Finite Group Databasebeta

← #29721331

Subgroups — #29721331

Order 15616.

828 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840044C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic
3#21454166C211yesmaximal, characteristic
4C4×#14519757C211yesmaximal, characteristic
5#21447258C212yesmaximal
6#21447025C212yesmaximal
7#21446577C211yesmaximal, characteristic, complemented
8#21446556C212yesmaximal, complemented
9#21446507C212yesmaximal, complemented
10#21445825C211yesmaximal, characteristic, complemented
11C61×#1988C211yesmaximal, characteristic, complemented
12C2×#14519757C2211yescharacteristic
13C2×#14519753—21no—
14C2×C4×Dic122C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16840044—24no—
17#16839818—24no—
18C2×#14519710C2212yes—
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519702C2212yes—
22C2×#14519700C2212yes—
23C2×#14519689C2212yes—
24C2×#14519687C2212yes—
25#16839669—21no—
26#16839667C2211yescharacteristic
27#16839661C2211yescharacteristic
28#16839658—21no—
29#16839658C2212yes—
30#16839647—21no—
31#16839636—21no—
32#16839632C2211yescharacteristic
33#16839619—21no—
34#16839619C2212yes—
35#16839619C2211yescharacteristic
36C4×C4⋊Dic61—21no—
37C4×#11758198—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40C2×C4⋊Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C4×C4⋊C4×C61C2211yescharacteristic
43C61×#340C2212yes—
44C61×#340C2212yes—
45C22×Dic122C2311yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519757—22no—
51#14519757—22no—
52#14519753—44no—
53C4×Dic122—22no—
54C4×Dic122—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2311yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×#11758196—21no—
62C2×#11758196—22no—
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—21no—
66C2×C4×Dic61—21no—
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61C2311yescharacteristic
70C42×D61—22no—
71C42×D61—22no—
72#14519710—24no—
73#14519708—24no—
74C4⋊#92418D414yes—
75#14519702—24no—
76#14519700—24no—
77#14519689—24no—
78#14519687—24no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82#14519684—21no—
83#14519684—22no—
84#14519684—22no—
85C22×Q8×C61—21no—
86C22×C8×C61C2312yes—
87C4⋊Q8×C61—22no—
88C4⋊Q8×C61—22no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61C2312yes—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C42×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96#95×C61—21no—
97C23×D61C2×D411yescharacteristic
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×Q8×C61—42no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2411yescharacteristic
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61C2×D411yescharacteristic
111C22×Dic61C2×D411yescharacteristic
112C22×Dic61C2×D411yescharacteristic
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—41no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122—21no—
128C2×Dic122—21no—
129C2×Dic122—21no—
130C2×Dic122#4112yes—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61#4112yes—
141C42×C61—42no—
142C42×C61C2×D412yes—
143C42×C61#4112yes—
144#11758198—24no—
145#11758198—44no—
146#11758198—42no—
147C4⋊Dic61—24no—
148C4⋊Dic61—21no—
149C4⋊Dic61—22no—
150C4⋊Dic61—22no—
151C4⋊Dic61—21no—
152C4⋊Dic61—42no—
153C4⋊Dic61—42no—
154C4⋊Dic61—42no—
155#11758196—44no—
156#11758196—42no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C4×Dic61—21no—
160C4×Dic61—21no—
161C4×Dic61—42no—
162C2×#92418—22no—
163C2×#92418—22no—
164C2×#92418C2×D414yes—
165C23×C61C22×D411yescharacteristic
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61#13712yes—
173Q8×C61—84no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—22no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61#12112yes—
186C2×C4×C61#12112yes—
187C2×C4×C61AGL(1,ℤ/8)12yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C612−1+411yescharacteristic
194C2×C4×C612−1+411yescharacteristic
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—22no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C4×D61—44no—
214C4×D61—41no—
215C4×D61—41no—
216C4×D61—41no—
217C4×D61—41no—
218C4×D61—44no—
219Dic122—22no—
220Dic122—22no—
221Dic122—41no—
222Dic122—41no—
223C8×C61—42no—
224#92418—24no—
225#24710—611nomaximal, Sylow
226C22×C61C2×#12111yescharacteristic
227C22×C61#52511yescharacteristic
228C22×C61#53611yescharacteristic
229C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C61C2×#13711yescharacteristic
231C22×C61C2×#13711yescharacteristic
232C22×C61#57511yescharacteristic
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—41no—
240C4×C61—22no—
241C4×C61—42no—
242C4×C61—81no—
243C4×C61—82no—
244C4×C61#46814yes—
245C4×C61#54711yescharacteristic
246C4×C61#54811yescharacteristic
247C4×C61#55211yescharacteristic
248C4×C61#55611yescharacteristic
249Dic61—42no—
250Dic61—41no—
251Dic61—41no—
252Dic61—41no—
253Dic61—82no—
254Dic61—81no—
255C2×#475—611no—
256C2×#432—611no—
257#2857—611no—
258C4×C22⋊Q8—611no—
259#2077—612no—
260#2036—612no—
261#1988D6111yescharacteristic, complemented
262#1947—611no—
263#1943—612no—
264#1935—612no—
265#1803—611no—
266C2×C61#241511yescharacteristic
267C2×C61C2×#46811yescharacteristic
268C2×C61#305611yescharacteristic
269C2×C61#311311yescharacteristic
270C2×C61#311511yescharacteristic
271C2×C61#323711yescharacteristic
272C2×C61#323911yescharacteristic
273D61—42no—
274C2×C4⋊Q8C2×D6111yescharacteristic
275C2×C22⋊Q8—611no—
276C2×C22⋊Q8—1221no—
277C2×C4×Q8—611no—
278C22×C42—611no—
279#475—1224no—
280#432—1224no—
281C2×#107—612no—
282C2×#106—612no—
283C2×C4⋊C8—611no—
284C2×C4⋊C8C2×D6111yescharacteristic
285C2×#103—612no—
286C2×#103—612no—
287C2×C22⋊C8—612no—
288C2×#97—612no—
289#391—611no—
290#391—1221no—
291#389—612no—
292#389—611no—
293#389—1221no—
294#387—1221no—
295#385—1221no—
296#384—611no—
297#380—611no—
298#380—612no—
299#380—1221no—
300C4×C4⋊C4—1221no—
301C4×C4⋊C4C2×D6111yescharacteristic
302C4×C22⋊C4—1221no—
303#340—612no—
304#340—612no—
305#340C2×D6112yes—
306#340C2×D6112yes—
307C61#2471011yescharacteristic, complemented, Sylow
308C22×Q8—21no—
309C22×Q8—611no—
310C23×C4—611no—
311C23×C4—611no—
312C23×C4—611no—
313C22×C8—612no—
314C22×C8C22×D6112yes—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—2444no—
320C4×Q8—1222no—
321C4×Q8—1222no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—611no—
324C2×C4⋊C4—612no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1221no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4C22×D6112yes—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6111yescharacteristic
334C2×C22⋊C4—1221no—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C42—21no—
338C2×C42—611no—
339C2×C42—611no—
340C2×C42—611no—
341C2×C42—1222no—
342C2×C42—1222no—
343C2×C42—1221no—
344C2×C42C22×D6111yescharacteristic
345#107—1224no—
346#106—1224no—
347C4⋊C8—22no—
348C4⋊C8—614no—
349#103—1224no—
350#103—1224no—
351C22⋊C8—1224no—
352#97—1224no—
353#95—21no—
354#95—1221no—
355#95—1221no—
356#95—1221no—
357#95—1221no—
358#95—1222no—
359#95—1222no—
360C24—611no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—612no—
365C2×Q8—1221no—
366C2×Q8—1221no—
367C2×Q8—1221no—
368C2×Q8—1221no—
369C2×Q8—1221no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—1222no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1222no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—2441no—
391C22×C4D4×D6111yescharacteristic
392C22×C4D4×D6111yescharacteristic
393C22×C4C23×D6111yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—614no—
398C2×C8—1222no—
399C2×C8—1222no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—42no—
406C4⋊C4—1224no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1222no—
411C4⋊C4—2442no—
412C4⋊C4—2444no—
413C4⋊C4—2442no—
414C4⋊C4—2442no—
415C4⋊C4—2442no—
416C4⋊C4#1175822612yes—
417C22⋊C4—1224no—
418C22⋊C4—2442no—
419C22⋊C4—2444no—
420C42—42no—
421C42—1221no—
422C42—1221no—
423C42—1222no—
424C42—1222no—
425C42—2442no—
426C42#1175822212yes—
427C42D4×D6112yes—
428C23—612no—
429C23—1221no—
430C23—1221no—
431C23—1221no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23C2×D4×D6111yescharacteristic
436Q8—84no—
437Q8—1222no—
438Q8—1222no—
439Q8—2441no—
440Q8—2441no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1222no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—2444no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2442no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4—2444no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451975012yes—
477C2×C4#1451975012yes—
478C2×C4#1451980112yes—
479C2×C4#1451980612yes—
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#41×D6111yescharacteristic
482C2×C4C2×#1175822211yescharacteristic
483C2×C4C2×#1175822611yescharacteristic
484C2×C4#1451982411yescharacteristic
485C2×C4#1451982511yescharacteristic
486C8—42no—
487C8—1224no—
488C22—1222no—
489C22—1222no—
490C22—1222no—
491C22—2441no—
492C22—2441no—
493C22—2441no—
494C22—2441no—
495C22C2×#1451975011yescharacteristic
496C22#1684027511yescharacteristic
497C22#1684033511yescharacteristic
498C22C2×#1451980111yescharacteristic
499C22#137×D6111yescharacteristic
500C22C2×#1451980611yescharacteristic
501C22#1684056111yescharacteristic
502C4—22no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—4881no—
507C4—4882no—
508C4—2441no—
509C4—2442no—
510C4—2441no—
511C4—2441no—
512C4#1684000714yes—
513C4#1684041511yescharacteristic
514C4#1684042011yescharacteristic
515C4#1684045311yescharacteristic
516C4#1684048511yescharacteristic
517C2—2442no—
518C2#2144925111yescharacteristic
519C2C2×#1684000711yescharacteristic
520C2#2145544711yescharacteristic
521C2#2145595111yescharacteristic
522C2#2145597211yescharacteristic
523C2#2145703211yescharacteristic
524C2#2145704811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1988C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1988D61
hypocenterC61#24710
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. C2×#16840044C2semidirect
7. #21446577C2semidirect
8. #21446556C2semidirect
9. #21446507C2semidirect
10. #21445825C2semidirect
11. C61×#1988C2semidirect
261. #1988D61semidirect
307. C61#24710semidirect

Maximal subgroups

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

subgroupindex
C2×#168400442
C2×#168398182
#214541662
C4×#145197572
#214472582
#214470252
#214465772
#214465562
#214465072
#214458252
C61×#19882
#2471061