Finite Group Databasebeta

← #29681393

Subgroups — #29681393

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
1C2×#41×Dic61C211yesmaximal, characteristic
2C2×#16839763C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839669C211yesmaximal, characteristic, complemented
5#21446162C214yesmaximal, complemented
6#21446155C214yesmaximal, complemented
7#21445539C211yesmaximal, characteristic, complemented
8#21445512C211yesmaximal, characteristic, complemented
9C22×#137×C61C211yesmaximal, characteristic
10C22×#11758194C2211yescharacteristic
11#41×Dic61—24no—
12#41×Dic61—24no—
13C2×Q8×Dic61C2212yes—
14C2×D4×Dic61—22no—
15C2×D4×Dic61C2212yes—
16C2×#14519736—21no—
17C2×#14519736C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C22×C4×Dic61—21no—
20C22×C4×Dic61C2211yescharacteristic
21#16839771—24no—
22#16839763—24no—
23C2×#14519701C2214yes—
24C2×#14519701C2212yes—
25C2×#14519698C2212yes—
26#16839669—21no—
27#16839669—21no—
28#16839669C2214yes—
29C2×#14519684—22no—
30#16839525C2214yes—
31#16839516C2214yes—
32C22×#41×C61C2211yescharacteristic
33C23×Q8×C61C2211yescharacteristic
34C2×#137×C61—24no—
35C2×#137×C61—24no—
36C2×#137×C61C418yescomplemented
37C22×Q16×C61C2212yes—
38C22×SD16×C61C2212yes—
39C22×M16×C61C2211yescharacteristic
40C23×Dic61—22no—
41C23×Dic61—21no—
42C23×Dic61D412yes—
43C2×#11758194—22no—
44C2×#11758194—22no—
45C2×#11758194—22no—
46C22×#92418C2312yes—
47Q8×Dic61—24no—
48Q8×Dic61—24no—
49D4×Dic61—216no—
50D4×Dic61—24no—
51D4×Dic61—24no—
52#14519736—28no—
53#14519736—22no—
54#14519736D414yes—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61C2312yes—
60C2×C4⋊Dic61C2312yes—
61C2×C22⋊Dic61—21no—
62C2×C22⋊Dic61—22no—
63C2×C4×Dic61—28no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61D411yescharacteristic
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61C2312yes—
71#14519701—24no—
72#14519701—28no—
73#14519698—24no—
74#14519684—24no—
75#14519684—24no—
76C2×#41×C61—22no—
77C2×#41×C61—22no—
78C2×#41×C61—22no—
79C2×#41×C61C2×C414yes—
80C2×#41×C61D414yes—
81C2×#41×C61D414yes—
82C22×Q8×C61—21no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—22no—
86C22×Q8×C61—22no—
87C22×Q8×C61—22no—
88C22×Q8×C61C2×C414yescomplemented
89C22×Q8×C61C2312yes—
90C22×Q8×C61C2311yescharacteristic
91C22×D4×C61—21no—
92C22×D4×C61C2311yescharacteristic
93C23×C4×C61—21no—
94C23×C4×C61—21no—
95C23×C4×C61C2311yescharacteristic
96#137×C61—232no—
97C2×Q16×C61—24no—
98C2×Q16×C61—24no—
99C2×Q16×C61C2×C418yescomplemented
100C2×SD16×C61—24no—
101C2×SD16×C61—24no—
102C2×SD16×C61C2×C418yescomplemented
103C2×M16×C61—22no—
104C2×M16×C61—22no—
105C2×M16×C61C2×C414yescomplemented
106C22×C8×C61C2312yes—
107C24×C61—21no—
108C24×C61C2×D411yescharacteristic
109#41×C61—28no—
110#41×C61—28no—
111#41×C61—28no—
112#41×C61C22⋊C4116yes—
113C2×Q8×C61—24no—
114C2×Q8×C61—21no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—28no—
119C2×Q8×C61—28no—
120C2×Q8×C61—22no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—42no—
125C2×Q8×C61—42no—
126C2×Q8×C61—44no—
127C2×Q8×C61—44no—
128C2×Q8×C61C22×C414yes—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61C2×D412yes—
131C2×Q8×C61C2×D412yes—
132C2×D4×C61—24no—
133C2×D4×C61—24no—
134C2×D4×C61—24no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61C22×C412yes—
139C2×D4×C61C2×D412yes—
140C2×D4×C61C2×D412yes—
141C22×C4×C61—21no—
142C22×C4×C61—24no—
143C22×C4×C61—24no—
144C22×C4×C61—24no—
145C22×C4×C61—24no—
146C22×C4×C61—22no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61C22×C412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D412yes—
154C22×C4×C61#4112yes—
155C22×C4×C61#4112yes—
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—28no—
158C22×Dic61—24no—
159C22×Dic61—22no—
160C22×Dic61—24no—
161C22×Dic61—22no—
162C22×Dic61—42no—
163C22×Dic61—44no—
164C22×Dic61—44no—
165C22×Dic61—44no—
166C22×Dic61—42no—
167C22×Dic61C2×D412yes—
168C22×Dic61C2×D412yes—
169Q16×C61—216no—
170SD16×C61—216no—
171M16×C61—28no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61C22×C414yes—
175C22⋊Dic61—28no—
176C22⋊Dic61—24no—
177C4⋊Dic61—28no—
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61—42no—
182C4⋊Dic61C2×D414yes—
183C4×Dic61—28no—
184C4×Dic61—22no—
185C4×Dic61—24no—
186C4×Dic61—22no—
187C4×Dic61—22no—
188C4×Dic61—22no—
189#11758194—44no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—22no—
197C23×C61—22no—
198C23×C61—42no—
199C23×C61—42no—
200C23×C61C4×D412yes—
201C23×C61#12012yes—
202C23×C61#12112yes—
203C23×C61C22×D411yescharacteristic
204Q8×C61—22no—
205Q8×C61—22no—
206Q8×C61—22no—
207Q8×C61—24no—
208Q8×C61—24no—
209Q8×C61—42no—
210Q8×C61—42no—
211Q8×C61—48no—
212Q8×C61C2×C22⋊C414yes—
213D4×C61—216no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—22no—
217D4×C61C2×C22⋊C414yes—
218C2×C4×C61—22no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—216no—
222C2×C4×C61—24no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—48no—
226C2×C4×C61C2×C22⋊C414yes—
227C2×C4×C61C4×D414yes—
228C2×C4×C61C4×D412yes—
229C2×C4×C61#12012yes—
230C2×C4×C61#12114yes—
231C2×C4×C61#12112yes—
232C2×C4×C61#12314yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—48no—
246C2×Dic61—416no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—44no—
251C2×Dic61—84no—
252C2×Dic61#13714yescomplemented
253C8×C61—24no—
254#92418—42no—
255#20927—611nomaximal, Sylow
256C22×C61—24no—
257C22×C61—24no—
258C22×C61—22no—
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—44no—
262C22×C61—44no—
263C22×C61—42no—
264C22×C61—42no—
265C22×C61#38614yes—
266C22×C61C2×C4×D411yescharacteristic
267C22×C61C2×#12011yescharacteristic
268C22×C61C2×#12111yescharacteristic
269C22×C61C2×#13712yes—
270C22×C61#57512yes—
271C4×C61—24no—
272C4×C61—44no—
273C4×C61#38614yes—
274C4×C61C22×C22⋊C411yescharacteristic
275C4×C61C2×C4×D411yescharacteristic
276C4×C61C2×#12111yescharacteristic
277C4×C61C2×#12311yescharacteristic
278Dic61—24no—
279Dic61—42no—
280Dic61—44no—
281Dic61—82no—
282C22×#137D6111yescharacteristic
283C2×C4×#41—611no—
284C2×#417—611no—
285C2×#419—611no—
286C2×#391—611no—
287#1878—614no—
288#1877—614no—
289#1736—611no—
290#1728—611no—
291C2×C61—24no—
292C2×C61—44no—
293C2×C61C2×#38611yescharacteristic
294C2×C61C4×#13712yes—
295C2×C61#300712yes—
296C2×C61#303911yescharacteristic
297C2×C61#304011yescharacteristic
298C22×#41C2×D6111yescharacteristic
299C23×Q8C2×D6111yescharacteristic
300C2×#137—24no—
301C2×#137—24no—
302C2×#137Dic6118yescomplemented
303C22×Q16C2×D6112yes—
304C22×SD16C2×D6112yes—
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—611no—
313C2×#117—1221no—
314C22×C4⋊C4—611no—
315C22×C42—611no—
316C22×C42—1221no—
317#419—1224no—
318#417—1224no—
319C2×#103—614no—
320C2×#103—612no—
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—22no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41C2×Dic6114yes—
333C2×#41C22⋊D6114yes—
334C2×#41C22⋊D6114yes—
335C22×Q8—21no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8—22no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8C2×Dic6114yescomplemented
342C22×Q8C22×D6112yes—
343C22×Q8C22×D6111yescharacteristic
344C22×D4—21no—
345C22×D4C22×D6111yescharacteristic
346C23×C4—21no—
347C23×C4—21no—
348C23×C4—612no—
349C23×C4—1222no—
350C23×C4—1221no—
351C23×C4C22×D6111yescharacteristic
352#137—232no—
353C2×Q16—24no—
354C2×Q16—24no—
355C2×Q16C2×Dic6118yescomplemented
356C2×SD16—24no—
357C2×SD16—24no—
358C2×SD16C2×Dic6118yescomplemented
359C2×M16—22no—
360C2×M16—22no—
361C2×M16—1222no—
362C2×M16—1222no—
363C2×M16—1222no—
364C2×M16C2×Dic6114yescomplemented
365C22×C8—612no—
366C22×C8C22×D6112yes—
367C4×Q8—1224no—
368C4×Q8—1224no—
369C4×D4—1224no—
370C4×D4—12216no—
371C4×D4—1224no—
372#117—614no—
373#117—1228no—
374#117—1222no—
375C2×C4⋊C4—612no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4—1222no—
378C2×C4⋊C4—1222no—
379C2×C4⋊C4—1222no—
380C2×C4⋊C4—1221no—
381C2×C22⋊C4—1222no—
382C2×C22⋊C4—1221no—
383C2×C42—611no—
384C2×C42—611no—
385C2×C42—612no—
386C2×C42—1222no—
387C2×C42—1228no—
388C2×C42—1221no—
389C2×C42—1222no—
390C2×C42—1222no—
391#103—1228no—
392#103—1224no—
393#102—1224no—
394#95—1224no—
395#95—1224no—
396C24—21no—
397C24D4×D6111yescharacteristic
398#41—28no—
399#41—28no—
400#41—28no—
401#41C22⋊Dic61116yes—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—22no—
406C2×Q8—24no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—28no—
412C2×Q8—28no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8C22×Dic6114yes—
418C2×Q8C22×Dic6112yes—
419C2×Q8C2×C22⋊D6112yes—
420C2×Q8C2×C22⋊D6112yes—
421C2×D4—24no—
422C2×D4—24no—
423C2×D4—24no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C22×Dic6112yes—
428C2×D4C2×C22⋊D6112yes—
429C2×D4C2×C22⋊D6112yes—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—612no—
439C22×C4—612no—
440C22×C4—1224no—
441C22×C4—1228no—
442C22×C4—1222no—
443C22×C4—1224no—
444C22×C4—1222no—
445C22×C4—2444no—
446C22×C4—2442no—
447C22×C4—2444no—
448C22×C4—2444no—
449C22×C4—2442no—
450C22×C4D4×D6112yes—
451C22×C4D4×D6111yescharacteristic
452C22×C4#1175822412yes—
453C22×C4#1175822412yes—
454C22×C4C22×Dic6112yes—
455C22×C4C2×C22⋊D6112yes—
456C22×C4C2×C22⋊D6112yes—
457C22×C4C23×D6111yescharacteristic
458Q16—216no—
459SD16—216no—
460M16—28no—
461M16—2444no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—1222no—
465C2×C8—1222no—
466C2×C8—1222no—
467C2×C8C22×Dic6114yes—
468C4⋊C4—614no—
469C4⋊C4—1228no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—2442no—
474C22⋊C4—1228no—
475C22⋊C4—1224no—
476C42—1228no—
477C42—1222no—
478C42—1222no—
479C42—1222no—
480C42—1222no—
481C42—1224no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—42no—
488C23—42no—
489C23D4×Dic6112yes—
490C23#1451974512yes—
491C23#1451974912yes—
492C23C2×D4×D6111yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—22no—
496Q8—24no—
497Q8—24no—
498Q8—42no—
499Q8—42no—
500Q8—48no—
501Q8C2×C22⋊Dic6114yes—
502D4—216no—
503D4—22no—
504D4—22no—
505D4—22no—
506D4C2×C22⋊Dic6114yes—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—216no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—48no—
515C2×C4—614no—
516C2×C4—1222no—
517C2×C4—1222no—
518C2×C4—1224no—
519C2×C4—1222no—
520C2×C4—1222no—
521C2×C4—1222no—
522C2×C4—4884no—
523C2×C4—2448no—
524C2×C4—24416no—
525C2×C4—2442no—
526C2×C4—2444no—
527C2×C4—2442no—
528C2×C4—2442no—
529C2×C4C2×C22⋊Dic6114yes—
530C2×C4D4×Dic6114yes—
531C2×C4D4×Dic6112yes—
532C2×C4#1451974512yes—
533C2×C4#1451974914yes—
534C2×C4#1451974912yes—
535C2×C4#1451975814yes—
536C2×C4C23×Dic6111yescharacteristic
537C2×C4C2×D4×D6111yescharacteristic
538C2×C4C22×C22⋊D6111yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C2×C4C2×#1175822411yescharacteristic
542C8—24no—
543C8—2442no—
544C22—24no—
545C22—22no—
546C22—22no—
547C22—24no—
548C22—24no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22#1683964214yes—
554C22C2×D4×Dic6111yescharacteristic
555C22C2×#1451974511yescharacteristic
556C22C2×#1451974911yescharacteristic
557C22#137×D6112yes—
558C22#1684056112yes—
559C4—24no—
560C4—44no—
561C4—1224no—
562C4—4882no—
563C4—2442no—
564C4—2444no—
565C4#1683964214yes—
566C4C22×C22⋊Dic6111yescharacteristic
567C4C2×D4×Dic6111yescharacteristic
568C4C2×#1451974911yescharacteristic
569C4C2×#1451975811yescharacteristic
570C2—24no—
571C2—44no—
572C2C2×#1683964211yescharacteristic
573C2#137×Dic6112yes—
574C2#2145507512yes—
575C2#2145530411yescharacteristic
576C2#2145531511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC22×#137×C61C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
cosocleC22×C4×C61C24
hypercenterC22×#137D61
hypocenterC61#20927
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
4. C2×#16839669C2semidirect
5. #21446162C2semidirect
6. #21446155C2semidirect
7. #21445539C2semidirect
8. #21445512C2semidirect
36. C2×#137×C61C4semidirect
88. C22×Q8×C61C2×C4semidirect
99. C2×Q16×C61C2×C4semidirect
102. C2×SD16×C61C2×C4semidirect
105. C2×M16×C61C2×C4semidirect
252. C2×Dic61#137semidirect
302. C2×#137Dic61semidirect
328. C61#20927semidirect
341. C22×Q8C2×Dic61semidirect
355. C2×Q16C2×Dic61semidirect
358. C2×SD16C2×Dic61semidirect
364. C2×M16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic612
C2×#168397632
C2×#168397712
C2×#168396692
#214461622
#214461552
#214455392
#214455122
C22×#137×C612
#2092761