Finite Group Databasebeta

← #29734183

Subgroups — #29734183

Order 15616.

1028 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×#16840485C211yesmaximal, characteristic
2C2×#16840124C211yesmaximal, characteristic
3C2×#16840044C211yesmaximal, characteristic, complemented
4#21447065C212yesmaximal, complemented
5#21447060C212yesmaximal
6#21446760C211yesmaximal, characteristic, complemented
7#21446751C211yesmaximal, characteristic
8#21446086C212yesmaximal, complemented
9#21446079C212yesmaximal
10#21445652C211yesmaximal, characteristic, complemented
11C61×#2567C211yesmaximal, characteristic, complemented
12#16840485—24no—
13#16840485—24no—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—21no—
16C2×#14519772—21no—
17C2×#14519765C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×#14519753—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519735—21no—
24#16840124—24no—
25#16840044—24no—
26C2×#14519708C2212yes—
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519700C2212yes—
29C2×#14519700C2212yes—
30C2×#14519700C2212yes—
31C2×#14519700C2212yes—
32C2×#14519697C2212yes—
33C2×#14519697C2212yes—
34C2×#14519688C2211yescharacteristic
35#16839524C2212yes—
36#16839517C2212yes—
37#16839517C2212yes—
38C2×C4⋊Q8×C61C2211yescharacteristic
39C2×C4×Q8×C61C2211yescharacteristic
40C61×#395—21no—
41C61×#395—21no—
42C61×#391—21no—
43C61×#389C2212yes—
44C61×#389C2212yes—
45C61×#384—21no—
46C4×C4⋊C4×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C22×Dic122C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—21no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519777—22no—
54#14519777—22no—
55#14519772—28no—
56#14519772—28no—
57#14519765—22no—
58#14519765—22no—
59#14519765—22no—
60#14519765D414yes—
61#14519753—28no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64C4×Dic122—22no—
65C4×Dic122D414yes—
66C4×D244—22no—
67C4×D244—22no—
68#14519735—28no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519708—24no—
82C4⋊#92418—22no—
83#14519700—24no—
84#14519700—24no—
85#14519700—24no—
86#14519700—24no—
87#14519697—24no—
88#14519697—24no—
89#14519688—22no—
90C22×Q8×C61—21no—
91C22×Q8×C61D411yescharacteristic
92C4⋊Q8×C61—22no—
93C4⋊Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—22no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2312yes—
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C2×C42×C61C2311yescharacteristic
107C2×C42×C61C2311yescharacteristic
108#95×C61—21no—
109#95×C61—22no—
110#95×C61—22no—
111C23×D61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—41no—
115C2×Q8×C61—42no—
116C2×Q8×C61—82no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2411yescharacteristic
126C22×Dic61—21no—
127C22×Dic61—21no—
128C22×Dic61—21no—
129C2×D244—22no—
130C2×D244—22no—
131C2×D244—21no—
132C2×D244C2×D412yes—
133C2×C4×D61—24no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—22no—
140C2×Dic122—21no—
141C2×Dic122—21no—
142C2×Dic122C2×D412yes—
143C2×Dic122C2×D412yes—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—42no—
151C4⋊C4×C61—82no—
152C4⋊C4×C61—82no—
153C4⋊C4×C61#4114yes—
154C42×C61—21no—
155C42×C61—21no—
156C42×C61—21no—
157C42×C61—21no—
158C42×C61—42no—
159C42×C61C2×D412yes—
160C42×C61C2×D412yes—
161#11758198—24no—
162#11758198—24no—
163#11758198—24no—
164#11758198—24no—
165#11758198—24no—
166C4⋊Dic61—24no—
167C4⋊Dic61—24no—
168C4⋊Dic61—21no—
169C4⋊Dic61C2×D412yes—
170#11758196—24no—
171#11758196—24no—
172C4×Dic61—24no—
173C4×Dic61—24no—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C2×#92418—22no—
178C2×#92418—22no—
179C2×#92418—22no—
180C23×C61C22×D411yescharacteristic
181C22×D61—22no—
182C22×D61—42no—
183C22×D61—41no—
184C22×D61—41no—
185C22×D61—42no—
186Q8×C61—44no—
187Q8×C61—84no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—24no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
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—42no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61#13712yes—
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C22×D411yescharacteristic
211C2×C4×C61C2×#4112yes—
212C2×C4×C612−1+412yes—
213C2×Dic61—22no—
214C2×Dic61—22no—
215C2×Dic61—22no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—41no—
219C2×Dic61—41no—
220D244—24no—
221D244—41no—
222D244—41no—
223C4×D61—48no—
224Dic122—22no—
225Dic122—22no—
226Dic122—24no—
227Dic122—41no—
228Dic122—41no—
229Dic122—42no—
230#92418—42no—
231#92418—42no—
232#25662—611nomaximal, Sylow
233C22×C61C2×#12011yescharacteristic
234C22×C61#54812yes—
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57712yes—
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—82no—
242C2×D61—82no—
243C4×C61—22no—
244C4×C61—22no—
245C4×C61—22no—
246C4×C61—44no—
247C4×C61—41no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—82no—
251C4×C61#53211yescharacteristic
252C4×C61#53611yescharacteristic
253C4×C61#54712yes—
254Dic61—42no—
255Dic61—42no—
256Dic61—42no—
257C2×#556—611no—
258C2×#489—611no—
259C2×#475—611no—
260#2567D6111yescharacteristic, complemented
261#2043—612no—
262#2042—612no—
263#1985—611no—
264#1983—611no—
265#1865—612no—
266#1864—612no—
267#1763—611no—
268C2×C61#243211yescharacteristic
269C2×C61#300711yescharacteristic
270C2×C61#301011yescharacteristic
271C2×C61#336912yes—
272C2×C61#337612yes—
273D61—82no—
274#556—1224no—
275#556—1224no—
276C2×C4⋊Q8—611no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C2×#126—1221no—
279C2×#126—1221no—
280C2×#124—611no—
281C2×#124—611no—
282C2×C22⋊Q8—1221no—
283C2×C4×Q8—611no—
284C2×C4×Q8—611no—
285C2×C4×Q8C2×D6111yescharacteristic
286C2×C4×D4—611no—
287C2×#117—1221no—
288#489—1224no—
289#475—1224no—
290C2×#106—612no—
291C2×C4⋊C8—611no—
292C2×#103—612no—
293C2×#103—612no—
294C2×#103—612no—
295C2×#103—612no—
296C2×#102—612no—
297C2×#102—612no—
298C2×#97—611no—
299#395—21no—
300#395—21no—
301#391—21no—
302#389C2×D6112yes—
303#389C2×D6112yes—
304#384—21no—
305C4×C4⋊C4C2×D6111yescharacteristic
306#340—612no—
307#336—612no—
308#336—612no—
309C61#2566211yescharacteristic, complemented, Sylow
310C22×Q8—21no—
311C22×Q8—611no—
312C22×Q8—611no—
313C22×Q8C22⋊D6111yescharacteristic
314C22×D4—611no—
315C23×C4—1221no—
316C22×C8—612no—
317C22×C8—612no—
318C4⋊Q8—22no—
319C4⋊Q8—22no—
320C4⋊Q8—1222no—
321C4⋊Q8—1222no—
322#126—1228no—
323#126—1228no—
324#124—614no—
325#124—1222no—
326#124—1222no—
327#124—1222no—
328C22⋊Q8—1228no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—614no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×Q8—1222no—
335C4×D4—1222no—
336C4×D4—1222no—
337#117—1228no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—611no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1221no—
346C2×C4⋊C4C22⋊D6111yescharacteristic
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C4⋊C4C22×D6112yes—
350C2×C4⋊C4C22×D6112yes—
351C2×C22⋊C4—1221no—
352C2×C22⋊C4—1221no—
353C2×C22⋊C4—1221no—
354C2×C22⋊C4—1221no—
355C2×C22⋊C4—1221no—
356C2×C42—21no—
357C2×C42—1221no—
358C2×C42—1221no—
359C2×C42C22×D6111yescharacteristic
360C2×C42C22×D6111yescharacteristic
361C2×C42C22×D6111yescharacteristic
362#106—1224no—
363C4⋊C8—1222no—
364#103—1224no—
365#103—1224no—
366#103—1224no—
367#103—1224no—
368#102—1224no—
369#102—1224no—
370#97—1222no—
371#95—21no—
372#95—22no—
373#95—22no—
374C24—1221no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—41no—
378C2×Q8—42no—
379C2×Q8—82no—
380C2×Q8—612no—
381C2×Q8—612no—
382C2×Q8—1221no—
383C2×Q8—1221no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×D4—612no—
389C2×D4—1221no—
390C2×D4—1222no—
391C2×D4—1222no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—1224no—
395C22×C4—1221no—
396C22×C4—1221no—
397C22×C4—1221no—
398C22×C4—2442no—
399C22×C4—2442no—
400C22×C4C2×C22⋊D6111yescharacteristic
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C23×D6111yescharacteristic
407C2×C8—1222no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1222no—
411C2×C8—1222no—
412C2×C8—1222no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4—612no—
423C4⋊C4—1224no—
424C4⋊C4—1224no—
425C4⋊C4—1224no—
426C4⋊C4—1221no—
427C4⋊C4—1224no—
428C4⋊C4#1175822614yes—
429C22⋊C4—1224no—
430C22⋊C4—1224no—
431C22⋊C4—1224no—
432C22⋊C4—1224no—
433C22⋊C4—1224no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—42no—
439C42—1224no—
440C42—1224no—
441C42D4×D6112yes—
442C42D4×D6112yes—
443C23—1222no—
444C23—2441no—
445C23—2441no—
446C23—2442no—
447C23—2442no—
448C23C22×C22⋊D6111yescharacteristic
449Q8—44no—
450Q8—84no—
451Q8—1222no—
452Q8—1222no—
453Q8—1224no—
454Q8—2441no—
455Q8—2441no—
456Q8—2442no—
457D4—1224no—
458D4—2441no—
459D4—2441no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—1222no—
479C2×C4—1222no—
480C2×C4—1222no—
481C2×C4—2448no—
482C2×C4—2441no—
483C2×C4—2441no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4#1451974512yes—
487C2×C4#1451974512yes—
488C2×C4#1451980812yes—
489C2×C4C2×D4×D6111yescharacteristic
490C2×C4C2×D4×D6111yescharacteristic
491C2×C4C2×#1175822612yes—
492C2×C4#1451982512yes—
493C8—2442no—
494C8—2442no—
495C22—4882no—
496C22—4882no—
497C22—2442no—
498C22—2442no—
499C22—2442no—
500C22C2×#1451974511yescharacteristic
501C22#1684042312yes—
502C22C2×#1451980811yescharacteristic
503C22#1684056011yescharacteristic
504C22#1684057312yes—
505C4—22no—
506C4—22no—
507C4—22no—
508C4—44no—
509C4—41no—
510C4—41no—
511C4—41no—
512C4—82no—
513C4—2442no—
514C4—2442no—
515C4—2442no—
516C4#1684031311yescharacteristic
517C4#1684033711yescharacteristic
518C4#1684041512yes—
519C2—4882no—
520C2#2144944411yescharacteristic
521C2#2145507711yescharacteristic
522C2#2145510311yescharacteristic
523C2#2145855912yes—
524C2#2145864612yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2567C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2567D61
hypocenterC61#25662
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
3. C2×#16840044C2semidirect
4. #21447065C2semidirect
6. #21446760C2semidirect
8. #21446086C2semidirect
10. #21445652C2semidirect
11. C61×#2567C2semidirect
260. #2567D61semidirect
309. C61#25662semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
C2×#168401242
C2×#168400442
#214470652
#214470602
#214467602
#214467512
#214460862
#214460792
#214456522
C61×#25672
#2566261