Finite Group Databasebeta

← #29696652

Subgroups — #29696652

Order 15616.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2C2×#16839891C211yesmaximal, characteristic, complemented
3#21448777C211yesmaximal, characteristic, complemented
4#21446793C211yesmaximal, characteristic, complemented
5#21446790C211yesmaximal, characteristic, complemented
6#21446778C211yesmaximal, characteristic, complemented
7#21446769C211yesmaximal, characteristic, complemented
8#21446765C211yesmaximal, characteristic, complemented
9#21445830C211yesmaximal, characteristic
10#21445828C211yesmaximal, characteristic
11#21445818C211yesmaximal, characteristic
12#21445647C211yesmaximal, characteristic
13#21445643C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#1987C211yesmaximal, characteristic, complemented
16C22×Q8×D61—21no—
17#16840427—24no—
18#16840427C418yescomplemented
19C2×#14519776C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519755—21no—
22C2×#14519755—21no—
23C2×#14519753—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×#14519739C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28#16839891—24no—
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839666—21no—
39#16839646—21no—
40#16839645—21no—
41#16839631C2211yescharacteristic
42#16839628C2211yescharacteristic
43#16839627C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4⋊Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C2×Q8×D61—28no—
66C2×Q8×D61—44no—
67C2×Q8×D61—44no—
68C22×D244C2311yescharacteristic
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519776—22no—
75#14519776—22no—
76#14519776C2×C414yes—
77#14519776C2×C414yes—
78#14519755—28no—
79#14519755—28no—
80#14519753—28no—
81C4×Dic122C2×C414yes—
82C4×Dic122D414yes—
83C4×D244—22no—
84C4×D244C2×C414yes—
85#14519739—22no—
86#14519739—22no—
87#14519739C2×C414yes—
88#14519739C2×C414yes—
89C2×C4⋊Dic61D411yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196D411yescharacteristic
98C2×#11758196D411yescharacteristic
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61C2311yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106#14519706—22no—
107#14519706—22no—
108#14519699—22no—
109#14519699—22no—
110#14519697—22no—
111#14519697Q814yescomplemented
112#14519688—22no—
113#14519687—22no—
114C8×Dic61—22no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121C22×Q8×C61—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4⋊Q8×C61—22no—
125C4⋊Q8×C61C2×C414yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133C4⋊C8×C61—22no—
134#95×C61—21no—
135C23×D61—21no—
136C2×Q8×C61—24no—
137C2×Q8×C61—42no—
138C2×Q8×C61—42no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153Q8×D61—432no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C22×C412yes—
158C2×D244C2×Q812yes—
159C2×D244#4112yes—
160C2×C4×D61—24no—
161C2×C4×D61—24no—
162C2×C4×D61—24no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—24no—
172C2×Dic122—24no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122C22⋊C414yes—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C4⋊C4×C61—24no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C4⋊C4×C61C22×C412yes—
192C4⋊C4×C61C2×Q812yescomplemented
193C42×C61—81no—
194C42×C61C22×C412yes—
195C42×C61C2×D412yes—
196#11758198—24no—
197#11758198—24no—
198#11758198—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61C22⋊C414yes—
205C4⋊Dic61C22×C412yes—
206#11758196—24no—
207#11758196—24no—
208#11758196—42no—
209#11758196—81no—
210#11758196C22⋊C414yes—
211#11758196C22⋊C414yes—
212C4×Dic61—42no—
213C4×Dic61—81no—
214C4×Dic61C22×C412yes—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418C2×Q812yescomplemented
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—22no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232Q8×C61—48no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
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×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61C2×C22⋊C412yes—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61AGL(1,ℤ/8)12yes—
277C2×Dic61AGL(1,ℤ/8)12yes—
278D244—22no—
279D244—22no—
280D244—22no—
281D244C4×Q814yes—
282C4×D61—48no—
283C4×D61—48no—
284C4×D61—48no—
285Dic122—28no—
286Dic122—48no—
287Dic122—48no—
288C8×C61—41no—
289C8×C61—41no—
290#92418—22no—
291#92418—41no—
292#22523—611nomaximal, Sylow
293C22×C61C22×C22⋊C411yescharacteristic
294C22×C61#54811yescharacteristic
295C22×C61D4×Q811yescharacteristic
296C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57811yescharacteristic
300C2×D61—24no—
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C4×C61—22no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61C2×C4×Q811yescharacteristic
314C4×C61#51911yescharacteristic
315C4×C61#54811yescharacteristic
316C4×C61D4×Q811yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—81no—
322Dic61—81no—
323Dic61#41814yes—
324C2×D4×Q8—611no—
325C2×#446—611no—
326#2355—611no—
327#1993—611no—
328#1993—611no—
329#1990—611no—
330#1988—611no—
331#1987—611no—
332#1987D6111yescharacteristic, complemented
333#1804—611no—
334#1804—611no—
335#1802—611no—
336#1762—611no—
337#1762—611no—
338C4×#102—611no—
339C2×C61Q8×C22⋊C411yescharacteristic
340C2×C61C2×#41811yescharacteristic
341C2×C61#289811yescharacteristic
342C2×C61#336711yescharacteristic
343C2×C61#337311yescharacteristic
344C2×C61#338411yescharacteristic
345C2×C61#338611yescharacteristic
346D61—44no—
347C23×Q8—1221no—
348D4×Q8—618no—
349D4×Q8—1224no—
350C2×C4⋊Q8—611no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8C2×D6111yescharacteristic
353C2×C22⋊Q8—1221no—
354C2×C22⋊Q8—1221no—
355C2×C22⋊Q8—1221no—
356C2×C4×Q8—611no—
357C2×C4×D4—611no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360#446—1224no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8C2×D6111yescharacteristic
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×#97—611no—
369C2×#97—611no—
370C2×C4×C8—611no—
371#391—1221no—
372#387—1221no—
373#387—1221no—
374#384—611no—
375#382—611no—
376#382—611no—
377C4×C4⋊C4—611no—
378C4×C4⋊C4—611no—
379C4×C4⋊C4—1221no—
380C4×C4⋊C4C2×D6111yescharacteristic
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#340C2×D6111yescharacteristic
393C61#2252311yescharacteristic, complemented, Sylow
394C22×Q8—21no—
395C22×Q8—611no—
396C22×Q8—1221no—
397C22×Q8—1221no—
398C22×Q8—1228no—
399C22×Q8—2444no—
400C22×Q8—2444no—
401C22×D4—611no—
402C23×C4—1221no—
403C23×C4—1221no—
404C23×C4—1221no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—22no—
410C4⋊Q8—614no—
411C4⋊Q8—614no—
412C4⋊Q8—1222no—
413C4⋊Q8—1222no—
414C4⋊Q8C4×D6114yes—
415C22⋊Q8—1228no—
416C22⋊Q8—1228no—
417C22⋊Q8—1228no—
418C4×Q8—614no—
419C4×Q8—614no—
420C4×D4—614no—
421C4×D4—614no—
422C4×D4—614no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—21no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42C22×D6111yescharacteristic
450C4⋊C8—22no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453#102—614no—
454#102—1222no—
455#102—1222no—
456#102—1222no—
457#97—1222no—
458#97—1222no—
459C4×C8—1222no—
460#95—21no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—1221no—
468C2×Q8—24no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—614no—
472C2×Q8—1222no—
473C2×Q8—1222no—
474C2×Q8—1224no—
475C2×Q8—1224no—
476C2×Q8—2442no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×Q8—24432no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1221no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—614no—
529C4⋊C4—614no—
530C4⋊C4—614no—
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—4881no—
540C4⋊C4—2442no—
541C4⋊C4C2×C4×D6112yes—
542C4⋊C4C2×C4×D6112yes—
543C4⋊C4Q8×D6112yescomplemented
544C22⋊C4—1224no—
545C22⋊C4—1224no—
546C22⋊C4—1224no—
547C42—81no—
548C42—612no—
549C42—612no—
550C42—612no—
551C42—612no—
552C42—4881no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42D4×D6112yes—
556C23—1222no—
557C23—1222no—
558C23—1222no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23C2×D4×D6111yescharacteristic
564Q8—48no—
565Q8—1228no—
566Q8—2448no—
567Q8—2448no—
568D4—614no—
569D4—1222no—
570D4—1222no—
571D4—1222no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
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—612no—
589C2×C4—612no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—2448no—
602C2×C4—2448no—
603C2×C4—2448no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4C22⋊C4×D6112yes—
613C2×C4C22⋊C4×D6112yes—
614C2×C4C22×C4×D6111yescharacteristic
615C2×C4C2×D4×D6111yescharacteristic
616C2×C4C2×Q8×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4#1451982411yescharacteristic
619C2×C4#1451982511yescharacteristic
620C8—41no—
621C8—41no—
622C8—1222no—
623C8—2441no—
624C22—1224no—
625C22—2442no—
626C22—2442no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22C2×C22⋊C4×D6111yescharacteristic
632C22#1684042211yescharacteristic
633C22#1684042511yescharacteristic
634C22AGL(1,ℤ/8)×D6111yescharacteristic
635C22AGL(1,ℤ/8)×D6111yescharacteristic
636C22#1684056811yescharacteristic
637C22#1684057511yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—81no—
644C4—614no—
645C4—1222no—
646C4—1222no—
647C4—1222no—
648C4—4881no—
649C4—4881no—
650C4—2442no—
651C4C4×Q8×D6111yescharacteristic
652C4#1684023811yescharacteristic
653C4#1684042011yescharacteristic
654C4#1684042711yescharacteristic
655C2—2444no—
656C2#2144865311yescharacteristic
657C2#418×D6111yescharacteristic
658C2#2145434711yescharacteristic
659C2#2145854011yescharacteristic
660C2#2145861411yescharacteristic
661C2#2145871311yescharacteristic
662C2#2145874311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1987C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1987D61
hypocenterC61#22523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
2. C2×#16839891C2semidirect
3. #21448777C2semidirect
4. #21446793C2semidirect
5. #21446790C2semidirect
6. #21446778C2semidirect
7. #21446769C2semidirect
8. #21446765C2semidirect
15. C61×#1987C2semidirect
18. #16840427C4semidirect
111. #14519697Q8semidirect
192. C4⋊C4×C61C2×Q8semidirect
223. C2×#92418C2×Q8semidirect
332. #1987D61semidirect
393. C61#22523semidirect
543. C4⋊C4Q8×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
C2×#168398912
#214487772
#214467932
#214467902
#214467782
#214467692
#214467652
#214458302
#214458282
#214458182
#214456472
#214456432
#214456332
C61×#19872
#2252361