Finite Group Databasebeta

← #29709398

Subgroups — #29709398

Order 15616.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839868C211yesmaximal, characteristic, complemented
2C2×#16839853C211yesmaximal, characteristic
3#21448529C211yesmaximal, characteristic
4#21448466C211yesmaximal, characteristic, complemented
5#21447298C211yesmaximal, characteristic, complemented
6#21447231C211yesmaximal, characteristic, complemented
7#21447197C211yesmaximal, characteristic, complemented
8#21447142C211yesmaximal, characteristic, complemented
9#21447136C211yesmaximal, characteristic
10#21447059C211yesmaximal, characteristic
11#21446915C211yesmaximal, characteristic
12#21445678C211yesmaximal, characteristic, complemented
13#21445646C211yesmaximal, characteristic
14#21445638C211yesmaximal, characteristic, complemented
15C61×#2065C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839868—24no—
23#16839853—24no—
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519687C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839667C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839656—21no—
39#16839656—21no—
40#16839649—21no—
41#16839643—21no—
42#16839634—21no—
43#16839634—21no—
44#16839631C2211yescharacteristic
45#16839629—21no—
46#16839627—21no—
47C4⋊C4×Dic61C2211yescharacteristic
48#16839608C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×Dic244—22no—
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×#11758200—22no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519764—22no—
73#14519764—22no—
74#14519742—22no—
75#14519742—22no—
76#14519742—22no—
77#14519742D414yes—
78D4×Dic61—22no—
79D4×Dic61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519708—22no—
96C4⋊#92418—22no—
97#14519702—22no—
98#14519702—22no—
99#14519700—22no—
100#14519700—22no—
101#14519700—22no—
102#14519698—22no—
103#14519687—22no—
104C8×Dic61—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684D411yescharacteristic
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×D4×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#106×C61—22no—
128C4⋊C8×C61—22no—
129#102×C61—22no—
130#102×C61—22no—
131#95×C61—21no—
132C24×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—81no—
151C22×Dic61—81no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C2×D412yes—
167C2×Dic122#4112yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61#4112yes—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—81no—
180C4⋊C4×C61C2×D412yes—
181C22⋊C4×C61—81no—
182C22⋊C4×C61—81no—
183C42×C61—21no—
184C42×C61—21no—
185C22⋊Dic61—24no—
186C22⋊Dic61—24no—
187C22⋊Dic61—42no—
188#11758200—28no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—81no—
194#11758196—24no—
195#11758196—42no—
196#11758196—81no—
197#11758196—81no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C4×Dic61—24no—
201C4×Dic61—42no—
202C4×Dic61—81no—
203C4×Dic61C2×D412yes—
204C4×Dic61#4112yes—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418#4112yes—
211Dic244—28no—
212C23×C61—22no—
213C23×C61—41no—
214C23×C61—41no—
215C23×C61—41no—
216C23×C61—41no—
217C23×C61—41no—
218C23×C61—41no—
219C23×C61C22×D411yescharacteristic
220D4×C61—22no—
221D4×C61—22no—
222D4×C61—41no—
223D4×C61—41no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—22no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—84no—
268C2×Dic61#12112yes—
269C2×Dic61#12112yes—
270Dic122—22no—
271Dic122—22no—
272Dic122—22no—
273Dic122—22no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—41no—
280#92418—22no—
281#92418—41no—
282#23694—611nomaximal, Sylow
283C22×C61—42no—
284C22×C61—42no—
285C22×C61—42no—
286C22×C61—81no—
287C22×C61—81no—
288C22×C61—81no—
289C22×C61—81no—
290C22×C61C2×#12111yescharacteristic
291C22×C61#52511yescharacteristic
292C22×C61#53211yescharacteristic
293C22×C61#57711yescharacteristic
294C22×C61#57711yescharacteristic
295C22×C61#57711yescharacteristic
296C22×C61#57811yescharacteristic
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—42no—
300C4×C61—81no—
301C4×C61—81no—
302C4×C61#52511yescharacteristic
303C4×C61#52511yescharacteristic
304C4×C61#52911yescharacteristic
305C4×C61#52911yescharacteristic
306Dic61—22no—
307Dic61—22no—
308Dic61—41no—
309Dic61—41no—
310Dic61—42no—
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314C2×#441—611no—
315C2×C4×SD16—611no—
316#2326—611no—
317#2320—611no—
318#2084—611no—
319#2072—611no—
320#2066—611no—
321#2065D6111yescharacteristic
322#2056—611no—
323#2055—611no—
324#2042—611no—
325#2015—611no—
326#1768—611no—
327#1762—611no—
328C4×#103—611no—
329C2×C61—82no—
330C2×C61#239211yescharacteristic
331C2×C61#304311yescharacteristic
332C2×C61#304611yescharacteristic
333C2×C61#312011yescharacteristic
334C2×C61#312511yescharacteristic
335C2×C61#317311yescharacteristic
336C2×C61#317811yescharacteristic
337C22×Q16—611no—
338C22×SD16—611no—
339C2×#124—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×D4—611no—
343#441—1224no—
344C4×SD16—1224no—
345C2×#106—611no—
346C2×#106C2×D6111yescharacteristic
347C2×C4⋊C8—611no—
348C2×C4⋊C8C2×D6111yescharacteristic
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#102—611no—
355C2×#102C2×D6111yescharacteristic
356C2×#102C2×D6111yescharacteristic
357C2×#97—611no—
358C2×C4×C8—611no—
359#391—611no—
360#391—611no—
361#391—611no—
362#390C2×D6111yescharacteristic
363#389—611no—
364#389C2×D6111yescharacteristic
365#388—1221no—
366#388—1221no—
367#387—1221no—
368#387—1221no—
369#385—1221no—
370#385—1221no—
371#384—611no—
372#383—1221no—
373#382—1221no—
374C4×C4⋊C4—611no—
375#376—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340C2×D6111yescharacteristic
380#336—611no—
381#336—611no—
382C61#2369411yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×D4C22×D6111yescharacteristic
386C23×C4—1221no—
387C2×Q16—1222no—
388C2×Q16—1222no—
389C2×Q16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8—611no—
394C22×C8—611no—
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397#124—1222no—
398#124—1222no—
399C4×Q8—614no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4—1221no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C4⋊C4C22×D6111yescharacteristic
415C2×C4⋊C4C22×D6111yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—1221no—
424C2×C42—1221no—
425C2×C42—1221no—
426C2×C42C22×D6111yescharacteristic
427#106—22no—
428#106—1222no—
429C4⋊C8—22no—
430C4⋊C8—1222no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#102—22no—
437#102—22no—
438#102—1222no—
439#97—1222no—
440C4×C8—1222no—
441#95—21no—
442#95—611no—
443#95—611no—
444#95—611no—
445#95—611no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454C24—21no—
455C2×Q8—612no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×Q8—1221no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4#1175822612yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4—4881no—
485C22×C4—4881no—
486C22×C4—2442no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—1228no—
491SD16—1228no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—612no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8#1175822412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—81no—
510C4⋊C4—1224no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—4881no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—2442no—
519C4⋊C4D4×D6112yes—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—1224no—
523C22⋊C4—1224no—
524C22⋊C4—2442no—
525C42—21no—
526C42—21no—
527C42—612no—
528C42—612no—
529C42—1224no—
530C42—1221no—
531C42—1221no—
532C42—4881no—
533C42—2442no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C2×D4×D6111yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—1222no—
545Q8—1222no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—1221no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1222no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1222no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
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×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4#41×D6111yescharacteristic
598C2×C4C2×#1175822411yescharacteristic
599C2×C4C2×#1175822611yescharacteristic
600C8—22no—
601C8—41no—
602C8—1222no—
603C8—2441no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#121×D6111yescharacteristic
612C22#1684027511yescharacteristic
613C22#1684030711yescharacteristic
614C22#1684056811yescharacteristic
615C22#1684057211yescharacteristic
616C22#1684057211yescharacteristic
617C22#1684057611yescharacteristic
618C4—42no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—1222no—
624C4—1222no—
625C4—4881no—
626C4—4881no—
627C4—2442no—
628C4—2442no—
629C4—2441no—
630C4—2441no—
631C4#1684027211yescharacteristic
632C4#1684027511yescharacteristic
633C4#1684029511yescharacteristic
634C4#1684029811yescharacteristic
635C2—82no—
636C2#2144906411yescharacteristic
637C2#2145533511yescharacteristic
638C2#2145537011yescharacteristic
639C2#2145603811yescharacteristic
640C2#2145610111yescharacteristic
641C2#2145643111yescharacteristic
642C2#2145648211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2065C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2065D61
hypocenterC61#23694
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. C2×#16839868C2semidirect
4. #21448466C2semidirect
5. #21447298C2semidirect
6. #21447231C2semidirect
7. #21447197C2semidirect
8. #21447142C2semidirect
12. #21445678C2semidirect
14. #21445638C2semidirect
382. C61#23694semidirect

Maximal subgroups

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

subgroupindex
C2×#168398682
C2×#168398532
#214485292
#214484662
#214472982
#214472312
#214471972
#214471422
#214471362
#214470592
#214469152
#214456782
#214456462
#214456382
C61×#20652
#2369461