Finite Group Databasebeta

← #29709165

Subgroups — #29709165

Order 15616.

924 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×#16839849C212yesmaximal
2#21448386C211yesmaximal, characteristic, complemented
3#21448277C211yesmaximal, characteristic
4#21447223C211yesmaximal, characteristic
5#21447195C212yesmaximal, complemented
6#21447138C212yesmaximal
7#21447065C211yesmaximal, characteristic, complemented
8#21446904C211yesmaximal, characteristic
9#21445667C211yesmaximal, characteristic, complemented
10#21445638C212yesmaximal, complemented
11C61×#2084C211yesmaximal, characteristic, complemented
12C22×#11758190C2212yes—
13C2×#14519777C2211yescharacteristic
14C2×#14519742C2212yes—
15C2×#14519739C2211yescharacteristic
16C2×#14519735—21no—
17#16839849—28no—
18C2×#14519706C2212yes—
19C2×#14519702C2212yes—
20C2×#14519700C2212yes—
21C2×#14519700C2212yes—
22C2×#14519697C2212yes—
23C2×C8×Dic61C2212yes—
24#16839667C2212yes—
25#16839664C2211yescharacteristic
26#16839663C2211yescharacteristic
27#16839661C2211yescharacteristic
28#16839658—21no—
29#16839658—21no—
30#16839654—21no—
31#16839652—21no—
32C4⋊C4×Dic61—21no—
33C4⋊C4×Dic61—21no—
34C4⋊C4×Dic61C2211yescharacteristic
35#16839609—21no—
36C42×Dic61C2211yescharacteristic
37#16839524C2212yes—
38#16839517C2211yescharacteristic
39C2×#106×C61C2212yes—
40C61×#389C2211yescharacteristic
41C61×#384C2211yescharacteristic
42C61×#340C2212yes—
43C61×#336C2211yescharacteristic
44C22×Dic122C2312yes—
45C22×D244C2311yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—24no—
48C2×#11758190—24no—
49C2×#11758190—24no—
50C22×#92418C2312yes—
51#14519777—24no—
52#14519742—24no—
53#14519742—24no—
54#14519739—22no—
55#14519739D414yes—
56#14519735—44no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61—21no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61C2311yescharacteristic
61C2×#11758196—22no—
62C2×#11758198—21no—
63C2×#11758198—21no—
64C2×#11758198—21no—
65C2×#11758198—21no—
66C2×#11758198—21no—
67C2×C4×Dic61—21no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61D411yescharacteristic
70C2×C4×Dic61D411yescharacteristic
71C2×C4×Dic61D411yescharacteristic
72C2×C4×Dic61D411yescharacteristic
73C2×C4×Dic61C2311yescharacteristic
74C2×C4×Dic61C2311yescharacteristic
75#14519706—24no—
76#14519702—24no—
77#14519700—24no—
78#14519700—24no—
79#14519697—24no—
80C8×Dic61—24no—
81#14519684—21no—
82#14519684—22no—
83#14519684—22no—
84#14519684—21no—
85C22×C8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2312yes—
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C42×C61C2311yescharacteristic
93#106×C61—24no—
94#95×C61—21no—
95C23×D61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2411yescharacteristic
102C22×Dic61—22no—
103C22×Dic61—21no—
104C22×Dic61C2×D411yescharacteristic
105C22×Dic61C2×D411yescharacteristic
106C22×Dic61C2×D411yescharacteristic
107C22×Dic61C2×D411yescharacteristic
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—21no—
111C2×D244C2×D412yes—
112C2×C4×D61—24no—
113C2×C4×D61—42no—
114C2×C4×D61—42no—
115C2×Dic122—22no—
116C2×Dic122—22no—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122#4114yes—
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61#4114yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—42no—
129C4⋊C4×C61—42no—
130C4⋊C4×C61—42no—
131C4⋊C4×C61C2×D412yes—
132C42×C61—22no—
133#11758198—24no—
134#11758198—42no—
135#11758198—42no—
136#11758198—42no—
137#11758198—42no—
138C4⋊Dic61—24no—
139C4⋊Dic61—24no—
140C4⋊Dic61—22no—
141C4⋊Dic61—42no—
142#11758196—44no—
143C4×Dic61—22no—
144C4×Dic61—22no—
145C4×Dic61—22no—
146C4×Dic61—41no—
147C4×Dic61—44no—
148C4×Dic61—42no—
149C4×Dic61SD1614yescomplemented
150C4×Dic61C2×D412yes—
151C4×Dic61#4112yes—
152C2×#92418—22no—
153C2×#92418—22no—
154C2×#92418—22no—
155#11758190—216no—
156C23×C61C22×D411yescharacteristic
157C22×D61—22no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—41no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61C2×SD1612yes—
178C2×C4×C61#13512yes—
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4112yes—
181C2×C4×C61C2×#4112yes—
182C2×C4×C61C2×#4111yescharacteristic
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—24no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61#12112yes—
197C2×Dic61#12112yes—
198C2×Dic61C2×SD1612yes—
199C2×Dic61C2×SD1612yes—
200C2×Dic61#13712yes—
201D244—24no—
202D244—42no—
203C4×D61—48no—
204Dic122—24no—
205Dic122—24no—
206Dic122—42no—
207Dic122—42no—
208C8×C61—24no—
209#92418—42no—
210#23677—611nomaximal, Sylow
211C22×C61C2×#12111yescharacteristic
212C22×C61#52511yescharacteristic
213C22×C61#53211yescharacteristic
214C22×C61C22×SD1611yescharacteristic
215C22×C61C2×#13511yescharacteristic
216C22×C61C2×#13711yescharacteristic
217C22×C61#57511yescharacteristic
218C2×D61—42no—
219C2×D61—42no—
220C2×D61—42no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C4×C61—22no—
226C4×C61—22no—
227C4×C61—42no—
228C4×C61—81no—
229C4×C61—82no—
230C4×C61#52512yes—
231C4×C61#52912yes—
232Dic61—22no—
233Dic61—22no—
234Dic61—44no—
235Dic61—41no—
236Dic61—81no—
237Dic61#46114yes—
238C2×C4×SD16—612no—
239C4×C4⋊Q8—611no—
240#2293—611no—
241#2084D6111yescharacteristic, complemented
242#2071—611no—
243#2066—612no—
244#2055—612no—
245#2043—611no—
246#2013—611no—
247C4×C4⋊C8—611no—
248C4×#103—612no—
249C2×C61#239211yescharacteristic
250C2×C61C2×#46111yescharacteristic
251C2×C61#303511yescharacteristic
252C2×C61#310311yescharacteristic
253C2×C61#310811yescharacteristic
254C2×C61#316711yescharacteristic
255C2×C61#317211yescharacteristic
256D61—82no—
257C22×SD16—612no—
258C2×C4⋊Q8—611no—
259C2×C4×Q8—612no—
260C2×C4×D4—611no—
261C2×#117—1221no—
262C4×SD16—1228no—
263C2×#106C2×D6112yes—
264C2×C4⋊C8—612no—
265C2×#103—612no—
266C2×#103—612no—
267C2×#103—612no—
268C2×#102—612no—
269C2×C4×C8—612no—
270#391—612no—
271#390—611no—
272#390—611no—
273#389—611no—
274#389—1221no—
275#389—1221no—
276#389C2×D6111yescharacteristic
277#388—1221no—
278#388—1221no—
279#384C2×D6111yescharacteristic
280C4×C4⋊C4—611no—
281C4×C4⋊C4—1221no—
282C4×C4⋊C4—1221no—
283C4×C22⋊C4—1221no—
284C43—611no—
285#340—612no—
286#340C2×D6112yes—
287#336—611no—
288#336C2×D6111yescharacteristic
289C61#2367711yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×D4—611no—
292C23×C4—1221no—
293C2×SD16—1224no—
294C2×SD16—1224no—
295C2×SD16—1224no—
296C22×C8—612no—
297C22×C8C22×D6112yes—
298C4⋊Q8—1224no—
299C4×Q8—1224no—
300C4×Q8—1224no—
301C4×D4—614no—
302C4×D4—1222no—
303#117—2444no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4C22×D6112yes—
313C2×C4⋊C4C22×D6111yescharacteristic
314C2×C4⋊C4C22×D6111yescharacteristic
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1221no—
319C2×C22⋊C4—1221no—
320C2×C42—611no—
321C2×C42—611no—
322C2×C42—611no—
323C2×C42—611no—
324C2×C42—611no—
325C2×C42—611no—
326C2×C42—1221no—
327C2×C42—1222no—
328C2×C42C22×D6111yescharacteristic
329#106—24no—
330C4⋊C8—1224no—
331#103—1224no—
332#103—1224no—
333#103—1224no—
334#102—1224no—
335C4×C8—1224no—
336#95—21no—
337#95—1221no—
338#95—1221no—
339#95—1222no—
340#95—1222no—
341C24—1221no—
342C2×Q8—614no—
343C2×Q8—1222no—
344C2×Q8—1222no—
345C2×Q8—1222no—
346C2×Q8—1222no—
347C2×Q8—1222no—
348C2×D4—612no—
349C2×D4—1222no—
350C2×D4—1221no—
351C2×D4—1222no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—1224no—
360C22×C4—1222no—
361C22×C4—1221no—
362C22×C4—2442no—
363C22×C4—2442no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C23×D6111yescharacteristic
367SD16—12216no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—1222no—
371C2×C8—1222no—
372C2×C8—1222no—
373C2×C8#1175822614yes—
374C4⋊C4—21no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C4⋊C4—1222no—
384C4⋊C4—2444no—
385C4⋊C4—2442no—
386C4⋊C4D4×D6112yes—
387C22⋊C4—1224no—
388C22⋊C4—2442no—
389C22⋊C4—2442no—
390C22⋊C4—2442no—
391C22⋊C4—2442no—
392C42—22no—
393C42—612no—
394C42—612no—
395C42—614no—
396C42—1222no—
397C42—1222no—
398C42—1222no—
399C42—2441no—
400C42—2442no—
401C42—2444no—
402C23—1222no—
403C23—2441no—
404C23—2441no—
405C23—2441no—
406C23—2441no—
407C23—2441no—
408C23—2441no—
409C23C2×D4×D6111yescharacteristic
410Q8—1224no—
411Q8—1224no—
412Q8—2442no—
413Q8—2442no—
414D4—1224no—
415D4—2442no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—612no—
430C2×C4—612no—
431C2×C4—612no—
432C2×C4—612no—
433C2×C4—612no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1221no—
437C2×C4—1221no—
438C2×C4—1224no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—2448no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4SD16×D6112yes—
449C2×C4#1451979812yes—
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4#41×D6112yes—
452C2×C4#41×D6111yescharacteristic
453C2×C4C2×#1175822612yes—
454C8—24no—
455C8—2442no—
456C22—4881no—
457C22—4881no—
458C22—4881no—
459C22—4881no—
460C22—2442no—
461C22—2442no—
462C22—2442no—
463C22#121×D6111yescharacteristic
464C22#1684027011yescharacteristic
465C22#1684031311yescharacteristic
466C22C2×SD16×D6111yescharacteristic
467C22C2×#1451979811yescharacteristic
468C22#137×D6111yescharacteristic
469C22#1684055811yescharacteristic
470C4—22no—
471C4—22no—
472C4—42no—
473C4—82no—
474C4—81no—
475C4—614no—
476C4—1222no—
477C4—1222no—
478C4—4881no—
479C4—2441no—
480C4—2444no—
481C4#1684027512yes—
482C4#1684029712yes—
483C2—4882no—
484C2#2144906011yescharacteristic
485C2#461×D6111yescharacteristic
486C2#2145526911yescharacteristic
487C2#2145584711yescharacteristic
488C2#2145590211yescharacteristic
489C2#2145638011yescharacteristic
490C2#2145642711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2084C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2084D61
hypocenterC61#23677
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
2. #21448386C2semidirect
5. #21447195C2semidirect
7. #21447065C2semidirect
9. #21445667C2semidirect
10. #21445638C2semidirect
11. C61×#2084C2semidirect
149. C4×Dic61SD16semidirect
241. #2084D61semidirect
289. C61#23677semidirect

Maximal subgroups

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

subgroupindex
C2×#168398492
#214483862
#214482772
#214472232
#214471952
#214471382
#214470652
#214469042
#214456672
#214456382
C61×#20842
#2367761