Finite Group Databasebeta

← #29696689

Subgroups — #29696689

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic
2C2×#16839890C211yesmaximal, characteristic, complemented
3#21448777C211yesmaximal, characteristic, complemented
4#21446792C211yesmaximal, characteristic, complemented
5#21446786C211yesmaximal, characteristic, complemented
6#21446782C211yesmaximal, characteristic, complemented
7#21446775C211yesmaximal, characteristic, complemented
8#21446773C211yesmaximal, characteristic, complemented
9#21446771C211yesmaximal, characteristic, complemented
10#21445829C211yesmaximal, characteristic
11#21445827C211yesmaximal, characteristic
12#21445817C211yesmaximal, characteristic
13#21445646C211yesmaximal, characteristic
14#102×Dic61C211yesmaximal, characteristic
15C61×#1762C211yesmaximal, characteristic
16#16840469—24no—
17#16840469C418yescomplemented
18C2×#14519777C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519756—21no—
22C2×#14519754—21no—
23C2×#14519754—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×D4×Dic61C2211yescharacteristic
26C2×D4×Dic61C2211yescharacteristic
27C22×C4⋊Dic61—21no—
28#16839890—24no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519706C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839666—21no—
37#16839646—21no—
38#16839645—21no—
39#16839631C2211yescharacteristic
40#16839628C2211yescharacteristic
41#16839627C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#97×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122D411yescharacteristic
64C23×Dic61—21no—
65C23×Dic61—21no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519777—22no—
69#14519777C2×C414yes—
70#14519768—22no—
71#14519768—22no—
72#14519768C2×C414yes—
73#14519768C2×C414yes—
74#14519756—28no—
75#14519754—28no—
76#14519754—28no—
77C4×Dic122C2×C414yes—
78C4×Dic122D414yes—
79D4×Dic61—22no—
80D4×Dic61—22no—
81D4×Dic61C2×C414yes—
82D4×Dic61C2×C414yes—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—28no—
87C2×C4⋊Dic61D411yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196D411yescharacteristic
97C2×#11758196D411yescharacteristic
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C4⋊#92418—22no—
105#14519706—22no—
106#14519706—22no—
107#14519698—22no—
108#14519698—22no—
109#14519687—22no—
110C8×Dic61—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117C22×D4×C61C2311yescharacteristic
118C23×C4×C61—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×D4×C61—22no—
122C4×D4×C61C2×C414yes—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C42×C61—21no—
128C2×C42×C61C2311yescharacteristic
129#102×C61—22no—
130#102×C61Q814yescomplemented
131#97×C61—22no—
132#95×C61—21no—
133C24×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C2×D4×C61C2×Q812yes—
139C2×D4×C61#4112yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—21no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—24no—
152C22×Dic61—24no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—81no—
156C22×Dic61—81no—
157C22×Dic61—81no—
158C22×Dic61—81no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×Dic122—24no—
164C2×Dic122—41no—
165C2×Dic122—41no—
166C2×Dic122—81no—
167C2×Dic122—81no—
168C2×Dic122C22⋊C414yes—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×Q812yes—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C412yes—
180C4⋊C4×C61C2×Q812yes—
181C22⋊C4×C61—24no—
182C42×C61—81no—
183C42×C61C22×C412yes—
184C42×C61C2×D412yes—
185C22⋊Dic61—24no—
186C22⋊Dic61—24no—
187C4⋊Dic61—24no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—24no—
191C4⋊Dic61—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61C22⋊C414yes—
197C4⋊Dic61C22×C412yes—
198C4⋊Dic61C22×C412yes—
199#11758196—24no—
200#11758196—24no—
201#11758196—42no—
202#11758196—81no—
203#11758196C22⋊C414yes—
204#11758196C22⋊C414yes—
205C4×Dic61—42no—
206C4×Dic61—81no—
207C4×Dic61C22×C412yes—
208C4×Dic61C22×C412yes—
209C4×Dic61C2×D412yes—
210C4×Dic61C2×D412yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C23×C61—22no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61—41no—
222C23×C61—41no—
223C23×C61—41no—
224C23×C61C22×D411yescharacteristic
225D4×C61—22no—
226D4×C61—22no—
227D4×C61—22no—
228D4×C61C4×Q814yes—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—48no—
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—41no—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C2×C22⋊C412yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×Q811yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×C4×C612−1+411yescharacteristic
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—84no—
272C2×Dic61—84no—
273C2×Dic61C2×C22⋊C412yes—
274C2×Dic61C2×C22⋊C412yes—
275C2×Dic61C2×C22⋊C412yes—
276C2×Dic61C2×C22⋊C412yes—
277Dic122—44no—
278Dic122—84no—
279C8×C61—22no—
280C8×C61—41no—
281#92418—41no—
282#92418—41no—
283#22526—611nomaximal, Sylow
284C22×C61—24no—
285C22×C61—42no—
286C22×C61—42no—
287C22×C61—42no—
288C22×C61—42no—
289C22×C61—42no—
290C22×C61—42no—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61#54811yescharacteristic
293C22×C61D4×Q811yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57511yescharacteristic
296C22×C61#57611yescharacteristic
297C22×C61#57711yescharacteristic
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—42no—
302C4×C61—81no—
303C4×C61C2×C4×Q811yescharacteristic
304C4×C61#51911yescharacteristic
305C4×C61#54811yescharacteristic
306C4×C61D4×Q811yescharacteristic
307Dic61—22no—
308Dic61—22no—
309Dic61—22no—
310Dic61—22no—
311Dic61—42no—
312Dic61—42no—
313Dic61—81no—
314Dic61—81no—
315C2×#554—611no—
316C2×#446—611no—
317#2355—611no—
318#1993—611no—
319#1992—611no—
320#1991—611no—
321#1989—611no—
322#1989—611no—
323#1988—611no—
324#1804—611no—
325#1804—611no—
326#1802—611no—
327#1762—611no—
328#1762D6111yescharacteristic
329C4×#102—611no—
330C2×C61—44no—
331C2×C61Q8×C22⋊C411yescharacteristic
332C2×C61#289611yescharacteristic
333C2×C61#289711yescharacteristic
334C2×C61#337011yescharacteristic
335C2×C61#337611yescharacteristic
336C2×C61#338811yescharacteristic
337C2×C61#338911yescharacteristic
338#554—618no—
339#554—1224no—
340C2×C4⋊Q8—611no—
341C2×#125—611no—
342C2×#125—611no—
343C2×C22⋊Q8—1221no—
344C2×C22⋊Q8—1221no—
345C2×C22⋊Q8—1221no—
346C2×C4×Q8—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4C2×D6111yescharacteristic
350C22×C4⋊C4—1221no—
351#446—1224no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102C2×D6111yescharacteristic
358C2×#102C2×D6111yescharacteristic
359C2×#97—611no—
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8—611no—
362#391—1221no—
363#387—1221no—
364#387—1221no—
365#384—611no—
366#382—611no—
367#382—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—1221no—
371C4×C4⋊C4C2×D6111yescharacteristic
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340C2×D6111yescharacteristic
383#340C2×D6111yescharacteristic
384C61#2252611yescharacteristic, complemented, Sylow
385C22×Q8—611no—
386C22×Q8—1221no—
387C22×D4C22×D6111yescharacteristic
388C23×C4—21no—
389C23×C4—1221no—
390C23×C4—1221no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C4⋊Q8—614no—
396C4⋊Q8—1222no—
397#125—614no—
398#125—614no—
399#125—1222no—
400#125—1222no—
401C22⋊Q8—1228no—
402C22⋊Q8—1228no—
403C22⋊Q8—1228no—
404C4×Q8—614no—
405C4×Q8—614no—
406C4×D4—22no—
407C4×D4—614no—
408C4×D4—614no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4C4×D6114yes—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1228no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—1221no—
437C2×C42—1221no—
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—1222no—
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442#102—22no—
443#102—1222no—
444#102—1222no—
445#102Dic12214yescomplemented
446#97—22no—
447#97—1222no—
448C4×C8—1222no—
449#95—21no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—21no—
457C2×Q8—614no—
458C2×Q8—1224no—
459C2×Q8—4881no—
460C2×Q8—4881no—
461C2×Q8—2441no—
462C2×Q8—2441no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×Dic12212yes—
467C2×D4C2×C4×D6112yes—
468C2×D4#1175822212yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—24no—
472C22×C4—21no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1224no—
480C22×C4—1224no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—4881no—
486C22×C4—4881no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4D4×D6111yescharacteristic
490C22×C4D4×D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8C2×Dic12212yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—614no—
509C4⋊C4—614no—
510C4⋊C4—614no—
511C4⋊C4—612no—
512C4⋊C4—612no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1224no—
516C4⋊C4—1224no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—1221no—
524C4⋊C4—4881no—
525C4⋊C4—2442no—
526C4⋊C4C2×Dic12212yes—
527C4⋊C4C2×C4×D6112yes—
528C22⋊C4—24no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C42—81no—
532C42—612no—
533C42—612no—
534C42—612no—
535C42—612no—
536C42—4881no—
537C42—2442no—
538C42C2×C4×D6112yes—
539C42D4×D6112yes—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×D4×D6111yescharacteristic
548Q8—4884no—
549Q8—2444no—
550D4—22no—
551D4—22no—
552D4—22no—
553D4C4×Dic12214yes—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—48no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—4884no—
583C2×C4—4884no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4C22⋊C4×D6112yes—
595C2×C4C22⋊C4×D6112yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4C22×Dic12211yescharacteristic
599C2×C4C2×#1175822211yescharacteristic
600C2×C4#1451982411yescharacteristic
601C2×C4#1451982411yescharacteristic
602C8—22no—
603C8—41no—
604C8—2441no—
605C8—2441no—
606C22—24no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22C2×C22⋊C4×D6111yescharacteristic
614C22#1684042111yescharacteristic
615C22D4×Dic12211yescharacteristic
616C22#1684055811yescharacteristic
617C22#1684055811yescharacteristic
618C22#1684056611yescharacteristic
619C22#1684057111yescharacteristic
620C4—22no—
621C4—22no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—1222no—
626C4—1222no—
627C4—1222no—
628C4—1222no—
629C4—4881no—
630C4—4881no—
631C4—2442no—
632C4—2442no—
633C4C2×C4×Dic12211yescharacteristic
634C4#1684023911yescharacteristic
635C4#1684042111yescharacteristic
636C4D4×Dic12211yescharacteristic
637C2—44no—
638C2C22⋊C4×Dic12211yescharacteristic
639C2#2145433311yescharacteristic
640C2#2145434311yescharacteristic
641C2#2145857711yescharacteristic
642C2#2145865111yescharacteristic
643C2#2145876911yescharacteristic
644C2#2145877811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1762C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1762D61
hypocenterC61#22526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1762

Complements

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

NH ≅ G/Ntype
2. C2×#16839890C2semidirect
3. #21448777C2semidirect
4. #21446792C2semidirect
5. #21446786C2semidirect
6. #21446782C2semidirect
7. #21446775C2semidirect
8. #21446773C2semidirect
9. #21446771C2semidirect
17. #16840469C4semidirect
130. #102×C61Q8semidirect
384. C61#22526semidirect
445. #102Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
C2×#168398902
#214487772
#214467922
#214467862
#214467822
#214467752
#214467732
#214467712
#214458292
#214458272
#214458172
#214456462
#102×Dic612
C61×#17622
#2252661