Finite Group Databasebeta

← #29733513

Subgroups — #29733513

Order 15616.

1438 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840313C211yesmaximal, characteristic
2C2×#16840135C211yesmaximal, characteristic, complemented
3C2×#16840114C211yesmaximal, characteristic, complemented
4#21454172C211yesmaximal, characteristic, complemented
5#21446825C214yesmaximal, complemented
6#21446624C211yesmaximal, characteristic, complemented
7#21446154C214yesmaximal, complemented
8C4×#14519689C211yesmaximal, characteristic, complemented
9C61×#1990C211yesmaximal, characteristic, complemented
10C22×#11758226—21no—
11C22×#11758190C2214yes—
12#16840313—24no—
13#16840313—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519774C2211yescharacteristic
16C2×#14519766—21no—
17C2×#14519757C2212yes—
18C2×#14519752—21no—
19C2×#14519752C2212yes—
20C2×#14519739—21no—
21C2×#14519739C2212yes—
22C2×C42×D61C2211yescharacteristic
23#16840135—24no—
24#16840114—24no—
25C2×#14519700C2214yes—
26C2×#14519697C2214yes—
27C2×#14519689C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839667C2212yes—
30#16839661—21no—
31#16839661—21no—
32#16839619—21no—
33#16839619—21no—
34#16839516C2212yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×C4×C8×C61C2211yescharacteristic
37C61×#384C2211yescharacteristic
38C61×#340C2214yes—
39C2×#11758226—28no—
40C2×#11758226—44no—
41C2×#11758226—44no—
42C22×Dic122C2312yes—
43C22×D244—22no—
44C22×D244—21no—
45C22×D244—21no—
46C22×D244C2312yes—
47C22×C4×D61—22no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C2×#11758190—28no—
52C2×#11758190—28no—
53C2×#11758190—44no—
54C22×#92418C2312yes—
55#14519777—22no—
56#14519777—22no—
57#14519774—22no—
58#14519774D414yes—
59#14519766—28no—
60#14519757—44no—
61#14519752—28no—
62#14519752—24no—
63#14519752—24no—
64#14519739—28no—
65#14519739—24no—
66#14519739—24no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—22no—
73C2×C4×Dic61D411yescharacteristic
74C2×C4×Dic61D411yescharacteristic
75C2×C4×Dic61C2311yescharacteristic
76C42×D61—22no—
77C42×D61—22no—
78#14519700—28no—
79#14519697—44no—
80#14519689—24no—
81C4×#92418—22no—
82#14519684—22no—
83#14519684—22no—
84C22×Q8×C61—21no—
85C22×C8×C61C2312yes—
86C4⋊Q8×C61—22no—
87C4⋊Q8×C61D414yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2312yes—
92C2×C4⋊C4×C61C2312yes—
93C2×C42×C61C2311yescharacteristic
94C4×C8×C61—22no—
95C23×D61—22no—
96C23×D61C2×D411yescharacteristic
97C2×Q8×C61—24no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2411yescharacteristic
105C22×Dic61—22no—
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C22×Dic61C2×D411yescharacteristic
109#11758226—432no—
110C2×D244—24no—
111C2×D244—24no—
112C2×D244—28no—
113C2×D244—24no—
114C2×D244—24no—
115C2×D244—22no—
116C2×D244—44no—
117C2×D244—44no—
118C2×D244—42no—
119C2×D244—42no—
120C2×D244—42no—
121C2×D244—42no—
122C2×D244C2×D414yes—
123C2×C4×D61—22no—
124C2×C4×D61—22no—
125C2×C4×D61—28no—
126C2×C4×D61—22no—
127C2×C4×D61—22no—
128C2×C4×D61—22no—
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61SD1614yescomplemented
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—24no—
139C2×Dic122—24no—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61#4114yescomplemented
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—42no—
148C4⋊C4×C61C2×D414yes—
149C42×C61C2×D412yes—
150C42×C61C2×D412yes—
151#11758198—28no—
152#11758198—28no—
153#11758198—44no—
154C4⋊Dic61—44no—
155C4⋊Dic61—42no—
156C4⋊Dic61—42no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C4×Dic61—21no—
160C4×Dic61—21no—
161C2×#92418—22no—
162C2×#92418—22no—
163C2×#92418—22no—
164#11758190—416no—
165C23×C61C22×D411yescharacteristic
166C22×D61—21no—
167C22×D61—21no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61—21no—
171C22×D61—21no—
172C22×D61—24no—
173C22×D61—42no—
174C22×D61—42no—
175C22×D61—42no—
176C22×D61—42no—
177C22×D61—42no—
178C22×D61—42no—
179C22×D61C2×SD1612yes—
180Q8×C61—48no—
181C2×C4×C61—24no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61#12012yes—
188C2×C4×C61#12012yes—
189C2×C4×C61C2×SD1612yes—
190C2×C4×C61#13512yes—
191C2×C4×C61AGL(1,ℤ/8)12yes—
192C2×C4×C61#13712yes—
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C2×#4112yes—
196C2×C4×C61H(5,𝔽2)12yes—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61C2×SD1612yes—
209D244—28no—
210D244—416no—
211D244—44no—
212D244—48no—
213D244—48no—
214C4×D61—28no—
215C4×D61—416no—
216C4×D61—42no—
217C4×D61—42no—
218C4×D61—44no—
219Dic122—44no—
220Dic122—44no—
221C8×C61—24no—
222#92418—42no—
223#25597—611nomaximal, Sylow
224C22×C61C2×#12011yescharacteristic
225C22×C61#53211yescharacteristic
226C22×C61#53511yescharacteristic
227C22×C61C22×SD1611yescharacteristic
228C22×C61C2×#13511yescharacteristic
229C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C61C2×#13711yescharacteristic
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—41no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—44no—
239C2×D61—44no—
240C2×D61—44no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61—82no—
244C2×D61—82no—
245C4×C61—44no—
246C4×C61—82no—
247C4×C61#45014yes—
248C4×C61#45214yes—
249C4×C61#54612yes—
250C4×C61#55012yes—
251Dic61—22no—
252Dic61—41no—
253Dic61—41no—
254Dic61—82no—
255C2×#532—611no—
256C2×#492—611no—
257C2×#487—611no—
258#2857—611no—
259#2000—614no—
260#1990D6111yescharacteristic, complemented
261#1957—611no—
262#1877—614no—
263C4×C22⋊C8—611no—
264C2×C61#243111yescharacteristic
265C2×C61C2×#45011yescharacteristic
266C2×C61C2×#45211yescharacteristic
267C2×C61#316711yescharacteristic
268C2×C61#318211yescharacteristic
269C2×C61#320611yescharacteristic
270C2×C61#320711yescharacteristic
271D61—42no—
272D61—84no—
273C22×#41—1221no—
274C22×SD16—614no—
275#532—1224no—
276#532—1224no—
277C2×C4⋊Q8—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×#127—611no—
280C2×#124—1221no—
281C2×C22⋊Q8—612no—
282C2×#121—612no—
283C2×#121—1221no—
284C2×C4×D4—612no—
285C2×C4×D4—1221no—
286C22×C42—611no—
287#492—1224no—
288#487—1224no—
289C2×#103—614no—
290C2×#102—614no—
291C2×C22⋊C8—612no—
292C2×C4×C8—611no—
293C2×C4×C8C2×D6111yescharacteristic
294#391—612no—
295#389—1221no—
296#389—1221no—
297#384C2×D6111yescharacteristic
298#380—1221no—
299#380—1221no—
300#340C2×D6114yes—
301#336—612no—
302C61#2559711yescharacteristic, complemented, Sylow
303C2×#41—1228no—
304C2×#41—2444no—
305C2×#41—2444no—
306C22×Q8—21no—
307C22×Q8—612no—
308C22×D4—612no—
309C22×D4—1222no—
310C22×D4—1221no—
311C22×D4—1221no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—1222no—
316C2×SD16—1228no—
317C2×SD16—1228no—
318C2×SD16—2444no—
319C22×C8—612no—
320C22×C8C22×D6112yes—
321C4⋊Q8—22no—
322C4⋊Q8—1222no—
323C4⋊Q8—1222no—
324C4⋊Q8D24414yes—
325#127—614no—
326#127—1222no—
327#124—1228no—
328C22⋊Q8—2444no—
329#121—1224no—
330#121—1228no—
331#121—1224no—
332C4×D4—1224no—
333C4×D4—1228no—
334C4×D4—1224no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—1222no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4C22×D6112yes—
342C2×C4⋊C4C22×D6112yes—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4—1222no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—611no—
349C2×C42—1222no—
350C2×C42—1222no—
351C2×C42C22×D6111yescharacteristic
352#103—1228no—
353#102—2444no—
354C22⋊C8—1224no—
355C4×C8—22no—
356C4×C8—1222no—
357#95—1222no—
358#95—1222no—
359C24—611no—
360C24—1222no—
361#41—24432no—
362C2×Q8—24no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—1222no—
366C2×Q8—1222no—
367C2×Q8—1224no—
368C2×Q8—1224no—
369C2×D4—614no—
370C2×D4—1228no—
371C2×D4—1222no—
372C2×D4—1224no—
373C2×D4—1224no—
374C2×D4—1224no—
375C2×D4—1224no—
376C2×D4—2444no—
377C2×D4—2444no—
378C2×D4—2442no—
379C2×D4—2442no—
380C2×D4—2442no—
381C2×D4—2442no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—611no—
385C22×C4—614no—
386C22×C4—611no—
387C22×C4—611no—
388C22×C4—1228no—
389C22×C4—1222no—
390C22×C4—1222no—
391C22×C4—1222no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—1222no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1222no—
399C22×C4—2444no—
400C22×C4—2444no—
401C22×C4D4×D6111yescharacteristic
402C22×C4D4×D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404SD16—24416no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8#1175822614yescomplemented
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—2444no—
417C4⋊C4—2442no—
418C4⋊C4—2442no—
419C4⋊C4C2×D24414yes—
420C22⋊C4—1228no—
421C22⋊C4—1228no—
422C22⋊C4—2444no—
423C42—1221no—
424C42—1221no—
425C42—1222no—
426C42—1222no—
427C42C2×D24412yes—
428C42D4×D6112yes—
429C23—612no—
430C23—1221no—
431C23—1221no—
432C23—1221no—
433C23—1221no—
434C23—1221no—
435C23—1221no—
436C23—1224no—
437C23—2442no—
438C23—2442no—
439C23—2442no—
440C23—2442no—
441C23—2442no—
442C23—2442no—
443C23C2×D4×D6111yescharacteristic
444Q8—48no—
445Q8—2444no—
446Q8—2444no—
447D4—1228no—
448D4—24416no—
449D4—2448no—
450D4—2444no—
451D4—2448no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—612no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1228no—
462C2×C4—1221no—
463C2×C4—1221no—
464C2×C4—1221no—
465C2×C4—1221no—
466C2×C4—1221no—
467C2×C4—1221no—
468C2×C4—24416no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2444no—
475C2×C4#1451974712yes—
476C2×C4#1451974712yes—
477C2×C4SD16×D6112yes—
478C2×C4#1451979812yes—
479C2×C4#1451980112yes—
480C2×C4#1451980912yes—
481C2×C4C2×D4×D6111yescharacteristic
482C2×C4C22×D24411yescharacteristic
483C2×C4C2×#1175822612yes—
484C2×C4#1451982312yes—
485C8—24no—
486C8—2442no—
487C22—1222no—
488C22—1222no—
489C22—1222no—
490C22—4882no—
491C22—4882no—
492C22—4882no—
493C22—4882no—
494C22—2444no—
495C22—2444no—
496C22—2444no—
497C22—2441no—
498C22—2441no—
499C22—2441no—
500C22—2441no—
501C22C2×#1451974711yescharacteristic
502C22#1684031311yescharacteristic
503C22#1684033011yescharacteristic
504C22C2×SD16×D6111yescharacteristic
505C22C2×#1451979811yescharacteristic
506C22C2×#1451980111yescharacteristic
507C22C2×#1451980911yescharacteristic
508C4—44no—
509C4—82no—
510C4—1222no—
511C4—4882no—
512C4—2441no—
513C4—2441no—
514C4#1683991414yes—
515C4#1683992714yes—
516C4#1684040312yes—
517C4#1684043412yes—
518C2—4884no—
519C2—2442no—
520C2#2144944011yescharacteristic
521C2C2×#1683991411yescharacteristic
522C2C2×#1683992711yescharacteristic
523C2#2145638011yescharacteristic
524C2#2145653911yescharacteristic
525C2#2145674511yescharacteristic
526C2#2145675311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1990C2
socleC23×C61C22×D4
prime radicalC2×#16840313C2
hypercenter#1990D61
hypocenterC61#25597
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#16840135C2semidirect
3. C2×#16840114C2semidirect
4. #21454172C2semidirect
5. #21446825C2semidirect
6. #21446624C2semidirect
7. #21446154C2semidirect
8. C4×#14519689C2semidirect
9. C61×#1990C2semidirect
135. C2×C4×D61SD16semidirect
142. C2×C8×C61#41semidirect
260. #1990D61semidirect
302. C61#25597semidirect
410. C2×C8#11758226semidirect

Maximal subgroups

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

subgroupindex
C2×#168403132
C2×#168401352
C2×#168401142
#214541722
#214468252
#214466242
#214461542
C4×#145196892
C61×#19902
#2559761