Finite Group Databasebeta

← #29711111

Subgroups — #29711111

Order 15616.

1734 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11758202C211yesmaximal, characteristic, complemented
2C2×#16839918C212yesmaximal, complemented
3C2×#16839917C211yesmaximal, characteristic, complemented
4#21449184C211yesmaximal, characteristic
5#21446919C212yesmaximal, complemented
6#21446827C212yesmaximal, complemented
7#21446243C211yesmaximal, characteristic, complemented
8#21446160C212yesmaximal, complemented
9#21446112C211yesmaximal, characteristic
10#21445754C211yesmaximal, characteristic, complemented
11C2×D4×Q8×C61C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758202—28no—
14C22×#11758202—24no—
15C22×#11758202C2212yes—
16C22×#11758202C2212yes—
17C22×#11758202C2212yes—
18C23×#92418C2211yescharacteristic
19C2×#14519756C2212yes—
20#16839918—28no—
21#16839917—24no—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519701C2212yes—
25C2×#14519700C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27C2×C22⋊#92418C2212yes—
28#16839682—21no—
29#16839674—21no—
30#16839668C2212yes—
31#16839642—21no—
32#16839634—21no—
33#16839622C2211yescharacteristic
34#16839524C2212yes—
35#16839517C2212yes—
36C23×Q8×C61C2211yescharacteristic
37C23×Q8×C61C2211yescharacteristic
38D4×Q8×C61—24no—
39D4×Q8×C61—24no—
40C2×C4⋊Q8×C61—21no—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61—21no—
43C2×C22⋊Q8×C61—21no—
44C2×C22⋊Q8×C61C2211yescharacteristic
45C2×C22⋊Q8×C61C2211yescharacteristic
46C2×C4×Q8×C61C2211yescharacteristic
47C2×C4×D4×C61—21no—
48C2×C4×D4×C61C2211yescharacteristic
49C22×Dic122—22no—
50C22×Dic122—24no—
51C22×Dic122—22no—
52C22×Dic122—21no—
53C22×Dic122—42no—
54C22×Dic122C2312yes—
55C23×Dic61—21no—
56C2×#11758202—232no—
57C2×#11758202—24no—
58C2×#11758202—24no—
59C2×#11758202—24no—
60C2×#11758202—24no—
61C2×#11758202—24no—
62C2×#11758202—24no—
63C2×#11758202—24no—
64C2×#11758202—42no—
65C2×#11758202D418yescomplemented
66C22×#92418—22no—
67C22×#92418—24no—
68C22×#92418C2312yes—
69C22×#92418C2312yes—
70#14519756—24no—
71#14519756—42no—
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—22no—
74C2×C22⋊Dic61—22no—
75C2×C22⋊Dic61—21no—
76#14519701—24no—
77#14519701—24no—
78#14519701—42no—
79#14519700—24no—
80C22⋊#92418—22no—
81C22⋊#92418—24no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61—22no—
87C22×Q8×C61—21no—
88C22×Q8×C61—21no—
89C22×Q8×C61—22no—
90C22×Q8×C61—24no—
91C22×Q8×C61—24no—
92C22×Q8×C61—42no—
93C22×Q8×C61—42no—
94C22×Q8×C61D414yes—
95C22×Q8×C61C2311yescharacteristic
96C22×Q8×C61C2311yescharacteristic
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61D411yescharacteristic
99C23×C4×C61—21no—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C23×C4×C61C2311yescharacteristic
103C4⋊Q8×C61—28no—
104C4⋊Q8×C61—22no—
105C4⋊Q8×C61—22no—
106C22⋊Q8×C61—28no—
107C22⋊Q8×C61—28no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C22⋊Q8×C61D414yes—
112C4×Q8×C61—22no—
113C4×Q8×C61—22no—
114C4×D4×C61—28no—
115C4×D4×C61—22no—
116C4×D4×C61—22no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C24×C61C2×D411yescharacteristic
131C24×C61C2×D411yescharacteristic
132C2×Q8×C61—24no—
133C2×Q8×C61—24no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—22no—
138C2×Q8×C61—28no—
139C2×Q8×C61—22no—
140C2×Q8×C61—22no—
141C2×Q8×C61—22no—
142C2×Q8×C61—22no—
143C2×Q8×C61—44no—
144C2×Q8×C61—44no—
145C2×Q8×C61—44no—
146C2×Q8×C61—48no—
147C2×Q8×C61—48no—
148C2×Q8×C61—44no—
149C2×Q8×C61—44no—
150C2×Q8×C61C2×D412yes—
151C2×Q8×C61C2×D412yes—
152C2×Q8×C61C2×D414yes—
153C2×Q8×C61C2×D412yes—
154C2×D4×C61—22no—
155C2×D4×C61—22no—
156C2×D4×C61Q1614yescomplemented
157C22×C4×C61—21no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—24no—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—21no—
167C22×C4×C61—22no—
168C22×C4×C61—22no—
169C22×C4×C61—44no—
170C22×C4×C61—44no—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61C2×D412yes—
175C22×C4×C61C2×D412yes—
176C22×C4×C61C2×D411yescharacteristic
177C22×C4×C61C2411yescharacteristic
178C22×Dic61—22no—
179C22×Dic61—22no—
180C22×Dic61—44no—
181C22×Dic61—42no—
182C2×Dic122—28no—
183C2×Dic122—22no—
184C2×Dic122—22no—
185C2×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122—48no—
189C2×Dic122—44no—
190C2×Dic122—44no—
191C2×Dic122—42no—
192C2×Dic122C2×D414yes—
193C4⋊C4×C61—24no—
194C4⋊C4×C61—24no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—24no—
197C4⋊C4×C61—24no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61C2×D412yes—
201C22⋊C4×C61—24no—
202C22⋊C4×C61—24no—
203C22⋊C4×C61—22no—
204C22⋊C4×C61—22no—
205C42×C61—24no—
206C42×C61—21no—
207C42×C61—21no—
208C22⋊Dic61—44no—
209C22⋊Dic61—42no—
210#11758202—216no—
211#11758202—216no—
212#11758202—48no—
213C4⋊Dic61—22no—
214C4⋊Dic61—22no—
215#11758196—44no—
216C2×#92418—28no—
217C2×#92418—22no—
218C2×#92418—22no—
219C2×#92418—22no—
220C2×#92418—22no—
221C2×#92418—22no—
222C2×#92418C2×D414yes—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—21no—
226C23×C61—22no—
227C23×C61—21no—
228C23×C61—22no—
229C23×C61#12012yes—
230C23×C61C2×Q1612yes—
231C23×C61C2×Q1612yescomplemented
232C23×C61#13712yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—22no—
235Q8×C61—24no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—24no—
239Q8×C61—48no—
240Q8×C61—48no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—42no—
244Q8×C61#12014yes—
245D4×C61—28no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—24no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—24no—
254C2×C4×C61—21no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—22no—
260C2×C4×C61—22no—
261C2×C4×C61—48no—
262C2×C4×C61—48no—
263C2×C4×C61—42no—
264C2×C4×C61—42no—
265C2×C4×C61—42no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—42no—
269C2×C4×C61#12012yes—
270C2×C4×C61C2×Q1612yescomplemented
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D412yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61H(5,𝔽2)12yes—
275C2×Dic61—24no—
276C2×Dic61—48no—
277C2×Dic61—42no—
278C2×Dic61—42no—
279C2×Dic61—42no—
280C2×Dic61—42no—
281C2×Dic61—42no—
282Dic122—24no—
283Dic122—24no—
284Dic122—48no—
285Dic122—42no—
286Dic122—42no—
287#92418—24no—
288#92418—42no—
289#23856—611nomaximal, Sylow
290C22×C61—22no—
291C22×C61—22no—
292C22×C61—24no—
293C22×C61—24no—
294C22×C61—22no—
295C22×C61—24no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—42no—
299C22×C61#45114yescomplemented
300C22×C61C2×#12011yescharacteristic
301C22×C61D4212yes—
302C22×C61C22×Q1611yescharacteristic
303C22×C61C2×#13711yescharacteristic
304C22×C61#57812yes—
305C4×C61—22no—
306C4×C61—24no—
307C4×C61—22no—
308C4×C61—41no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—42no—
313C4×C61C2×#12011yescharacteristic
314C4×C61#53411yescharacteristic
315C4×C61D4212yes—
316Dic61—44no—
317Dic61—82no—
318C23×Q16—611no—
319C2×D4×Q8D6111yescharacteristic
320C2×#451—612no—
321C2×#451—611no—
322#2409—611no—
323#2016—612no—
324#2001—612no—
325#1892—611no—
326#1878—612no—
327#1869—611no—
328#1787—611no—
329C2×C61—24no—
330C2×C61—42no—
331C2×C61#240311yescharacteristic
332C2×C61C2×#45111yescharacteristic
333C2×C61#301111yescharacteristic
334C2×C61#327012yes—
335C2×C61D4×Q1612yes—
336C23×Q8—611no—
337C23×Q8C2×D6111yescharacteristic
338C23×Q8C2×D6111yescharacteristic
339C22×Q16—612no—
340C22×Q16—612no—
341C22×Q16—612no—
342C22×Q16—1228no—
343C22×Q16—1224no—
344C23×C8—611no—
345D4×Q8—24no—
346D4×Q8—24no—
347C2×C4⋊Q8—21no—
348C2×C4⋊Q8C2×D6111yescharacteristic
349C2×C22⋊Q8—21no—
350C2×C22⋊Q8—21no—
351C2×C22⋊Q8—612no—
352C2×C22⋊Q8C2×D6111yescharacteristic
353C2×C22⋊Q8C2×D6111yescharacteristic
354C2×C4×Q8C2×D6111yescharacteristic
355C2×C4×D4—21no—
356C2×C4×D4C2×D6111yescharacteristic
357#451—1228no—
358#451—1224no—
359C2×#103—612no—
360C2×#103—612no—
361C2×#103—612no—
362C2×#103—612no—
363C2×C22⋊C8—612no—
364C2×C22⋊C8—611no—
365#395—1221no—
366#393—1221no—
367#391—612no—
368#386—1221no—
369#385—1221no—
370#380—611no—
371#340—612no—
372#336—612no—
373C61#2385611yescharacteristic, complemented, Sylow
374C22×Q8—22no—
375C22×Q8—22no—
376C22×Q8—21no—
377C22×Q8—21no—
378C22×Q8—22no—
379C22×Q8—24no—
380C22×Q8—24no—
381C22×Q8—42no—
382C22×Q8—42no—
383C22×Q8—612no—
384C22×Q8—1224no—
385C22×Q8—1221no—
386C22×Q8—1222no—
387C22×Q8—1222no—
388C22×Q8—2442no—
389C22×Q8C22⋊D6114yes—
390C22×Q8C22×D6111yescharacteristic
391C22×Q8C22×D6111yescharacteristic
392C22×Q8C22×D6111yescharacteristic
393C22×D4C22⋊D6111yescharacteristic
394C23×C4—21no—
395C23×C4—21no—
396C23×C4—1221no—
397C23×C4C22×D6111yescharacteristic
398C23×C4C22×D6111yescharacteristic
399C2×Q16—618no—
400C2×Q16—1224no—
401C2×Q16—1224no—
402C2×Q16—12232no—
403C2×Q16—1224no—
404C2×Q16—1224no—
405C2×Q16—1224no—
406C2×Q16—1224no—
407C2×Q16—1224no—
408C2×Q16—2442no—
409C22×C8—612no—
410C22×C8—612no—
411C22×C8—1224no—
412C22×C8—1222no—
413C4⋊Q8—28no—
414C4⋊Q8—22no—
415C4⋊Q8—22no—
416C22⋊Q8—28no—
417C22⋊Q8—28no—
418C22⋊Q8—22no—
419C22⋊Q8—22no—
420C22⋊Q8—22no—
421C22⋊Q8—1224no—
422C22⋊Q8—2442no—
423C22⋊Q8C22⋊D6114yes—
424C4×Q8—22no—
425C4×Q8—22no—
426C4×D4—28no—
427C4×D4—22no—
428C4×D4—22no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—612no—
435C2×C4⋊C4—1222no—
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—1222no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4C22⋊D6111yescharacteristic
443C2×C22⋊C4C22⋊D6111yescharacteristic
444C2×C42—21no—
445C2×C42C22×D6111yescharacteristic
446#103—1224no—
447#103—1224no—
448#103—1224no—
449#103—2442no—
450C22⋊C8—1224no—
451C22⋊C8—1222no—
452#95—1221no—
453#95—1222no—
454#95—1222no—
455C24C2×C22⋊D6111yescharacteristic
456C24C2×C22⋊D6111yescharacteristic
457C2×Q8—24no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—24no—
463C2×Q8—28no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—44no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—48no—
472C2×Q8—48no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—614no—
476C2×Q8—1228no—
477C2×Q8—1222no—
478C2×Q8—1222no—
479C2×Q8—1222no—
480C2×Q8—1222no—
481C2×Q8—1222no—
482C2×Q8—2444no—
483C2×Q8—2448no—
484C2×Q8—2444no—
485C2×Q8—2442no—
486C2×Q8D4×D6114yes—
487C2×Q8C2×C22⋊D6112yes—
488C2×Q8C2×C22⋊D6112yes—
489C2×Q8C2×C22⋊D6112yes—
490C2×D4—22no—
491C2×D4—22no—
492C2×D4#1175820214yescomplemented
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—2444no—
510C22×C4—2442no—
511C22×C4C2×C22⋊D6111yescharacteristic
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6112yes—
514C22×C4C2×C22⋊D6112yes—
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518Q16—12216no—
519Q16—12216no—
520Q16—2448no—
521C2×C8—614no—
522C2×C8—1228no—
523C2×C8—1222no—
524C2×C8—1222no—
525C2×C8—1222no—
526C2×C8—1222no—
527C2×C8—1222no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—21no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—2444no—
538C4⋊C4C2×C22⋊D6112yes—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—2444no—
544C22⋊C4—2442no—
545C42—24no—
546C42—21no—
547C42—21no—
548C23—22no—
549C23—22no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23#1451974612yes—
555C23C2×#1175820212yes—
556C23C2×#1175820212yescomplemented
557C23#1451980812yes—
558C23C22×C22⋊D6111yescharacteristic
559Q8—22no—
560Q8—24no—
561Q8—22no—
562Q8—22no—
563Q8—24no—
564Q8—48no—
565Q8—48no—
566Q8—41no—
567Q8—41no—
568Q8—42no—
569Q8—1224no—
570Q8—1224no—
571Q8—2448no—
572Q8—2442no—
573Q8—2442no—
574Q8#1451974514yes—
575D4—28no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—1224no—
600C2×C4—2442no—
601C2×C4—2442no—
602C2×C4—2442no—
603C2×C4—2448no—
604C2×C4—2442no—
605C2×C4—2442no—
606C2×C4#1451974612yes—
607C2×C4C2×#1175820212yescomplemented
608C2×C4C2×D4×D6112yes—
609C2×C4C22×C22⋊D6111yescharacteristic
610C2×C4C22×C22⋊D6111yescharacteristic
611C2×C4#1451982212yes—
612C8—1224no—
613C8—2442no—
614C22—22no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—24no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22#1683991814yescomplemented
624C22C2×#1451974611yescharacteristic
625C22D4×C22⋊D6112yes—
626C22C22×#1175820211yescharacteristic
627C22C2×#1451980811yescharacteristic
628C22#1684057712yes—
629C4—22no—
630C4—22no—
631C4—24no—
632C4—41no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—42no—
637C4—4882no—
638C4—2444no—
639C4C2×#1451974511yescharacteristic
640C4#1684032111yescharacteristic
641C4D4×C22⋊D6112yes—
642C2—24no—
643C2—42no—
644C2#2144912911yescharacteristic
645C2C2×#1683991811yescharacteristic
646C2#2145511411yescharacteristic
647C2#2145739012yes—
648C2D4×#1175820212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×D4×Q8×C61C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenterC2×D4×Q8D61
hypocenterC61#23856
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C23×#11758202C2semidirect
2. C2×#16839918C2semidirect
3. C2×#16839917C2semidirect
5. #21446919C2semidirect
6. #21446827C2semidirect
7. #21446243C2semidirect
8. #21446160C2semidirect
10. #21445754C2semidirect
65. C2×#11758202D4semidirect
156. C2×D4×C61Q16semidirect
231. C23×C61C2×Q16semidirect
270. C2×C4×C61C2×Q16semidirect
299. C22×C61#451semidirect
373. C61#23856semidirect
492. C2×D4#11758202semidirect
556. C23C2×#11758202semidirect
607. C2×C4C2×#11758202semidirect
623. C22#16839918semidirect

Maximal subgroups

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

subgroupindex
C23×#117582022
C2×#168399182
C2×#168399172
#214491842
#214469192
#214468272
#214462432
#214461602
#214461122
#214457542
C2×D4×Q8×C612
#2385661