Finite Group Databasebeta

← #29716217

Subgroups — #29716217

Order 15616.

1012 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×#16840066C211yesmaximal, characteristic, complemented
2C2×#16840054C211yesmaximal, characteristic, complemented
3#21448501C211yesmaximal, characteristic
4#21447154C211yesmaximal, characteristic, complemented
5#21447144C211yesmaximal, characteristic, complemented
6#21446967C211yesmaximal, characteristic, complemented
7#21446957C211yesmaximal, characteristic, complemented
8#21446777C211yesmaximal, characteristic, complemented
9#21446772C211yesmaximal, characteristic, complemented
10#21446131C211yesmaximal, characteristic
11#21446126C211yesmaximal, characteristic
12#21446018C211yesmaximal, characteristic
13#21446007C211yesmaximal, characteristic
14#21445803C211yesmaximal, characteristic
15C2×C61×#544C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18#16840066—24no—
19#16840054—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839660C2211yescharacteristic
34#16839657—21no—
35#16839656—21no—
36#16839650—21no—
37#16839643—21no—
38#16839622C2211yescharacteristic
39C4×C4⋊Dic61C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C61×#544—24no—
51C61×#544—24no—
52C2×C4⋊Q8×C61—21no—
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4⋊Q8×C61C2211yescharacteristic
55C2×#125×C61—21no—
56C2×#125×C61C2211yescharacteristic
57C2×#125×C61C2211yescharacteristic
58C2×C22⋊Q8×C61—21no—
59C2×C22⋊Q8×C61—21no—
60C2×#117×C61C2211yescharacteristic
61C2×#117×C61C2211yescharacteristic
62C22×C4⋊C4×C61C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519756—22no—
69#14519756Q814yescomplemented
70C4×Dic122—22no—
71C4×Dic122—22no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61—21no—
82#14519710—22no—
83#14519710—22no—
84#14519710—22no—
85#14519708—22no—
86C4⋊#92418—22no—
87C4⋊#92418—22no—
88#14519700—22no—
89#14519700—22no—
90#14519700—22no—
91#14519700—22no—
92C22⋊#92418—22no—
93C22⋊#92418—22no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99C22×Q8×C61—21no—
100C22×Q8×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C4⋊Q8×C61—28no—
104C4⋊Q8×C61—22no—
105C4⋊Q8×C61—22no—
106C4⋊Q8×C61—22no—
107C4⋊Q8×C61D414yes—
108#125×C61—28no—
109#125×C61—22no—
110#125×C61—22no—
111#125×C61—22no—
112#125×C61D414yes—
113C22⋊Q8×C61—28no—
114C22⋊Q8×C61—28no—
115#117×C61—22no—
116#117×C61—22no—
117#117×C61—22no—
118#117×C61—22no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61—22no—
127C2×C4⋊C4×C61—22no—
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C4⋊C4×C61D411yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61D411yescharacteristic
136C2×C22⋊C4×C61D411yescharacteristic
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139C2×C42×C61C2311yescharacteristic
140C24×C61C2×D411yescharacteristic
141C2×Q8×C61—24no—
142C2×Q8×C61—24no—
143C2×Q8×C61—81no—
144C2×Q8×C61—81no—
145C2×Q8×C61—81no—
146C2×Q8×C61—81no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—24no—
156C22×C4×C61—81no—
157C22×C4×C61—81no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×Q812yes—
164C22×C4×C61#4112yes—
165C22×C4×C61C2411yescharacteristic
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C2×Q812yes—
176C4⋊C4×C61—24no—
177C4⋊C4×C61—24no—
178C4⋊C4×C61—24no—
179C4⋊C4×C61—24no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—24no—
183C4⋊C4×C61—22no—
184C4⋊C4×C61—22no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C2×D412yes—
190C4⋊C4×C61C2×D412yes—
191C4⋊C4×C61C2×D412yes—
192C4⋊C4×C61C2×D412yes—
193C22⋊C4×C61—24no—
194C22⋊C4×C61—22no—
195C22⋊C4×C61—22no—
196C42×C61—24no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61C2×D412yes—
200C42×C61C2×D412yes—
201C22⋊Dic61—24no—
202C22⋊Dic61—81no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61C2×Q812yes—
209#11758196—24no—
210#11758196—42no—
211#11758196—81no—
212C4×Dic61—42no—
213C4×Dic61—81no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61C22⋊Q812yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—84no—
235Q8×C61—84no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—84no—
265C2×C4×C61#12012yes—
266C2×C4×C61#12012yes—
267C2×C4×C61C22⋊Q812yes—
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×Q811yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C612−1+411yescharacteristic
273C2×C4×C612−1+411yescharacteristic
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286Dic122—22no—
287Dic122—22no—
288Dic122—41no—
289Dic122—41no—
290#92418—41no—
291#92418—41no—
292#92418—41no—
293#92418—41no—
294#24311—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C2×#12011yescharacteristic
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61#53611yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C22×C61#57811yescharacteristic
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—42no—
317C4×C61—42no—
318C4×C61#54811yescharacteristic
319C4×C61#54811yescharacteristic
320C4×C61D4×Q811yescharacteristic
321C4×C61D4×Q811yescharacteristic
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326C2×#544D6111yescharacteristic
327C2×#479—611no—
328C2×#477—611no—
329#2323—611no—
330#2059—611no—
331#2057—611no—
332#2026—611no—
333#2024—611no—
334#1989—611no—
335#1988—611no—
336#1873—611no—
337#1872—611no—
338#1854—611no—
339#1852—611no—
340#1798—611no—
341C2×C61—42no—
342C2×C61#240911yescharacteristic
343C2×C61#301111yescharacteristic
344C2×C61#301211yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#308311yescharacteristic
347C2×C61#324211yescharacteristic
348C2×C61#324311yescharacteristic
349#544—24no—
350#544—24no—
351C2×C4⋊Q8—21no—
352C2×C4⋊Q8C2×D6111yescharacteristic
353C2×C4⋊Q8C2×D6111yescharacteristic
354C2×#125—21no—
355C2×#125C2×D6111yescharacteristic
356C2×#125C2×D6111yescharacteristic
357C2×C22⋊Q8—21no—
358C2×C22⋊Q8—21no—
359C2×C22⋊Q8—611no—
360C2×C4×Q8—611no—
361C2×#117C2×D6111yescharacteristic
362C2×#117C2×D6111yescharacteristic
363C22×C4⋊C4C2×D6111yescharacteristic
364#479—1224no—
365#477—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—611no—
369C2×#106—611no—
370C2×C4⋊C8—611no—
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378#391—611no—
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#380—611no—
385C4×C4⋊C4—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#336—611no—
395#336—611no—
396C61#2431111yescharacteristic, complemented, Sylow
397C22×Q8—21no—
398C22×Q8—21no—
399C22×Q8—611no—
400C23×C4—21no—
401C23×C4C22×D6111yescharacteristic
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—611no—
406C4⋊Q8—28no—
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8—22no—
410C4⋊Q8C22⋊D6114yes—
411#125—28no—
412#125—22no—
413#125—22no—
414#125—22no—
415#125C22⋊D6114yes—
416C22⋊Q8—28no—
417C22⋊Q8—28no—
418C22⋊Q8—614no—
419C22⋊Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422#117—22no—
423#117—22no—
424#117—22no—
425#117—22no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—22no—
434C2×C4⋊C4—22no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4C22⋊D6111yescharacteristic
442C2×C4⋊C4C22⋊D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4C22⋊D6111yescharacteristic
451C2×C22⋊C4C22⋊D6111yescharacteristic
452C2×C42—21no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42C22×D6111yescharacteristic
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#106—1222no—
461C4⋊C8—1222no—
462C4⋊C8—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467C22⋊C8—1222no—
468C22⋊C8—1222no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24C2×C22⋊D6111yescharacteristic
475C2×Q8—24no—
476C2×Q8—24no—
477C2×Q8—81no—
478C2×Q8—81no—
479C2×Q8—81no—
480C2×Q8—81no—
481C2×Q8—612no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—24no—
496C22×C4—81no—
497C22×C4—81no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4Q8×D6112yes—
503C22×C4#1175822612yes—
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C2×C22⋊D6111yescharacteristic
506C22×C4C2×C22⋊D6111yescharacteristic
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—24no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—612no—
536C4⋊C4—1221no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—2442no—
544C4⋊C4C2×C22⋊D6112yes—
545C4⋊C4C2×C22⋊D6112yes—
546C4⋊C4C2×C22⋊D6112yes—
547C4⋊C4C2×C22⋊D6112yes—
548C22⋊C4—24no—
549C22⋊C4—22no—
550C22⋊C4—22no—
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C42—24no—
554C42—21no—
555C42—21no—
556C42—4881no—
557C42—2442no—
558C42C2×C22⋊D6112yes—
559C42C2×C22⋊D6112yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975312yes—
567C23C22×C22⋊D6111yescharacteristic
568Q8—84no—
569Q8—84no—
570Q8—1222no—
571Q8—1222no—
572Q8—2441no—
573Q8—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
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—84no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—2441no—
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#1451974612yes—
616C2×C4#1451974612yes—
617C2×C4#1451975312yes—
618C2×C4C22×C22⋊D6111yescharacteristic
619C2×C4C22×C22⋊D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4C2×#1175822611yescharacteristic
622C2×C4#1451982511yescharacteristic
623C2×C4#1451982511yescharacteristic
624C8—2441no—
625C8—2441no—
626C8—2441no—
627C8—2441no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1451974611yescharacteristic
636C22C2×#1451975311yescharacteristic
637C22#1684033611yescharacteristic
638C22#1684056511yescharacteristic
639C22#1684057311yescharacteristic
640C22#1684057711yescharacteristic
641C22#1684057711yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—4881no—
652C4—4881no—
653C4—2442no—
654C4—2442no—
655C4#1684042311yescharacteristic
656C4#1684042311yescharacteristic
657C4Q8×C22⋊D6111yescharacteristic
658C4Q8×C22⋊D6111yescharacteristic
659C2—42no—
660C2#2144917811yescharacteristic
661C2#2145511411yescharacteristic
662C2#2145512311yescharacteristic
663C2#2145565511yescharacteristic
664C2#2145566611yescharacteristic
665C2#2145708711yescharacteristic
666C2#2145710211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#544C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#544D61
hypocenterC61#24311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
1. C2×#16840066C2semidirect
2. C2×#16840054C2semidirect
4. #21447154C2semidirect
5. #21447144C2semidirect
6. #21446967C2semidirect
7. #21446957C2semidirect
8. #21446777C2semidirect
9. #21446772C2semidirect
69. #14519756Q8semidirect
396. C61#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#168400662
C2×#168400542
#214485012
#214471542
#214471442
#214469672
#214469572
#214467772
#214467722
#214461312
#214461262
#214460182
#214460072
#214458032
C2×C61×#5442
#2431161