Finite Group Databasebeta

← #29730146

Subgroups — #29730146

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840057C211yesmaximal, characteristic, complemented
2C2×C8×D244C211yesmaximal, characteristic
3#21450988C211yesmaximal, characteristic
4#21450737C211yesmaximal, characteristic
5#21447251C212yesmaximal, complemented
6#21447020C212yesmaximal, complemented
7#21446780C211yesmaximal, characteristic, complemented
8#21446577C211yesmaximal, characteristic, complemented
9#21446239C212yesmaximal, complemented
10#21445847C212yesmaximal, complemented
11C4×#103×C61C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519753C2212yes—
15C2×C4×D244C2211yescharacteristic
16#16840057—24no—
17C8×D244—24no—
18C2×#14519710C2212yes—
19C2×#14519710C2212yes—
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2212yes—
22C2×#14519689C2212yes—
23#16839684—22no—
24#16839675—22no—
25#16839664—21no—
26#16839664—21no—
27#16839660—21no—
28#16839632C2211yescharacteristic
29#16839628—21no—
30#16839619C2212yes—
31#16839619C2212yes—
32#16839619C2212yes—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2212yes—
36C2×C4×Q8×C61C2211yescharacteristic
37C2×#103×C61C2211yescharacteristic
38C2×#103×C61C2211yescharacteristic
39C2×C4×C8×C61C2211yescharacteristic
40C4×C4⋊C4×C61C2211yescharacteristic
41C61×#340C2211yescharacteristic
42C61×#340C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244D411yescharacteristic
45C22×C4×D61C2312yes—
46C22×#92418C2312yes—
47C2×C8×D61—24no—
48C2×C8×D61—24no—
49C2×C8×D61—42no—
50#14519777—22no—
51#14519777—22no—
52#14519753—24no—
53#14519753—42no—
54C4×D244—22no—
55C4×D244—22no—
56C2×C4⋊Dic61—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61D411yescharacteristic
60C2×C4⋊Dic61C2312yes—
61C2×C4⋊Dic61C2312yes—
62C2×#11758196—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×#11758198—22no—
67C2×#11758198D412yes—
68#14519710—24no—
69#14519710—42no—
70C4⋊#92418D414yes—
71#14519701—24no—
72#14519689—24no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76C22×Q8×C61C2311yescharacteristic
77C22×C8×C61C2311yescharacteristic
78C22×C8×C61C2311yescharacteristic
79C4×Q8×C61—22no—
80C4×Q8×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C4⋊C4×C61C2311yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C42×C61—21no—
86C2×C42×C61—21no—
87C2×C42×C61C2311yescharacteristic
88#103×C61—22no—
89#103×C61—22no—
90C4×C8×C61—22no—
91#95×C61—21no—
92C23×D61C2×D412yes—
93C2×Q8×C61—21no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×Q8×C61—21no—
97C2×Q8×C61—21no—
98C2×Q8×C61#4112yes—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2411yescharacteristic
106C22×Dic61—22no—
107C22×Dic61—22no—
108C22×Dic61C2×D412yes—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244Q1614yes—
112C2×C4×D61—22no—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×Dic122—42no—
119C2×Dic122—81no—
120C2×Dic122—81no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—21no—
126C2×C8×C61#4112yes—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—42no—
134C42×C61—42no—
135C42×C61—42no—
136C42×C61C2×D412yes—
137C42×C61#4112yes—
138#11758198—24no—
139#11758198—44no—
140#11758198—82no—
141#11758198—82no—
142#11758198—82no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61—42no—
149C4⋊Dic61—42no—
150C4⋊Dic61—81no—
151#11758196—44no—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418C2×D414yes—
155C8×D61—48no—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—22no—
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—22no—
163C22×D61C2×Q1614yes—
164Q8×C61—22no—
165Q8×C61—22no—
166Q8×C61—41no—
167Q8×C61—41no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61#12112yes—
183C2×C4×C61#12112yes—
184C2×C4×C61C2×Q1612yes—
185C2×C4×C61#13512yes—
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61C2×#4111yescharacteristic
189C2×C4×C61C2×#4111yescharacteristic
190C2×C4×C61H(5,𝔽2)11yescharacteristic
191C2×C4×C612−1+411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201D244—28no—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—42no—
205C4×D61—42no—
206Dic122—84no—
207C8×C61—22no—
208C8×C61—41no—
209#92418—24no—
210#25340—611nomaximal, Sylow
211C22×C61C2×#12111yescharacteristic
212C22×C61#54611yescharacteristic
213C22×C61#54711yescharacteristic
214C22×C61C22×Q1611yescharacteristic
215C22×C61C2×#13511yescharacteristic
216C22×C61#57611yescharacteristic
217C22×C61#57711yescharacteristic
218C2×D61—24no—
219C2×D61—24no—
220C2×D61—24no—
221C2×D61—42no—
222C2×D61—42no—
223C2×D61—42no—
224C2×D61—42no—
225C4×C61—22no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61—81no—
230C4×C61#46714yes—
231C4×C61#52911yescharacteristic
232C4×C61#54211yescharacteristic
233C4×C61#54611yescharacteristic
234C4×C61#54811yescharacteristic
235Dic61—42no—
236Dic61—82no—
237Dic61—82no—
238C2×#477—611no—
239C2×C8×D4—611no—
240#2563—611no—
241#2538—611no—
242#2076—612no—
243#2036—612no—
244#1990—611no—
245#1947—611no—
246#1892—612no—
247#1808—612no—
248C4×#103D6111yescharacteristic, complemented
249C2×C61#242711yescharacteristic
250C2×C61C2×#46711yescharacteristic
251C2×C61#306211yescharacteristic
252C2×C61#329911yescharacteristic
253C2×C61#330511yescharacteristic
254C2×C61#333811yescharacteristic
255C2×C61#334011yescharacteristic
256D61—44no—
257C23×C8—612no—
258C2×C4⋊Q8—611no—
259C2×C22⋊Q8—612no—
260C2×C4×Q8C2×D6111yescharacteristic
261C2×C4×D4—611no—
262#477—1224no—
263C8×D4—1224no—
264C2×#107—612no—
265C2×#107—612no—
266C2×C4⋊C8—611no—
267C2×#103—612no—
268C2×#103C2×D6111yescharacteristic
269C2×#103C2×D6111yescharacteristic
270C2×C22⋊C8—612no—
271C2×C4×C8C2×D6111yescharacteristic
272#396—1222no—
273#393—1222no—
274#390—1221no—
275#390—1221no—
276#389—1221no—
277#384—611no—
278#382—1221no—
279#380—612no—
280#380—612no—
281#380—612no—
282C4×C4⋊C4C2×D6111yescharacteristic
283#340—612no—
284#340—612no—
285#340—612no—
286#340C2×D6111yescharacteristic
287#340C2×D6111yescharacteristic
288C61#2534011yescharacteristic, complemented, Sylow
289C22×Q8—1221no—
290C22×Q8C22×D6111yescharacteristic
291C22×D4—611no—
292C23×C4—612no—
293C22×C8—612no—
294C22×C8—1224no—
295C22×C8—1224no—
296C22×C8—2442no—
297C22×C8C22×D6111yescharacteristic
298C22×C8C22×D6111yescharacteristic
299C4⋊Q8—1222no—
300C4⋊Q8—1222no—
301C22⋊Q8—1224no—
302C22⋊Q8—2442no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×D4—1222no—
306C4×D4—1222no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—611no—
309C2×C4⋊C4—612no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1222no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C22⋊C4—612no—
319C2×C22⋊C4—1222no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42C22×D6111yescharacteristic
326#107—1224no—
327#107—2442no—
328C4⋊C8—614no—
329#103—22no—
330#103—22no—
331#103—1224no—
332C22⋊C8—1224no—
333C4×C8—22no—
334#95—21no—
335#95—1222no—
336#95—1222no—
337#95—1222no—
338C24—612no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—4881no—
345C2×Q8—4881no—
346C2×Q8—2442no—
347C2×Q8#1175822212yes—
348C2×D4—614no—
349C2×D4—1222no—
350C2×D4—1222no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—612no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—614no—
373C2×C8—1222no—
374C2×C8—1222no—
375C2×C8—2448no—
376C2×C8#1175822212yes—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—42no—
384C4⋊C4—1222no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—4881no—
390C4⋊C4—2444no—
391C4⋊C4—2442no—
392C4⋊C4—2442no—
393C22⋊C4—1224no—
394C22⋊C4—4882no—
395C22⋊C4—4882no—
396C22⋊C4—4882no—
397C22⋊C4—2444no—
398C42—42no—
399C42—42no—
400C42#1175822212yes—
401C42D4×D6112yes—
402C23—614no—
403C23—1222no—
404C23—1222no—
405C23—1222no—
406C23—1222no—
407C23—1222no—
408C23—1222no—
409C23C2×D4×D6111yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414Q8—4884no—
415D4—1228no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—1222no—
431C2×C4—1222no—
432C2×C4—1222no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4#1451975012yes—
444C2×C4#1451975012yes—
445C2×C4Q16×D6112yes—
446C2×C4#1451979812yes—
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C2×#1175822211yescharacteristic
449C2×C4C2×#1175822211yescharacteristic
450C2×C4C2×#1175822211yescharacteristic
451C2×C4#1451982311yescharacteristic
452C2×C4#1451982411yescharacteristic
453C8—22no—
454C8—41no—
455C8—1224no—
456C22—1224no—
457C22—1224no—
458C22—1224no—
459C22—2442no—
460C22—2442no—
461C22—2442no—
462C22—2442no—
463C22C2×#1451975011yescharacteristic
464C22#1684040811yescharacteristic
465C22#1684041711yescharacteristic
466C22C2×Q16×D6111yescharacteristic
467C22C2×#1451979811yescharacteristic
468C22#1684056411yescharacteristic
469C22#1684056911yescharacteristic
470C4—22no—
471C4—42no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—4882no—
476C4—4882no—
477C4—2442no—
478C4#1684000214yes—
479C4#1684029611yescharacteristic
480C4#1684038311yescharacteristic
481C4#1684040811yescharacteristic
482C4#1684042111yescharacteristic
483C2—2444no—
484C2#2144939611yescharacteristic
485C2C2×#1684000211yescharacteristic
486C2#2145550211yescharacteristic
487C2#2145769711yescharacteristic
488C2#2145777111yescharacteristic
489C2#2145821311yescharacteristic
490C2#2145823411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC4×#103D61
hypocenterC61#25340
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16840057C2semidirect
5. #21447251C2semidirect
6. #21447020C2semidirect
7. #21446780C2semidirect
8. #21446577C2semidirect
9. #21446239C2semidirect
10. #21445847C2semidirect
11. C4×#103×C61C2semidirect
248. C4×#103D61semidirect
288. C61#25340semidirect

Maximal subgroups

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

subgroupindex
C2×#168400572
C2×C8×D2442
#214509882
#214507372
#214472512
#214470202
#214467802
#214465772
#214462392
#214458472
C4×#103×C612
#2534061