Finite Group Databasebeta

← #29689705

Subgroups — #29689705

Order 15616.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840297C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic, complemented
3#382×D61C211yesmaximal, characteristic, complemented
4#21446785C211yesmaximal, characteristic, complemented
5#21446630C211yesmaximal, characteristic, complemented
6#21446628C211yesmaximal, characteristic, complemented
7#21446135C212yesmaximal
8#21446125C212yesmaximal, complemented
9#21445833C212yesmaximal, complemented
10#21445828C212yesmaximal, complemented
11C61×#1991C211yesmaximal, characteristic, complemented
12#16840297—24no—
13#16840297C418yescomplemented
14C2×#14519772—21no—
15C2×#14519765C2211yescharacteristic
16C2×#14519762—21no—
17C2×#14519757C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×#14519735—21no—
23C2×C4⋊C4×D61C2212yes—
24C2×C42×D61C2211yescharacteristic
25#16839813—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519700C2212yes—
29C2×#14519699C2212yes—
30C2×#14519697C2212yes—
31C2×#14519689C2212yes—
32C2×C4×#92418C2211yescharacteristic
33#16839628C2211yescharacteristic
34#16839627C2212yes—
35#16839626—21no—
36#16839626—21no—
37#95×D61—21no—
38#95×D61—21no—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4×C8×C61C2211yescharacteristic
43C61×#382C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122C2311yescharacteristic
47C22×D244—21no—
48C22×D244C2311yescharacteristic
49C22×C4×D61—21no—
50C22×C4×D61—22no—
51C22×C4×D61D411yescharacteristic
52C22×C4×D61D411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×#92418C2312yes—
55#14519772—28no—
56#14519765—22no—
57#14519765C2×C414yescomplemented
58#14519762—28no—
59#14519757—22no—
60#14519757C2×C414yes—
61#14519752—22no—
62#14519752C2×C414yes—
63#14519742—22no—
64#14519742C2×C414yescomplemented
65#14519739—28no—
66#14519739—22no—
67#14519739C2×C414yescomplemented
68#14519735—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61C2312yes—
71C2×#11758196—21no—
72C2×#11758196—22no—
73C4⋊C4×D61—42no—
74C4⋊C4×D61—42no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×#11758198—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61D411yescharacteristic
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61C2311yescharacteristic
84C42×D61C2×C414yes—
85C42×D61D414yes—
86#14519706—24no—
87#14519702—42no—
88#14519700—42no—
89#14519699—42no—
90#14519697—42no—
91#14519689—24no—
92C4×#92418—22no—
93#14519684—21no—
94#14519684—21no—
95#14519684—22no—
96#14519684—22no—
97C22×C8×C61C2312yes—
98#125×C61—22no—
99#125×C61C2×C414yescomplemented
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C4⋊C4×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C4×C8×C61—22no—
107#95×C61—21no—
108#95×C61—21no—
109C23×D61—21no—
110C23×D61C2×D411yescharacteristic
111C22×C4×C61—21no—
112C22×C4×C61—21no—
113C22×C4×C61—22no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2411yescharacteristic
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61C2×D411yescharacteristic
120C22×Dic61C2×D411yescharacteristic
121C22×Dic61C2×D411yescharacteristic
122C2×D244—22no—
123C2×D244—22no—
124C2×D244—24no—
125C2×D244—21no—
126C2×D244—44no—
127C2×D244C22×C412yes—
128C2×C4×D61—24no—
129C2×C4×D61—22no—
130C2×C4×D61—22no—
131C2×C4×D61—24no—
132C2×C4×D61—42no—
133C2×C4×D61—44no—
134C2×C4×D61—44no—
135C2×C4×D61—44no—
136C2×C4×D61—44no—
137C2×C4×D61C22⋊C414yes—
138C2×C4×D61C22⋊C414yes—
139C2×C4×D61C22×C412yes—
140C2×C4×D61C2×D412yes—
141C2×Dic122—22no—
142C2×Dic122—22no—
143C2×Dic122—21no—
144C2×Dic122C22×C412yes—
145C2×C8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61—22no—
148C4⋊C4×C61—24no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61C22×C412yes—
155C4⋊C4×C61C22×C412yes—
156C42×C61C22×C412yes—
157C42×C61C2×D412yes—
158#11758198—24no—
159#11758198—24no—
160#11758198—24no—
161#11758198—24no—
162#11758198—24no—
163C4⋊Dic61—24no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166#11758196—24no—
167#11758196—44no—
168C4×Dic61—24no—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22⋊C414yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61C22×D411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—22no—
180C22×D61—22no—
181C22×D61—41no—
182C22×D61—41no—
183C22×D61—42no—
184C22×D61—42no—
185C22×D61C2×C22⋊C412yes—
186C22×D61#13514yes—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61C2×C22⋊C412yes—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61H(5,𝔽2)12yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—22no—
205C2×Dic61—22no—
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61C2×C22⋊C412yes—
213C2×Dic61C2×C22⋊C412yes—
214C2×Dic61C2×C22⋊C412yes—
215D244—24no—
216D244—48no—
217D244—42no—
218C4×D61—28no—
219C4×D61—24no—
220C4×D61—24no—
221C4×D61—44no—
222C4×D61—48no—
223C4×D61—88no—
224Dic122—24no—
225Dic122—42no—
226C8×C61—42no—
227#92418—42no—
228#21861—611nomaximal, Sylow
229C22×C61C22×C22⋊C411yescharacteristic
230C22×C61#53511yescharacteristic
231C22×C61#53611yescharacteristic
232C22×C61C2×#13512yes—
233C22×C61#57512yes—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—24no—
237C2×D61—24no—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—82no—
242C2×D61—82no—
243C2×D61#41614yes—
244C4×C61—22no—
245C4×C61—22no—
246C4×C61—42no—
247C4×C61—42no—
248C4×C61—82no—
249C4×C61#52012yes—
250C4×C61#53712yes—
251Dic61—22no—
252Dic61—22no—
253Dic61—22no—
254Dic61—42no—
255Dic61—82no—
256C2×#529—611no—
257C2×#431—611no—
258C2×#382—611no—
259#1991—611no—
260#1991D6111yescharacteristic, complemented
261#1958—611no—
262#1958—611no—
263#1874—612no—
264#1872—612no—
265#1805—612no—
266#1804—612no—
267C2×C61#231111yescharacteristic
268C2×C61C2×#41611yescharacteristic
269C2×C61#289611yescharacteristic
270C2×C61#321112yes—
271C2×C61#323212yes—
272D61—24no—
273D61—82no—
274#529—618no—
275#529—1224no—
276C2×#126—1221no—
277C2×#125C2×D6111yescharacteristic
278C2×#124—611no—
279C2×#123—1221no—
280C2×C22⋊Q8—611no—
281C2×#121—611no—
282C2×C4×Q8—611no—
283C2×C4×D4—611no—
284C2×C4×D4—1221no—
285C2×#117—1221no—
286C22×C4⋊C4—612no—
287C22×C42—611no—
288#431—1224no—
289C2×C4⋊C8—612no—
290C2×#103—612no—
291C2×#103—612no—
292C2×#102—612no—
293C2×#102—612no—
294C2×C22⋊C8—612no—
295C2×C4×C8—611no—
296C2×C4×C8C2×D6111yescharacteristic
297#382—612no—
298#382—611no—
299#382—1221no—
300#382—1221no—
301#382C2×D6111yescharacteristic
302C2×#95—1221no—
303C2×#95—1221no—
304#340—612no—
305#340—612no—
306#340C2×D6112yes—
307#340C2×D6112yes—
308C61#2186111yescharacteristic, complemented, Sylow
309C22×Q8—611no—
310C22×D4—611no—
311C22×D4—1221no—
312C23×C4—611no—
313C23×C4—611no—
314C23×C4—611no—
315C23×C4—1221no—
316C23×C4—1222no—
317C22×C8—612no—
318C22×C8C22×D6112yes—
319#126—1228no—
320#125—22no—
321#125C4×D6114yescomplemented
322#124—614no—
323#124—1222no—
324#123—1228no—
325C22⋊Q8—614no—
326C22⋊Q8—1222no—
327#121—614no—
328#121—1222no—
329C4×Q8—614no—
330C4×Q8—1222no—
331C4×D4—614no—
332C4×D4—1222no—
333C4×D4—1228no—
334#117—1228no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1222no—
341C2×C4⋊C4—2442no—
342C2×C4⋊C4—2442no—
343C2×C4⋊C4C22×D6112yes—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—611no—
354C2×C42—614no—
355C2×C42—614no—
356C2×C42—1221no—
357C2×C42C22×D6111yescharacteristic
358C4⋊C8—1224no—
359#103—2442no—
360#103—2442no—
361#102—2442no—
362#102—2442no—
363C22⋊C8—1224no—
364C4×C8—22no—
365C4×C8—1222no—
366#95—21no—
367#95—21no—
368#95—1222no—
369#95—1222no—
370#95—1221no—
371#95—1221no—
372C24—611no—
373C24—1221no—
374C2×Q8—612no—
375C2×Q8—1221no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×D4—612no—
379C2×D4—1224no—
380C2×D4—1221no—
381C2×D4—1222no—
382C2×D4—1222no—
383C2×D4—2444no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—611no—
388C22×C4—611no—
389C22×C4—611no—
390C22×C4—612no—
391C22×C4—612no—
392C22×C4—614no—
393C22×C4—614no—
394C22×C4—1224no—
395C22×C4—1224no—
396C22×C4—1222no—
397C22×C4—1222no—
398C22×C4—1221no—
399C22×C4—1222no—
400C22×C4—2442no—
401C22×C4—2444no—
402C22×C4—2444no—
403C22×C4—2444no—
404C22×C4—2444no—
405C22×C4D4×D6111yescharacteristic
406C22×C4D4×D6111yescharacteristic
407C22×C4C23×D6111yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—1224no—
421C4⋊C4—1224no—
422C4⋊C4—1222no—
423C4⋊C4—1222no—
424C4⋊C4—2444no—
425C4⋊C4C2×C4×D6112yes—
426C4⋊C4C2×C4×D6112yes—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C42—612no—
433C42—612no—
434C42—614no—
435C42—614no—
436C42—1224no—
437C42C2×C4×D6112yes—
438C42D4×D6112yes—
439C23—612no—
440C23—614no—
441C23—1222no—
442C23—1221no—
443C23—1221no—
444C23—1222no—
445C23—2442no—
446C23—2442no—
447C23—2441no—
448C23—2441no—
449C23C2×D4×D6111yescharacteristic
450Q8—1224no—
451Q8—2442no—
452D4—1224no—
453D4—2448no—
454D4—2442no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—612no—
465C2×C4—612no—
466C2×C4—612no—
467C2×C4—1228no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1221no—
471C2×C4—1221no—
472C2×C4—1224no—
473C2×C4—1224no—
474C2×C4—1222no—
475C2×C4—4888no—
476C2×C4—2444no—
477C2×C4—2448no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2442no—
481C2×C4—2441no—
482C2×C4—2441no—
483C2×C4C22⋊C4×D6112yes—
484C2×C4C22⋊C4×D6112yes—
485C2×C4C22×C4×D6111yescharacteristic
486C2×C4C2×D4×D6111yescharacteristic
487C2×C4#1451982312yes—
488C2×C4#1451982512yes—
489C8—42no—
490C8—2442no—
491C22—614no—
492C22—1224no—
493C22—1224no—
494C22—1222no—
495C22—1222no—
496C22—4882no—
497C22—4882no—
498C22—2442no—
499C22—2442no—
500C22—2442no—
501C22C2×C22⋊C4×D6111yescharacteristic
502C22#1684033011yescharacteristic
503C22#1684033711yescharacteristic
504C22#135×D6112yes—
505C22#1684055912yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—42no—
510C4—82no—
511C4—1222no—
512C4—1222no—
513C4—1222no—
514C4—4882no—
515C4—2442no—
516C4#1684024412yes—
517C4#1684034212yes—
518C2—1224no—
519C2—4882no—
520C2#2144841011yescharacteristic
521C2#416×D6111yescharacteristic
522C2#2145433411yescharacteristic
523C2#2145678512yes—
524C2#2145698812yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1991C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenter#1991D61
hypocenterC61#21861
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
2. C2×#16839813C2semidirect
3. #382×D61C2semidirect
4. #21446785C2semidirect
5. #21446630C2semidirect
6. #21446628C2semidirect
8. #21446125C2semidirect
9. #21445833C2semidirect
10. #21445828C2semidirect
11. C61×#1991C2semidirect
13. #16840297C4semidirect
57. #14519765C2×C4semidirect
64. #14519742C2×C4semidirect
67. #14519739C2×C4semidirect
99. #125×C61C2×C4semidirect
260. #1991D61semidirect
308. C61#21861semidirect
321. #125C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402972
C2×#168398132
#382×D612
#214467852
#214466302
#214466282
#214461352
#214461252
#214458332
#214458282
C61×#19912
#2186161