163C2×C4×D61—41no—
164C2×C4×D61—41no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61C2×D412yes—
171C2×C4×D61C2×D412yes—
172C2×C4×D61C2×Q812yes—
173C2×C4×D61C2×Q812yes—
174C2×C4×D61C2×Q812yes—
175C2×C4×D61#4112yes—
176C2×Dic122—24no—
177C2×Dic122—24no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61C2×Q812yes—
187C2×C8×C61C2×Q812yes—
188C4⋊C4×C61—24no—
189C4⋊C4×C61—24no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—42no—
196C4⋊C4×C61C2×D412yes—
197C4⋊C4×C61C2×D412yes—
198C4⋊C4×C61C2×Q812yes—
199C42×C61—21no—
200C42×C61—21no—
201#11758198—24no—
202#11758198—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61C2×D412yes—
210C4⋊Dic61C2×D412yes—
211C4⋊Dic61C2×Q812yes—
212#11758196—24no—
213#11758196—24no—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—42no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61—24no—
225C4×Dic61—24no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418C2×Q812yes—
231C2×#92418C2×Q812yes—
232#11758189—28no—
233#11758189—28no—
234C23×C61C22×D411yescharacteristic
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61#12012yes—
240C22×D61C22⋊Q812yes—
241C22×D61C4⋊Q812yes—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×Q811yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C22×Q811yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61#12012yes—
283C2×Dic61C22⋊Q812yes—
284C2×Dic61C4⋊Q812yes—
285C2×Dic61AGL(1,ℤ/8)12yes—
286C2×Dic61#13712yes—
287C4×D61—44no—
288C4×D61—44no—
289C4×D61—48no—
290C4×D61—84no—
291C4×D61C22⋊Q814yes—
292C4×D61C22⋊Q814yes—
293C4×D61C4⋊Q814yes—
294C4×D61C4⋊Q814yes—
295Dic122—48no—
296Dic122—48no—
297C8×C61—22no—
298C8×C61—22no—
299#92418—22no—
300#92418—22no—
301#23554—611nomaximal, Sylow
302C22×C61C2×#12011yescharacteristic
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61C2×C4⋊Q811yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C22×C61#57511yescharacteristic
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61#39514yes—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61—41no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61C2×C22⋊Q811yescharacteristic
323C4×C61C2×C22⋊Q811yescharacteristic
324C4×C61C2×C4⋊Q811yescharacteristic
325C4×C61C2×C4⋊Q811yescharacteristic
326Dic61—22no—
327Dic61—22no—
328Dic61—42no—
329Dic61—42no—
330Dic61—42no—
331Dic61—81no—
332Dic61#39514yes—
333C2×#533—611no—
334C2×#428—611no—
335C2×#428—611no—
336C2×#389—611no—
337#2059—611no—
338#2059—611no—
339#2059—611no—
340#2059D6111yescharacteristic, complemented
341#2058—611no—
342#2058—611no—
343#2057—611no—
344#2057—611no—
345#1736—611no—
346#1736—611no—
347#1728—611no—
348C2×C61C2×#39511yescharacteristic
349C2×C61#300611yescharacteristic
350C2×C61#300711yescharacteristic
351C2×C61#307711yescharacteristic
352C2×C61#307811yescharacteristic
353C2×C61#316511yescharacteristic
354C2×C61#316611yescharacteristic
355D61—24no—
356C22×M16—611no—
357C22×M16—611no—
358#533—1224no—
359#533—1224no—
360C2×C4⋊Q8—611no—
361C2×C4⋊Q8—611no—
362C2×#125—611no—
363C2×#125C2×D6111yescharacteristic
364C2×C22⋊Q8—1221no—
365C2×C22⋊Q8—1221no—
366C2×C4×Q8—1221no—
367C2×C4×Q8—1221no—
368C2×#117—611no—
369C2×#117—611no—
370C22×C4⋊C4—611no—
371C22×C4⋊C4—611no—
372C22×C4⋊C4—1221no—
373C22×C42—611no—
374#428—1224no—
375#428—1224no—
376C2×#107—611no—
377C2×#107—611no—
378C2×#107—611no—
379C2×#107C2×D6111yescharacteristic
380C2×#106—611no—
381C2×#106—611no—
382C2×#106—611no—
383C2×#106C2×D6111yescharacteristic
384#389—611no—
385#389—611no—
386#389—611no—
387#389—1221no—
388#389—1221no—
389#389C2×D6111yescharacteristic
390C2×#95—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340C2×D6111yescharacteristic
398#340C2×D6111yescharacteristic
399#336—611no—
400#336—611no—
401#336—611no—
402#336C2×D6111yescharacteristic
403C61#2355411yescharacteristic, complemented, Sylow
404C22×Q8—1221no—
405C22×Q8—1221no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—611no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×M16—614no—
412C2×M16—614no—
413C2×M16—1222no—
414C2×M16—1222no—
415C2×M16—1222no—
416C2×M16—1222no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8C22×D6111yescharacteristic
420C22×C8C22×D6111yescharacteristic
421C4⋊Q8—1222no—
422C4⋊Q8—1222no—
423C4⋊Q8—1222no—
424C4⋊Q8—1222no—
425#125—22no—
426#125—22no—
427#125—1222no—
428#125—1222no—
429C22⋊Q8—1228no—
430C22⋊Q8—1228no—
431C4×Q8—1228no—
432C4×Q8—1228no—
433#117—614no—
434#117—614no—
435#117—1222no—
436#117—1222no—
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—21no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—614no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4—2444no—
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C4⋊C4C22×D6111yescharacteristic
460C2×C4⋊C4C22×D6111yescharacteristic
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—1222no—
468C2×C42—1222no—
469C2×C42—1221no—
470C2×C42—1221no—
471C2×C42C22×D6111yescharacteristic
472#107—22no—
473#107—1222no—
474#107—1222no—
475#107—1222no—
476#106—22no—
477#106—1222no—
478#106—1222no—
479#106—1222no—
480#95—21no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484C24—611no—
485C2×Q8—1224no—
486C2×Q8—1224no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×Q8—2442no—
490C2×Q8—2442no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—612no—
500C22×C4—612no—
501C22×C4—612no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1224no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—2441no—
514C22×C4—2441no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4—2442no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523M16—1228no—
524M16—1228no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—612no—
530C2×C8—612no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8Q8×D6112yes—
536C2×C8Q8×D6112yes—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—42no—
545C4⋊C4—612no—
546C4⋊C4—612no—
547C4⋊C4—612no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4D4×D6112yes—
563C4⋊C4D4×D6112yes—
564C4⋊C4Q8×D6112yes—
565C22⋊C4—1224no—
566C22⋊C4—1224no—
567C42—21no—
568C42—21no—
569C42—1222no—
570C42—1222no—
571C42—1224no—
572C42—1224no—
573C42—1221no—
574C42—1221no—
575C23—612no—
576C23—612no—
577C23—612no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23C2×D4×D6111yescharacteristic
583Q8—2448no—
584Q8—2448no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—612no—
606C2×C4—614no—
607C2×C4—614no—
608C2×C4—614no—
609C2×C4—614no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—4884no—
620C2×C4—2444no—
621C2×C4—2444no—
622C2×C4—2448no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4C2×D4×D6111yescharacteristic
633C2×C4C2×D4×D6111yescharacteristic
634C2×C4C2×Q8×D6111yescharacteristic
635C2×C4C2×Q8×D6111yescharacteristic
636C2×C4C2×Q8×D6111yescharacteristic
637C2×C4#41×D6111yescharacteristic
638C8—22no—
639C8—22no—
640C8—1222no—
641C8—1222no—
642C22—614no—
643C22—1222no—
644C22—1222no—
645C22—1222no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22#120×D6111yescharacteristic
650C22C22⋊Q8×D6111yescharacteristic
651C22C4⋊Q8×D6111yescharacteristic
652C22AGL(1,ℤ/8)×D6111yescharacteristic
653C22#137×D6111yescharacteristic
654C22#1684055911yescharacteristic
655C22#1684055911yescharacteristic
656C4—41no—
657C4—42no—
658C4—41no—
659C4—42no—
660C4—42no—
661C4—81no—
662C4—614no—
663C4—1222no—
664C4—1222no—
665C4—4881no—
666C4—2442no—
667C4—2442no—
668C4—2442no—
669C4C22⋊Q8×D6111yescharacteristic
670C4C22⋊Q8×D6111yescharacteristic
671C4C4⋊Q8×D6111yescharacteristic
672C4C4⋊Q8×D6111yescharacteristic
673C2—1224no—
674C2#395×D6111yescharacteristic
675C2#2145506811yescharacteristic
676C2#2145507611yescharacteristic
677C2#2145561411yescharacteristic
678C2#2145562211yescharacteristic
679C2#2145636411yescharacteristic
680C2#2145636711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2059C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2059D61
hypocenterC61#23554
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
5. #21447158C2semidirect
6. #21447157C2semidirect
7. #21447154C2semidirect
8. #21447151C2semidirect
9. #21447150C2semidirect
10. #21447146C2semidirect
11. #21447145C2semidirect
15. C61×#2059C2semidirect
74. C2×#11758189Q8semidirect
75. C2×#11758189Q8semidirect
109. C4⋊C4×D61Q8semidirect
340. #2059D61semidirect
403. C61#23554semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168398032
C2×#168398012
#389×D612
#214471582
#214471572
#214471542
#214471512
#214471502
#214471462
#214471452
#214455372
#214455112
C61×#20592
#2355461