Finite Group Databasebeta

← #29730142

Subgroups — #29730142

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×#16839827C211yesmaximal, characteristic
2#21450993C211yesmaximal, characteristic
3#21450737C211yesmaximal, characteristic
4#21447248C212yesmaximal, complemented
5#21447021C212yesmaximal, complemented
6#21446781C211yesmaximal, characteristic, complemented
7#21446578C211yesmaximal, characteristic, complemented
8#21446239C212yesmaximal, complemented
9#21445845C212yesmaximal, complemented
10C4×#14519701C211yesmaximal, characteristic, complemented
11C2×C61×#477C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13C2×#14519753C2212yes—
14C2×C4×D244C2211yescharacteristic
15#16839827—24no—
16C2×#14519711C2212yes—
17C2×#14519701C2211yescharacteristic
18C2×#14519701C2211yescharacteristic
19C2×#14519689C2212yes—
20C2×C4×#92418C2211yescharacteristic
21#16839684—22no—
22#16839673—22no—
23#16839664—21no—
24#16839663—21no—
25#16839660—21no—
26#16839632C2211yescharacteristic
27#16839627—21no—
28#16839619C2212yes—
29#16839619C2212yes—
30#16839619C2212yes—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839525C2212yes—
33#16839524C2211yescharacteristic
34#16839524C2211yescharacteristic
35#16839524C2212yes—
36#16839524C2212yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#477—24no—
40C2×#107×C61C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×#103×C61C2212yes—
43C22×D244D411yescharacteristic
44C22×C4×D61C2312yes—
45C22×#92418C2311yescharacteristic
46C22×#92418C2311yescharacteristic
47C2×C8×D61—24no—
48C2×C8×D61—24no—
49C2×C8×D61—42no—
50#14519753—24no—
51#14519753—42no—
52C4×D244—22no—
53C4×D244—22no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D411yescharacteristic
57C2×C4⋊Dic61C2312yes—
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×#11758198—22no—
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×#11758198D412yes—
66C2×C4×Dic61—21no—
67#14519711—42no—
68#14519701—22no—
69#14519701—22no—
70#14519689—24no—
71C4×#92418—22no—
72#14519684—21no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×C8×C61C2312yes—
79C4⋊Q8×C61—22no—
80C4⋊Q8×C61—22no—
81C4×Q8×C61—22no—
82C4×Q8×C61—22no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61C2312yes—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C42×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#107×C61—24no—
90C4⋊C8×C61D414yes—
91#103×C61—24no—
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—42no—
99C2×Q8×C61—81no—
100C2×Q8×C61—81no—
101C2×Q8×C61#4112yes—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—22no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C22×Dic61—21no—
109C22×Dic61—21no—
110C22×Dic61—22no—
111C22×Dic61C2×D412yes—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244Q1614yes—
115C2×C4×D61—22no—
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—22no—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61C2×D414yes—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C42×C61—42no—
131C42×C61C2×D412yes—
132C42×C61#4112yes—
133#11758198—24no—
134#11758198—44no—
135#11758198—82no—
136#11758198—82no—
137#11758198—82no—
138C4⋊Dic61—22no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61—42no—
146C4⋊Dic61—81no—
147#11758196—44no—
148C4×Dic61—42no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418#4112yes—
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—
168Q8×C61—84no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—42no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—41no—
180C2×C4×C61#12112yes—
181C2×C4×C61#12112yes—
182C2×C4×C61C2×Q1612yes—
183C2×C4×C61#13512yes—
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×C4×C61C2×#4111yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C2×C4×C61H(5,𝔽2)11yescharacteristic
189C2×C4×C612−1+411yescharacteristic
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—42no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202D244—28no—
203C4×D61—42no—
204C4×D61—42no—
205C4×D61—42no—
206C4×D61—42no—
207C8×C61—24no—
208#92418—22no—
209#92418—41no—
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—82no—
229C4×C61#46714yes—
230C4×C61#52911yescharacteristic
231C4×C61#54211yescharacteristic
232C4×C61#54611yescharacteristic
233C4×C61#54811yescharacteristic
234Dic61—42no—
235Dic61—81no—
236Dic61—82no—
237Dic61—81no—
238C2×#477D6111yescharacteristic, complemented
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×#103—611no—
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⋊Q8C2×D6111yescharacteristic
259C2×C22⋊Q8—612no—
260C2×C4×Q8C2×D6111yescharacteristic
261C2×C4×D4—611no—
262#477—24no—
263C8×D4—1224no—
264C2×#107—612no—
265C2×#107C2×D6112yes—
266C2×C4⋊C8C2×D6111yescharacteristic
267C2×#103—611no—
268C2×#103—611no—
269C2×#103C2×D6112yes—
270C2×C22⋊C8—612no—
271C2×C4×C8—611no—
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⋊C4—611no—
283#340—612no—
284#340—611no—
285#340—612no—
286#340—611no—
287#340—612no—
288C61#2534011yescharacteristic, complemented, Sylow
289C22×Q8—21no—
290C22×Q8C22×D6111yescharacteristic
291C22×D4—611no—
292C23×C4—612no—
293C22×C8—611no—
294C22×C8—611no—
295C22×C8—1224no—
296C22×C8—1224no—
297C22×C8—2442no—
298C22×C8C22×D6112yes—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
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—21no—
309C2×C4⋊C4—611no—
310C2×C4⋊C4—612no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—1221no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C4⋊C4C22×D6112yes—
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—1221no—
325C2×C42C22×D6111yescharacteristic
326#107—24no—
327#107—2442no—
328C4⋊C8C22⋊D6114yes—
329#103—24no—
330#103—1222no—
331#103—1222no—
332C22⋊C8—1224no—
333C4×C8—1222no—
334#95—1222no—
335#95—1222no—
336#95—1222no—
337#95—1221no—
338C24—612no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—42no—
345C2×Q8—81no—
346C2×Q8—81no—
347C2×Q8#1175822612yes—
348C2×D4—614no—
349C2×D4—1222no—
350C2×D4—1222no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—612no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—612no—
370C2×C8—1221no—
371C2×C8—1221no—
372C2×C8—1221no—
373C2×C8—1221no—
374C2×C8—1221no—
375C2×C8—2448no—
376C2×C8C2×C22⋊D6114yes—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—4881no—
391C4⋊C4—2444no—
392C4⋊C4—2442no—
393C22⋊C4—1224no—
394C22⋊C4—4882no—
395C22⋊C4—4882no—
396C22⋊C4—4882no—
397C22⋊C4—2444no—
398C42—42no—
399C42—2442no—
400C42#1175822412yes—
401C42C2×C22⋊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—84no—
415D4—1228no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4—2442no—
433C2×C4—2441no—
434C2×C4—2442no—
435C2×C4—2441no—
436C2×C4—2442no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4#1451974812yes—
444C2×C4#1451974812yes—
445C2×C4Q16×D6112yes—
446C2×C4#1451980012yes—
447C2×C4C22×C22⋊D6111yescharacteristic
448C2×C4#41×D6111yescharacteristic
449C2×C4C2×#1175822411yescharacteristic
450C2×C4C2×#1175822611yescharacteristic
451C2×C4#1451982311yescharacteristic
452C2×C4#1451982511yescharacteristic
453C8—24no—
454C8—1222no—
455C8—2441no—
456C22—1224no—
457C22—1224no—
458C22—1224no—
459C22—2442no—
460C22—2442no—
461C22—2442no—
462C22—2442no—
463C22C2×#1451974811yescharacteristic
464C22#1684040311yescharacteristic
465C22#1684041411yescharacteristic
466C22C2×Q16×D6111yescharacteristic
467C22C2×#1451980011yescharacteristic
468C22#1684056611yescharacteristic
469C22#1684057211yescharacteristic
470C4—22no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4—4881no—
475C4—4881no—
476C4—4882no—
477C4—2442no—
478C4#1684000014yes—
479C4#1684029811yescharacteristic
480C4#1684038511yescharacteristic
481C4#1684040011yescharacteristic
482C4#1684042311yescharacteristic
483C2—2444no—
484C2#2144939211yescharacteristic
485C2C2×#1684000011yescharacteristic
486C2#2145549311yescharacteristic
487C2#2145769211yescharacteristic
488C2#2145776611yescharacteristic
489C2#2145821011yescharacteristic
490C2#2145823511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#477C2
socleC23×C61C22×D4
prime radicalC22×D244D4
hypercenterC2×#477D61
hypocenterC61#25340
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
4. #21447248C2semidirect
5. #21447021C2semidirect
6. #21446781C2semidirect
7. #21446578C2semidirect
8. #21446239C2semidirect
9. #21445845C2semidirect
10. C4×#14519701C2semidirect
11. C2×C61×#477C2semidirect
238. C2×#477D61semidirect
288. C61#25340semidirect

Maximal subgroups

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

subgroupindex
C2×#168398272
#214509932
#214507372
#214472482
#214470212
#214467812
#214465782
#214462392
#214458452
C4×#145197012
C2×C61×#4772
#2534061