Finite Group Databasebeta

← #30623970

Subgroups — #30623970

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
1C2×#16840741C212yesmaximal, complemented
2#21462101C212yesmaximal, complemented
3#21462098C212yesmaximal, complemented
4#21462094C214yesmaximal, complemented
5#21462090C214yesmaximal, complemented
6#21459148C211yesmaximal, characteristic
7C2×#14519856—22no—
8#16840741—22no—
9#16840741—22no—
10#16840741C2214yes—
11#16840741C2214yescomplemented
12C2×#14519836C2214yescomplemented
13C2×#14519835—22no—
14#16840668C2218yescomplemented
15#16840666C2218yescomplemented
16C22×#11758222C2211yescharacteristic
17#16840340—22no—
18#16840336C411yescharacteristic, complemented
19#16840336C2211yescharacteristic
20#16840333C412yescomplemented
21#16840329C411yescharacteristic, complemented
22#16840329C2211yescharacteristic
23#16840326C2212yes—
24#16840321—22no—
25#16840308C412yescomplemented
26#16840308C2212yes—
27#16840273—22no—
28#16840269—22no—
29C2×#14519736C411yescharacteristic, complemented
30C22×#41×C61C411yescharacteristic, 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×#11758222—42no—
43C2×#11758222—42no—
44C2×#11758222C2×C411yescharacteristic
45C2×#11758222C2×C411yescharacteristic
46C2×#11758222D412yes—
47C2×#11758222D412yes—
48C2×#11758222C2311yescharacteristic
49C2×#11758222C2311yescharacteristic
50C22×Dic122C2×C411yescharacteristic
51C22×D244C2311yescharacteristic
52C22×C22⋊D61D412yes—
53C22×C4×D61D412yes—
54#14519775C2×C412yescomplemented
55#14519775C2×C412yescomplemented
56#14519773C2×C412yes—
57#14519773C2312yes—
58#14519766C2×C412yes—
59#14519766C2312yes—
60#14519764C2×C412yescomplemented
61#14519764C2×C412yescomplemented
62#14519760—22no—
63#14519760—22no—
64#14519758—24no—
65#14519756—24no—
66#14519753—22no—
67#14519753—22no—
68#14519751—24no—
69#14519749—24no—
70#14519749—24no—
71#14519748—22no—
72#14519748—22no—
73#14519746—22no—
74#14519746—22no—
75#14519745—24no—
76C4×C22⋊D61—24no—
77C4×C22⋊D61—24no—
78#14519736—21no—
79#14519736—21no—
80#14519736C2×C412yes—
81#14519736C2×C411yescharacteristic, complemented
82#14519736C2×C411yescharacteristic, complemented
83C2×C4⋊Dic61—21no—
84C2×C22⋊Dic61—21no—
85C2×C4×Dic61C2×C412yescomplemented
86C2×#41×C61—21no—
87C2×#41×C61—21no—
88C2×#41×C61—42no—
89C2×#41×C61—42no—
90C2×#41×C61C2×C412yes—
91C2×#41×C61C2×C412yescomplemented
92C22×Q8×C61C2×C411yescharacteristic, complemented
93C22×D4×C61—21no—
94C22×D4×C61C2×C411yescharacteristic, complemented
95C23×C4×C61—21no—
96C23×C4×C61C2×C411yescharacteristic
97C24×C61—21no—
98C24×C61C22⋊C411yescharacteristic
99C23×D61C2×D412yes—
100C22×#6923—44no—
101#41×C61—44no—
102#41×C61—44no—
103#41×C61—42no—
104#41×C61—42no—
105#41×C61—42no—
106#41×C61—42no—
107C2×Q8×C61—21no—
108C2×Q8×C61—42no—
109C2×Q8×C61C22×C412yes—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—21no—
113C2×D4×C61—42no—
114C2×D4×C61—44no—
115C2×D4×C61C22×C412yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—42no—
122C22×C4×C61—42no—
123C22×C4×C61C22⋊C411yescharacteristic
124C22×C4×C61C22⋊C411yescharacteristic
125C22×C4×C61C22⋊C411yescharacteristic
126C22×C4×C61C22×C411yescharacteristic
127C22×C4×C61C22×C411yescharacteristic
128C22×C4×C61C22×C411yescharacteristic
129C2×C22⋊D61—24no—
130C2×C22⋊D61—24no—
131C2×C22⋊D61—42no—
132C2×C22⋊D61—42no—
133C2×C22⋊D61C22⋊C412yes—
134C2×C22⋊D61C22⋊C412yes—
135C2×C22⋊D61C2×D412yes—
136C2×C22⋊D61C2×D412yes—
137C22×Dic61—22no—
138C22×Dic61C22⋊C412yes—
139#11758222—42no—
140#11758222—41no—
141#11758222—41no—
142#11758222—42no—
143#11758222—42no—
144#11758222—42no—
145#11758222—42no—
146#11758222—41no—
147#11758222—42no—
148#11758222—41no—
149C2×D244—42no—
150C2×D244C22×C411yescharacteristic
151C2×D244C2×D412yes—
152C2×D244C2411yescharacteristic
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—22no—
156C2×C4×D61—22no—
157C2×C4×D61C22⋊C412yes—
158C2×C4×D61C22⋊C412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×Dic122—42no—
162C2×Dic122C22⋊C412yes—
163C2×Dic122C22×C411yescharacteristic
164C2×Dic122C22×C411yescharacteristic
165C22⋊#6923—28no—
166C22⋊#6923—44no—
167C4⋊#6923—44no—
168C4⋊#6923—44no—
169C4×#6923—44no—
170C4×#6923—44no—
171C22⋊Dic61—22no—
172C22⋊Dic61—21no—
173C22⋊Dic61—21no—
174C22⋊Dic61—24no—
175C22⋊Dic61—24no—
176#11758198—24no—
177#11758198—24no—
178C4⋊Dic61—22no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181#11758196—24no—
182#11758196—24no—
183C4×Dic61—22no—
184C4×Dic61C22×C412yes—
185C4×Dic61C22×C412yes—
186C23×C61—22no—
187C23×C61—22no—
188C23×C61—42no—
189C23×C61—82no—
190C23×C61—81no—
191C23×C61C2×C22⋊C411yescharacteristic
192C23×C61C2×C22⋊C411yescharacteristic
193C23×C61C2×C22⋊C411yescharacteristic
194C22×D61—22no—
195C22×D61—22no—
196C22×D61—82no—
197C22×D61C2×C22⋊C412yes—
198C22×D61C22×D412yes—
199C2×#6923—48no—
200C2×#6923—88no—
201Q8×C61—42no—
202Q8×C61—42no—
203D4×C61—42no—
204D4×C61—44no—
205D4×C61—44no—
206D4×C61—42no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—22no—
215C2×C4×C61—44no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61#9912yes—
220C2×C4×C61#9912yes—
221C2×C4×C61C2×C22⋊C411yescharacteristic
222C2×C4×C61C2×C22⋊C411yescharacteristic
223C2×C4×C61C2×C22⋊C411yescharacteristic
224C2×C4×C61C23×C411yescharacteristic
225C22⋊D61—42no—
226C22⋊D61—42no—
227C22⋊D61—42no—
228C22⋊D61—42no—
229C22⋊D61—84no—
230C22⋊D61—84no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—82no—
236C2×Dic61C2×C22⋊C412yes—
237C2×Dic61C2×C22⋊C412yes—
238D244—41no—
239D244—42no—
240D244—41no—
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249Dic122—42no—
250Dic122—41no—
251Dic122—41no—
252#33358—611nomaximal, Sylow
253C22×C61—21no—
254C22×C61—21no—
255C22×C61—41no—
256C22×C61—41no—
257C22×C61—41no—
258C22×C61—44no—
259C22×C61—42no—
260C22×C61—82no—
261C22×C61—84no—
262C22×C61C2×#9912yes—
263C22×C61C22×C22⋊C411yescharacteristic
264C2×D61—22no—
265C2×D61—22no—
266C2×D61—44no—
267C2×D61—42no—
268C2×D61—42no—
269C2×D61—82no—
270#6923—88no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—42no—
274C4×C61C2×#9911yescharacteristic
275C4×C61C2×#9912yes—
276C4×C61C2×#9911yescharacteristic
277Dic61—22no—
278Dic61—22no—
279Dic61—42no—
280Dic61—42no—
281#3449—611no—
282C2×#410—612no—
283#2129—612no—
284#2128—612no—
285#2127—614no—
286#2126—614no—
287C2×C61—21no—
288C2×C61—22no—
289C2×C61—41no—
290C2×C61—41no—
291C2×C61—82no—
292C2×C61C22×#9911yescharacteristic
293D61—42no—
294D61—42no—
295C22×#41—611no—
296C22×#41#692311yescharacteristic, complemented
297#537—1222no—
298#536—612no—
299#536—611no—
300#536—611no—
301#535—612no—
302#535—611no—
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—21no—
321C2×#41—21no—
322C2×#41—42no—
323C2×#41—42no—
324C2×#41—612no—
325C2×#41—612no—
326C2×#41—611no—
327C2×#41—611no—
328C2×#41—611no—
329C2×#41—611no—
330C2×#41—2442no—
331C2×#41—2442no—
332C2×#41C2×#692312yes—
333C2×#41C2×#692312yescomplemented
334C22×Q8—611no—
335C22×Q8C2×#692311yescharacteristic, complemented
336C22×D4—21no—
337C22×D4—612no—
338C22×D4—611no—
339C22×D4C2×#692311yescharacteristic, complemented
340C23×C4—21no—
341C23×C4—612no—
342C23×C4C2×#692311yescharacteristic
343C4⋊Q8—612no—
344C4⋊Q8—612no—
345#127—612no—
346#127—612no—
347#124—612no—
348#124—612no—
349#124—612no—
350#124—612no—
351#123—1222no—
352#123—1222no—
353#123—1224no—
354C22⋊Q8—1224no—
355C22⋊Q8—1222no—
356C22⋊Q8—1222no—
357#121—1224no—
358#121—1222no—
359#121—1222no—
360#121—1224no—
361#121—1224no—
362#120—1224no—
363#120—1222no—
364#120—1222no—
365C4×D4—1224no—
366C4×D4—1224no—
367#117—612no—
368#117—611no—
369#117—611no—
370#117—1221no—
371#117—1221no—
372#117—1224no—
373#117—1224no—
374#117—2442no—
375#117—2442no—
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—
388C24C22⋊#692311yescharacteristic
389#41—44no—
390#41—44no—
391#41—42no—
392#41—42no—
393#41—42no—
394#41—42no—
395#41—2442no—
396#41—2442no—
397#41—2442no—
398#41—2442no—
399#41—2442no—
400#41—2441no—
401#41—2441no—
402#41—2442no—
403#41—2441no—
404#41—2441no—
405C2×Q8—21no—
406C2×Q8—42no—
407C2×Q8—612no—
408C2×Q8—611no—
409C2×Q8—611no—
410C2×Q8—2442no—
411C2×Q8C22×#692312yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—612no—
418C2×D4—612no—
419C2×D4—612no—
420C2×D4—612no—
421C2×D4—612no—
422C2×D4—611no—
423C2×D4—611no—
424C2×D4—1224no—
425C2×D4—1224no—
426C2×D4—2442no—
427C2×D4—2442no—
428C2×D4—2442no—
429C2×D4C22×#692312yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—612no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—612no—
441C22×C4—612no—
442C22×C4—1222no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1222no—
446C22×C4—1222no—
447C22×C4—2444no—
448C22×C4C22⋊#692311yescharacteristic
449C22×C4C22⋊#692311yescharacteristic
450C22×C4C22⋊#692311yescharacteristic
451C22×C4C22×#692311yescharacteristic
452C22×C4C22×#692311yescharacteristic
453C22×C4C22×#692311yescharacteristic
454C4⋊C4—1224no—
455C4⋊C4—1224no—
456C4⋊C4—1222no—
457C4⋊C4—1221no—
458C4⋊C4—1221no—
459C4⋊C4—2444no—
460C4⋊C4—2444no—
461C22⋊C4—1224no—
462C22⋊C4—1224no—
463C22⋊C4—1224no—
464C22⋊C4—1224no—
465C22⋊C4—1221no—
466C22⋊C4—1221no—
467C22⋊C4—1222no—
468C22⋊C4—1228no—
469C22⋊C4—2444no—
470C42—612no—
471C42—612no—
472C42—1222no—
473C42—2444no—
474C42—2444no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—81no—
479C23—82no—
480C23—612no—
481C23—612no—
482C23—1222no—
483C23—1222no—
484C23—4882no—
485C23C2×C22⋊#692311yescharacteristic
486C23C2×C22⋊#692311yescharacteristic
487C23C2×C22⋊#692311yescharacteristic
488Q8—42no—
489Q8—42no—
490Q8—2442no—
491Q8—2441no—
492Q8—2441no—
493D4—42no—
494D4—44no—
495D4—44no—
496D4—42no—
497D4—4884no—
498D4—4884no—
499D4—2442no—
500D4—2441no—
501D4—2441no—
502D4—2442no—
503D4—2442no—
504D4—2442no—
505D4—2442no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—612no—
519C2×C4—612no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1222no—
528C2×C4—1222no—
529C2×C4—1222no—
530C2×C4—1222no—
531C2×C4—1222no—
532C2×C4—4888no—
533C2×C4—4882no—
534C2×C4—2448no—
535C2×C4#1451983512yes—
536C2×C4#1451983512yes—
537C2×C4C2×C22⋊#692311yescharacteristic
538C2×C4C2×C22⋊#692311yescharacteristic
539C2×C4C2×C22⋊#692311yescharacteristic
540C2×C4C23×#692311yescharacteristic
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—44no—
547C22—42no—
548C22—82no—
549C22—84no—
550C22—1222no—
551C22—1222no—
552C22—4882no—
553C22—2442no—
554C22—2442no—
555C22—2444no—
556C22C2×#1451983512yes—
557C22C22×C22⋊#692311yescharacteristic
558C4—21no—
559C4—21no—
560C4—42no—
561C4—1222no—
562C4—1222no—
563C4—4888no—
564C4—2442no—
565C4—2442no—
566C4C2×#1451983511yescharacteristic
567C4C2×#1451983512yes—
568C4C2×#1451983511yescharacteristic
569C2—21no—
570C2—22no—
571C2—41no—
572C2—41no—
573C2—82no—
574C2—2442no—
575C2—2442no—
576C2C22×#1451983511yescharacteristic
subgroupG/H
centerC4C2×#14519835
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC22×#41×C61C4
socleC2×C61C22×#99
prime radical#21459148C2
cosocleC2×D244C24
hypercenterC22×#41#6923
hypocenterC61#33358
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16840741C2semidirect
2. #21462101C2semidirect
3. #21462098C2semidirect
4. #21462094C2semidirect
5. #21462090C2semidirect
11. #16840741C22semidirect
12. C2×#14519836C22semidirect
14. #16840668C22semidirect
15. #16840666C22semidirect
18. #16840336C4semidirect
20. #16840333C4semidirect
21. #16840329C4semidirect
25. #16840308C4semidirect
29. C2×#14519736C4semidirect
30. C22×#41×C61C4semidirect
54. #14519775C2×C4semidirect
55. #14519775C2×C4semidirect
60. #14519764C2×C4semidirect
61. #14519764C2×C4semidirect
81. #14519736C2×C4semidirect
82. #14519736C2×C4semidirect
85. C2×C4×Dic61C2×C4semidirect
91. C2×#41×C61C2×C4semidirect
92. C22×Q8×C61C2×C4semidirect
94. C22×D4×C61C2×C4semidirect
296. C22×#41#6923semidirect
319. C61#33358semidirect
333. C2×#41C2×#6923semidirect
335. C22×Q8C2×#6923semidirect
339. C22×D4C2×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407412
#214621012
#214620982
#214620942
#214620902
#214591482
#3335861