Finite Group Databasebeta

← #29718432

Subgroups — #29718432

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519700C211yesmaximal, characteristic, complemented
2#21449478C211yesmaximal, characteristic
3#21448696C211yesmaximal, characteristic, complemented
4#21447195C212yesmaximal, complemented
5#21447059C212yesmaximal, complemented
6#21446750C212yesmaximal, complemented
7#21446452C212yesmaximal, complemented
8#21446248C211yesmaximal, characteristic
9#21445996C211yesmaximal, characteristic, complemented
10#21445755C211yesmaximal, characteristic, complemented
11C2×C61×#481C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519764C2212yes—
15C2×#14519706C2212yes—
16C2×#14519702C2212yes—
17C2×#14519702C2212yes—
18C2×#14519700—24no—
19C2×#14519700C2212yes—
20C2×#14519700C2212yes—
21C2×#14519698C2212yes—
22#16839674—22no—
23#16839667C2212yes—
24#16839661C2212yes—
25#16839648—22no—
26#16839642—21no—
27#16839642—21no—
28#16839622—21no—
29#16839622C2211yescharacteristic
30C4⋊C4×Dic61C2212yes—
31C4×C22⋊Dic61—21no—
32C4×C22⋊Dic61C2211yescharacteristic
33#16839524C2212yes—
34#16839516C2212yes—
35C2×#121×C61C2211yescharacteristic
36C22×C4⋊C4×C61C2211yescharacteristic
37C61×#481—24no—
38C2×#106×C61C2211yescharacteristic
39C2×#106×C61C2211yescharacteristic
40C2×#102×C61C2211yescharacteristic
41C2×#102×C61C2211yescharacteristic
42C2×C22⋊C8×C61C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122—42no—
48C22×Dic122C2312yes—
49C23×Dic61—21no—
50C22×#92418—22no—
51C22×#92418—42no—
52C22×#92418C2312yes—
53#14519764—24no—
54#14519764—42no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61C2312yes—
57C2×C22⋊Dic61—21no—
58C2×C22⋊Dic61—22no—
59C2×C22⋊Dic61—22no—
60C2×C22⋊Dic61D411yescharacteristic
61C2×C22⋊Dic61D411yescharacteristic
62C2×C4×Dic61—22no—
63C2×C4×Dic61C2312yes—
64#14519706—42no—
65#14519702—24no—
66#14519702—42no—
67#14519700—24no—
68#14519700—24no—
69#14519698—42no—
70#14519684—21no—
71#14519684—22no—
72#14519684—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684D412yes—
76C22×D4×C61—21no—
77C22×D4×C61C2311yescharacteristic
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80C22×C8×C61C2311yescharacteristic
81C22×C8×C61C2311yescharacteristic
82#121×C61—22no—
83#121×C61—22no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—22no—
86C2×C4⋊C4×C61D414yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—21no—
91#106×C61—22no—
92#106×C61—22no—
93#102×C61—22no—
94#102×C61—22no—
95C22⋊C8×C61—22no—
96C24×C61—21no—
97C24×C61C2×D411yescharacteristic
98C2×D4×C61—21no—
99C2×D4×C61—21no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—42no—
103C2×D4×C61—42no—
104C2×D4×C61—42no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—24no—
110C22×C4×C61—21no—
111C22×C4×C61—82no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D412yes—
114C22×C4×C61#4112yes—
115C22×C4×C61C2411yescharacteristic
116C22×Dic61—22no—
117C22×Dic61—22no—
118C22×Dic61—44no—
119C22×Dic61—42no—
120C22×Dic61C2×D412yes—
121C22×Dic61C2×D412yes—
122C2×Dic122—28no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—48no—
129C2×Dic122—44no—
130C2×Dic122—44no—
131C2×Dic122—42no—
132C2×Dic122#4114yes—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—42no—
145C22⋊Dic61—22no—
146C22⋊Dic61—22no—
147C22⋊Dic61—44no—
148C22⋊Dic61—44no—
149C22⋊Dic61—42no—
150C4⋊Dic61—22no—
151C4⋊Dic61—22no—
152C4⋊Dic61—82no—
153C4×Dic61—22no—
154C4×Dic61—22no—
155C4×Dic61—82no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—44no—
160C23×C61—22no—
161C23×C61—22no—
162C23×C61—41no—
163C23×C61—42no—
164C23×C61—41no—
165C23×C61—41no—
166C23×C61—42no—
167C23×C61#12112yes—
168C23×C61#13512yes—
169C23×C61#13712yes—
170C23×C61C22×D411yescharacteristic
171D4×C61—41no—
172D4×C61—41no—
173D4×C61—42no—
174D4×C61—84no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—84no—
187C2×C4×C61#12112yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4112yes—
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)12yes—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—24no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—48no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205Dic122—24no—
206Dic122—24no—
207Dic122—48no—
208Dic122—42no—
209Dic122—42no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—42no—
213#24483—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—22no—
216C22×C61—24no—
217C22×C61—24no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61—81no—
221C22×C61—82no—
222C22×C61—82no—
223C22×C61#46414yes—
224C22×C61C2×#12111yescharacteristic
225C22×C61#54612yes—
226C22×C61C2×#13511yescharacteristic
227C22×C61C2×#13711yescharacteristic
228C22×C61#57712yes—
229C4×C61—22no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—81no—
233C4×C61#52511yescharacteristic
234C4×C61#53511yescharacteristic
235C4×C61#55112yes—
236Dic61—44no—
237Dic61—42no—
238Dic61—42no—
239Dic61—82no—
240C2×#481D6111yescharacteristic
241C22×#103—611no—
242#2436—611no—
243#2345—611no—
244#2066—612no—
245#2042—612no—
246#1983—612no—
247#1927—612no—
248#1893—611no—
249#1850—611no—
250#1787—611no—
251C2×C61—24no—
252C2×C61—161no—
253C2×C61#241111yescharacteristic
254C2×C61C2×#46411yescharacteristic
255C2×C61#304211yescharacteristic
256C2×C61#330112yes—
257C2×C61#331612yes—
258C23×Q8—611no—
259C23×C8—611no—
260C2×#124—612no—
261C2×#121C2×D6111yescharacteristic
262C22×C4⋊C4C2×D6111yescharacteristic
263#481—24no—
264C2×#106C2×D6111yescharacteristic
265C2×#106C2×D6111yescharacteristic
266C2×C4⋊C8—612no—
267C2×#103—612no—
268C2×#103—612no—
269C2×#103—612no—
270C2×#103—612no—
271C2×#103—1224no—
272C2×#102—612no—
273C2×#102C2×D6111yescharacteristic
274C2×#102C2×D6111yescharacteristic
275C2×C22⋊C8C2×D6111yescharacteristic
276#393—1222no—
277#391—612no—
278#389—612no—
279#387—1222no—
280#386—1221no—
281#386—1221no—
282#380—611no—
283#380—1221no—
284C4×C4⋊C4—612no—
285C4×C22⋊C4—611no—
286C4×C22⋊C4—1221no—
287#340—612no—
288#336—612no—
289C61#2448311yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×Q8—1224no—
292C22×Q8—1221no—
293C22×Q8—1222no—
294C22×Q8—1222no—
295C22×Q8—2442no—
296C22×D4—21no—
297C22×D4C22×D6111yescharacteristic
298C23×C4—21no—
299C23×C4—1221no—
300C23×C4C22×D6111yescharacteristic
301C22×C8—612no—
302C22×C8—1222no—
303C22×C8—2442no—
304C22×C8C22×D6111yescharacteristic
305C22×C8C22×D6111yescharacteristic
306#124—1224no—
307#124—2442no—
308#121—22no—
309#121—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4C22⋊D6114yes—
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—611no—
320C2×C22⋊C4—611no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1221no—
324C2×C42—612no—
325C2×C42—1222no—
326#106—22no—
327#106—22no—
328C4⋊C8—2442no—
329#103—1224no—
330#103—1224no—
331#103—1224no—
332#103—2442no—
333#102—22no—
334#102—22no—
335#102—2442no—
336C22⋊C8—22no—
337#95—612no—
338#95—1222no—
339#95—1222no—
340#95—1222no—
341#95—1222no—
342#95—1221no—
343C24—21no—
344C24D4×D6111yescharacteristic
345C2×Q8—614no—
346C2×Q8—1228no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—1222no—
352C2×Q8—2444no—
353C2×Q8—2448no—
354C2×Q8—2444no—
355C2×Q8—2442no—
356C2×D4—21no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—82no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—1222no—
373C22×C4—1222no—
374C22×C4—2444no—
375C22×C4—2442no—
376C22×C4D4×D6111yescharacteristic
377C22×C4#1175822412yes—
378C22×C4C2×C22⋊D6112yes—
379C22×C4C23×D6111yescharacteristic
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—1222no—
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—2444no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—4882no—
396C4⋊C4C2×C22⋊D6112yes—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—42no—
399C22⋊C4—1222no—
400C22⋊C4—1222no—
401C22⋊C4—2444no—
402C22⋊C4—2444no—
403C22⋊C4—2442no—
404C42—1222no—
405C42—1222no—
406C42—4882no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23#1451974912yes—
415C23#1451979812yes—
416C23#1451980612yes—
417C23C2×D4×D6111yescharacteristic
418Q8—1224no—
419Q8—1224no—
420Q8—2448no—
421Q8—2442no—
422Q8—2442no—
423D4—41no—
424D4—41no—
425D4—42no—
426D4—84no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—84no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1224no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2448no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451974912yes—
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4#41×D6112yes—
455C2×C4C2×#1175822411yescharacteristic
456C2×C4#1451982312yes—
457C8—41no—
458C8—41no—
459C8—2442no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#1683998214yes—
470C22C2×#1451974911yescharacteristic
471C22#1684040412yes—
472C22C2×#1451979811yescharacteristic
473C22C2×#1451980611yescharacteristic
474C22#1684057212yes—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—4882no—
480C4—2442no—
481C4—2442no—
482C4—2444no—
483C4#1684026911yescharacteristic
484C4#1684032611yescharacteristic
485C4#1684043512yes—
486C2—24no—
487C2—161no—
488C2#2144920511yescharacteristic
489C2C2×#1683998211yescharacteristic
490C2#2145533311yescharacteristic
491C2#2145771912yes—
492C2#2145792812yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#481C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenterC2×#481D61
hypocenterC61#24483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C22×#14519700C2semidirect
3. #21448696C2semidirect
4. #21447195C2semidirect
5. #21447059C2semidirect
6. #21446750C2semidirect
7. #21446452C2semidirect
9. #21445996C2semidirect
10. #21445755C2semidirect
289. C61#24483semidirect

Maximal subgroups

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

subgroupindex
C22×#145197002
#214494782
#214486962
#214471952
#214470592
#214467502
#214464522
#214462482
#214459962
#214457552
C2×C61×#4812
#2448361