Finite Group Databasebeta

← #30623973

Subgroups — #30623973

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#21462103C212yesmaximal, complemented
3#21462098C212yesmaximal, complemented
4#21462095C214yesmaximal, complemented
5#21462091C214yesmaximal, complemented
6#21459150C211yesmaximal, characteristic
7C2×#14519856—22no—
8#16840743C2214yescomplemented
9#16840742—22no—
10#16840741—22no—
11#16840741C2214yes—
12D61⋊#99—22no—
13D61⋊#99C2214yescomplemented
14#16840669C2218yescomplemented
15#16840666C2218yescomplemented
16C22×#11758224C411yescharacteristic, complemented
17C2×#41×D61C2211yescharacteristic
18#16840341—22no—
19#16840338C2211yescharacteristic
20#16840333C411yescharacteristic, complemented
21#16840332C412yescomplemented
22#16840329C2211yescharacteristic
23#16840325C2212yes—
24#16840322—22no—
25#16840311C2212yes—
26#16840310C412yescomplemented
27#16840272—22no—
28#16840268—22no—
29#117×D61C411yescharacteristic, complemented
30C61×#535C411yescharacteristic, 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×#11758226C2311yescharacteristic
43C2×#11758222D412yes—
44C2×#11758222C2311yescharacteristic
45C2×#11758224—21no—
46C2×#11758224—21no—
47C2×#11758224—42no—
48C2×#11758224—42no—
49C2×#11758224C2×C412yes—
50C2×#11758224C2×C411yescharacteristic
51C2×#11758224C2×C412yescomplemented
52C2×#11758224D412yes—
53#41×D61—42no—
54#41×D61—42no—
55C22×Dic122C2×C411yescharacteristic, complemented
56C2×Q8×D61C2×C411yescharacteristic
57C22×C22⋊D61—21no—
58C2×D4×D61D412yes—
59C2×D4×D61C2311yescharacteristic
60C23×Dic61—21no—
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61D412yes—
63#14519777C2×C412yescomplemented
64#14519775C2×C412yescomplemented
65#14519773C2312yes—
66#14519765C2312yes—
67#14519764C2×C412yes—
68#14519764C2×C412yescomplemented
69#14519761—22no—
70#14519759—24no—
71#14519758—22no—
72#14519757—22no—
73#14519756—22no—
74#14519754—24no—
75#14519750—24no—
76#14519749—24no—
77#14519748—24no—
78#14519748—22no—
79#14519746—22no—
80#14519745—24no—
81D4×Dic61—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×D4×C61C2×C411yescharacteristic, complemented
92#127×C61C2×C412yes—
93#124×C61C2×C412yescomplemented
94#121×C61—22no—
95#120×C61—22no—
96#117×C61C2×C411yescharacteristic, complemented
97C24×C61—21no—
98C23×D61C22⋊C411yescharacteristic
99C23×D61C2×D412yes—
100C22×#6923—44no—
101#41×C61—42no—
102C2×Q8×C61C22×C411yescharacteristic
103C2×D4×C61—24no—
104C2×D4×C61—21no—
105C2×D4×C61—42no—
106C2×D4×C61C22⋊C412yes—
107C2×D4×C61C22×C412yes—
108C2×D4×C61C22×C411yescharacteristic
109C22×C4×C61C22⋊C412yes—
110C22×C4×C61C22×C411yescharacteristic
111C2×C22⋊D61—22no—
112C2×C22⋊D61—22no—
113C2×C22⋊D61—24no—
114C2×C22⋊D61—44no—
115C2×C22⋊D61C22⋊C412yes—
116C2×C22⋊D61C2×D412yes—
117C2×C22⋊D61C2×D412yes—
118C22×Dic61—21no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—42no—
122C22×Dic61—42no—
123C22×Dic61C22⋊C411yescharacteristic
124C22×Dic61C22⋊C412yes—
125#11758226—41no—
126#11758226—41no—
127Q8×D61—42no—
128#11758224—41no—
129#11758224—41no—
130#11758224—42no—
131#11758224—42no—
132#11758224—44no—
133#11758224—44no—
134#11758224—42no—
135#11758224—42no—
136#11758224—42no—
137#11758224—42no—
138D4×D61—42no—
139D4×D61—42no—
140D4×D61—42no—
141#11758222—42no—
142#11758222—42no—
143#11758222—42no—
144C2×D244C2×D412yes—
145C2×D244C2411yescharacteristic
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×D61C22⋊C411yescharacteristic
154C2×C4×D61C22⋊C412yes—
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C22×C411yescharacteristic
158C2×C4×D61C2×D412yes—
159C2×C4×D61C2×D412yes—
160C2×Dic122—21no—
161C2×Dic122—42no—
162C2×Dic122C22⋊C412yes—
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—
172C22⋊C4×C61—21no—
173C22⋊C4×C61—24no—
174C42×C61C22×C412yes—
175C22⋊Dic61—21no—
176C22⋊Dic61—24no—
177C22⋊Dic61—24no—
178#11758198—22no—
179#11758198—24no—
180C4⋊Dic61—21no—
181C4⋊Dic61—24no—
182#11758196—24no—
183#11758196—22no—
184C4×Dic61—22no—
185C4×Dic61C22×C412yes—
186C23×C61—22no—
187C23×C61—42no—
188C23×C61—82no—
189C23×C61C2×C22⋊C412yes—
190C23×C61C2×C22⋊C411yescharacteristic
191C22×D61—22no—
192C22×D61—22no—
193C22×D61—22no—
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—
202D4×C61—42no—
203D4×C61—41no—
204D4×C61—42no—
205D4×C61—42no—
206D4×C61—84no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61C2×C22⋊C412yes—
212C2×C4×C61C2×C22⋊C411yescharacteristic
213C2×C4×C61C23×C411yescharacteristic
214C22⋊D61—42no—
215C22⋊D61—42no—
216C22⋊D61—42no—
217C22⋊D61—44no—
218C22⋊D61—44no—
219C22⋊D61—84no—
220C2×Dic61—21no—
221C2×Dic61—21no—
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—22no—
225C2×Dic61—22no—
226C2×Dic61—44no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61C2×C22⋊C411yescharacteristic
231C2×Dic61C2×C22⋊C412yes—
232C2×Dic61C2×C22⋊C411yescharacteristic
233D244—42no—
234D244—41no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—22no—
239C4×D61—22no—
240C4×D61—22no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—21no—
244C4×D61—21no—
245C4×D61—82no—
246C4×D61#9912yes—
247C4×D61#9912yes—
248Dic122—42no—
249Dic122—41no—
250Dic122—42no—
251Dic122—42no—
252#33358—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—22no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61—44no—
258C22×C61—42no—
259C22×C61—84no—
260C22×C61C22×C22⋊C411yescharacteristic
261C2×D61—21no—
262C2×D61—22no—
263C2×D61—41no—
264C2×D61—44no—
265C2×D61—41no—
266C2×D61—42no—
267C2×D61—82no—
268C2×D61—82no—
269C2×D61C2×#9912yes—
270#6923—88no—
271C4×C61—21no—
272C4×C61—22no—
273C4×C61—42no—
274C4×C61C2×#9912yes—
275Dic61—21no—
276Dic61—22no—
277Dic61—42no—
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×C61—82no—
291C2×C61C22×#9911yescharacteristic
292D61—22no—
293D61—42no—
294D61—41no—
295C22×#41—611no—
296C22×#41—611no—
297#537—1222no—
298#536—611no—
299#536—612no—
300#536—611no—
301#535—611no—
302#535—612no—
303#535#692311yescharacteristic, complemented
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×Q8—611no—
336C22×D4—612no—
337C22×D4—611no—
338C22×D4—1221no—
339C22×D4C2×#692311yescharacteristic, complemented
340C23×C4—612no—
341C23×C4—611no—
342C23×C4—1221no—
343C4⋊Q8—612no—
344C4⋊Q8—612no—
345#127—612no—
346#127C2×#692312yes—
347#124—612no—
348#124—612no—
349#124—612no—
350#124C2×#692312yescomplemented
351#123—1224no—
352#123—1222no—
353#123—1222no—
354C22⋊Q8—1224no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357#121—22no—
358#121—1224no—
359#121—1222no—
360#121—1224no—
361#121—1224no—
362#120—22no—
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—21no—
387C24—612no—
388C24—611no—
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—612no—
406C2×Q8—611no—
407C2×Q8—612no—
408C2×Q8—1221no—
409C2×Q8—2442no—
410C2×Q8—2442no—
411C2×Q8C22×#692311yescharacteristic
412C2×D4—24no—
413C2×D4—21no—
414C2×D4—42no—
415C2×D4—612no—
416C2×D4—612no—
417C2×D4—612no—
418C2×D4—612no—
419C2×D4—611no—
420C2×D4—1224no—
421C2×D4—1222no—
422C2×D4—1222no—
423C2×D4—2442no—
424C2×D4—2442no—
425C2×D4—2442no—
426C2×D4—2444no—
427C2×D4C22⋊#692312yes—
428C2×D4C22×#692312yes—
429C2×D4C22×#692311yescharacteristic
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—612no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—611no—
436C22×C4—612no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—1222no—
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—1221no—
449C22×C4—2442no—
450C22×C4—2442no—
451C22×C4—2444no—
452C22×C4C22⋊#692312yes—
453C22×C4C22×#692311yescharacteristic
454C4⋊C4—21no—
455C4⋊C4—1222no—
456C4⋊C4—1224no—
457C4⋊C4—1224no—
458C4⋊C4—1221no—
459C4⋊C4—2444no—
460C4⋊C4—2444no—
461C22⋊C4—21no—
462C22⋊C4—24no—
463C22⋊C4—1222no—
464C22⋊C4—1224no—
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—22no—
476C23—42no—
477C23—82no—
478C23—611no—
479C23—611no—
480C23—612no—
481C23—1222no—
482C23—1222no—
483C23—1222no—
484C23—4881no—
485C23—4882no—
486C23C2×C22⋊#692311yescharacteristic
487C23C2×C22⋊#692312yes—
488Q8—41no—
489Q8—2442no—
490Q8—2441no—
491Q8—2442no—
492Q8—2442no—
493D4—42no—
494D4—41no—
495D4—42no—
496D4—42no—
497D4—84no—
498D4—4884no—
499D4—2442no—
500D4—2441no—
501D4—2442no—
502D4—2442no—
503D4—2442no—
504D4—2444no—
505D4—2444no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—611no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—612no—
514C2×C4—611no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1221no—
523C2×C4—1221no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—4882no—
532C2×C4—4888no—
533C2×C4—2442no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2448no—
537C2×C4—2442no—
538C2×C4C2×C22⋊#692312yes—
539C2×C4C2×C22⋊#692311yescharacteristic
540C2×C4C23×#692311yescharacteristic
541C22—21no—
542C22—22no—
543C22—42no—
544C22—41no—
545C22—44no—
546C22—42no—
547C22—84no—
548C22—612no—
549C22—1222no—
550C22—1221no—
551C22—4882no—
552C22—4882no—
553C22—2441no—
554C22—2441no—
555C22—2442no—
556C22—2444no—
557C22C22×C22⋊#692311yescharacteristic
558C4—22no—
559C4—21no—
560C4—42no—
561C4—611no—
562C4—611no—
563C4—1221no—
564C4—1222no—
565C4—4888no—
566C4—2442no—
567C4—2442no—
568C4D61⋊#9912yes—
569C2—21no—
570C2—41no—
571C2—42no—
572C2—82no—
573C2—1222no—
574C2—2442no—
575C2—2441no—
576C2C2×D61⋊#9911yescharacteristic
subgroupG/H
centerC2C2×D61⋊#99
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#535C4
socleC2×C61C22×#99
prime radical#21459150C2
cosocleC2×D244C24
hypercenter#535#6923
hypocenterC61#33358
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. D61⋊#410C2semidirect
2. #21462103C2semidirect
3. #21462098C2semidirect
4. #21462095C2semidirect
5. #21462091C2semidirect
8. #16840743C22semidirect
13. D61⋊#99C22semidirect
14. #16840669C22semidirect
15. #16840666C22semidirect
16. C22×#11758224C4semidirect
20. #16840333C4semidirect
21. #16840332C4semidirect
26. #16840310C4semidirect
29. #117×D61C4semidirect
30. C61×#535C4semidirect
51. C2×#11758224C2×C4semidirect
55. C22×Dic122C2×C4semidirect
63. #14519777C2×C4semidirect
64. #14519775C2×C4semidirect
68. #14519764C2×C4semidirect
83. #14519736C2×C4semidirect
89. C42×D61C2×C4semidirect
91. C22×D4×C61C2×C4semidirect
93. #124×C61C2×C4semidirect
96. #117×C61C2×C4semidirect
303. #535#6923semidirect
319. C61#33358semidirect
339. C22×D4C2×#6923semidirect
350. #124C2×#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
#214621032
#214620982
#214620952
#214620912
#214591502
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