Finite Group Databasebeta

← #29731606

Subgroups — #29731606

Order 15616.

834 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840159C211yesmaximal, characteristic, complemented
2C2×#16840051C211yesmaximal, characteristic
3#21450771C211yesmaximal, characteristic
4#21448905C211yesmaximal, characteristic, complemented
5#21447299C211yesmaximal, characteristic, complemented
6#21447194C211yesmaximal, characteristic
7#21447190C211yesmaximal, characteristic
8#21446769C211yesmaximal, characteristic, complemented
9#21446580C211yesmaximal, characteristic, complemented
10#21446579C211yesmaximal, characteristic, complemented
11#21446536C211yesmaximal, characteristic, complemented
12#21446447C211yesmaximal, characteristic
13#21446445C211yesmaximal, characteristic
14#21445633C211yesmaximal, characteristic
15C61×#2084C211yesmaximal, characteristic, complemented
16C2×#14519776C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519755—21no—
20C2×#14519739C2211yescharacteristic
21#16840159—24no—
22#16840051—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839697—21no—
36#16839697—21no—
37#16839667—21no—
38#16839666—21no—
39#16839664C2211yescharacteristic
40#16839663C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839657—21no—
43#16839631C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758196C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839613—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#106×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#384C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776—22no—
69#14519776—22no—
70#14519776D414yes—
71#14519755—44no—
72#14519755—44no—
73#14519739—22no—
74#14519739—22no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95#14519709—22no—
96#14519708—22no—
97#14519708—22no—
98#14519708—22no—
99C4⋊#92418—22no—
100#14519706—22no—
101#14519706—22no—
102#14519699—22no—
103#14519699Q814yescomplemented
104#14519697—22no—
105#14519697—22no—
106C8×Dic61—22no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123#106×C61—22no—
124#106×C61—22no—
125#95×C61—21no—
126C23×D61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—21no—
137C22×Dic61C2×D411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139C22×Dic61C2×D411yescharacteristic
140C22×Dic61C2×D411yescharacteristic
141C2×D244—21no—
142C2×D244—21no—
143C2×D244—21no—
144C2×D244—21no—
145C2×D244—21no—
146C2×D244C2×Q812yes—
147C2×C4×D61—42no—
148C2×C4×D61—42no—
149C2×C4×D61—42no—
150C2×Dic122—24no—
151C2×Dic122—42no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—81no—
155C2×Dic122—81no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61C2×Q812yes—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—81no—
172C4⋊C4×C61C2×D412yes—
173C42×C61—21no—
174C42×C61—21no—
175#11758198—42no—
176#11758198—42no—
177#11758198—42no—
178#11758198—42no—
179#11758198—42no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61C2×D412yes—
185C4⋊Dic61C2×Q812yes—
186#11758196—24no—
187#11758196—22no—
188#11758196—42no—
189#11758196—42no—
190#11758196—42no—
191#11758196—81no—
192#11758196SD1614yes—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—42no—
196C4×Dic61—81no—
197C4×Dic61—81no—
198C4×Dic61C2×D412yes—
199C4×Dic61#4112yes—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418#4112yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213C22×D61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C2×SD1612yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×Q811yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61#12112yes—
255C2×Dic61#12112yes—
256C2×Dic61C2×SD1612yes—
257C2×Dic61C2×SD1612yes—
258C2×Dic61AGL(1,ℤ/8)12yes—
259D244—22no—
260D244—22no—
261D244—41no—
262D244—41no—
263C4×D61—84no—
264Dic122—84no—
265Dic122—84no—
266C8×C61—22no—
267C8×C61—41no—
268#92418—22no—
269#92418—41no—
270#25451—611nomaximal, Sylow
271C22×C61C2×#12111yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61D4×Q811yescharacteristic
274C22×C61C22×SD1611yescharacteristic
275C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C61#57711yescharacteristic
277C22×C61#57711yescharacteristic
278C2×D61—42no—
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—81no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C4×C61—22no—
286C4×C61—41no—
287C4×C61—42no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61—81no—
291C4×C61#53311yescharacteristic
292C4×C61#54111yescharacteristic
293C4×C61#54811yescharacteristic
294C4×C61D4×Q811yescharacteristic
295Dic61—22no—
296Dic61—22no—
297Dic61—41no—
298Dic61—42no—
299Dic61—42no—
300Dic61—81no—
301Dic61#46014yes—
302C2×#498—611no—
303C2×#476—611no—
304#2542—611no—
305#2369—611no—
306#2084—611no—
307#2084D6111yescharacteristic, complemented
308#2065—611no—
309#2065—611no—
310#1988—611no—
311#1948—611no—
312#1948—611no—
313#1939—611no—
314#1926—611no—
315#1926—611no—
316C4×#102—611no—
317C2×C61#242911yescharacteristic
318C2×C61C2×#46011yescharacteristic
319C2×C61#304211yescharacteristic
320C2×C61#333511yescharacteristic
321C2×C61#335411yescharacteristic
322C2×C61Q8×SD1611yescharacteristic
323C2×C61#338411yescharacteristic
324D61—82no—
325C2×C4⋊Q8—611no—
326C2×C4⋊Q8—611no—
327C2×C22⋊Q8—1221no—
328C2×C22⋊Q8—1221no—
329C2×C4×D4—611no—
330#498—1224no—
331#476—1224no—
332C2×#106—611no—
333C2×#106—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×#106C2×D6111yescharacteristic
337C2×#106C2×D6111yescharacteristic
338C2×C4⋊C8—611no—
339C2×C4⋊C8—611no—
340C2×C4⋊C8—611no—
341C2×#102—611no—
342C2×#102—611no—
343C2×#102—611no—
344C2×#102—611no—
345C2×C4×C8—611no—
346#399—1221no—
347#399—1221no—
348#391—1221no—
349#391—1221no—
350#390—611no—
351#390—611no—
352#389—611no—
353#389—1221no—
354#389C2×D6111yescharacteristic
355#384—611no—
356#384C2×D6111yescharacteristic
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359C4×C4⋊C4—611no—
360C4×C4⋊C4—611no—
361C4×C4⋊C4—1221no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#340C2×D6111yescharacteristic
368#336—611no—
369#336C2×D6111yescharacteristic
370C61#2545111yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C22×Q8—1221no—
373C22×D4—611no—
374C23×C4—1221no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C4⋊Q8—614no—
380C4⋊Q8—1222no—
381C4⋊Q8—1222no—
382C4⋊Q8—1222no—
383C22⋊Q8—2444no—
384C22⋊Q8—2444no—
385C4×D4—1222no—
386C4×D4—1222no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C22×D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415#106—22no—
416#106—22no—
417#106—1222no—
418#106—1222no—
419#106—1222no—
420#106—1222no—
421C4⋊C8—1222no—
422C4⋊C8—1222no—
423C4⋊C8—1222no—
424#102—614no—
425#102—1222no—
426#102—1222no—
427#102—1222no—
428C4×C8—1222no—
429#95—21no—
430#95—611no—
431#95—611no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435#95—1221no—
436C24—1221no—
437C2×Q8—1224no—
438C2×Q8—4881no—
439C2×Q8—4881no—
440C2×Q8—2442no—
441C2×Q8—2442no—
442C2×Q8—2442no—
443C2×D4—612no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—2442no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4D4×D6111yescharacteristic
464C22×C4D4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—612no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8Q8×D6112yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—614no—
489C4⋊C4—612no—
490C4⋊C4—612no—
491C4⋊C4—1222no—
492C4⋊C4—1224no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—4881no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4D4×D6112yes—
502C22⋊C4—2442no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C22⋊C4—2442no—
506C22⋊C4—2442no—
507C42—21no—
508C42—21no—
509C42—612no—
510C42—612no—
511C42—1221no—
512C42—1221no—
513C42—4881no—
514C42—4881no—
515C42—2442no—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23C2×D4×D6111yescharacteristic
524Q8—4884no—
525Q8—4884no—
526D4—1222no—
527D4—1222no—
528D4—2441no—
529D4—2441no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—612no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1222no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1222no—
561C2×C4—4884no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4SD16×D6112yes—
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4#41×D6111yescharacteristic
574C2×C4#1451982411yescharacteristic
575C2×C4#1451982511yescharacteristic
576C8—22no—
577C8—41no—
578C8—1222no—
579C8—2441no—
580C22—4881no—
581C22—4881no—
582C22—4881no—
583C22—4881no—
584C22—2442no—
585C22—2442no—
586C22—2442no—
587C22#121×D6111yescharacteristic
588C22#1684041011yescharacteristic
589C22#1684042711yescharacteristic
590C22C2×SD16×D6111yescharacteristic
591C22AGL(1,ℤ/8)×D6111yescharacteristic
592C22#1684056811yescharacteristic
593C22#1684057111yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—614no—
601C4—1222no—
602C4—1222no—
603C4—4881no—
604C4—2442no—
605C4—2442no—
606C4—2441no—
607C4#1684031811yescharacteristic
608C4#1684037111yescharacteristic
609C4#1684042211yescharacteristic
610C4#1684042511yescharacteristic
611C2—4882no—
612C2#2144941311yescharacteristic
613C2#460×D6111yescharacteristic
614C2#2145532611yescharacteristic
615C2#2145816911yescharacteristic
616C2#2145840711yescharacteristic
617C2#2145867611yescharacteristic
618C2#2145871511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2084C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2084D61
hypocenterC61#25451
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
1. C2×#16840159C2semidirect
4. #21448905C2semidirect
5. #21447299C2semidirect
8. #21446769C2semidirect
9. #21446580C2semidirect
10. #21446579C2semidirect
11. #21446536C2semidirect
15. C61×#2084C2semidirect
103. #14519699Q8semidirect
307. #2084D61semidirect
370. C61#25451semidirect

Maximal subgroups

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

subgroupindex
C2×#168401592
C2×#168400512
#214507712
#214489052
#214472992
#214471942
#214471902
#214467692
#214465802
#214465792
#214465362
#214464472
#214464452
#214456332
C61×#20842
#2545161