Finite Group Databasebeta

← #29718433

Subgroups — #29718433

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
1C2×#16840081C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21449473C211yesmaximal, characteristic
4#21448698C211yesmaximal, characteristic, complemented
5#21447198C212yesmaximal, complemented
6#21447056C212yesmaximal, complemented
7#21446748C212yesmaximal, complemented
8#21446455C212yesmaximal, complemented
9#21446245C211yesmaximal, characteristic
10#21445996C211yesmaximal, characteristic, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13C2×#14519766C2212yes—
14C2×#14519751C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16840081—24no—
17C2×#14519709C2211yescharacteristic
18C2×#14519709C2211yescharacteristic
19C2×#14519706C2212yes—
20C2×#14519701—24no—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519697C2212yes—
26#16839673—22no—
27#16839669C2212yes—
28#16839658C2212yes—
29#16839647—22no—
30#16839641—21no—
31#16839641—21no—
32#16839620—21no—
33C4⋊C4×Dic61C2212yes—
34C22⋊C4×Dic61—21no—
35C4×C22⋊Dic61C2211yescharacteristic
36#16839524C2212yes—
37#16839516C2212yes—
38C23×Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C2×#103×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×D244C2311yescharacteristic
44C22×C22⋊D61—21no—
45C23×Dic61—21no—
46C22×#92418—22no—
47C22×#92418—42no—
48C22×#92418C2312yes—
49#14519766—24no—
50#14519766—42no—
51#14519751—22no—
52#14519751—22no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61D414yes—
56C2×C4⋊Dic61C2311yescharacteristic
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×#11758196—22no—
60C2×C22⋊Dic61D411yescharacteristic
61C2×C22⋊Dic61D411yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C2×C4×Dic61—22no—
66C2×C4×Dic61C2312yes—
67#14519709—22no—
68#14519709—22no—
69#14519706—42no—
70#14519701—24no—
71#14519701—24no—
72#14519699—22no—
73#14519699—22no—
74#14519697—42no—
75#14519684—22no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79#14519684D412yes—
80C22×Q8×C61—21no—
81C22×Q8×C61—24no—
82C22×Q8×C61—22no—
83C22×Q8×C61—22no—
84C22×Q8×C61—42no—
85C22×Q8×C61C2312yes—
86C23×C4×C61—21no—
87C23×C4×C61C2311yescharacteristic
88C22×C8×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2312yes—
91C2×C22⋊C4×C61—21no—
92#103×C61—24no—
93#103×C61—42no—
94C22⋊C8×C61—22no—
95#95×C61—21no—
96C24×C61C2×D411yescharacteristic
97C23×D61—21no—
98C2×Q8×C61—22no—
99C2×Q8×C61—28no—
100C2×Q8×C61—22no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—44no—
105C2×Q8×C61—48no—
106C2×Q8×C61—44no—
107C2×Q8×C61—42no—
108C2×Q8×C61#4114yes—
109C22×C4×C61—22no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—44no—
114C22×C4×C61—42no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61#4112yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—42no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61—24no—
125C22×Dic61—21no—
126C22×Dic61—82no—
127C22×Dic61C2×D412yes—
128C22×Dic61C2×D412yes—
129C2×D244—21no—
130C2×D244—21no—
131C2×D244—22no—
132C2×D244—22no—
133C2×C8×C61—21no—
134C2×C8×C61—21no—
135C2×C8×C61—22no—
136C2×C8×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C22⋊C4×C61—42no—
140C22⋊Dic61—22no—
141C22⋊Dic61—22no—
142#11758198—44no—
143#11758198—44no—
144#11758198—42no—
145C4⋊Dic61—24no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61C2×D412yes—
151C4⋊Dic61C2×D412yes—
152#11758196—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#12112yes—
163C23×C61#13512yes—
164C23×C61#13712yes—
165C23×C61C22×D411yescharacteristic
166C22×D61—41no—
167C22×D61—42no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—42no—
171Q8×C61—24no—
172Q8×C61—24no—
173Q8×C61—48no—
174Q8×C61—42no—
175Q8×C61—42no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—24no—
181C2×C4×C61—48no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61#12112yes—
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C2×#4112yes—
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61H(5,𝔽2)12yes—
192C22⋊D61—84no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—84no—
207D244—41no—
208D244—41no—
209D244—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#46414yes—
219C22×C61C2×#12111yescharacteristic
220C22×C61#54612yes—
221C22×C61C2×#13511yescharacteristic
222C22×C61C2×#13711yescharacteristic
223C22×C61#57712yes—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—82no—
228C2×D61—82no—
229C4×C61—22no—
230C4×C61—44no—
231C4×C61—82no—
232C4×C61#52511yescharacteristic
233C4×C61#53511yescharacteristic
234C4×C61#55112yes—
235Dic61—42no—
236Dic61—42no—
237Dic61—42no—
238Dic61—42no—
239Dic61—81no—
240C2×#481—611no—
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#1787D6111yescharacteristic, complemented
251C2×C61—24no—
252C2×C61#241111yescharacteristic
253C2×C61C2×#46411yescharacteristic
254C2×C61#304211yescharacteristic
255C2×C61#330112yes—
256C2×C61#331612yes—
257D61—161no—
258C23×Q8C2×D6111yescharacteristic
259C23×C8—611no—
260C2×#124—612no—
261C2×#121—611no—
262C22×C4⋊C4—611no—
263#481—1224no—
264C2×#106—611no—
265C2×#106—611no—
266C2×C4⋊C8—612no—
267C2×#103—612no—
268C2×#103—612no—
269C2×#103—1224no—
270C2×#103C2×D6112yes—
271C2×#103C2×D6112yes—
272C2×#102—611no—
273C2×#102—611no—
274C2×#102—612no—
275C2×C22⋊C8C2×D6111yescharacteristic
276#393—1222no—
277#391—612no—
278#389—612no—
279#387—1222no—
280#386—1221no—
281#386—1221no—
282#380—1221no—
283#380C2×D6111yescharacteristic
284C4×C4⋊C4—612no—
285C4×C22⋊C4—611no—
286C4×C22⋊C4—1221no—
287#340—612no—
288#336—612no—
289C61#2448311yescharacteristic, complemented, Sylow
290C22×Q8—21no—
291C22×Q8—24no—
292C22×Q8—22no—
293C22×Q8—22no—
294C22×Q8—42no—
295C22×Q8C22×D6112yes—
296C22×D4—611no—
297C22×D4—1221no—
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—1222no—
309#121—1222no—
310C2×C4⋊C4—614no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4C22×D6112yes—
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—1222no—
327#106—1222no—
328C4⋊C8—2442no—
329#103—24no—
330#103—42no—
331#103—1224no—
332#103—1224no—
333#102—1222no—
334#102—1222no—
335#102—2442no—
336C22⋊C8—22no—
337#95—21no—
338#95—612no—
339#95—1222no—
340#95—1222no—
341#95—1222no—
342#95—1222no—
343C24—1221no—
344C24D4×D6111yescharacteristic
345C2×Q8—22no—
346C2×Q8—28no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—44no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—42no—
355C2×Q8#1175822414yes—
356C2×D4—1221no—
357C2×D4—1222no—
358C2×D4—1221no—
359C2×D4—1222no—
360C2×D4—2442no—
361C2×D4—2442no—
362C2×D4—2442no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—612no—
370C22×C4—612no—
371C22×C4—1224no—
372C22×C4—1221no—
373C22×C4—1221no—
374C22×C4—1221no—
375C22×C4—4882no—
376C22×C4#1175822212yes—
377C22×C4D4×D6112yes—
378C22×C4D4×D6111yescharacteristic
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—22no—
389C4⋊C4—22no—
390C4⋊C4—612no—
391C4⋊C4—612no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1224no—
395C4⋊C4—1221no—
396C4⋊C4—1221no—
397C4⋊C4—4882no—
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—2441no—
410C23—2441no—
411C23—2441no—
412C23—2442no—
413C23—2442no—
414C23#1451975012yes—
415C23#1451979812yes—
416C23#1451980912yes—
417C23C2×D4×D6111yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—48no—
421Q8—42no—
422Q8—42no—
423D4—4884no—
424D4—2441no—
425D4—2441no—
426D4—2442no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—48no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—4884no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451975012yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×#1175822412yes—
455C2×C4C2×#1175822211yescharacteristic
456C2×C4#1451982212yes—
457C8—41no—
458C8—41no—
459C8—2442no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—4882no—
465C22—4882no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22#1683998714yes—
470C22C2×#1451975011yescharacteristic
471C22#1684039912yes—
472C22C2×#1451979811yescharacteristic
473C22C2×#1451980911yescharacteristic
474C22#1684057112yes—
475C4—22no—
476C4—44no—
477C4—82no—
478C4—4881no—
479C4—2442no—
480C4—2442no—
481C4—2442no—
482C4—2442no—
483C4#1684026811yescharacteristic
484C4#1684032711yescharacteristic
485C4#1684043812yes—
486C2—24no—
487C2—9761no—
488C2#2144920611yescharacteristic
489C2C2×#1683998711yescharacteristic
490C2#2145533011yescharacteristic
491C2#2145772512yes—
492C2#2145793412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1787D61
hypocenterC61#24483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
4. #21448698C2semidirect
5. #21447198C2semidirect
6. #21447056C2semidirect
7. #21446748C2semidirect
8. #21446455C2semidirect
10. #21445996C2semidirect
11. C61×#1787C2semidirect
250. #1787D61semidirect
289. C61#24483semidirect

Maximal subgroups

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

subgroupindex
C2×#168400812
C22×#145197012
#214494732
#214486982
#214471982
#214470562
#214467482
#214464552
#214462452
#214459962
C61×#17872
#2448361