Finite Group Databasebeta

← #29743077

Subgroups — #29743077

Order 15616.

900 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic
2C2×#16839849C211yesmaximal, characteristic
3#21449654C211yesmaximal, characteristic, complemented
4#21448720C211yesmaximal, characteristic
5#21447359C211yesmaximal, characteristic, complemented
6#21447298C211yesmaximal, characteristic, complemented
7#21447197C211yesmaximal, characteristic, complemented
8#21446546C211yesmaximal, characteristic, complemented
9#21446379C211yesmaximal, characteristic, complemented
10#21446154C211yesmaximal, characteristic
11#21446152C211yesmaximal, characteristic
12#21446088C211yesmaximal, characteristic, complemented
13#21446081C211yesmaximal, characteristic
14#21446077C211yesmaximal, characteristic
15C61×#2058C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519742C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20C2×#14519739C2211yescharacteristic
21C2×C4⋊C4×D61—21no—
22#16840049—24no—
23#16839849—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839695—21no—
36#16839681—21no—
37#16839669—21no—
38#16839668C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839643—21no—
44#16839636—21no—
45#16839633—21no—
46#16839631C2211yescharacteristic
47#16839621—21no—
48C4×#11758196C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4⋊Q8×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C2×#11758190—22no—
67C2×#11758190—22no—
68C2×#11758190Q814yescomplemented
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519742—22no—
72#14519742—41no—
73C4×D244—22no—
74C4×D244—41no—
75#14519739—22no—
76#14519739—22no—
77#14519739—22no—
78#14519739—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×#11758196D411yescharacteristic
89C4⋊C4×D61—44no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519708—22no—
97#14519708—22no—
98#14519708—22no—
99C4⋊#92418—22no—
100#14519702—41no—
101#14519700—22no—
102#14519700—22no—
103#14519699—22no—
104#14519697—22no—
105#14519697D414yes—
106C8×Dic61—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61—21no—
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4⋊Q8×C61—22no—
120C4⋊Q8×C61—41no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#106×C61—22no—
129#106×C61—41no—
130#95×C61—21no—
131C23×D61—21no—
132C2×Q8×C61—42no—
133C2×Q8×C61—42no—
134C2×Q8×C61—81no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2411yescharacteristic
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244C2×D412yes—
153C2×D244C2×Q812yes—
154C2×D244#4112yes—
155C2×C4×D61—24no—
156C2×C4×D61—42no—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C2×Q812yes—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61C2×Q812yes—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61C2×D412yes—
185C4⋊C4×C61#4112yes—
186C42×C61—21no—
187C42×C61—21no—
188#11758198—24no—
189#11758198—42no—
190#11758198—42no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—81no—
197#11758196—41no—
198#11758196—42no—
199#11758196—42no—
200#11758196—81no—
201#11758196—81no—
202#11758196SD1614yescomplemented
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418C2×D412yes—
214#11758190—28no—
215C23×C61C22×D411yescharacteristic
216C22×D61—22no—
217C22×D61—22no—
218C22×D61—22no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223Q8×C61—84no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
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×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61C2×SD1612yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—22no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
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×Dic61C2×SD1612yes—
268C2×Dic61C2×SD1612yes—
269D244—22no—
270D244—22no—
271D244—22no—
272D244C22⋊Q814yes—
273C4×D61—48no—
274C4×D61—84no—
275C4×D61—84no—
276Dic122—22no—
277Dic122—22no—
278Dic122—41no—
279Dic122—41no—
280C8×C61—22no—
281C8×C61—41no—
282#92418—22no—
283#92418—41no—
284#26376—611nomaximal, Sylow
285C22×C61#52511yescharacteristic
286C22×C61D4211yescharacteristic
287C22×C61D4×Q811yescharacteristic
288C22×C61C22×SD1611yescharacteristic
289C22×C61#57511yescharacteristic
290C22×C61#57711yescharacteristic
291C22×C61#57711yescharacteristic
292C2×D61—24no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C2×D61—42no—
299C4×C61—22no—
300C4×C61—42no—
301C4×C61—41no—
302C4×C61—42no—
303C4×C61—81no—
304C4×C61—81no—
305C4×C61C2×C22⋊Q811yescharacteristic
306C4×C61#53911yescharacteristic
307C4×C61#54711yescharacteristic
308C4×C61#55411yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—42no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316C2×#476—611no—
317C2×C4×SD16—611no—
318#2451—611no—
319D4×C4⋊C4—611no—
320#2096—611no—
321#2084—611no—
322#2066—611no—
323#2058D6111yescharacteristic, complemented
324#1941—611no—
325#1916—611no—
326#1877—611no—
327#1877—611no—
328#1865—611no—
329#1864—611no—
330#1864—611no—
331C2×C61#244911yescharacteristic
332C2×C61#310111yescharacteristic
333C2×C61#312511yescharacteristic
334C2×C61#326711yescharacteristic
335C2×C61D4×SD1611yescharacteristic
336C2×C61Q8×SD1611yescharacteristic
337C2×C61#338811yescharacteristic
338D61—44no—
339C22×SD16—611no—
340C2×C4⋊Q8C2×D6111yescharacteristic
341C2×C4×Q8—611no—
342C2×C4×D4—611no—
343C2×C4×D4—611no—
344C2×C4×D4—611no—
345C22×C4⋊C4—1221no—
346#476—1224no—
347C4×SD16—1224no—
348C2×#106—611no—
349C2×#106—611no—
350C2×#106—611no—
351C2×#106C2×D6111yescharacteristic
352C2×#106C2×D6111yescharacteristic
353C2×C4⋊C8—611no—
354C2×#103—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×C4×C8—611no—
361#398—1221no—
362#395—1221no—
363#391—611no—
364#391—611no—
365#391—1221no—
366#389—611no—
367#389—611no—
368#389—611no—
369#389C2×D6111yescharacteristic
370#387—1221no—
371#385—1221no—
372#385—1221no—
373#384—611no—
374#380—1221no—
375C4×C4⋊C4—611no—
376#340—611no—
377#340—611no—
378#340C2×D6111yescharacteristic
379#340C2×D6111yescharacteristic
380#336—611no—
381#336—611no—
382#336—611no—
383#336C2×D6111yescharacteristic
384C61#2637611yescharacteristic, complemented, Sylow
385C22×Q8—21no—
386C22×Q8—611no—
387C22×D4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4—1221no—
391C2×SD16—614no—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C22×C8—611no—
395C22×C8—611no—
396C22×C8C22×D6111yescharacteristic
397C22×C8C22×D6111yescharacteristic
398C4⋊Q8—22no—
399C4⋊Q8—41no—
400C4×Q8—1222no—
401C4×Q8—2441no—
402C4×D4—1222no—
403C4×D4—1222no—
404C4×D4—1222no—
405C4×D4—1222no—
406C4×D4—2441no—
407C4×D4—2441no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—2444no—
421C2×C4⋊C4C22×D6111yescharacteristic
422C2×C4⋊C4C22×D6111yescharacteristic
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C42—611no—
429C2×C42—611no—
430C2×C42—1221no—
431C2×C42C22×D6111yescharacteristic
432#106—22no—
433#106—41no—
434#106—1222no—
435#106—1222no—
436#106—1222no—
437C4⋊C8—1222no—
438#103—1222no—
439#103—1222no—
440#103—2441no—
441#102—614no—
442#102—1222no—
443#102—1222no—
444C4×C8—1222no—
445#95—21no—
446#95—611no—
447#95—611no—
448#95—1221no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455C24—1221no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—81no—
459C2×Q8—612no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—1221no—
463C2×Q8—1221no—
464C2×Q8—1221no—
465C2×D4—612no—
466C2×D4—612no—
467C2×D4—612no—
468C2×D4—1221no—
469C2×D4—1221no—
470C2×D4—1221no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—1224no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—4881no—
484C22×C4—4881no—
485C22×C4—2442no—
486C22×C4—2442no—
487C22×C4—2442no—
488C22×C4—2442no—
489C22×C4—2442no—
490C22×C4—2442no—
491C22×C4D4×D6111yescharacteristic
492C22×C4D4×D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494SD16—1228no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—612no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8Q8×D6112yes—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—614no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—4881no—
521C4⋊C4—4881no—
522C4⋊C4—4881no—
523C4⋊C4—2441no—
524C4⋊C4—2442no—
525C4⋊C4—2442no—
526C4⋊C4—2442no—
527C4⋊C4D4×D6112yes—
528C4⋊C4#1175822612yes—
529C22⋊C4—1224no—
530C22⋊C4—2442no—
531C22⋊C4—2442no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
539C23—1222no—
540C23—1222no—
541C23—1222no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23C2×D4×D6111yescharacteristic
547Q8—84no—
548Q8—1222no—
549Q8—1222no—
550Q8—2441no—
551Q8—2441no—
552D4—614no—
553D4—1222no—
554D4—1222no—
555D4—1222no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—612no—
574C2×C4—612no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1222no—
586C2×C4—4884no—
587C2×C4—4884no—
588C2×C4—2448no—
589C2×C4—2441no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4SD16×D6112yes—
598C2×C4C2×D4×D6111yescharacteristic
599C2×C4C2×Q8×D6111yescharacteristic
600C2×C4#41×D6111yescharacteristic
601C2×C4C2×#1175822611yescharacteristic
602C2×C4#1451982211yescharacteristic
603C2×C4#1451982511yescharacteristic
604C8—22no—
605C8—41no—
606C8—1222no—
607C8—2441no—
608C22—1224no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22—2442no—
614C22—2442no—
615C22#1684027511yescharacteristic
616C22#1684039411yescharacteristic
617C22#1684042711yescharacteristic
618C22C2×SD16×D6111yescharacteristic
619C22#1684055811yescharacteristic
620C22#1684056811yescharacteristic
621C22#1684057111yescharacteristic
622C4—22no—
623C4—41no—
624C4—42no—
625C4—42no—
626C4—81no—
627C4—81no—
628C4—1222no—
629C4—1222no—
630C4—4881no—
631C4—4881no—
632C4—2442no—
633C4—2441no—
634C4—2441no—
635C4C22⋊Q8×D6111yescharacteristic
636C4#1684035811yescharacteristic
637C4#1684041511yescharacteristic
638C4#1684047311yescharacteristic
639C2—2444no—
640C2#2144962111yescharacteristic
641C2#2145582911yescharacteristic
642C2#2145610311yescharacteristic
643C2#2145734211yescharacteristic
644C2#2145750811yescharacteristic
645C2#2145867611yescharacteristic
646C2#2145875911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2058C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2058D61
hypocenterC61#26376
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
3. #21449654C2semidirect
5. #21447359C2semidirect
6. #21447298C2semidirect
7. #21447197C2semidirect
8. #21446546C2semidirect
9. #21446379C2semidirect
12. #21446088C2semidirect
15. C61×#2058C2semidirect
68. C2×#11758190Q8semidirect
202. #11758196SD16semidirect
323. #2058D61semidirect
384. C61#26376semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
C2×#168398492
#214496542
#214487202
#214473592
#214472982
#214471972
#214465462
#214463792
#214461542
#214461522
#214460882
#214460812
#214460772
C61×#20582
#2637661