Finite Group Databasebeta

← #29707119

Subgroups — #29707119

Order 15616.

1938 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×#16840336C211yesmaximal, characteristic
3C2×#16839771C211yesmaximal, characteristic, complemented
4C2×#16839759C211yesmaximal, characteristic
5#21448405C211yesmaximal, characteristic, complemented
6#21446917C214yesmaximal, complemented
7#21446910C214yesmaximal, complemented
8#21445515C211yesmaximal, characteristic, complemented
9C23×Q16×C61C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11C2×#14519808—24no—
12C2×#14519808—24no—
13C2×#14519808—24no—
14C22×#11758202C2212yes—
15C22×#11758201C2212yes—
16C22×#11758194C2211yescharacteristic
17#16840336—24no—
18#16840336—24no—
19C2×#14519775C2212yes—
20C2×#14519766C2212yes—
21C2×#14519760—22no—
22C2×#14519753—22no—
23C2×#14519736C2211yescharacteristic
24C2×#14519736C2211yescharacteristic
25#16839771—24no—
26#16839759—24no—
27C2×#14519702C2212yes—
28C2×#14519701C2214yes—
29C2×#14519699C2212yes—
30#16839669—22no—
31#16839669C2214yes—
32#16839625—22no—
33#16839516C2214yes—
34C23×Q8×C61C2211yescharacteristic
35C23×Q8×C61C2211yescharacteristic
36C22×Q16×C61—24no—
37C22×Q16×C61—24no—
38C22×Q16×C61—24no—
39C22×Q16×C61C2214yes—
40C23×C8×C61C2211yescharacteristic
41C2×#11758222—22no—
42C2×#11758222—22no—
43C2×#11758222—22no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222D414yes—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51#14519808—232no—
52C2×#11758202—24no—
53C2×#11758202—24no—
54C2×#11758202—24no—
55C2×#11758201—24no—
56C2×#11758201—24no—
57C2×#11758201—24no—
58C2×#11758194—22no—
59C2×#11758194—22no—
60C2×#11758194D414yes—
61C22×#92418C2312yes—
62#14519775—24no—
63#14519775—24no—
64#14519766—24no—
65#14519766—24no—
66#14519760—216no—
67#14519753—216no—
68#14519736—22no—
69#14519736—22no—
70#14519736—22no—
71#14519736—22no—
72C2×C4⋊Dic61D412yes—
73C2×C4⋊Dic61C2312yes—
74C2×#11758196—24no—
75C2×C22⋊Dic61—21no—
76C2×C22⋊Dic61D412yes—
77C2×#11758198—24no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61C2312yes—
80#14519702—24no—
81#14519701—28no—
82#14519699—24no—
83#14519684—28no—
84C22×Q8×C61—21no—
85C22×Q8×C61—22no—
86C22×Q8×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61—22no—
90C22×Q8×C61—22no—
91C22×Q8×C61—22no—
92C22×Q8×C61—22no—
93C22×Q8×C61—22no—
94C22×Q8×C61—22no—
95C22×Q8×C61—22no—
96C22×Q8×C61D414yes—
97C22×Q8×C61D414yes—
98C22×Q8×C61C2312yes—
99C22×Q8×C61C2312yes—
100C23×C4×C61—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103C2×Q16×C61—232no—
104C2×Q16×C61—28no—
105C2×Q16×C61—28no—
106C2×Q16×C61—28no—
107C22×C8×C61—22no—
108C22×C8×C61—22no—
109C22×C8×C61D414yes—
110C22×C8×C61C2312yes—
111C24×C61C2×D411yescharacteristic
112C23×D61—21no—
113C2×Q8×C61—24no—
114C2×Q8×C61—24no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—28no—
118C2×Q8×C61—28no—
119C2×Q8×C61—28no—
120C2×Q8×C61—28no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—22no—
127C2×Q8×C61—22no—
128C2×Q8×C61—22no—
129C2×Q8×C61—42no—
130C2×Q8×C61—44no—
131C2×Q8×C61—42no—
132C2×Q8×C61—44no—
133C2×Q8×C61—44no—
134C2×Q8×C61—44no—
135C2×Q8×C61—42no—
136C2×Q8×C61—42no—
137C2×Q8×C61C2×D414yes—
138C2×Q8×C61C2×D414yes—
139C22×C4×C61—24no—
140C22×C4×C61—24no—
141C22×C4×C61—22no—
142C22×C4×C61—22no—
143C22×C4×C61—42no—
144C22×C4×C61—42no—
145C22×C4×C61—42no—
146C22×C4×C61—42no—
147C22×C4×C61C2×D412yes—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2×D412yes—
153C22×C4×C61#4112yes—
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—24no—
156C2×C22⋊D61—42no—
157C2×C22⋊D61—42no—
158C22×Dic61—21no—
159C22×Dic61—22no—
160C22×Dic61C2×D414yes—
161#11758222—28no—
162#11758222—28no—
163#11758222—44no—
164#11758222—44no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×D244—21no—
170C2×D244C2×D412yes—
171C2×C4×D61—24no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C2×D412yes—
180Q16×C61—232no—
181C2×C8×C61—28no—
182C2×C8×C61—22no—
183C2×C8×C61—22no—
184C2×C8×C61C2×D414yes—
185C22⋊Dic61—24no—
186C22⋊Dic61—42no—
187#11758202—216no—
188#11758201—216no—
189#11758198—216no—
190C4⋊Dic61—24no—
191C4⋊Dic61—22no—
192C4⋊Dic61—22no—
193#11758196—216no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61—42no—
197#11758194—28no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418C2×D414yes—
201C23×C61—24no—
202C23×C61#12012yes—
203C23×C61#12112yes—
204C23×C61#12712yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—44no—
210Q8×C61—24no—
211Q8×C61—24no—
212Q8×C61—24no—
213Q8×C61—24no—
214Q8×C61—42no—
215Q8×C61—42no—
216Q8×C61—42no—
217Q8×C61—42no—
218Q8×C61—48no—
219Q8×C61—48no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—42no—
223C2×C4×C61—42no—
224C2×C4×C61—42no—
225C2×C4×C61—42no—
226C2×C4×C61—48no—
227C2×C4×C61—48no—
228C2×C4×C61#12014yes—
229C2×C4×C61#12014yes—
230C2×C4×C61#12012yes—
231C2×C4×C61#12114yes—
232C2×C4×C61#12114yes—
233C2×C4×C61#12112yes—
234C2×C4×C61#12712yes—
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C22⋊D61—48no—
242C2×Dic61—24no—
243C2×Dic61—22no—
244C2×Dic61—28no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—42no—
248C2×Dic61—44no—
249D244—22no—
250D244—22no—
251D244—41no—
252D244—41no—
253C4×D61—48no—
254Dic122—22no—
255Dic122—22no—
256Dic122—41no—
257Dic122—41no—
258C8×C61—24no—
259#92418—24no—
260#23425—611nomaximal, Sylow
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—24no—
264C22×C61#39214yes—
265C22×C61C2×#12011yescharacteristic
266C22×C61C2×#12111yescharacteristic
267C22×C61C2×#12711yescharacteristic
268C22×C61#57814yes—
269C2×D61—42no—
270C2×D61—44no—
271C2×D61—82no—
272C2×D61—82no—
273C4×C61—44no—
274C4×C61—44no—
275C4×C61#39214yes—
276C4×C61C2×#12011yescharacteristic
277C4×C61C2×#12011yescharacteristic
278C4×C61C2×#12111yescharacteristic
279C4×C61C2×#12111yescharacteristic
280Dic61—42no—
281Dic61—44no—
282Dic61—82no—
283C22×#137—611no—
284C23×Q16D6111yescharacteristic, complemented
285C2×#536—611no—
286C2×#419—611no—
287C2×#416—611no—
288#2310—611no—
289#2016—614no—
290#2015—614no—
291#1729—611no—
292C2×C61—24no—
293C2×C61C2×#39211yescharacteristic
294C2×C61#301112yes—
295C2×C61#304412yes—
296C2×C61#315612yes—
297D61—82no—
298C22×#41—611no—
299C23×Q8C2×D6111yescharacteristic
300C23×Q8C2×D6111yescharacteristic
301C2×#137—1224no—
302C2×#137—1224no—
303C2×#137—1224no—
304C22×Q16—24no—
305C22×Q16—24no—
306C22×Q16—24no—
307C22×Q16—612no—
308C22×Q16C2×D6114yes—
309C22×SD16—612no—
310C22×M16—611no—
311C23×C8C2×D6111yescharacteristic
312#536—1224no—
313#536—1224no—
314C2×C4⋊Q8—612no—
315C2×#124—612no—
316C2×#123—1222no—
317C2×C22⋊Q8—1222no—
318C2×#117—611no—
319C2×#117—611no—
320#419—1224no—
321#416—1224no—
322C2×#103—614no—
323C2×#103—612no—
324C2×#102—612no—
325#391—614no—
326#391—1222no—
327#381—1222no—
328#336—614no—
329C61#2342511yescharacteristic, complemented, Sylow
330C2×#41—614no—
331C2×#41—1222no—
332C2×#41—1222no—
333C2×#41—1222no—
334C2×#41—1222no—
335C2×#41—1222no—
336C22×Q8—21no—
337C22×Q8—22no—
338C22×Q8—21no—
339C22×Q8—22no—
340C22×Q8—22no—
341C22×Q8—22no—
342C22×Q8—22no—
343C22×Q8—22no—
344C22×Q8—22no—
345C22×Q8—22no—
346C22×Q8—22no—
347C22×Q8—22no—
348C22×Q8—611no—
349C22×Q8C22⋊D6114yes—
350C22×Q8C22⋊D6114yes—
351C22×Q8C22×D6112yes—
352C22×Q8C22×D6112yes—
353C22×D4—611no—
354C22×D4—1221no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—1221no—
358C23×C4C22×D6111yescharacteristic
359#137—12232no—
360C2×Q16—232no—
361C2×Q16—28no—
362C2×Q16—28no—
363C2×Q16—28no—
364C2×Q16—1224no—
365C2×Q16—1224no—
366C2×Q16—1224no—
367C2×SD16—1224no—
368C2×SD16—1224no—
369C2×SD16—1224no—
370C2×M16—614no—
371C2×M16—1222no—
372C2×M16—1222no—
373C22×C8—22no—
374C22×C8—22no—
375C22×C8—612no—
376C22×C8C22⋊D6114yes—
377C22×C8C22×D6112yes—
378C4⋊Q8—1224no—
379C4⋊Q8—1224no—
380#124—1224no—
381#124—1224no—
382#123—12216no—
383C22⋊Q8—12216no—
384#117—1222no—
385#117—1222no—
386#117—1222no—
387#117—1222no—
388C2×C4⋊C4—612no—
389C2×C4⋊C4—612no—
390C2×C4⋊C4—1224no—
391C2×C22⋊C4—612no—
392C2×C22⋊C4—1224no—
393C2×C22⋊C4—1221no—
394C2×C42—612no—
395C2×C42—1221no—
396#103—1228no—
397#103—1224no—
398#102—1224no—
399#95—1228no—
400C24—1221no—
401C24D4×D6111yescharacteristic
402#41—1228no—
403#41—1228no—
404#41—2444no—
405#41—2444no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—42no—
423C2×Q8—44no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—42no—
427C2×Q8—44no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—612no—
431C2×Q8—1221no—
432C2×Q8—1221no—
433C2×Q8—1221no—
434C2×Q8—1221no—
435C2×Q8—1221no—
436C2×Q8C2×C22⋊D6114yes—
437C2×Q8C2×C22⋊D6114yes—
438C2×D4—612no—
439C2×D4—1224no—
440C2×D4—1221no—
441C2×D4—1221no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—2442no—
446C2×D4—2442no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—614no—
456C22×C4—1221no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4D4×D6112yes—
462C22×C4D4×D6112yes—
463C22×C4D4×D6111yescharacteristic
464C22×C4#1175822412yes—
465C22×C4C2×C22⋊D6112yes—
466C22×C4C2×C22⋊D6112yes—
467C22×C4C2×C22⋊D6112yes—
468C22×C4C23×D6111yescharacteristic
469Q16—232no—
470Q16—12216no—
471SD16—12216no—
472M16—1228no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—28no—
476C2×C8—614no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8C2×C22⋊D6114yes—
480C4⋊C4—12216no—
481C4⋊C4—1224no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C22⋊C4—12216no—
485C22⋊C4—1224no—
486C22⋊C4—2442no—
487C42—1222no—
488C42—1222no—
489C42—2442no—
490C23—24no—
491C23—1222no—
492C23—2441no—
493C23—2441no—
494C23—2444no—
495C23#1451974512yes—
496C23#1451974912yes—
497C23#1451977312yes—
498C23C2×D4×D6111yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—24no—
502Q8—24no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—42no—
507Q8—48no—
508Q8—48no—
509Q8—1222no—
510Q8—1222no—
511Q8—2441no—
512Q8—2441no—
513D4—1222no—
514D4—1222no—
515D4—2448no—
516D4—2441no—
517D4—2441no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—1224no—
527C2×C4—1222no—
528C2×C4—1228no—
529C2×C4—2442no—
530C2×C4—2444no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4—2448no—
534C2×C4#1451974514yes—
535C2×C4#1451974512yes—
536C2×C4#1451974614yes—
537C2×C4#1451974814yes—
538C2×C4#1451974914yes—
539C2×C4#1451974912yes—
540C2×C4#1451977312yes—
541C2×C4C2×D4×D6111yescharacteristic
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4C22×C22⋊D6111yescharacteristic
545C2×C4C22×C22⋊D6111yescharacteristic
546C2×C4C2×#1175822411yescharacteristic
547C8—24no—
548C8—1224no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—4882no—
553C22—4882no—
554C22—2444no—
555C22—2442no—
556C22#1683967014yes—
557C22C2×#1451974511yescharacteristic
558C22C2×#1451974911yescharacteristic
559C22C2×#1451977311yescharacteristic
560C22#1684057514yes—
561C4—44no—
562C4—44no—
563C4—4882no—
564C4—2444no—
565C4—2442no—
566C4#1683967014yes—
567C4C2×#1451974511yescharacteristic
568C4C2×#1451974611yescharacteristic
569C4C2×#1451974811yescharacteristic
570C4C2×#1451974911yescharacteristic
571C2—24no—
572C2—4882no—
573C2C2×#1683967011yescharacteristic
574C2#2145511012yes—
575C2#2145534612yes—
576C2#2145630812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC23×Q16×C61C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenterC23×Q16D61
hypocenterC61#23425
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
3. C2×#16839771C2semidirect
5. #21448405C2semidirect
6. #21446917C2semidirect
7. #21446910C2semidirect
8. #21445515C2semidirect
9. C23×Q16×C61C2semidirect
284. C23×Q16D61semidirect
329. C61#23425semidirect

Maximal subgroups

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

subgroupindex
C22×#145198082
C2×#168403362
C2×#168397712
C2×#168397592
#214484052
#214469172
#214469102
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C23×Q16×C612
#2342561