Finite Group Databasebeta

← #29681387

Subgroups — #29681387

Order 15616.

1830 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
1C22×#14519808C211yesmaximal, characteristic
2C2×#16840230C211yesmaximal, characteristic
3C2×#16839764C211yesmaximal, characteristic
4C2×#16839669C211yesmaximal, characteristic, complemented
5#21446157C214yesmaximal, complemented
6#21446150C214yesmaximal, complemented
7#21445540C211yesmaximal, characteristic, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C2×C61×#419C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—24no—
12C2×#14519808—24no—
13C2×#14519808C418yescomplemented
14C22×#11758202C2212yes—
15C22×#11758201C2212yes—
16C22×#11758194C2211yescharacteristic
17#16840230—24no—
18#16840230—24no—
19C2×#14519742C2212yes—
20C2×#14519739C2212yes—
21C2×#14519737—22no—
22C2×#14519734—21no—
23C22×C4⋊Dic61C2211yescharacteristic
24C22×C4×Dic61C2211yescharacteristic
25C2×C42×D61—21no—
26#16839764—24no—
27C2×#14519702C2212yes—
28C2×#14519699C2212yes—
29#16839669—21no—
30#16839669—21no—
31#16839669C2214yes—
32C2×#14519684—22no—
33#16839524C2214yes—
34#16839516C2214yes—
35C23×Q8×C61C2211yescharacteristic
36C22×M16×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#419—24no—
39C2×#103×C61C2214yes—
40C2×#11758222—22no—
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222C2×C414yes—
44C2×#11758222D414yes—
45C2×#11758222D414yes—
46C22×Dic122C2311yescharacteristic
47C22×D244C2311yescharacteristic
48C22×C22⋊D61—21no—
49C23×Dic61—21no—
50C23×Dic61D412yes—
51C22×C4×D61—21no—
52C22×C4×D61—22no—
53#14519808—232no—
54C2×#11758202—24no—
55C2×#11758202—24no—
56C2×#11758202C2×C418yescomplemented
57C2×#11758201—24no—
58C2×#11758201—24no—
59C2×#11758201C2×C418yescomplemented
60C2×#11758194—22no—
61C2×#11758194—22no—
62C2×#11758194C2×C414yescomplemented
63C22×#92418C2312yes—
64#14519742—24no—
65#14519742—24no—
66#14519739—24no—
67#14519739—24no—
68#14519737—216no—
69#14519734—28no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—22no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—22no—
75C2×#11758198—22no—
76C2×C4×Dic61—22no—
77C2×C4×Dic61—22no—
78C2×C4×Dic61—22no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2312yes—
82C42×D61—28no—
83#14519702—24no—
84#14519699—24no—
85#14519684—24no—
86#14519684—24no—
87C22×Q8×C61—21no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61—22no—
93C22×Q8×C61C2×C414yescomplemented
94C22×Q8×C61C2312yes—
95C23×C4×C61—21no—
96C23×C4×C61C2311yescharacteristic
97C2×M16×C61—22no—
98C2×M16×C61—22no—
99C2×M16×C61—22no—
100C22×C8×C61C2312yes—
101#117×C61—22no—
102#117×C61D414yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—21no—
106#103×C61—28no—
107C24×C61C2×D411yescharacteristic
108C23×D61—21no—
109C2×Q8×C61—24no—
110C2×Q8×C61—22no—
111C2×Q8×C61—28no—
112C2×Q8×C61—28no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—42no—
118C2×Q8×C61—42no—
119C2×Q8×C61—44no—
120C2×Q8×C61—44no—
121C2×Q8×C61C22×C414yes—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C22×C412yes—
128C22×C4×C61C2×D412yes—
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D412yes—
132C22×C4×C61#4112yes—
133C22×C4×C61#4112yes—
134C22×C4×C61C2411yescharacteristic
135C2×C22⋊D61—24no—
136C2×C22⋊D61—24no—
137C2×C22⋊D61—24no—
138C22×Dic61—21no—
139C22×Dic61—24no—
140C22×Dic61—24no—
141C22×Dic61—22no—
142C22×Dic61—42no—
143C22×Dic61—44no—
144C22×Dic61—42no—
145C22×Dic61C2×D412yes—
146C22×Dic61C2×D412yes—
147#11758222—28no—
148#11758222—28no—
149#11758222—28no—
150#11758222C22⋊C4116yes—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×D244C2×D412yes—
156C2×D244C2×D412yes—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—28no—
161C2×C4×D61—44no—
162C2×C4×D61—44no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122C22×C412yes—
167C2×Dic122C2×D412yes—
168C2×Dic122C2×D412yes—
169M16×C61—44no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C2×D414yes—
175C22⋊C4×C61—24no—
176C42×C61—24no—
177#11758202—216no—
178#11758201—216no—
179#11758198—28no—
180C4⋊Dic61—22no—
181C4⋊Dic61—22no—
182C4⋊Dic61—42no—
183#11758196—28no—
184C4×Dic61—28no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189#11758194—28no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418C22×C414yes—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61C4×D412yes—
196C23×C61#12012yes—
197C23×C61#12112yes—
198C23×C61C22×D411yescharacteristic
199C22×D61—22no—
200C22×D61—22no—
201C22×D61—22no—
202C22×D61—42no—
203C22×D61—42no—
204Q8×C61—24no—
205Q8×C61—24no—
206Q8×C61—42no—
207Q8×C61—42no—
208Q8×C61—48no—
209C2×C4×C61—24no—
210C2×C4×C61—24no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—48no—
216C2×C4×C61C2×C22⋊C414yes—
217C2×C4×C61C4×D414yes—
218C2×C4×C61C4×D412yes—
219C2×C4×C61#12012yes—
220C2×C4×C61#12114yes—
221C2×C4×C61#12112yes—
222C2×C4×C61#12314yes—
223C2×C4×C61C23×C411yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C2×#4111yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C22⋊D61—216no—
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—48no—
239C2×Dic61—42no—
240C2×Dic61—44no—
241C2×Dic61—84no—
242C2×Dic61#13714yescomplemented
243D244—22no—
244D244—22no—
245D244—22no—
246D244C2×C22⋊C414yes—
247C4×D61—216no—
248C4×D61—416no—
249Dic122—22no—
250Dic122—22no—
251Dic122—22no—
252Dic122C2×C22⋊C414yes—
253C8×C61—42no—
254#92418—24no—
255#20927—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61#38614yes—
261C22×C61C2×C4×D411yescharacteristic
262C22×C61C2×#12011yescharacteristic
263C22×C61C2×#12111yescharacteristic
264C22×C61C2×#13712yes—
265C22×C61#57512yes—
266C2×D61—24no—
267C2×D61—44no—
268C2×D61—44no—
269C2×D61—42no—
270C2×D61—42no—
271C4×C61—44no—
272C4×C61—44no—
273C4×C61#38614yes—
274C4×C61C22×C22⋊C411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#12311yescharacteristic
278Dic61—24no—
279Dic61—24no—
280Dic61—42no—
281Dic61—82no—
282C22×#137—611no—
283C2×C4×#41—611no—
284C2×#417—611no—
285C2×#419D6111yescharacteristic, complemented
286C2×#391—611no—
287#1878—614no—
288#1877—614no—
289#1736—611no—
290#1728—611no—
291C2×C61—24no—
292C2×C61C2×#38611yescharacteristic
293C2×C61C4×#13712yes—
294C2×C61#300712yes—
295C2×C61#303911yescharacteristic
296C2×C61#304011yescharacteristic
297D61—44no—
298C22×#41—611no—
299C23×Q8C2×D6111yescharacteristic
300C2×#137—618no—
301C2×#137—1224no—
302C2×#137—1224no—
303C22×Q16—612no—
304C22×SD16—612no—
305C22×M16—611no—
306C22×M16C2×D6111yescharacteristic
307C4×#41—1224no—
308C4×#41—1224no—
309C2×C4×Q8—612no—
310C2×C4×D4—612no—
311C2×C4×D4—1222no—
312C2×#117—1221no—
313C2×#117C2×D6111yescharacteristic
314C22×C4⋊C4—611no—
315C22×C42—611no—
316C22×C42—1221no—
317#419—24no—
318#417—1224no—
319C2×#103—612no—
320C2×#103C2×D6114yes—
321C2×#102—612no—
322#391—614no—
323#391—1221no—
324#391—1221no—
325C2×#95—1222no—
326#340—614no—
327#336—614no—
328C61#2092711yescharacteristic, complemented, Sylow
329C2×#41—614no—
330C2×#41—614no—
331C2×#41—614no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—611no—
342C22×Q8C4×D6114yescomplemented
343C22×Q8C22×D6112yes—
344C22×D4—611no—
345C22×D4—1221no—
346C23×C4—21no—
347C23×C4—612no—
348C23×C4—1222no—
349C23×C4—1221no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352#137—12232no—
353C2×Q16—618no—
354C2×Q16—1224no—
355C2×Q16—1224no—
356C2×SD16—618no—
357C2×SD16—1224no—
358C2×SD16—1224no—
359C2×M16—22no—
360C2×M16—22no—
361C2×M16—22no—
362C2×M16—614no—
363C2×M16—1222no—
364C2×M16—1222no—
365C22×C8—612no—
366C22×C8C22×D6112yes—
367C4×Q8—1224no—
368C4×Q8—1224no—
369C4×D4—12216no—
370C4×D4—1224no—
371C4×D4—1224no—
372#117—22no—
373#117—1228no—
374#117C22⋊D6114yes—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—1222no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1221no—
380C2×C4⋊C4C22×D6112yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—1222no—
383C2×C42—21no—
384C2×C42—611no—
385C2×C42—611no—
386C2×C42—612no—
387C2×C42—1228no—
388C2×C42—1222no—
389C2×C42—1222no—
390C2×C42—1222no—
391#103—28no—
392#103—1224no—
393#102—1224no—
394#95—1224no—
395#95—1224no—
396C24—1221no—
397C24D4×D6111yescharacteristic
398#41—6116no—
399#41—1228no—
400#41—1228no—
401#41—1228no—
402C2×Q8—22no—
403C2×Q8—24no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—42no—
413C2×Q8—44no—
414C2×Q8—612no—
415C2×Q8—612no—
416C2×Q8—612no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8C2×C4×D6114yes—
421C2×D4—612no—
422C2×D4—612no—
423C2×D4—612no—
424C2×D4—1224no—
425C2×D4—1224no—
426C2×D4—1224no—
427C2×D4—1221no—
428C2×D4—1221no—
429C2×D4—1221no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—1228no—
438C22×C4—1224no—
439C22×C4—1221no—
440C22×C4—1224no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—2444no—
446C22×C4—2444no—
447C22×C4—2442no—
448C22×C4—2444no—
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
458Q16—12216no—
459SD16—12216no—
460M16—44no—
461M16—1228no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—614no—
466C2×C8—1222no—
467C2×C8—1222no—
468C4⋊C4—22no—
469C4⋊C4—1228no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—2442no—
473C4⋊C4C2×C22⋊D6114yes—
474C22⋊C4—24no—
475C22⋊C4—1228no—
476C42—24no—
477C42—1228no—
478C42—1222no—
479C42—1222no—
480C42—1222no—
481C42—1222no—
482C23—22no—
483C23—22no—
484C23—1222no—
485C23—1222no—
486C23—1222no—
487C23—2442no—
488C23—2442no—
489C23#1451973712yes—
490C23#1451974512yes—
491C23#1451975012yes—
492C23C2×D4×D6111yescharacteristic
493Q8—24no—
494Q8—24no—
495Q8—42no—
496Q8—42no—
497Q8—48no—
498Q8—614no—
499Q8—1222no—
500Q8—1222no—
501Q8—1222no—
502D4—614no—
503D4—12216no—
504D4—1222no—
505D4—1222no—
506D4—1222no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—48no—
514C2×C4—614no—
515C2×C4—1222no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1222no—
519C2×C4—1222no—
520C2×C4—12216no—
521C2×C4—1222no—
522C2×C4—1222no—
523C2×C4—1222no—
524C2×C4—4884no—
525C2×C4—24416no—
526C2×C4—2448no—
527C2×C4—2442no—
528C2×C4—2444no—
529C2×C4C22⋊C4×D6114yes—
530C2×C4#1451973712yes—
531C2×C4C4×C22⋊D6114yes—
532C2×C4#1451974512yes—
533C2×C4#1451974814yes—
534C2×C4#1451975012yes—
535C2×C4#1451975914yes—
536C2×C4C22×C4×D6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C2×C4C2×#1175822211yescharacteristic
542C8—42no—
543C8—1224no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—1224no—
549C22—2444no—
550C22—2442no—
551C22—2442no—
552C22—2444no—
553C22#1683963814yes—
554C22C2×#1451973711yescharacteristic
555C22C2×#1451974511yescharacteristic
556C22C2×#1451975011yescharacteristic
557C22#137×D6112yes—
558C22#1684056112yes—
559C4—44no—
560C4—44no—
561C4—1224no—
562C4—1224no—
563C4—4882no—
564C4—2442no—
565C4#1683963814yes—
566C4C2×C22⋊C4×D6111yescharacteristic
567C4C2×C4×C22⋊D6111yescharacteristic
568C4C2×#1451974811yescharacteristic
569C4C2×#1451975911yescharacteristic
570C2—24no—
571C2—2444no—
572C2C2×#1683963811yescharacteristic
573C2#2145463412yes—
574C2#2145507512yes—
575C2#2145530511yescharacteristic
576C2#2145531611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#419C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
cosocleC22×C4×C61C24
hypercenterC2×#419D61
hypocenterC61#20927
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
4. C2×#16839669C2semidirect
5. #21446157C2semidirect
6. #21446150C2semidirect
7. #21445540C2semidirect
8. #21445512C2semidirect
9. C2×C61×#419C2semidirect
13. C2×#14519808C4semidirect
56. C2×#11758202C2×C4semidirect
59. C2×#11758201C2×C4semidirect
62. C2×#11758194C2×C4semidirect
93. C22×Q8×C61C2×C4semidirect
242. C2×Dic61#137semidirect
285. C2×#419D61semidirect
328. C61#20927semidirect
342. C22×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168402302
C2×#168397642
C2×#168396692
#214461572
#214461502
#214455402
#214455122
C2×C61×#4192
#2092761