Finite Group Databasebeta

← #29708129

Subgroups — #29708129

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#544×D61C211yesmaximal, characteristic
2#21454406C212yesmaximal
3#21448444C211yesmaximal, characteristic
4#21447169C211yesmaximal, characteristic, complemented
5#21447167C212yesmaximal, complemented
6#21447165C211yesmaximal, characteristic, complemented
7#21447163C212yesmaximal, complemented
8#21447161C211yesmaximal, characteristic, complemented
9#343×D61C211yesmaximal, characteristic
10#21445527C212yesmaximal
11C61×#2061C211yesmaximal, characteristic, complemented
12C2×M16×D61C2212yes—
13#16840393C2211yescharacteristic
14#16840392—22no—
15#16840391—22no—
16#16840390C2212yes—
17#16840389—22no—
18C4⋊Q8×D61—22no—
19#125×D61—22no—
20C22⋊Q8×D61—22no—
21C2×#14519735C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23#117×D61C2212yes—
24C2×C4⋊C4×D61C2211yescharacteristic
25#16839805C2212yes—
26#16839804C2212yes—
27#16839803—22no—
28#16839802C2212yes—
29#16839801—22no—
30#16839799—22no—
31#16839797—22no—
32#16839659—22no—
33#16839636—22no—
34#16839632C2211yescharacteristic
35#16839631C2211yescharacteristic
36#16839627C2211yescharacteristic
37#16839531C2211yescharacteristic
38#16839530C2212yes—
39#16839523C2212yes—
40#16839522C2212yes—
41#16839521C2212yes—
42C61×#544C2211yescharacteristic
43C61×#428C2212yes—
44C61×#382C2211yescharacteristic
45C61×#343C2211yescharacteristic
46C61×#339C2212yes—
47C2×Q8×D61—22no—
48C22×C4×D61D412yes—
49C22×C4×D61C2311yescharacteristic
50C2×#11758194C2312yes—
51C2×#11758189—22no—
52M16×D61—24no—
53M16×D61Q818yescomplemented
54C2×C8×D61—22no—
55#14519777—22no—
56#14519776—24no—
57#14519776—22no—
58#14519776—22no—
59#14519775—24no—
60#14519769—22no—
61#14519768—22no—
62#14519768—22no—
63#14519767—24no—
64#14519767—24no—
65#14519757—22no—
66#14519756—22no—
67#14519755—24no—
68#14519754—24no—
69#14519753—22no—
70#14519736C2312yes—
71#14519735—22no—
72#14519735—22no—
73#14519735D412yes—
74#14519735D412yes—
75#14519734—22no—
76#14519734—22no—
77#14519734Q814yescomplemented
78#14519733—22no—
79C2×C4⋊Dic61—22no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—22no—
84C2×#11758196D412yes—
85C4⋊C4×D61—22no—
86C4⋊C4×D61—24no—
87C4⋊C4×D61—22no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61D412yes—
90C4⋊C4×D61D412yes—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C22⋊C4×D61—22no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C42×D61—22no—
97#14519711—22no—
98#14519710—22no—
99#14519710—22no—
100#14519709—22no—
101#14519708—22no—
102#14519708—22no—
103#14519684—22no—
104C2×M16×C61C2312yes—
105C4⋊Q8×C61—22no—
106#125×C61—22no—
107C22⋊Q8×C61—22no—
108#117×C61C2312yes—
109C2×C4⋊C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C42×C61C2311yescharacteristic
112#107×C61—22no—
113#106×C61—22no—
114#95×C61—22no—
115C23×D61C2×D411yescharacteristic
116C2×Q8×C61—22no—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D412yes—
120C22×C4×C61C2411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61C2×D412yes—
124C22×Dic61C2×D411yescharacteristic
125Q8×D61—84no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—82no—
131C2×C4×D61C2×D412yes—
132C2×C4×D61C2×D412yes—
133C2×C4×D61C2×Q814yes—
134C2×C4×D61C2×Q812yes—
135C2×C4×D61#4112yes—
136C2×Dic122—24no—
137C2×Dic122—22no—
138M16×C61—22no—
139M16×C61C2×Q814yescomplemented
140C2×C8×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61C2×D412yes—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—22no—
148C42×C61—22no—
149C42×C61—21no—
150C42×C61C2×Q812yescomplemented
151C22⋊Dic61—22no—
152#11758198—22no—
153#11758198—22no—
154#11758198—24no—
155#11758198—41no—
156#11758198—41no—
157C4⋊Dic61—22no—
158C4⋊Dic61—24no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—82no—
162C4⋊Dic61C2×D412yes—
163C4⋊Dic61C2×D412yes—
164#11758196—24no—
165#11758196—24no—
166#11758196—24no—
167#11758196—24no—
168#11758196—22no—
169#11758196—22no—
170#11758196—24no—
171#11758196—82no—
172C4×Dic61—22no—
173C4×Dic61—21no—
174C4×Dic61—22no—
175C4×Dic61—22no—
176C4×Dic61—42no—
177C4×Dic61C2×Q812yescomplemented
178#11758194—22no—
179#11758194C2×Q814yescomplemented
180C2×#92418—22no—
181#11758189—28no—
182C8×D61—28no—
183C23×C61C22×D411yescharacteristic
184C22×D61—41no—
185C22×D61—41no—
186C22×D61#12011yescharacteristic
187C22×D61#12011yescharacteristic
188C22×D61C22⋊Q812yes—
189C22×D61C4⋊Q812yes—
190Q8×C61—82no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61C22×D412yes—
201C2×C4×C61C22×Q812yes—
202C2×C4×C61C22×Q811yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61#12012yes—
214C2×Dic61C22⋊Q812yes—
215C2×Dic61C4⋊Q812yes—
216C4×D61—44no—
217C4×D61—44no—
218C4×D61—44no—
219C4×D61—82no—
220C4×D61C22⋊Q814yes—
221C4×D61C22⋊Q814yes—
222C4×D61C4⋊Q814yes—
223C4×D61C4⋊Q814yes—
224Dic122—82no—
225Dic122—84no—
226C8×C61—24no—
227#92418—24no—
228#23561—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61C2×#12011yescharacteristic
232C22×C61C2×C22⋊Q811yescharacteristic
233C22×C61C2×C4⋊Q811yescharacteristic
234C2×D61—21no—
235C2×D61—21no—
236C2×D61—21no—
237C2×D61—21no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61#39512yes—
241C2×D61#39512yes—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61—42no—
245C4×C61—42no—
246C4×C61—81no—
247C4×C61C2×C22⋊Q812yes—
248C4×C61C2×C4⋊Q812yes—
249Dic61—42no—
250Dic61—42no—
251Dic61—42no—
252Dic61—42no—
253Dic61—81no—
254Dic61#39514yes—
255C2×#544—611no—
256#2911—612no—
257#2317—611no—
258#2061—612no—
259#2061—611no—
260#2061D6111yescharacteristic, complemented
261#2060—611no—
262#2060—612no—
263#2060—611no—
264C2×#343—611no—
265#1733—612no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61C2×#39511yescharacteristic
269C2×C61#301911yescharacteristic
270C2×C61#302011yescharacteristic
271D61—22no—
272D61—22no—
273C22×M16—612no—
274#544—612no—
275#544—611no—
276#544—1222no—
277#544—1222no—
278#544—1222no—
279#544C2×D6111yescharacteristic
280C2×C4⋊Q8—1222no—
281C2×#125—1222no—
282C2×C22⋊Q8—1222no—
283C2×#117—611no—
284C2×#117—612no—
285C2×#117—611no—
286C22×C4⋊C4—611no—
287#428—612no—
288#428—612no—
289#428—612no—
290#428—1222no—
291#428—1222no—
292#428C2×D6112yes—
293#427—1222no—
294#427—1222no—
295#389—1222no—
296#385—1222no—
297#384—611no—
298#384—611no—
299#382—611no—
300#382C2×D6111yescharacteristic
301#343—612no—
302#343—611no—
303#343C2×D6111yescharacteristic
304#339—612no—
305#339—612no—
306#339—612no—
307#339C2×D6112yes—
308C61#2356111yescharacteristic, complemented, Sylow
309C22×Q8—1222no—
310C23×C4—611no—
311C23×C4—612no—
312C2×M16—612no—
313C2×M16—618no—
314C2×M16—1222no—
315C2×M16—1224no—
316C2×M16C22×D6112yes—
317C22×C8—1222no—
318C4⋊Q8—22no—
319C4⋊Q8—1222no—
320C4⋊Q8—1222no—
321C4⋊Q8—1224no—
322C4⋊Q8—1224no—
323C4⋊Q8—1222no—
324#125—22no—
325#125—1222no—
326#125—1222no—
327#125—1224no—
328#125—1224no—
329#125—1222no—
330C22⋊Q8—22no—
331C22⋊Q8—1224no—
332C22⋊Q8—1224no—
333C22⋊Q8—1222no—
334C22⋊Q8—1222no—
335C22⋊Q8—1222no—
336#117—614no—
337#117—612no—
338#117—612no—
339#117—612no—
340#117—1222no—
341#117—1222no—
342#117—1222no—
343#117—1222no—
344#117—1222no—
345#117C22×D6112yes—
346C2×C4⋊C4—612no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—612no—
349C2×C4⋊C4—611no—
350C2×C4⋊C4—611no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4—1222no—
353C2×C4⋊C4—1222no—
354C2×C4⋊C4—1224no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1222no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4C22×D6111yescharacteristic
359C2×C4⋊C4C22×D6111yescharacteristic
360C2×C22⋊C4—1222no—
361C2×C22⋊C4—1221no—
362C2×C22⋊C4—1221no—
363C2×C42—611no—
364C2×C42—1222no—
365C2×C42—1221no—
366C2×C42C22×D6111yescharacteristic
367#107—22no—
368#107—1222no—
369#107—1222no—
370#107—1222no—
371#106—22no—
372#106—1222no—
373#106—1222no—
374#106—1222no—
375#95—22no—
376#95—1222no—
377C24—611no—
378C2×Q8—22no—
379C2×Q8—1224no—
380C2×Q8—1222no—
381C2×Q8—4884no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—611no—
385C22×C4—612no—
386C22×C4—612no—
387C22×C4—612no—
388C22×C4—614no—
389C22×C4—612no—
390C22×C4—612no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4—1222no—
394C22×C4—1222no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—4882no—
398C22×C4D4×D6112yes—
399C22×C4C23×D6111yescharacteristic
400M16—22no—
401M16—614no—
402M16—1228no—
403M16—1222no—
404M16Q8×D6114yescomplemented
405C2×C8—22no—
406C2×C8—1228no—
407C2×C8—1222no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—612no—
413C4⋊C4—612no—
414C4⋊C4—1224no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C4⋊C4—1224no—
421C4⋊C4—1222no—
422C4⋊C4—1224no—
423C4⋊C4—1222no—
424C4⋊C4—1222no—
425C4⋊C4—4882no—
426C4⋊C4—4882no—
427C4⋊C4D4×D6112yes—
428C4⋊C4D4×D6112yes—
429C22⋊C4—22no—
430C22⋊C4—1224no—
431C22⋊C4—1222no—
432C22⋊C4—1222no—
433C22⋊C4—1222no—
434C22⋊C4—2441no—
435C22⋊C4—2441no—
436C42—22no—
437C42—21no—
438C42—612no—
439C42—1222no—
440C42—1222no—
441C42—1222no—
442C42—1221no—
443C42—2442no—
444C42Q8×D6112yescomplemented
445C23—611no—
446C23—611no—
447C23—612no—
448C23—612no—
449C23—2441no—
450C23—2441no—
451C23C2×D4×D6111yescharacteristic
452Q8—82no—
453Q8—4882no—
454Q8—4884no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—614no—
468C2×C4—614no—
469C2×C4—614no—
470C2×C4—614no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—1222no—
474C2×C4—1222no—
475C2×C4—4882no—
476C2×C4—2444no—
477C2×C4—2444no—
478C2×C4—2444no—
479C2×C4—2442no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4—2442no—
484C2×C4C2×D4×D6112yes—
485C2×C4C2×Q8×D6112yes—
486C2×C4C2×Q8×D6111yescharacteristic
487C2×C4#41×D6111yescharacteristic
488C8—24no—
489C8—1224no—
490C22—21no—
491C22—21no—
492C22—612no—
493C22—612no—
494C22—1221no—
495C22—1221no—
496C22—1221no—
497C22—1221no—
498C22—2442no—
499C22—2442no—
500C22#120×D6111yescharacteristic
501C22C22⋊Q8×D6111yescharacteristic
502C22C4⋊Q8×D6111yescharacteristic
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—614no—
509C4—4881no—
510C4—2442no—
511C4—2442no—
512C4—2442no—
513C4—2442no—
514C4C22⋊Q8×D6112yes—
515C4C4⋊Q8×D6112yes—
516C2—21no—
517C2—21no—
518C2—1222no—
519C2—1222no—
520C2#395×D6111yescharacteristic
521C2#2145517711yescharacteristic
522C2#2145518011yescharacteristic
subgroupG/H
centerC22#120×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2061C2
socleC22×C61C2×#120
prime radicalC23×D61C2×D4
hypercenter#2061D61
hypocenterC61#23561
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
4. #21447169C2semidirect
5. #21447167C2semidirect
6. #21447165C2semidirect
7. #21447163C2semidirect
8. #21447161C2semidirect
11. C61×#2061C2semidirect
53. M16×D61Q8semidirect
77. #14519734Q8semidirect
139. M16×C61C2×Q8semidirect
150. C42×C61C2×Q8semidirect
177. C4×Dic61C2×Q8semidirect
179. #11758194C2×Q8semidirect
260. #2061D61semidirect
308. C61#23561semidirect
404. M16Q8×D61semidirect
444. C42Q8×D61semidirect

Maximal subgroups

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

subgroupindex
#544×D612
#214544062
#214484442
#214471692
#214471672
#214471652
#214471632
#214471612
#343×D612
#214455272
C61×#20612
#2356161