Finite Group Databasebeta

← #29678531

Subgroups — #29678531

Order 15616.

1130 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
1C2×#16839864C211yesmaximal, characteristic
2C2×#16839861C211yesmaximal, characteristic
3C2×#16839849C211yesmaximal, characteristic
4#21448183C211yesmaximal, characteristic, complemented
5#21448179C211yesmaximal, characteristic
6#21445719C211yesmaximal, characteristic, complemented
7#107×Dic61C211yesmaximal, characteristic, complemented
8#21445648C211yesmaximal, characteristic, complemented
9#21445643C211yesmaximal, characteristic
10C4×#14519700C211yesmaximal, characteristic, complemented
11#21445638C211yesmaximal, characteristic, complemented
12C4×#14519697C211yesmaximal, characteristic
13#21445633C211yesmaximal, characteristic
14#21445507C211yesmaximal, characteristic, complemented
15C61×#1776C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839864C418yescomplemented
24#16839861C418yescomplemented
25#16839849C418yescomplemented
26C2×#14519711C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×C4×#92418C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38C4⋊C4×Dic61C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4×#11758196—21no—
43C4×#11758196—21no—
44#16839613—21no—
45C4×#11758198—21no—
46C4×#11758198—21no—
47#16839609—21no—
48#16839608C2211yescharacteristic
49#16839607—21no—
50#95×D61—21no—
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190C2×C414yes—
68C2×#11758190C2×C414yes—
69C2×#11758190C2×C414yes—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C4×Dic122—22no—
73C4×Dic122C2×C414yescomplemented
74#14519742—22no—
75#14519742—22no—
76#14519742C2×C414yescomplemented
77#14519742C2×C414yescomplemented
78C4×D244—22no—
79C4×D244C2×C414yes—
80#14519739—22no—
81#14519739—22no—
82#14519739C2×C414yes—
83#14519739C2×C414yes—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519711C2×C414yescomplemented
102#14519702C2×C414yescomplemented
103#14519700C2×C414yescomplemented
104#14519700C2×C414yescomplemented
105#14519699C2×C414yes—
106#14519697C2×C414yes—
107#14519697C2×C414yes—
108#14519687—22no—
109#14519687C2×C414yescomplemented
110C4×#92418—22no—
111C8×Dic61—22no—
112C8×Dic61C2×C414yescomplemented
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#107×C61C2×C414yescomplemented
131#106×C61C2×C414yescomplemented
132#97×C61C2×C414yescomplemented
133#95×C61—21no—
134#95×C61—21no—
135C23×D61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244C22×C412yes—
154C2×D244C22×C412yes—
155C2×D244C22×C412yes—
156C2×C4×D61—24no—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—81no—
160C2×C4×D61—81no—
161C2×C4×D61—81no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C412yes—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61C22×C412yes—
174C2×C8×C61C22×C412yes—
175C2×C8×C61C22×C412yes—
176C2×C8×C61C22×C412yes—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61C22×C412yes—
182C4⋊C4×C61C22×C412yes—
183C4⋊C4×C61C22×C412yes—
184C4⋊C4×C61C22×C412yes—
185C42×C61—81no—
186C42×C61—81no—
187C42×C61C22×C412yes—
188C42×C61#4112yes—
189#11758198—24no—
190#11758198—24no—
191#11758198—24no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61C22×C412yes—
195C4⋊Dic61C22×C412yes—
196#11758196—24no—
197#11758196—24no—
198#11758196—24no—
199C4×Dic61—24no—
200C4×Dic61—24no—
201C4×Dic61—24no—
202C4×Dic61—81no—
203C4×Dic61—81no—
204C4×Dic61—81no—
205C4×Dic61—81no—
206C4×Dic61C22×C412yes—
207C4×Dic61C22×C412yes—
208C4×Dic61#4112yes—
209C4×Dic61#4112yes—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C22×C412yes—
215C2×#92418C22×C412yes—
216#11758190C42116yescomplemented
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
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×C61C4×D412yes—
242C2×C4×C61C4×D412yes—
243C2×C4×C61C23×C411yescharacteristic
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C2×#4111yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61C4×D412yes—
267C2×Dic61C4×D412yes—
268C2×Dic61C4×D412yes—
269C2×Dic61C4×D412yes—
270D244—22no—
271D244—22no—
272D244—22no—
273D244C2×C4214yes—
274C4×D61—84no—
275C4×D61—84no—
276C4×D61—84no—
277Dic122—22no—
278Dic122—22no—
279Dic122—22no—
280Dic122C2×C4214yes—
281C8×C61—22no—
282C8×C61C2×C4214yes—
283#92418—22no—
284#92418—22no—
285#20504—611nomaximal, Sylow
286C22×C61C2×C4×D411yescharacteristic
287C22×C61C2×C4×D411yescharacteristic
288C22×C61C2×C4×D411yescharacteristic
289C22×C61#57611yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57711yescharacteristic
292C22×C61#57811yescharacteristic
293C2×D61—24no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—42no—
306C4×C61C22×C4211yescharacteristic
307C4×C61C4×#4111yescharacteristic
308C4×C61C4×#4111yescharacteristic
309C4×C61C4×#4111yescharacteristic
310Dic61—22no—
311Dic61—22no—
312Dic61—22no—
313Dic61—22no—
314Dic61—24no—
315Dic61—42no—
316Dic61—42no—
317C2×#440—611no—
318C2×#440—611no—
319C2×C4×SD16—611no—
320#2276—611no—
321#2275—611no—
322#1776—611no—
323#1776D6111yescharacteristic, complemented
324C4×#107—611no—
325#1763—611no—
326#1762—611no—
327C4×#103—611no—
328C4×#103—611no—
329C4×#102—611no—
330C4×#102—611no—
331#1726—611no—
332C2×C61C42×D411yescharacteristic
333C2×C61#294611yescharacteristic
334C2×C61#294611yescharacteristic
335C2×C61#294711yescharacteristic
336C2×C61#294711yescharacteristic
337C2×C61#294911yescharacteristic
338C2×C61#295111yescharacteristic
339D61—44no—
340C22×SD16—611no—
341C2×C4×Q8—611no—
342C2×C4×Q8—611no—
343C2×C4×Q8—611no—
344C2×C4×D4—611no—
345C2×C4×D4—611no—
346C2×C4×D4—611no—
347#440—618no—
348#440—618no—
349C4×SD16—618no—
350C2×#107—611no—
351C2×#107C2×D6111yescharacteristic
352C2×#106C2×D6111yescharacteristic
353C2×#103—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#97—611no—
360C2×#97—611no—
361C2×#97C2×D6111yescharacteristic
362C2×C4×C8—611no—
363C2×C4×C8—611no—
364C2×C4×C8—611no—
365C4×C4⋊C4—611no—
366C4×C4⋊C4—611no—
367C4×C4⋊C4—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—1221no—
370C4×C4⋊C4—1221no—
371C4×C4⋊C4—1221no—
372C4×C4⋊C4C2×D6111yescharacteristic
373C4×C4⋊C4C2×D6111yescharacteristic
374C4×C22⋊C4—1221no—
375C4×C22⋊C4—1221no—
376C4×C22⋊C4—1221no—
377#376—611no—
378#376—1221no—
379C2×#95—1221no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340C2×D6111yescharacteristic
385#340C2×D6111yescharacteristic
386C61#2050411yescharacteristic, complemented, Sylow
387C22×Q8—611no—
388C22×D4—611no—
389C23×C4—1221no—
390C23×C4—1221no—
391C23×C4—1221no—
392C2×SD16—614no—
393C2×SD16—614no—
394C2×SD16—614no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399C4×Q8—614no—
400C4×Q8—614no—
401C4×Q8—614no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×D4—614no—
406C4×D4—614no—
407C4×D4—614no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C4⋊C4C22×D6111yescharacteristic
418C2×C4⋊C4C22×D6111yescharacteristic
419C2×C4⋊C4C22×D6111yescharacteristic
420C2×C22⋊C4—1221no—
421C2×C22⋊C4—1221no—
422C2×C22⋊C4—1221no—
423C2×C42—21no—
424C2×C42—21no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—1221no—
428C2×C42—1221no—
429C2×C42—1221no—
430C2×C42—1221no—
431C2×C42—1221no—
432C2×C42—1221no—
433C2×C42—1221no—
434C2×C42C22×D6111yescharacteristic
435#107—614no—
436#107C4×D6114yescomplemented
437#106C4×D6114yescomplemented
438#103—614no—
439#103—614no—
440#103—614no—
441#102—614no—
442#102—614no—
443#102—614no—
444#97—614no—
445#97—1222no—
446#97C4×D6114yescomplemented
447C4×C8—614no—
448C4×C8—1222no—
449C4×C8—1222no—
450#95—21no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—611no—
455#95—611no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460C24—1221no—
461C2×Q8—612no—
462C2×Q8—612no—
463C2×Q8—612no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×D4—612no—
468C2×D4—612no—
469C2×D4—612no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—1224no—
482C22×C4—1224no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1224no—
487C22×C4—4881no—
488C22×C4—4881no—
489C22×C4—4881no—
490C22×C4—4881no—
491C22×C4—4881no—
492C22×C4—4881no—
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4C23×D6111yescharacteristic
496SD16—6116no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—612no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8C2×C4×D6112yes—
506C2×C8C2×C4×D6112yes—
507C2×C8C2×C4×D6112yes—
508C2×C8C2×C4×D6112yes—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—1224no—
516C4⋊C4—1224no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4C2×C4×D6112yes—
521C4⋊C4C2×C4×D6112yes—
522C4⋊C4C2×C4×D6112yes—
523C4⋊C4C2×C4×D6112yes—
524C22⋊C4—1224no—
525C22⋊C4—1224no—
526C22⋊C4—1224no—
527C42—81no—
528C42—81no—
529C42—612no—
530C42—612no—
531C42—612no—
532C42—612no—
533C42—1224no—
534C42—1224no—
535C42—1224no—
536C42—4881no—
537C42—4881no—
538C42—4881no—
539C42—4881no—
540C42C2×C4×D6112yes—
541C42#1175822612yes—
542C23—1222no—
543C23—1222no—
544C23—1222no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23—2441no—
549C23C2×D4×D6111yescharacteristic
550Q8—614no—
551Q8—1222no—
552Q8—1222no—
553Q8—1222no—
554D4—614no—
555D4—1222no—
556D4—1222no—
557D4—1222no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—612no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—612no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—4884no—
592C2×C4—4884no—
593C2×C4—4884no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4#1451973912yes—
599C2×C4#1451973912yes—
600C2×C4C22×C4×D6111yescharacteristic
601C2×C4C22×C4×D6111yescharacteristic
602C2×C4C22×C4×D6111yescharacteristic
603C2×C4#41×D6111yescharacteristic
604C2×C4#41×D6111yescharacteristic
605C2×C4C2×#1175822611yescharacteristic
606C8—22no—
607C8—1222no—
608C8—1222no—
609C8C42×D6114yes—
610C22—1224no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22—2442no—
616C22—2442no—
617C22C4×D4×D6111yescharacteristic
618C22C4×D4×D6111yescharacteristic
619C22C2×#1451973911yescharacteristic
620C22#1684056611yescharacteristic
621C22#1684057111yescharacteristic
622C22#1684057211yescharacteristic
623C22#1684057811yescharacteristic
624C4—22no—
625C4—22no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—1222no—
631C4—1222no—
632C4—1222no—
633C4—1222no—
634C4—1224no—
635C4—2442no—
636C4—2442no—
637C4C2×C42×D6111yescharacteristic
638C4#1684023011yescharacteristic
639C4C4×#1175822611yescharacteristic
640C4C4×#1175822611yescharacteristic
641C2—2444no—
642C2C4×#1451973911yescharacteristic
643C2#2145464811yescharacteristic
644C2#2145464911yescharacteristic
645C2#2145466011yescharacteristic
646C2#2145466111yescharacteristic
647C2#2145467511yescharacteristic
648C2#2145469111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC4×C61C22×C42
FrattiniC22×C4C23×D61
FittingC61×#1776C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1776D61
hypocenterC61#20504
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1776

Complements

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

NH ≅ G/Ntype
4. #21448183C2semidirect
6. #21445719C2semidirect
7. #107×Dic61C2semidirect
8. #21445648C2semidirect
10. C4×#14519700C2semidirect
11. #21445638C2semidirect
14. #21445507C2semidirect
15. C61×#1776C2semidirect
23. #16839864C4semidirect
24. #16839861C4semidirect
25. #16839849C4semidirect
73. C4×Dic122C2×C4semidirect
76. #14519742C2×C4semidirect
77. #14519742C2×C4semidirect
101. #14519711C2×C4semidirect
102. #14519702C2×C4semidirect
103. #14519700C2×C4semidirect
104. #14519700C2×C4semidirect
109. #14519687C2×C4semidirect
112. C8×Dic61C2×C4semidirect
130. #107×C61C2×C4semidirect
131. #106×C61C2×C4semidirect
132. #97×C61C2×C4semidirect
216. #11758190C42semidirect
323. #1776D61semidirect
386. C61#20504semidirect
436. #107C4×D61semidirect
437. #106C4×D61semidirect
446. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×#168398612
C2×#168398492
#214481832
#214481792
#214457192
#107×Dic612
#214456482
#214456432
C4×#145197002
#214456382
C4×#145196972
#214456332
#214455072
C61×#17762
#2050461