Finite Group Databasebeta

← #30623975

Subgroups — #30623975

Order 15616.

1252 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D61⋊#410C212yesmaximal, complemented
2#21462101C212yesmaximal, complemented
3#21462100C212yesmaximal, complemented
4#21462097C214yesmaximal, complemented
5#21462093C214yesmaximal, complemented
6#21459151C211yesmaximal, characteristic
7C2×#14519856—22no—
8#16840743C2214yescomplemented
9#16840742—22no—
10#16840741—22no—
11#16840741C2214yes—
12D61⋊#99—22no—
13D61⋊#99C2214yescomplemented
14#16840668C2218yescomplemented
15#16840667C2218yescomplemented
16C22×#11758226C411yescharacteristic, complemented
17C2×#41×D61C2211yescharacteristic
18#16840342—22no—
19#16840337C412yescomplemented
20#16840336C2211yescharacteristic
21#16840331C2211yescharacteristic
22#16840330C2212yes—
23#16840326C411yescharacteristic, complemented
24#16840323—22no—
25#16840313C2212yes—
26#16840311C412yescomplemented
27#16840275—22no—
28#16840270—22no—
29#117×D61C411yescharacteristic, complemented
30C61×#536C411yescharacteristic, complemented
31#14519856—24no—
32#14519856—24no—
33#14519856—42no—
34#14519856—42no—
35C2×C4⋊#6923—42no—
36C2×C22⋊#6923—24no—
37C2×C4×#6923—42no—
38#14519836—24no—
39#14519836C2318yes—
40#14519835—24no—
41#14519835—24no—
42C2×#11758226—21no—
43C2×#11758226—21no—
44C2×#11758226—42no—
45C2×#11758226—42no—
46C2×#11758226C2×C412yes—
47C2×#11758226C2×C412yescomplemented
48C2×#11758226C2311yescharacteristic
49C2×#11758222D412yes—
50C2×#11758222C2311yescharacteristic
51C2×#11758224C2×C411yescharacteristic
52C2×#11758224D412yes—
53#41×D61—42no—
54#41×D61—42no—
55C2×Q8×D61C2×C411yescharacteristic
56C22×D244—21no—
57C22×D244C2×C411yescharacteristic, complemented
58C2×D4×D61D412yes—
59C2×D4×D61C2311yescharacteristic
60C22×C4×D61—21no—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61D412yes—
63#14519775C2×C412yescomplemented
64#14519774C2312yes—
65#14519773C2×C412yes—
66#14519766C2×C412yescomplemented
67#14519766C2312yes—
68#14519765C2×C412yescomplemented
69#14519762—24no—
70#14519760—22no—
71#14519755—24no—
72#14519753—22no—
73#14519752—24no—
74#14519752—22no—
75#14519751—22no—
76#14519750—24no—
77#14519750—24no—
78#14519747—24no—
79#14519747—22no—
80#14519745—22no—
81#14519739—24no—
82#14519737—24no—
83#14519736C2×C411yescharacteristic, complemented
84#14519735—21no—
85#14519734C2×C412yes—
86#14519733—21no—
87C4⋊C4×D61—21no—
88C22⋊C4×D61—21no—
89C42×D61C2×C412yescomplemented
90C2×#41×C61C2×C411yescharacteristic
91C22×Q8×C61C2×C411yescharacteristic, complemented
92C4⋊Q8×C61C2×C412yescomplemented
93#124×C61C2×C412yes—
94#123×C61—22no—
95C22⋊Q8×C61—22no—
96#117×C61C2×C411yescharacteristic, complemented
97C23×D61—21no—
98C23×D61C22⋊C411yescharacteristic
99C23×D61C2×D412yes—
100C22×#6923—44no—
101#41×C61—42no—
102C2×Q8×C61—21no—
103C2×Q8×C61—42no—
104C2×Q8×C61C22×C412yes—
105C2×Q8×C61C22×C411yescharacteristic
106C2×D4×C61C22⋊C412yes—
107C2×D4×C61C22×C411yescharacteristic
108C22×C4×C61—21no—
109C22×C4×C61C22⋊C412yes—
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61—24no—
112C2×C22⋊D61C22⋊C412yes—
113C2×C22⋊D61C2×D412yes—
114C2×C22⋊D61C2×D412yes—
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C22⋊C412yes—
117#11758226—41no—
118#11758226—41no—
119#11758226—44no—
120#11758226—44no—
121#11758226—42no—
122#11758226—42no—
123#11758226—42no—
124#11758226—42no—
125Q8×D61—42no—
126#11758224—41no—
127#11758224—41no—
128#11758224—42no—
129#11758224—42no—
130D4×D61—42no—
131D4×D61—42no—
132D4×D61—42no—
133#11758222—42no—
134#11758222—42no—
135#11758222—42no—
136C2×D244—22no—
137C2×D244—22no—
138C2×D244—21no—
139C2×D244—24no—
140C2×D244—44no—
141C2×D244—42no—
142C2×D244C22×C412yes—
143C2×D244C2×D412yes—
144C2×D244C2411yescharacteristic
145C2×C4×D61—22no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—42no—
155C2×C4×D61—42no—
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C412yes—
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22×C411yescharacteristic
160C2×C4×D61C22×C411yescharacteristic
161C2×C4×D61C2×D412yes—
162C2×C4×D61C2×D412yes—
163C2×Dic122C22⋊C412yes—
164C2×Dic122C22×C411yescharacteristic
165C22⋊#6923—28no—
166C22⋊#6923—44no—
167C4⋊#6923—44no—
168C4⋊#6923—44no—
169C4×#6923—44no—
170C4×#6923—44no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—24no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—24no—
175C42×C61C22×C412yes—
176C22⋊Dic61—21no—
177#11758198—22no—
178#11758198—24no—
179#11758198—24no—
180#11758198—24no—
181C4⋊Dic61—21no—
182#11758196—22no—
183#11758196—24no—
184C4×Dic61—22no—
185C4×Dic61C22×C412yes—
186C23×C61C2×C22⋊C411yescharacteristic
187C23×C61C2×C22⋊C412yes—
188C22×D61—22no—
189C22×D61—22no—
190C22×D61—22no—
191C22×D61—22no—
192C22×D61—42no—
193C22×D61—82no—
194C22×D61—82no—
195C22×D61—81no—
196C22×D61C2×C22⋊C411yescharacteristic
197C22×D61C2×C22⋊C411yescharacteristic
198C22×D61C22×D412yes—
199C2×#6923—48no—
200C2×#6923—88no—
201Q8×C61—41no—
202Q8×C61—42no—
203Q8×C61—42no—
204D4×C61—42no—
205D4×C61—41no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—42no—
212C2×C4×C61C2×C22⋊C412yes—
213C2×C4×C61C2×C22⋊C411yescharacteristic
214C2×C4×C61C23×C411yescharacteristic
215C22⋊D61—42no—
216C22⋊D61—42no—
217C22⋊D61—42no—
218C22⋊D61—84no—
219C2×Dic61—21no—
220C2×Dic61—21no—
221C2×Dic61—22no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61C2×C22⋊C412yes—
225C2×Dic61C2×C22⋊C411yescharacteristic
226C2×Dic61C2×C22⋊C411yescharacteristic
227D244—42no—
228D244—41no—
229D244—42no—
230D244—44no—
231D244—44no—
232D244—42no—
233D244—84no—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—21no—
243C4×D61—21no—
244C4×D61—44no—
245C4×D61—42no—
246C4×D61—42no—
247C4×D61—82no—
248C4×D61#9912yes—
249C4×D61#9912yes—
250Dic122—42no—
251Dic122—41no—
252#33358—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—22no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61C22×C22⋊C411yescharacteristic
258C2×D61—21no—
259C2×D61—22no—
260C2×D61—41no—
261C2×D61—44no—
262C2×D61—41no—
263C2×D61—42no—
264C2×D61—44no—
265C2×D61—42no—
266C2×D61—82no—
267C2×D61—82no—
268C2×D61—84no—
269C2×D61C2×#9912yes—
270#6923—88no—
271C4×C61—21no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61—42no—
275C4×C61C2×#9912yes—
276Dic61—21no—
277Dic61—22no—
278Dic61—42no—
279Dic61C2×#9911yescharacteristic
280Dic61C2×#9911yescharacteristic
281#3449—611no—
282C2×#410—612no—
283#2129—612no—
284#2128—612no—
285#2127—614no—
286#2126—614no—
287C2×C61—21no—
288C2×C61—42no—
289C2×C61—41no—
290C2×C61C22×#9911yescharacteristic
291D61—22no—
292D61—42no—
293D61—41no—
294D61—82no—
295C22×#41—611no—
296C22×#41—611no—
297#537—1222no—
298#536—611no—
299#536—612no—
300#536#692311yescharacteristic, complemented
301#535—611no—
302#535—612no—
303#535—611no—
304#534—1222no—
305#532—612no—
306#532—612no—
307#525—1222no—
308#525—1222no—
309C2×#117—611no—
310C2×#117—1222no—
311#410—614no—
312#410—614no—
313#410—1222no—
314#410—1222no—
315C2×#99—614no—
316C2×#99—1222no—
317#354—618no—
318#353—618no—
319C61#3335811yescharacteristic, complemented, Sylow
320C2×#41—611no—
321C2×#41—612no—
322C2×#41—611no—
323C2×#41—612no—
324C2×#41—611no—
325C2×#41—612no—
326C2×#41—612no—
327C2×#41—1221no—
328C2×#41—1221no—
329C2×#41—2442no—
330C2×#41—2442no—
331C2×#41—2442no—
332C2×#41—2442no—
333C2×#41C2×#692311yescharacteristic
334C22×Q8—611no—
335C22×Q8C2×#692311yescharacteristic, complemented
336C22×D4—612no—
337C22×D4—611no—
338C22×D4—611no—
339C22×D4—1221no—
340C23×C4—612no—
341C23×C4—611no—
342C23×C4—1221no—
343C4⋊Q8—612no—
344C4⋊Q8C2×#692312yescomplemented
345#127—612no—
346#127—612no—
347#124—612no—
348#124—612no—
349#124—612no—
350#124C2×#692312yes—
351#123—22no—
352#123—1224no—
353#123—1222no—
354C22⋊Q8—22no—
355C22⋊Q8—1224no—
356C22⋊Q8—1222no—
357#121—1224no—
358#121—1224no—
359#121—1222no—
360#121—1224no—
361#121—1222no—
362#120—1222no—
363#120—1224no—
364#120—1222no—
365C4×D4—1224no—
366C4×D4—1224no—
367#117—611no—
368#117—612no—
369#117—1221no—
370#117—1221no—
371#117—1224no—
372#117—1224no—
373#117—2442no—
374#117—2442no—
375#117C2×#692311yescharacteristic, complemented
376C2×C4⋊C4—1221no—
377C2×C4⋊C4—2442no—
378C2×C22⋊C4—1221no—
379C2×C22⋊C4—1224no—
380C2×C42—612no—
381C2×C42—2442no—
382#99—618no—
383#99—1224no—
384#99—1224no—
385#99—1224no—
386C24—612no—
387C24—611no—
388C24—1221no—
389#41—42no—
390#41—2442no—
391#41—2442no—
392#41—2442no—
393#41—2442no—
394#41—2442no—
395#41—2442no—
396#41—2442no—
397#41—2441no—
398#41—2441no—
399#41—2441no—
400#41—2441no—
401#41—2444no—
402#41—2444no—
403#41—2442no—
404#41—2442no—
405C2×Q8—21no—
406C2×Q8—42no—
407C2×Q8—612no—
408C2×Q8—611no—
409C2×Q8—2442no—
410C2×Q8C22×#692312yes—
411C2×Q8C22×#692311yescharacteristic
412C2×D4—612no—
413C2×D4—612no—
414C2×D4—612no—
415C2×D4—612no—
416C2×D4—611no—
417C2×D4—612no—
418C2×D4—1224no—
419C2×D4—1222no—
420C2×D4—1222no—
421C2×D4—1224no—
422C2×D4—1221no—
423C2×D4—2442no—
424C2×D4—2444no—
425C2×D4—2442no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4C22⋊#692312yes—
429C2×D4C22×#692311yescharacteristic
430C22×C4—21no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—611no—
437C22×C4—612no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2444no—
452C22×C4C22⋊#692312yes—
453C22×C4C22×#692311yescharacteristic
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—1222no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—2444no—
460C4⋊C4—2444no—
461C22⋊C4—21no—
462C22⋊C4—24no—
463C22⋊C4—1224no—
464C22⋊C4—1222no—
465C22⋊C4—1224no—
466C22⋊C4—1221no—
467C22⋊C4—1224no—
468C22⋊C4—1228no—
469C22⋊C4—2444no—
470C42—612no—
471C42—1222no—
472C42—2444no—
473C42—2444no—
474C42C22×#692312yes—
475C23—611no—
476C23—611no—
477C23—612no—
478C23—1222no—
479C23—1222no—
480C23—1222no—
481C23—1222no—
482C23—4882no—
483C23—4881no—
484C23—4882no—
485C23—2442no—
486C23C2×C22⋊#692312yes—
487C23C2×C22⋊#692311yescharacteristic
488Q8—41no—
489Q8—42no—
490Q8—42no—
491Q8—2442no—
492Q8—2441no—
493D4—42no—
494D4—41no—
495D4—4884no—
496D4—4884no—
497D4—2442no—
498D4—2441no—
499D4—2442no—
500D4—2444no—
501D4—2444no—
502D4—2442no—
503D4—2442no—
504D4—2442no—
505D4—2442no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—611no—
513C2×C4—612no—
514C2×C4—612no—
515C2×C4—612no—
516C2×C4—611no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1221no—
520C2×C4—1221no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1221no—
525C2×C4—1221no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—4882no—
533C2×C4—4888no—
534C2×C4—2442no—
535C2×C4—2442no—
536C2×C4—2444no—
537C2×C4—2448no—
538C2×C4C2×C22⋊#692312yes—
539C2×C4C2×C22⋊#692311yescharacteristic
540C2×C4C23×#692311yescharacteristic
541C22—21no—
542C22—22no—
543C22—42no—
544C22—41no—
545C22—612no—
546C22—1222no—
547C22—1221no—
548C22—4882no—
549C22—4882no—
550C22—4884no—
551C22—2441no—
552C22—2444no—
553C22—2441no—
554C22—2442no—
555C22—2444no—
556C22—2442no—
557C22C22×C22⋊#692311yescharacteristic
558C4—22no—
559C4—21no—
560C4—42no—
561C4—42no—
562C4—611no—
563C4—611no—
564C4—1221no—
565C4—1222no—
566C4—4888no—
567C4—2442no—
568C4D61⋊#9912yes—
569C2—21no—
570C2—41no—
571C2—42no—
572C2—1222no—
573C2—4882no—
574C2—2442no—
575C2—2441no—
576C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#536C4
socleC2×C61C22×#99
prime radical#21459151C2
cosocleC2×D244C24
hypercenter#536#6923
hypocenterC61#33358
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
1. D61⋊#410C2semidirect
2. #21462101C2semidirect
3. #21462100C2semidirect
4. #21462097C2semidirect
5. #21462093C2semidirect
8. #16840743C22semidirect
13. D61⋊#99C22semidirect
14. #16840668C22semidirect
15. #16840667C22semidirect
16. C22×#11758226C4semidirect
19. #16840337C4semidirect
23. #16840326C4semidirect
26. #16840311C4semidirect
29. #117×D61C4semidirect
30. C61×#536C4semidirect
47. C2×#11758226C2×C4semidirect
57. C22×D244C2×C4semidirect
63. #14519775C2×C4semidirect
66. #14519766C2×C4semidirect
68. #14519765C2×C4semidirect
83. #14519736C2×C4semidirect
89. C42×D61C2×C4semidirect
91. C22×Q8×C61C2×C4semidirect
92. C4⋊Q8×C61C2×C4semidirect
96. #117×C61C2×C4semidirect
300. #536#6923semidirect
319. C61#33358semidirect
335. C22×Q8C2×#6923semidirect
344. C4⋊Q8C2×#6923semidirect
375. #117C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4102
#214621012
#214621002
#214620972
#214620932
#214591512
#3335861