Finite Group Databasebeta

← #29730747

Subgroups — #29730747

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839989C212yesmaximal, complemented
2C2×#16839967C211yesmaximal, characteristic, complemented
3C2×D4×#92418C211yesmaximal, characteristic, complemented
4#21450772C211yesmaximal, characteristic, complemented
5#21447015C212yesmaximal, complemented
6#21446822C212yesmaximal, complemented
7#21446620C211yesmaximal, characteristic, complemented
8#21446580C211yesmaximal, characteristic, complemented
9#21446240C212yesmaximal, complemented
10C4×#14519698C211yesmaximal, characteristic, complemented
11C61×#2537C211yesmaximal, characteristic
12C22×#11758200C2212yes—
13C23×#92418C2212yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519756C2212yes—
16C2×#14519756C2212yes—
17#16839989—28no—
18#16839967—24no—
19D4×#92418—24no—
20C2×#14519708C2212yes—
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519700C2212yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×C22⋊#92418C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839698—22no—
30#16839665—21no—
31#16839631—21no—
32C4×C4⋊Dic61C2211yescharacteristic
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36C2×#121×C61—21no—
37C2×#121×C61C2212yes—
38C2×C4×D4×C61C2211yescharacteristic
39C2×C4×D4×C61C2211yescharacteristic
40C61×#394—22no—
41C61×#384C2211yescharacteristic
42C61×#380—21no—
43C61×#380C2212yes—
44C22×Dic122C2312yes—
45C2×#11758200—24no—
46C2×#11758200—24no—
47C2×#11758200—42no—
48C22×#92418—24no—
49C22×#92418—24no—
50C22×#92418—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2311yescharacteristic
53C22×#92418C2312yes—
54#14519777—22no—
55#14519777—22no—
56#14519756—24no—
57#14519756—24no—
58#14519756—42no—
59#14519756—42no—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2311yescharacteristic
63C2×C4⋊Dic61C2311yescharacteristic
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×C22⋊Dic61—22no—
67C2×C22⋊Dic61—22no—
68C2×C4×Dic61—21no—
69#14519708—24no—
70C4⋊#92418D414yescomplemented
71#14519700—24no—
72#14519700—24no—
73#14519700—42no—
74#14519698—22no—
75#14519698D414yescomplemented
76C22⋊#92418—24no—
77C4×#92418—22no—
78#14519684—21no—
79C22×D4×C61—22no—
80C22×D4×C61D411yescharacteristic
81C22×D4×C61C2311yescharacteristic
82C23×C4×C61—21no—
83C23×C4×C61C2312yes—
84#121×C61—24no—
85#121×C61—42no—
86#121×C61—44no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C4⋊C4×C61C2312yes—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C22⋊C4×C61—21no—
99C2×C22⋊C4×C61D412yes—
100C2×C42×C61C2311yescharacteristic
101#95×C61—22no—
102C24×C61—21no—
103C24×C61C2×D412yes—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—21no—
108C2×D4×C61—21no—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—44no—
112C2×D4×C61—44no—
113C2×D4×C61—82no—
114C2×D4×C61SD1614yescomplemented
115C2×D4×C61C2×D412yes—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—42no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—22no—
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61—81no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—44no—
149C22⋊C4×C61—44no—
150C22⋊C4×C61—42no—
151C42×C61C2×D412yes—
152C42×C61#4112yes—
153C22⋊Dic61—44no—
154C22⋊Dic61—44no—
155#11758200—48no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—42no—
159C4⋊Dic61—42no—
160C4⋊Dic61C2×D412yes—
161C4⋊Dic61#4112yes—
162#11758196—44no—
163#11758196—44no—
164C4×Dic61—42no—
165C2×#92418—21no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—22no—
169C2×#92418—21no—
170C2×#92418—21no—
171C2×#92418—21no—
172C2×#92418—48no—
173C2×#92418C2×D414yes—
174C2×#92418C2×D412yes—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—41no—
185C23×C61—41no—
186C23×C61—41no—
187C23×C61—41no—
188C23×C61C2×SD1614yes—
189C23×C61C22×D411yescharacteristic
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—28no—
193D4×C61—41no—
194D4×C61—41no—
195D4×C61—88no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—41no—
206C2×C4×C61—42no—
207C2×C4×C61—41no—
208C2×C4×C61—42no—
209C2×C4×C61—41no—
210C2×C4×C61—42no—
211C2×C4×C61—84no—
212C2×C4×C61#12112yes—
213C2×C4×C61#12112yes—
214C2×C4×C61C2×SD1612yes—
215C2×C4×C61#13512yes—
216C2×C4×C61C22×D411yescharacteristic
217C2×C4×C61C22×D411yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×C4×C61C2×#4111yescharacteristic
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×Dic61—22no—
223C2×Dic61—22no—
224C2×Dic61—41no—
225C2×Dic61—41no—
226C2×Dic61—41no—
227C2×Dic61—41no—
228C2×Dic61—42no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231Dic122—42no—
232Dic122—42no—
233Dic122—42no—
234Dic122—42no—
235#92418—22no—
236#92418—24no—
237#92418—41no—
238#25385—611nomaximal, Sylow
239C22×C61—24no—
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—42no—
243C22×C61—42no—
244C22×C61—42no—
245C22×C61—42no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—42no—
249C22×C61—81no—
250C22×C61—81no—
251C22×C61—81no—
252C22×C61—81no—
253C22×C61C2×#12111yescharacteristic
254C22×C61D4211yescharacteristic
255C22×C61#54611yescharacteristic
256C22×C61C22×SD1611yescharacteristic
257C22×C61C2×#13511yescharacteristic
258C22×C61#57711yescharacteristic
259C22×C61#57811yescharacteristic
260C4×C61—22no—
261C4×C61—42no—
262C4×C61—82no—
263C4×C61—81no—
264C4×C61#46514yescomplemented
265C4×C61#53211yescharacteristic
266C4×C61#54011yescharacteristic
267C4×C61D4211yescharacteristic
268C4×C61#55011yescharacteristic
269Dic61—42no—
270Dic61—42no—
271Dic61—82no—
272C2×#465—612no—
273C2×#461—611no—
274C2×C8×D4—611no—
275#2542—611no—
276#2537D6111yescharacteristic
277#2035—612no—
278#2000—612no—
279#1956—611no—
280#1948—611no—
281#1892—612no—
282C4×#102—611no—
283C2×C61—44no—
284C2×C61—82no—
285C2×C61#242811yescharacteristic
286C2×C61C2×#46511yescharacteristic
287C2×C61#306311yescharacteristic
288C2×C61D4×SD1611yescharacteristic
289C2×C61#328311yescharacteristic
290C2×C61#329811yescharacteristic
291C2×C61#330311yescharacteristic
292C22×SD16—612no—
293C23×C8—612no—
294C2×C4⋊Q8—611no—
295C2×C22⋊Q8—612no—
296C2×C22⋊Q8—612no—
297C2×#121—21no—
298C2×#121C2×D6112yes—
299C2×C4×D4C2×D6111yescharacteristic
300C2×C4×D4C2×D6111yescharacteristic
301#465—1228no—
302#461—1224no—
303C8×D4—1224no—
304C2×#106—612no—
305C2×C4⋊C8—611no—
306C2×#103—612no—
307C2×#103—612no—
308C2×#103—612no—
309C2×#102—611no—
310C2×#102—611no—
311C2×C22⋊C8—612no—
312C2×C4×C8—611no—
313#399—1222no—
314#394—22no—
315#390—1221no—
316#384—1221no—
317#384C2×D6111yescharacteristic
318#380—21no—
319#380C2×D6112yes—
320C4×C4⋊C4—611no—
321#340—611no—
322#340—612no—
323#340—611no—
324C61#2538511yescharacteristic, complemented, Sylow
325C22×Q8—612no—
326C22×D4—22no—
327C22×D4C22⋊D6111yescharacteristic
328C22×D4C22×D6111yescharacteristic
329C23×C4—21no—
330C23×C4C22×D6112yes—
331C2×SD16—1224no—
332C2×SD16—1224no—
333C2×SD16—2442no—
334C22×C8—611no—
335C22×C8—612no—
336C22×C8—611no—
337C22×C8—1224no—
338C22×C8—1224no—
339C22×C8—2442no—
340C4⋊Q8—1222no—
341C4⋊Q8—1222no—
342C22⋊Q8—1224no—
343C22⋊Q8—1224no—
344C22⋊Q8—2442no—
345C22⋊Q8—2442no—
346#121—24no—
347#121—42no—
348#121—44no—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—611no—
356C2×C4⋊C4—611no—
357C2×C4⋊C4—1222no—
358C2×C4⋊C4—1222no—
359C2×C4⋊C4—1221no—
360C2×C4⋊C4—1221no—
361C2×C4⋊C4C22⋊D6111yescharacteristic
362C2×C4⋊C4C22×D6111yescharacteristic
363C2×C4⋊C4C22×D6112yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—1222no—
368C2×C22⋊C4—1222no—
369C2×C22⋊C4C22⋊D6112yes—
370C2×C42—1221no—
371C2×C42C22×D6111yescharacteristic
372#106—1224no—
373C4⋊C8—614no—
374#103—1224no—
375#103—1224no—
376#103—2442no—
377#102—614no—
378#102—1222no—
379C22⋊C8—1224no—
380C4×C8—1222no—
381#95—22no—
382#95—1221no—
383C24—21no—
384C24C2×C22⋊D6112yes—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—1222no—
390C2×Q8—1222no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—82no—
401C2×D4#1175820014yescomplemented
402C2×D4D4×D6112yes—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—1221no—
415C22×C4—1221no—
416C22×C4—1222no—
417C22×C4C2×C22⋊D6112yes—
418C22×C4C2×C22⋊D6111yescharacteristic
419C22×C4C2×C22⋊D6111yescharacteristic
420C22×C4C23×D6111yescharacteristic
421SD16—2448no—
422C2×C8—614no—
423C2×C8—612no—
424C2×C8—1221no—
425C2×C8—1222no—
426C2×C8—1221no—
427C2×C8—1222no—
428C2×C8—1221no—
429C2×C8—1221no—
430C2×C8—1221no—
431C2×C8—2448no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—81no—
439C4⋊C4—612no—
440C4⋊C4—612no—
441C4⋊C4—1221no—
442C4⋊C4—1221no—
443C4⋊C4—2444no—
444C4⋊C4—2444no—
445C4⋊C4—2442no—
446C4⋊C4—2442no—
447C22⋊C4—24no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—2444no—
452C22⋊C4—2444no—
453C42—2442no—
454C42D4×D6112yes—
455C42#1175822412yes—
456C23—22no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23C2×#1175820014yes—
470C23C22×C22⋊D6111yescharacteristic
471Q8—2442no—
472Q8—2442no—
473Q8—2442no—
474Q8—2442no—
475D4—28no—
476D4—22no—
477D4—22no—
478D4—41no—
479D4—41no—
480D4—88no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—1222no—
498C2×C4—1222no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4—2441no—
503C2×C4—2441no—
504C2×C4—2441no—
505C2×C4—2441no—
506C2×C4#1451974912yes—
507C2×C4#1451974912yes—
508C2×C4C2×#1175820012yes—
509C2×C4#1451979712yes—
510C2×C4C2×D4×D6111yescharacteristic
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4#41×D6111yescharacteristic
513C2×C4C2×#1175822411yescharacteristic
514C2×C4#1451982211yescharacteristic
515C2×C4#1451982311yescharacteristic
516C8—1222no—
517C8—1224no—
518C8—2441no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22C2×#1451974911yescharacteristic
534C22D4×C22⋊D6111yescharacteristic
535C22#1684040011yescharacteristic
536C22C22×#1175820011yescharacteristic
537C22C2×#1451979711yescharacteristic
538C22#1684057311yescharacteristic
539C22#1684057711yescharacteristic
540C4—22no—
541C4—42no—
542C4—81no—
543C4—82no—
544C4—4882no—
545C4—2442no—
546C4—2442no—
547C4#1683998914yescomplemented
548C4#1684030711yescharacteristic
549C4#1684036511yescharacteristic
550C4#1684039611yescharacteristic
551C4#1684043011yescharacteristic
552C2—44no—
553C2—82no—
554C2#2144940911yescharacteristic
555C2C2×#1683998911yescharacteristic
556C2#2145551211yescharacteristic
557C2D4×#1175820011yescharacteristic
558C2#2145753711yescharacteristic
559C2#2145767811yescharacteristic
560C2#2145774111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#2537C2
socleC23×C61C22×D4
hypercenter#2537D61
hypocenterC61#25385
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×#16839989C2semidirect
2. C2×#16839967C2semidirect
3. C2×D4×#92418C2semidirect
4. #21450772C2semidirect
5. #21447015C2semidirect
6. #21446822C2semidirect
7. #21446620C2semidirect
8. #21446580C2semidirect
9. #21446240C2semidirect
10. C4×#14519698C2semidirect
70. C4⋊#92418D4semidirect
75. #14519698D4semidirect
114. C2×D4×C61SD16semidirect
264. C4×C61#465semidirect
324. C61#25385semidirect
401. C2×D4#11758200semidirect
547. C4#16839989semidirect

Maximal subgroups

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

subgroupindex
C2×#168399892
C2×#168399672
C2×D4×#924182
#214507722
#214470152
#214468222
#214466202
#214465802
#214462402
C4×#145196982
C61×#25372
#2538561