Finite Group Databasebeta

← #29647148

Subgroups — #29647148

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840296C211yesmaximal, characteristic
2C4×#14519736C211yesmaximal, characteristic, complemented
3#21444345C214yesmaximal, complemented
4#21444340C212yesmaximal, complemented
5#21444336C212yesmaximal
6#21444306C214yesmaximal, complemented
7C61×#2920C211yesmaximal, characteristic, complemented
8#16840296—21no—
9#16840296—21no—
10#16840296—21no—
11#16840296—21no—
12#16840296C412yescomplemented
13#16840296C412yescomplemented
14#16840296C2212yes—
15#16840296C2212yes—
16C2×#14519771—21no—
17C2×#14519769C411yescharacteristic, complemented
18C2×#14519765C411yescharacteristic, complemented
19C2×#14519760—21no—
20C2×#14519756C2211yescharacteristic
21C2×#14519751C2211yescharacteristic
22C2×C4×Dic122C411yescharacteristic, complemented
23C2×C4×D244C411yescharacteristic, complemented
24C2×C4×C22⋊D61—21no—
25C2×#14519736—21no—
26C2×#14519734—21no—
27C4×C4⋊Dic61—21no—
28C4×C4⋊Dic61C2212yescomplemented
29C4×C22⋊Dic61—21no—
30C4×C22⋊Dic61C2212yes—
31#16839608C2212yescomplemented
32C42×Dic61—21no—
33#16839500C2218yescomplemented
34#16839497C2218yes—
35C22×C42×C61C2211yescharacteristic
36C61×#432—21no—
37C61×#432C2212yescomplemented
38C61×#431—21no—
39C61×#431C2212yescomplemented
40C61×#423—21no—
41C61×#407C2212yes—
42C4×M16×C61—21no—
43C22×Dic122C2×C411yescharacteristic
44C22×D244C2×C411yescharacteristic
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47#14519771—22no—
48#14519771—22no—
49#14519771—22no—
50#14519771—22no—
51#14519769—21no—
52#14519769—21no—
53#14519769C2×C412yes—
54#14519769C2×C412yes—
55#14519765—21no—
56#14519765—21no—
57#14519765C2×C412yes—
58#14519765C2×C412yes—
59#14519760—22no—
60#14519760—22no—
61#14519760—22no—
62#14519760—22no—
63#14519756—21no—
64#14519756—21no—
65#14519756C2×C412yes—
66#14519756C2312yes—
67#14519751—21no—
68#14519751—21no—
69#14519751C2×C412yes—
70#14519751C2312yes—
71C4×Dic122—21no—
72C4×Dic122—21no—
73C4×Dic122C2×C412yes—
74C4×Dic122C2×C412yes—
75C4×D244—21no—
76C4×D244—21no—
77C4×D244C2×C412yes—
78C4×D244C2×C412yes—
79C4×C22⋊D61—22no—
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82C4×C22⋊D61—22no—
83#14519736—28no—
84#14519734—22no—
85#14519734—22no—
86#14519734—22no—
87#14519734—22no—
88C2×C4⋊Dic61—22no—
89C2×C4⋊Dic61C2×C411yescharacteristic
90C2×C4⋊Dic61C2×C411yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×C22⋊Dic61—22no—
96C2×C22⋊Dic61—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—22no—
105#14519684—22no—
106#14519684C2314yes—
107C23×C4×C61D411yescharacteristic
108C23×C4×C61D411yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C2×M16×C61—22no—
111C2×C42×C61C2×C411yescharacteristic
112C2×C42×C61C2×C411yescharacteristic
113C2×C42×C61C2×C411yescharacteristic
114C2×C42×C61C2×C411yescharacteristic
115C2×C42×C61D411yescharacteristic
116C2×C42×C61D411yescharacteristic
117C2×C42×C61D411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—22no—
124C4⋊C8×C61—22no—
125C22⋊C8×C61—22no—
126C22⋊C8×C61C2314yes—
127#97×C61—22no—
128C4×C8×C61—22no—
129C24×C61C2×D411yescharacteristic
130C23×D61—21no—
131C22×C4×C61—24no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—22no—
153C2×C22⋊D61—22no—
154C2×C22⋊D61—44no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—24no—
159C2×D244—21no—
160C2×D244—42no—
161C2×D244C22×C412yes—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—44no—
165C2×Dic122—21no—
166C2×Dic122—42no—
167C2×Dic122C22×C412yes—
168M16×C61—44no—
169C2×C8×C61—24no—
170C42×C61C22⋊C411yescharacteristic
171C42×C61C22⋊C411yescharacteristic
172C42×C61C22⋊C411yescharacteristic
173C42×C61C22⋊C411yescharacteristic
174C42×C61C22⋊C411yescharacteristic
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22×C411yescharacteristic
179C42×C61C22×C411yescharacteristic
180C42×C61C22×C411yescharacteristic
181C42×C61C22×C411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C22⋊Dic61—28no—
187C22⋊Dic61—22no—
188C22⋊Dic61—22no—
189#11758198—22no—
190#11758198—22no—
191#11758198—22no—
192#11758198—22no—
193#11758198—22no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197C4⋊Dic61—28no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61C22×C412yes—
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210C4×Dic61—28no—
211C4×Dic61—22no—
212C4×Dic61—22no—
213C4×Dic61—44no—
214C4×Dic61—44no—
215C23×C61—22no—
216C23×C61—22no—
217C23×C61—41no—
218C23×C61C2×C22⋊C411yescharacteristic
219C23×C61C2×C22⋊C411yescharacteristic
220C23×C61C22×D411yescharacteristic
221C22×D61—22no—
222C22×D61—42no—
223C22×D61—42no—
224C22×D61—42no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61#10414yes—
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C22⋊D61—44no—
256C22⋊D61—44no—
257C2×Dic61—28no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—44no—
262C2×Dic61—42no—
263C2×Dic61—42no—
264C2×Dic61—44no—
265C2×Dic61—42no—
266D244—42no—
267D244—42no—
268C4×D61—44no—
269C4×D61—44no—
270Dic122—42no—
271Dic122—42no—
272C8×C61—44no—
273#17537—611nomaximal, Sylow
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—21no—
277C22×C61—21no—
278C22×C61—21no—
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—44no—
282C22×C61C2×#10412yes—
283C22×C61C22×C22⋊C411yescharacteristic
284C22×C61#57512yes—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—42no—
288C2×D61—84no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—21no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61#41012yes—
299C4×C61C2×#10412yes—
300C4×C61C2×#10412yes—
301Dic61—48no—
302Dic61—42no—
303Dic61—42no—
304Dic61—42no—
305C2×#529—611no—
306#2920D6111yescharacteristic, complemented
307C4×#117—611no—
308#1509—614no—
309#1508—612no—
310#1507—612no—
311#1500—614no—
312C2×C61—22no—
313C2×C61—22no—
314C2×C61—41no—
315C2×C61C2×#41011yescharacteristic
316C2×C61#289611yescharacteristic
317C2×C61C22×#10411yescharacteristic
318D61—82no—
319#529—612no—
320#529—612no—
321#529—612no—
322#529—612no—
323#529—1221no—
324#529—1221no—
325#529—1221no—
326#529—1221no—
327C2×#126—1221no—
328C2×#125—611no—
329C2×#124—611no—
330C2×#123—1221no—
331C2×C22⋊Q8—611no—
332C2×#121—611no—
333C2×C4×Q8—611no—
334C2×C4×D4—611no—
335C2×C4×D4—1221no—
336C2×#117—1221no—
337C2×#117—1221no—
338C22×C42C2×D6111yescharacteristic
339#432—21no—
340#432C2×D6112yescomplemented
341#431—21no—
342#431C2×D6112yescomplemented
343#423—21no—
344#407C2×D6112yes—
345C4×M16—21no—
346C4×C4⋊C4—612no—
347C4×C4⋊C4—1221no—
348C4×C22⋊C4—612no—
349C4×C22⋊C4—1221no—
350#376—612no—
351C43—1221no—
352#328—618no—
353#327—618no—
354C61#1753711yescharacteristic, complemented, Sylow
355C22×Q8—611no—
356C22×D4—611no—
357C22×D4—1221no—
358C23×C4—1221no—
359C23×C4D24411yescharacteristic
360C23×C4D24411yescharacteristic
361C23×C4C22×D6111yescharacteristic
362C2×M16—22no—
363#126—1222no—
364#126—1222no—
365#126—1222no—
366#126—1222no—
367#125—612no—
368#125—612no—
369#125—1221no—
370#125—1221no—
371#124—612no—
372#124—612no—
373#124—1221no—
374#124—1221no—
375#123—1222no—
376#123—1222no—
377#123—1222no—
378#123—1222no—
379C22⋊Q8—612no—
380C22⋊Q8—612no—
381C22⋊Q8—1221no—
382C22⋊Q8—1221no—
383#121—612no—
384#121—612no—
385#121—1221no—
386#121—1221no—
387C4×Q8—612no—
388C4×Q8—612no—
389C4×Q8—1221no—
390C4×Q8—1221no—
391C4×D4—612no—
392C4×D4—612no—
393C4×D4—1222no—
394C4×D4—1222no—
395C4×D4—1222no—
396C4×D4—1222no—
397C4×D4—1221no—
398C4×D4—1221no—
399#117—1228no—
400#117—1222no—
401#117—1222no—
402#117—1222no—
403#117—1222no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1222no—
410C2×C4⋊C4—1221no—
411C2×C22⋊C4—1222no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C42—1221no—
418C2×C42—1222no—
419C2×C42—1222no—
420C2×C42—1222no—
421C2×C42C4×D6111yescharacteristic
422C2×C42C4×D6111yescharacteristic
423C2×C42C4×D6111yescharacteristic
424C2×C42C4×D6111yescharacteristic
425C2×C42D24411yescharacteristic
426C2×C42D24411yescharacteristic
427C2×C42C22⋊D6111yescharacteristic
428C2×C42C22⋊D6111yescharacteristic
429C2×C42C22⋊D6111yescharacteristic
430C2×C42C22⋊D6111yescharacteristic
431C2×C42C22×D6111yescharacteristic
432C2×C42C22×D6111yescharacteristic
433C4⋊C8—22no—
434C4⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8C22×D6114yes—
437#97—22no—
438C4×C8—22no—
439#95—614no—
440#95—1222no—
441C24—1221no—
442C24C2×D24411yescharacteristic
443C2×Q8—612no—
444C2×Q8—1221no—
445C2×Q8—2442no—
446C2×D4—612no—
447C2×D4—1222no—
448C2×D4—1222no—
449C2×D4—1221no—
450C2×D4—2442no—
451C2×D4—2444no—
452C22×C4—24no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—1224no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1222no—
460C22×C4—1222no—
461C22×C4—2444no—
462C22×C4#1175819811yescharacteristic
463C22×C4#1175819811yescharacteristic
464C22×C4#1175819811yescharacteristic
465C22×C4#1175819811yescharacteristic
466C22×C4#1175819811yescharacteristic
467C22×C4#1175819811yescharacteristic
468C22×C4#1175819811yescharacteristic
469C22×C4#1175819811yescharacteristic
470C22×C4C2×C4×D6111yescharacteristic
471C22×C4C2×C4×D6111yescharacteristic
472C22×C4C2×D24411yescharacteristic
473C22×C4C2×D24411yescharacteristic
474C22×C4C2×D24411yescharacteristic
475C22×C4C2×D24411yescharacteristic
476C22×C4C2×D24411yescharacteristic
477C22×C4C2×C22⋊D6111yescharacteristic
478C22×C4C2×C22⋊D6111yescharacteristic
479C22×C4C23×D6111yescharacteristic
480M16—44no—
481C2×C8—24no—
482C4⋊C4—612no—
483C4⋊C4—612no—
484C4⋊C4—1228no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C22⋊C4—1228no—
496C22⋊C4—1222no—
497C22⋊C4—1222no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—1222no—
501C22⋊C4—1222no—
502C22⋊C4—1222no—
503C22⋊C4—1222no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C42—1228no—
507C42—1222no—
508C42—1222no—
509C42—2444no—
510C42—2444no—
511C42#1175819811yescharacteristic
512C42#1175819811yescharacteristic
513C42#1175819811yescharacteristic
514C42#1175819811yescharacteristic
515C42#1175819811yescharacteristic
516C42#1175819811yescharacteristic
517C42#1175819811yescharacteristic
518C42#1175819811yescharacteristic
519C42C2×C4×D6111yescharacteristic
520C42C2×C4×D6111yescharacteristic
521C42C2×C4×D6111yescharacteristic
522C42C2×C4×D6111yescharacteristic
523C42C2×C22⋊D6111yescharacteristic
524C42C2×C22⋊D6111yescharacteristic
525C42C2×C22⋊D6111yescharacteristic
526C42C2×C22⋊D6111yescharacteristic
527C23—22no—
528C23—22no—
529C23—41no—
530C23—1222no—
531C23—2442no—
532C23—2442no—
533C23—2442no—
534C23C2×#1175819811yescharacteristic
535C23C2×#1175819811yescharacteristic
536C23C22×D24411yescharacteristic
537Q8—2442no—
538Q8—2442no—
539D4—2442no—
540D4—2444no—
541D4—2444no—
542D4—2442no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—1228no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—2444no—
565C2×C4—2442no—
566C2×C4—2442no—
567C2×C4—2442no—
568C2×C4—2444no—
569C2×C4—2444no—
570C2×C4—2444no—
571C2×C4#1451970314yes—
572C2×C4C2×#1175819811yescharacteristic
573C2×C4C2×#1175819811yescharacteristic
574C2×C4C2×#1175819811yescharacteristic
575C2×C4C2×#1175819811yescharacteristic
576C2×C4C2×#1175819811yescharacteristic
577C2×C4C2×#1175819811yescharacteristic
578C2×C4C2×#1175819811yescharacteristic
579C2×C4C2×#1175819811yescharacteristic
580C2×C4C2×#1175819811yescharacteristic
581C2×C4C2×#1175819811yescharacteristic
582C2×C4C22×C4×D6111yescharacteristic
583C2×C4C22×C22⋊D6111yescharacteristic
584C8—44no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—44no—
593C22—4884no—
594C22—2442no—
595C22—2442no—
596C22—2442no—
597C22C2×#1451970312yes—
598C22C22×#1175819811yescharacteristic
599C22#1684056212yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—2448no—
610C4—2442no—
611C4—2442no—
612C4—2442no—
613C4#1683973512yes—
614C4C2×#1451970312yes—
615C4C2×#1451970312yes—
616C2—22no—
617C2—22no—
618C2—41no—
619C2—4882no—
620C2C2×#1683973511yescharacteristic
621C2#2145433911yescharacteristic
622C2C22×#1451970311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2920C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#2920D61
hypocenterC61#17537
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ #2920

Complements

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

NH ≅ G/Ntype
2. C4×#14519736C2semidirect
3. #21444345C2semidirect
4. #21444340C2semidirect
6. #21444306C2semidirect
7. C61×#2920C2semidirect
12. #16840296C4semidirect
13. #16840296C4semidirect
17. C2×#14519769C4semidirect
18. C2×#14519765C4semidirect
22. C2×C4×Dic122C4semidirect
23. C2×C4×D244C4semidirect
28. C4×C4⋊Dic61C22semidirect
31. #16839608C22semidirect
33. #16839500C22semidirect
37. C61×#432C22semidirect
39. C61×#431C22semidirect
306. #2920D61semidirect
340. #432C2×D61semidirect
342. #431C2×D61semidirect
354. C61#17537semidirect

Maximal subgroups

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

subgroupindex
C2×#168402962
C4×#145197362
#214443452
#214443402
#214443362
#214443062
C61×#29202
#1753761