Finite Group Databasebeta

← #29683710

Subgroups — #29683710

Order 15616.

1526 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×#16840336C211yesmaximal, characteristic
2C2×#16839761C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic
5C2×#16839717C211yesmaximal, characteristic, complemented
6#21448443C211yesmaximal, characteristic, complemented
7#21446637C212yesmaximal, complemented
8#21446632C212yesmaximal, complemented
9#21446627C212yesmaximal, complemented
10#21446619C212yesmaximal, complemented
11C22×#103×C61C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C22×#11758194C2211yescharacteristic
14#16840336—24no—
15#16840336C418yescomplemented
16C2×#14519775C2212yes—
17C2×#14519766C2212yes—
18C2×#14519760—22no—
19C2×#14519753—22no—
20C2×#14519736C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839771—24no—
23#16839761—24no—
24#16839759—24no—
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519701C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519697C2212yes—
33#16839717—24no—
34C2×#14519687C2212yes—
35C2×C8×Dic61C2212yes—
36#16839658—22no—
37#16839635—22no—
38#16839631C2212yes—
39#16839627C2212yes—
40C23×Q8×C61C2211yescharacteristic
41C23×C8×C61C2211yescharacteristic
42C22×C4⋊C4×C61C2211yescharacteristic
43C2×#103×C61—24no—
44C2×#103×C61C2212yes—
45C2×#103×C61C2212yes—
46C2×#11758222—22no—
47C2×#11758222—22no—
48C2×#11758222—22no—
49C2×#11758222—22no—
50C2×#11758222—22no—
51C2×#11758222C2×C414yes—
52C22×Dic122C2311yescharacteristic
53C22×D244C2311yescharacteristic
54C22×C22⋊D61—21no—
55C22×C4×D61—21no—
56C2×#11758194—22no—
57C2×#11758194—22no—
58C2×#11758194D414yes—
59C22×#92418C2312yes—
60#14519775—24no—
61#14519775C2×C418yescomplemented
62#14519766—24no—
63#14519766C2×C418yescomplemented
64#14519760—216no—
65#14519753—216no—
66#14519736—22no—
67#14519736—22no—
68#14519736C2×C414yes—
69#14519736D414yes—
70C2×C4⋊Dic61D412yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—24no—
74C2×#11758196—24no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61D412yes—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80#14519702—22no—
81#14519702—22no—
82#14519701—24no—
83#14519701—24no—
84#14519700—24no—
85#14519699—22no—
86#14519699—22no—
87#14519697—24no—
88#14519687—24no—
89C8×Dic61—24no—
90#14519684—24no—
91C22×Q8×C61—21no—
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61—22no—
97C22×Q8×C61C2×C414yes—
98C22×Q8×C61C2312yes—
99C23×C4×C61—21no—
100C23×C4×C61—21no—
101C23×C4×C61C2311yescharacteristic
102C22×C8×C61—22no—
103C22×C8×C61—22no—
104C22×C8×C61D414yes—
105C22×C8×C61C2311yescharacteristic
106C22×C8×C61C2311yescharacteristic
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61C2312yes—
111#103×C61—24no—
112#103×C61—24no—
113C24×C61C2×D411yescharacteristic
114C23×D61—21no—
115C2×Q8×C61—24no—
116C2×Q8×C61—22no—
117C2×Q8×C61—28no—
118C2×Q8×C61—28no—
119C2×Q8×C61—22no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—44no—
124C2×Q8×C61—44no—
125C2×Q8×C61—42no—
126C2×Q8×C61—42no—
127C2×Q8×C61C22×C414yes—
128C22×C4×C61—24no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—82no—
135C22×C4×C61C22×C412yes—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D412yes—
139C22×C4×C61C2×D412yes—
140C22×C4×C61#4112yes—
141C22×C4×C61#4112yes—
142C22×C4×C61C2411yescharacteristic
143C2×C22⋊D61—24no—
144C2×C22⋊D61—42no—
145C2×C22⋊D61—42no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D414yes—
150#11758222—28no—
151#11758222—28no—
152#11758222—44no—
153#11758222—44no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C22×C412yes—
160C2×C4×D61—24no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122C22×C412yes—
169C2×C8×C61—28no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61C2×D412yes—
175C2×C8×C61C2×D412yes—
176C4⋊C4×C61—22no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—42no—
179C22⋊Dic61—42no—
180C22⋊Dic61C22⋊C418yes—
181#11758198—216no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61C22⋊C418yes—
187#11758196—216no—
188#11758196—84no—
189C4×Dic61—42no—
190C4×Dic61C22×C414yes—
191C4×Dic61C2×D414yes—
192#11758194—28no—
193C2×#92418—22no—
194C2×#92418—22no—
195C2×#92418C2×D414yes—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61C2×C22⋊C412yes—
199C23×C61#12412yes—
200C23×C61#12712yes—
201C23×C61C22×D411yescharacteristic
202C22×D61—22no—
203C22×D61—41no—
204C22×D61—41no—
205C22×D61—42no—
206C22×D61—42no—
207Q8×C61—24no—
208Q8×C61—24no—
209Q8×C61—42no—
210Q8×C61—42no—
211Q8×C61—48no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—48no—
216C2×C4×C61—42no—
217C2×C4×C61—44no—
218C2×C4×C61—84no—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C4×D414yes—
221C2×C4×C61C4×D414yes—
222C2×C4×C61#12114yes—
223C2×C4×C61#12114yes—
224C2×C4×C61#12412yes—
225C2×C4×C61#12712yes—
226C2×C4×C61C23×C411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61C22×D411yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61C2×#4111yescharacteristic
232C22⋊D61—48no—
233C2×Dic61—24no—
234C2×Dic61—24no—
235C2×Dic61—22no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239C2×Dic61—41no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61C2×C22⋊C418yes—
243D244—22no—
244D244—22no—
245D244—41no—
246D244—41no—
247C4×D61—48no—
248Dic122—22no—
249Dic122—22no—
250Dic122—41no—
251Dic122—41no—
252C8×C61—22no—
253C8×C61—22no—
254#92418—24no—
255#21205—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61#39014yes—
261C22×C61C22×C22⋊C411yescharacteristic
262C22×C61C2×#12411yescharacteristic
263C22×C61C2×#12711yescharacteristic
264C22×C61#57712yes—
265C22×C61#57812yes—
266C2×D61—44no—
267C2×D61—42no—
268C2×D61—82no—
269C2×D61—81no—
270C2×D61—81no—
271C4×C61—44no—
272C4×C61—82no—
273C4×C61#39014yes—
274C4×C61C2×C4×D411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#12111yescharacteristic
278Dic61—28no—
279Dic61—42no—
280Dic61—81no—
281Dic61—81no—
282C2×#536—611no—
283C2×#417—611no—
284C2×#419—611no—
285C2×#416—611no—
286C22×#103D6111yescharacteristic, complemented
287C2×#405—611no—
288#2317—611no—
289#1960—612no—
290#1959—612no—
291#1958—612no—
292#1956—612no—
293C2×C61—24no—
294C2×C61C2×#39011yescharacteristic
295C2×C61#289812yes—
296C2×C61#313712yes—
297C2×C61#315411yescharacteristic
298C2×C61#315611yescharacteristic
299D61—82no—
300C22×#41—611no—
301C23×Q8C2×D6111yescharacteristic
302C22×M16—611no—
303C23×C8C2×D6111yescharacteristic
304#536—618no—
305#536—1224no—
306C2×C4⋊Q8—612no—
307C2×#124—612no—
308C2×#123—1222no—
309C2×C22⋊Q8—1222no—
310C2×#117—611no—
311C2×#117—611no—
312C22×C4⋊C4C2×D6111yescharacteristic
313#419—1224no—
314#417—1224no—
315#416—1224no—
316C2×#103—24no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#103—611no—
320C2×#103—611no—
321C2×#103—612no—
322C2×#103C2×D6112yes—
323C2×#103C2×D6112yes—
324C2×#102—611no—
325C2×#102—611no—
326C2×#102—612no—
327#405—1224no—
328C2×#97—612no—
329C2×C4×C8—612no—
330#389—1222no—
331#385—1222no—
332#384—612no—
333#382—612no—
334C61#2120511yescharacteristic, complemented, Sylow
335C2×#41—614no—
336C2×#41—1222no—
337C2×#41—1222no—
338C2×#41—1222no—
339C2×#41—1222no—
340C2×#41—1222no—
341C22×Q8—21no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—611no—
348C22×Q8C4×D6114yes—
349C22×Q8C22×D6112yes—
350C22×D4—611no—
351C22×D4—1221no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1221no—
355C23×C4C22×D6111yescharacteristic
356C2×M16—614no—
357C2×M16—1222no—
358C2×M16—1222no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—612no—
362C22×C8C22⋊D6114yes—
363C22×C8C22×D6111yescharacteristic
364C22×C8C22×D6111yescharacteristic
365C4⋊Q8—618no—
366C4⋊Q8—1224no—
367#124—618no—
368#124—1224no—
369#123—12216no—
370C22⋊Q8—12216no—
371#117—614no—
372#117—614no—
373#117—1222no—
374#117—1222no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—612no—
379C2×C4⋊C4—611no—
380C2×C4⋊C4—611no—
381C2×C4⋊C4—1224no—
382C2×C4⋊C4—1224no—
383C2×C4⋊C4C22×D6112yes—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—1224no—
386C2×C22⋊C4—1221no—
387C2×C42—612no—
388C2×C42—1221no—
389#103—24no—
390#103—24no—
391#103—1224no—
392#103—1224no—
393#103—1224no—
394#103—1222no—
395#103—1222no—
396#102—1224no—
397#102—1222no—
398#102—1222no—
399#97—1224no—
400C4×C8—1224no—
401#95—1224no—
402C24—1221no—
403C24D4×D6111yescharacteristic
404#41—1228no—
405#41—1228no—
406#41—2444no—
407#41—2444no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—28no—
415C2×Q8—28no—
416C2×Q8—44no—
417C2×Q8—44no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—612no—
421C2×Q8—1221no—
422C2×Q8—1221no—
423C2×Q8—1221no—
424C2×Q8—1221no—
425C2×Q8—1221no—
426C2×Q8C2×C4×D6114yes—
427C2×D4—612no—
428C2×D4—1224no—
429C2×D4—1221no—
430C2×D4—1221no—
431C2×D4—1221no—
432C2×D4—1221no—
433C2×D4—1221no—
434C2×D4—2442no—
435C2×D4—2442no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—82no—
443C22×C4—614no—
444C22×C4—1221no—
445C22×C4—1224no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—2442no—
449C22×C4—2442no—
450C22×C4C2×C4×D6112yes—
451C22×C4#1175822212yes—
452C22×C4D4×D6112yes—
453C22×C4D4×D6112yes—
454C22×C4D4×D6111yescharacteristic
455C22×C4#1175822412yes—
456C22×C4C2×C22⋊D6112yes—
457C22×C4C23×D6111yescharacteristic
458M16—1228no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—28no—
464C2×C8—614no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8C2×C22⋊D6112yes—
468C2×C8C2×C22⋊D6112yes—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—42no—
472C4⋊C4—618no—
473C4⋊C4—12216no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—4884no—
479C22⋊C4—618no—
480C22⋊C4—12216no—
481C22⋊C4—2442no—
482C42—614no—
483C42—614no—
484C42—2442no—
485C23—22no—
486C23—22no—
487C23—1222no—
488C23—2441no—
489C23—2441no—
490C23—2442no—
491C23—2442no—
492C23C22⋊C4×D6112yes—
493C23#1451976312yes—
494C23#1451977312yes—
495C23C2×D4×D6111yescharacteristic
496Q8—24no—
497Q8—24no—
498Q8—42no—
499Q8—42no—
500Q8—48no—
501Q8—1222no—
502Q8—1222no—
503Q8—2441no—
504Q8—2441no—
505D4—1222no—
506D4—1222no—
507D4—2448no—
508D4—2441no—
509D4—2441no—
510C2×C4—24no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—84no—
517C2×C4—618no—
518C2×C4—1224no—
519C2×C4—1224no—
520C2×C4—1222no—
521C2×C4—2441no—
522C2×C4—2441no—
523C2×C4—2442no—
524C2×C4—2442no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4—2448no—
528C2×C4C22⋊C4×D6112yes—
529C2×C4#1451973714yes—
530C2×C4C4×C22⋊D6114yes—
531C2×C4#1451974914yes—
532C2×C4#1451975014yes—
533C2×C4#1451976312yes—
534C2×C4#1451977312yes—
535C2×C4C22×C4×D6111yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C2×#1175822411yescharacteristic
540C2×C4C2×#1175822211yescharacteristic
541C8—22no—
542C8—22no—
543C8—1224no—
544C22—24no—
545C22—24no—
546C22—22no—
547C22—22no—
548C22—4882no—
549C22—4881no—
550C22—4881no—
551C22—2444no—
552C22—2442no—
553C22#1683966214yes—
554C22C2×C22⋊C4×D6111yescharacteristic
555C22C2×#1451976311yescharacteristic
556C22C2×#1451977311yescharacteristic
557C22#1684056812yes—
558C22#1684057512yes—
559C4—44no—
560C4—82no—
561C4—1228no—
562C4—4881no—
563C4—4881no—
564C4—2442no—
565C4#1683966214yes—
566C4C2×#1451973711yescharacteristic
567C4C2×C4×C22⋊D6111yescharacteristic
568C4C2×#1451974911yescharacteristic
569C4C2×#1451975011yescharacteristic
570C2—24no—
571C2—4882no—
572C2C2×#1683966211yescharacteristic
573C2#2145434712yes—
574C2#2145618212yes—
575C2#2145628911yescharacteristic
576C2#2145630811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC22×#103D61
hypocenterC61#21205
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. C2×#16839717C2semidirect
6. #21448443C2semidirect
7. #21446637C2semidirect
8. #21446632C2semidirect
9. #21446627C2semidirect
10. #21446619C2semidirect
11. C22×#103×C61C2semidirect
15. #16840336C4semidirect
61. #14519775C2×C4semidirect
63. #14519766C2×C4semidirect
286. C22×#103D61semidirect
334. C61#21205semidirect

Maximal subgroups

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

subgroupindex
C2×#168403362
C2×#168397612
C2×#168397712
C2×#168397592
C2×#168397172
#214484432
#214466372
#214466322
#214466272
#214466192
C22×#103×C612
#2120561