Finite Group Databasebeta

← #29709316

Subgroups — #29709316

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×#16839864C211yesmaximal, characteristic
2C2×C4×#11758190C211yesmaximal, characteristic
3#21448538C211yesmaximal, characteristic, complemented
4#21448463C211yesmaximal, characteristic
5#21447302C211yesmaximal, characteristic, complemented
6#21447234C211yesmaximal, characteristic
7#21447213C211yesmaximal, characteristic, complemented
8#21447199C211yesmaximal, characteristic, complemented
9#21447137C211yesmaximal, characteristic
10#21447137C211yesmaximal, characteristic
11#21447066C211yesmaximal, characteristic, complemented
12#21446904C211yesmaximal, characteristic
13#21445650C211yesmaximal, characteristic, complemented
14C4×#14519702C211yesmaximal, characteristic, complemented
15C61×#1768C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519777C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22#16839864—24no—
23C4×#11758190—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839651—21no—
41#16839651—21no—
42#16839647—21no—
43#16839647—21no—
44#16839647—21no—
45#16839647—21no—
46#16839628C2211yescharacteristic
47#16839624—21no—
48#16839621—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#376C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758190—22no—
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190—22no—
70C2×#11758190—22no—
71#14519777—22no—
72#14519777—22no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C4×D244—22no—
78C4×D244D414yes—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98#14519711—22no—
99#14519709—22no—
100#14519702—22no—
101#14519702—22no—
102#14519702—22no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519699—22no—
107#14519699—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C4⋊C8×C61—22no—
125C4⋊C8×C61—22no—
126#97×C61—22no—
127C4×C8×C61—22no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C2×D412yes—
153C2×C4×D61—24no—
154C2×C4×D61—81no—
155C2×C4×D61—81no—
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×Dic122#4112yes—
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—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61#4112yes—
179C2×C8×C61#4112yes—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D412yes—
185C42×C61#4112yes—
186#11758198—24no—
187#11758198—81no—
188#11758198—81no—
189#11758198—81no—
190#11758198—81no—
191C4⋊Dic61—24no—
192C4⋊Dic61—24no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61C2×D412yes—
203#11758196—42no—
204#11758196—42no—
205#11758196—81no—
206#11758196—81no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—81no—
210#11758190—28no—
211#11758190—28no—
212C23×C61C22×D411yescharacteristic
213C22×D61—22no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
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—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61#12112yes—
237C2×C4×C61#12112yes—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61C2×#4111yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265D244—22no—
266D244—22no—
267D244—41no—
268D244—41no—
269C4×D61—84no—
270Dic122—22no—
271Dic122—22no—
272Dic122—22no—
273Dic122—22no—
274Dic122—41no—
275Dic122—41no—
276Dic122—41no—
277Dic122—41no—
278C8×C61—22no—
279C8×C61—22no—
280C8×C61—41no—
281C8×C61—41no—
282#23689—611nomaximal, Sylow
283C22×C61C2×#12111yescharacteristic
284C22×C61#52511yescharacteristic
285C22×C61#53211yescharacteristic
286C22×C61#57711yescharacteristic
287C22×C61#57711yescharacteristic
288C22×C61#57711yescharacteristic
289C22×C61#57811yescharacteristic
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C4×C61—22no—
298C4×C61—22no—
299C4×C61—41no—
300C4×C61—41no—
301C4×C61—41no—
302C4×C61—41no—
303C4×C61#52511yescharacteristic
304C4×C61#52511yescharacteristic
305C4×C61#52911yescharacteristic
306C4×C61#52911yescharacteristic
307Dic61—42no—
308Dic61—42no—
309Dic61—42no—
310Dic61—81no—
311Dic61—81no—
312Dic61—81no—
313Dic61—81no—
314C2×#440—611no—
315C2×C4×SD16—611no—
316#2327—611no—
317#2319—611no—
318#2085—611no—
319#2073—611no—
320#2068—611no—
321#2066—611no—
322#2055—611no—
323#2055—611no—
324#2043—611no—
325#2013—611no—
326#1768D6111yescharacteristic, complemented
327#1763—611no—
328C4×#103—611no—
329C2×C61#239211yescharacteristic
330C2×C61#304211yescharacteristic
331C2×C61#304611yescharacteristic
332C2×C61#312111yescharacteristic
333C2×C61#312611yescharacteristic
334C2×C61#317411yescharacteristic
335C2×C61#317811yescharacteristic
336D61—82no—
337C22×SD16—611no—
338C22×SD16—611no—
339C2×C4⋊Q8—611no—
340C2×C4×Q8—611no—
341C2×C4×Q8—611no—
342C2×C4×D4—611no—
343#440—1224no—
344C4×SD16—1224no—
345C2×#107—611no—
346C2×#106—611no—
347C2×C4⋊C8C2×D6111yescharacteristic
348C2×C4⋊C8C2×D6111yescharacteristic
349C2×#103—611no—
350C2×#103—611no—
351C2×#103—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#97C2×D6111yescharacteristic
358C2×C4×C8C2×D6111yescharacteristic
359#391—611no—
360#391—611no—
361#390—611no—
362#390—611no—
363#389—611no—
364#389—611no—
365#388—1221no—
366#388—1221no—
367#387—1221no—
368#387—1221no—
369#387—1221no—
370#387—1221no—
371#382—611no—
372#381—1221no—
373#380—1221no—
374C4×C4⋊C4—611no—
375#376C2×D6111yescharacteristic
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#336C2×D6111yescharacteristic
381#336C2×D6111yescharacteristic
382C61#2368911yescharacteristic, complemented, Sylow
383C22×Q8—611no—
384C22×Q8—611no—
385C22×D4—611no—
386C23×C4—1221no—
387C2×SD16—1222no—
388C2×SD16—1222no—
389C2×SD16—1222no—
390C2×SD16—1222no—
391C2×SD16—1222no—
392C2×SD16—1222no—
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C22×C8C22×D6111yescharacteristic
396C22×C8C22×D6111yescharacteristic
397C4⋊Q8—1222no—
398C4⋊Q8—1222no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1222no—
402C4×Q8—1222no—
403C4×D4—614no—
404C4×D4—1222no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C22⋊C4—1221no—
420C2×C22⋊C4—1221no—
421C2×C42—1221no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42C22×D6111yescharacteristic
425C2×C42C22×D6111yescharacteristic
426C2×C42C22×D6111yescharacteristic
427#107—1222no—
428#106—1222no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—1222no—
435#103—1222no—
436#103—1222no—
437#102—1222no—
438#102—1222no—
439#97—22no—
440C4×C8—22no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446#95—1221no—
447#95—1221no—
448#95—1221no—
449#95—1221no—
450#95D24411yescharacteristic
451#95D24411yescharacteristic
452#95D24411yescharacteristic
453#95D24411yescharacteristic
454C24—1221no—
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—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1224no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—4881no—
482C22×C4—4881no—
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C23×D6111yescharacteristic
490SD16—1228no—
491SD16—1228no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8#1175822212yes—
503C2×C8#1175822212yes—
504C4⋊C4—612no—
505C4⋊C4—1224no—
506C4⋊C4—1224no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—2442no—
519C4⋊C4—2442no—
520C22⋊C4—1224no—
521C22⋊C4—4881no—
522C22⋊C4—4881no—
523C22⋊C4—4881no—
524C22⋊C4—4881no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—4881no—
530C42—2442no—
531C42—2442no—
532C42D4×D6112yes—
533C42#1175822612yes—
534C23—1222no—
535C23—2441no—
536C23—2441no—
537C23—2441no—
538C23—2441no—
539C23—2441no—
540C23—2441no—
541C23C22×D24411yescharacteristic
542Q8—1222no—
543Q8—1222no—
544Q8—1222no—
545Q8—1222no—
546Q8—2441no—
547Q8—2441no—
548Q8—2441no—
549Q8—2441no—
550D4—1222no—
551D4—1222no—
552D4—2441no—
553D4—2441no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—4884no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
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#1451975212yes—
593C2×C4#1451975212yes—
594C2×C4C2×D4×D6111yescharacteristic
595C2×C4#41×D6111yescharacteristic
596C2×C4#41×D6111yescharacteristic
597C2×C4C2×#1175822211yescharacteristic
598C2×C4C2×#1175822211yescharacteristic
599C2×C4C2×#1175822611yescharacteristic
600C8—22no—
601C8—22no—
602C8—41no—
603C8—41no—
604C22—4881no—
605C22—4881no—
606C22—4881no—
607C22—4881no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22C2×#1451975211yescharacteristic
612C22#1684027411yescharacteristic
613C22#1684031211yescharacteristic
614C22#1684057411yescharacteristic
615C22#1684057411yescharacteristic
616C22#1684057411yescharacteristic
617C22#1684057911yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—4881no—
625C4—4881no—
626C4—4881no—
627C4—4881no—
628C4—2442no—
629C4—2442no—
630C4—2442no—
631C4#1684027111yescharacteristic
632C4#1684027111yescharacteristic
633C4#1684029911yescharacteristic
634C4#1684029911yescharacteristic
635C2—4882no—
636C2#2144905911yescharacteristic
637C2#2145533211yescharacteristic
638C2#2145537111yescharacteristic
639C2#2145605011yescharacteristic
640C2#2145611111yescharacteristic
641C2#2145644711yescharacteristic
642C2#2145648711yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1768C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1768D61
hypocenterC61#23689
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
3. #21448538C2semidirect
5. #21447302C2semidirect
7. #21447213C2semidirect
8. #21447199C2semidirect
11. #21447066C2semidirect
13. #21445650C2semidirect
14. C4×#14519702C2semidirect
15. C61×#1768C2semidirect
326. #1768D61semidirect
382. C61#23689semidirect

Maximal subgroups

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

subgroupindex
C2×#168398642
C2×C4×#117581902
#214485382
#214484632
#214473022
#214472342
#214472132
#214471992
#214471372
#214470662
#214469042
#214456502
C4×#145197022
C61×#17682
#2368961