Finite Group Databasebeta

← #29721341

Subgroups — #29721341

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×#16840057C211yesmaximal, characteristic, complemented
2C2×#16839818C211yesmaximal, characteristic
3#21454172C211yesmaximal, characteristic
4C4×#14519753C211yesmaximal, characteristic
5#21447246C212yesmaximal
6#21447013C212yesmaximal
7#21446766C211yesmaximal, characteristic, complemented
8#21446580C211yesmaximal, characteristic, complemented
9#21446561C212yesmaximal, complemented
10#21446511C212yesmaximal, complemented
11C61×#1803C211yesmaximal, characteristic, complemented
12C2×#14519777C2211yescharacteristic
13C2×#14519757—21no—
14C2×#14519753C2211yescharacteristic
15C2×C42×D61C2211yescharacteristic
16#16840057—24no—
17#16839818—24no—
18C2×#14519710C2212yes—
19C2×#14519708C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519689C2212yes—
23C2×#14519687C2212yes—
24#16839669C2211yescharacteristic
25#16839667—21no—
26#16839661—21no—
27#16839661C2212yes—
28#16839658C2211yescharacteristic
29#16839643—21no—
30#16839633—21no—
31#16839619—21no—
32#16839619C2211yescharacteristic
33#16839619C2212yes—
34C4×#11758196—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36C4×#11758198—21no—
37#16839525C2212yes—
38#16839525C2212yes—
39#16839524C2212yes—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C61×#384C2211yescharacteristic
44C61×#340C2212yes—
45C22×Dic122—21no—
46C22×C4×D61D411yescharacteristic
47C22×C4×D61D411yescharacteristic
48C22×C4×D61C2311yescharacteristic
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52#14519757—44no—
53#14519753—22no—
54#14519753—22no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—21no—
61C2×#11758196—21no—
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—21no—
65C2×C4×Dic61—21no—
66C2×C4×Dic61D411yescharacteristic
67C2×C4×Dic61D411yescharacteristic
68C2×C4×Dic61C2311yescharacteristic
69C42×D61—22no—
70C42×D61—22no—
71#14519710—24no—
72#14519708—24no—
73C4⋊#92418D414yes—
74#14519701—24no—
75#14519689—24no—
76#14519687—24no—
77#14519684—21no—
78#14519684—21no—
79#14519684—21no—
80#14519684—21no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61C2311yescharacteristic
85C22×C8×C61C2312yes—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2312yes—
93C2×C42×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—22no—
96#103×C61—24no—
97C23×D61C2×D411yescharacteristic
98C2×Q8×C61—21no—
99C2×Q8×C61—21no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61#4112yes—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—22no—
112C22×Dic61—21no—
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C4×D61—21no—
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—41no—
128C2×Dic122—42no—
129C2×Dic122—42no—
130C2×Dic122—42no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—42no—
140C4⋊C4×C61—42no—
141C42×C61—42no—
142C42×C61C2×D412yes—
143C42×C61#4112yes—
144#11758198—24no—
145#11758198—42no—
146#11758198—44no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—44no—
153C4⋊Dic61—42no—
154C4⋊Dic61#4112yes—
155#11758196—24no—
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×D61AGL(1,ℤ/8)12yes—
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—41no—
176Q8×C61—41no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61#12112yes—
189C2×C4×C61#12112yes—
190C2×C4×C61#13712yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61C2×#4111yescharacteristic
196C2×C4×C612−1+411yescharacteristic
197C2×C4×C612−1+411yescharacteristic
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—22no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—21no—
207C2×Dic61—21no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—42no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C4×D61—44no—
217C4×D61—41no—
218C4×D61—41no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—44no—
222Dic122—84no—
223C8×C61—42no—
224#92418—24no—
225#24711—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—82no—
243C4×C61—81no—
244C4×C61#47014yes—
245C4×C61#54711yescharacteristic
246C4×C61#54811yescharacteristic
247C4×C61#55211yescharacteristic
248C4×C61#55611yescharacteristic
249Dic61—42no—
250Dic61—41no—
251Dic61—41no—
252Dic61—41no—
253Dic61—81no—
254Dic61—82no—
255C2×#477—611no—
256C2×#432—611no—
257#2857—611no—
258C4×C22⋊Q8—611no—
259#2075—612no—
260#2035—612no—
261#1987—611no—
262#1948—611no—
263#1944—612no—
264#1935—612no—
265#1803D6111yescharacteristic, complemented
266C2×C61#241511yescharacteristic
267C2×C61C2×#47011yescharacteristic
268C2×C61#305511yescharacteristic
269C2×C61#311211yescharacteristic
270C2×C61#311611yescharacteristic
271C2×C61#323711yescharacteristic
272C2×C61#323811yescharacteristic
273D61—42no—
274C2×C4⋊Q8—611no—
275C2×C22⋊Q8—611no—
276C2×C22⋊Q8—1221no—
277C2×C4×Q8C2×D6111yescharacteristic
278C22×C42—611no—
279#477—1224no—
280#432—1224no—
281C2×#107—612no—
282C2×#106—612no—
283C2×C4⋊C8—611no—
284C2×C4⋊C8C2×D6111yescharacteristic
285C2×#103—612no—
286C2×#103C2×D6112yes—
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#384C2×D6111yescharacteristic
297#380—611no—
298#380—612no—
299#380—1221no—
300C4×C4⋊C4—611no—
301C4×C4⋊C4—1221no—
302C4×C22⋊C4—1221no—
303#340—612no—
304#340—612no—
305#340—612no—
306#340C2×D6112yes—
307C61#2471111yescharacteristic, complemented, Sylow
308C22×Q8—1221no—
309C22×Q8C22×D6111yescharacteristic
310C23×C4—611no—
311C23×C4—611no—
312C23×C4—611no—
313C22×C8—612no—
314C22×C8C22×D6112yes—
315C4⋊Q8—1222no—
316C4⋊Q8—1222no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—2444no—
320C4×Q8—22no—
321C4×Q8—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—612no—
327C2×C4⋊C4—611no—
328C2×C4⋊C4—1221no—
329C2×C4⋊C4—1222no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4C22×D6111yescharacteristic
333C2×C4⋊C4C22×D6112yes—
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—24no—
350#103—1224no—
351C22⋊C8—1224no—
352#97—1224no—
353#95—1221no—
354#95—1221no—
355#95—1221no—
356#95—1221no—
357#95—1222no—
358#95—1222no—
359#95—1221no—
360C24—611no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—2442no—
367C2×Q8—2442no—
368C2×Q8—2442no—
369C2×Q8#1175822412yes—
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—1222no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
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—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—612no—
408C4⋊C4—1224no—
409C4⋊C4—1221no—
410C4⋊C4—1222no—
411C4⋊C4—1222no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—2442no—
415C4⋊C4—2444no—
416C4⋊C4—2442no—
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—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—4884no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
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—2442no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4—2444no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4#1451975012yes—
477C2×C4#1451975012yes—
478C2×C4#1451980612yes—
479C2×C4#1451980912yes—
480C2×C4C2×D4×D6111yescharacteristic
481C2×C4#41×D6111yescharacteristic
482C2×C4C2×#1175822411yescharacteristic
483C2×C4C2×#1175822211yescharacteristic
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#1684027211yescharacteristic
497C22#1684033511yescharacteristic
498C22AGL(1,ℤ/8)×D6111yescharacteristic
499C22C2×#1451980611yescharacteristic
500C22C2×#1451980911yescharacteristic
501C22#1684056111yescharacteristic
502C4—22no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—4882no—
507C4—4881no—
508C4—2441no—
509C4—2441no—
510C4—2441no—
511C4—2442no—
512C4#1684001914yes—
513C4#1684041411yescharacteristic
514C4#1684041811yescharacteristic
515C4#1684044711yescharacteristic
516C4#1684048011yescharacteristic
517C2—2442no—
518C2#2144925011yescharacteristic
519C2C2×#1684001911yescharacteristic
520C2#2145544111yescharacteristic
521C2#2145594011yescharacteristic
522C2#2145598011yescharacteristic
523C2#2145703211yescharacteristic
524C2#2145704411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1803D61
hypocenterC61#24711
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#16840057C2semidirect
7. #21446766C2semidirect
8. #21446580C2semidirect
9. #21446561C2semidirect
10. #21446511C2semidirect
11. C61×#1803C2semidirect
265. #1803D61semidirect
307. C61#24711semidirect

Maximal subgroups

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

subgroupindex
C2×#168400572
C2×#168398182
#214541722
C4×#145197532
#214472462
#214470132
#214467662
#214465802
#214465612
#214465112
C61×#18032
#2471161