Finite Group Databasebeta

← #29718411

Subgroups — #29718411

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840076C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21449473C211yesmaximal, characteristic
4#21448690C211yesmaximal, characteristic, complemented
5#21447212C212yesmaximal, complemented
6#21447056C212yesmaximal, complemented
7#21446748C212yesmaximal, complemented
8#21446462C212yesmaximal, complemented
9#21446245C211yesmaximal, characteristic
10#21445996C211yesmaximal, characteristic, complemented
11C61×#1787C211yesmaximal, characteristic, complemented
12C23×#92418C2211yescharacteristic
13C2×#14519766C2212yes—
14C2×#14519751C2211yescharacteristic
15C2×#14519733—21no—
16C22×C4⋊Dic61C2211yescharacteristic
17#16840076—24no—
18C2×#14519711C2211yescharacteristic
19C2×#14519711C2211yescharacteristic
20C2×#14519706C2212yes—
21C2×#14519701—24no—
22C2×#14519701C2212yes—
23C2×#14519701C2212yes—
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519697C2212yes—
27#16839673—22no—
28#16839669C2212yes—
29#16839658C2212yes—
30#16839645—22no—
31#16839641—21no—
32#16839641—21no—
33#16839623—21no—
34C4⋊C4×Dic61C2212yes—
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—
53#14519733—44no—
54C2×C4⋊Dic61—21no—
55C2×C4⋊Dic61—22no—
56C2×C4⋊Dic61D414yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C4⋊Dic61C2311yescharacteristic
60C2×#11758196—22no—
61C2×C22⋊Dic61D411yescharacteristic
62C2×C22⋊Dic61D411yescharacteristic
63C2×#11758198—21no—
64C2×#11758198—22no—
65C2×#11758198—22no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61C2312yes—
68#14519711—22no—
69#14519711—22no—
70#14519706—42no—
71#14519701—24no—
72#14519701—24no—
73#14519699—22no—
74#14519699—22no—
75#14519697—42no—
76#14519684—22no—
77#14519684—22no—
78#14519684—22no—
79#14519684—22no—
80#14519684D412yes—
81C22×Q8×C61—21no—
82C22×Q8×C61—24no—
83C22×Q8×C61—22no—
84C22×Q8×C61—22no—
85C22×Q8×C61—42no—
86C22×Q8×C61C2312yes—
87C23×C4×C61—21no—
88C23×C4×C61C2311yescharacteristic
89C22×C8×C61C2311yescharacteristic
90C22×C8×C61C2311yescharacteristic
91C2×C4⋊C4×C61C2312yes—
92C2×C22⋊C4×C61—21no—
93#103×C61—24no—
94#103×C61—42no—
95C22⋊C8×C61—22no—
96#95×C61—21no—
97C24×C61C2×D411yescharacteristic
98C23×D61—21no—
99C2×Q8×C61—22no—
100C2×Q8×C61—28no—
101C2×Q8×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61—44no—
106C2×Q8×C61—48no—
107C2×Q8×C61—44no—
108C2×Q8×C61—42no—
109C2×Q8×C61#4114yes—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—44no—
115C22×C4×C61—42no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61#4112yes—
119C22×C4×C61C2411yescharacteristic
120C2×C22⋊D61—42no—
121C2×C22⋊D61—42no—
122C2×C22⋊D61—42no—
123C22×Dic61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61C2×D412yes—
130C22×Dic61C2×D412yes—
131C2×D244—21no—
132C2×D244—21no—
133C2×D244—22no—
134C2×D244—22no—
135C2×C8×C61—21no—
136C2×C8×C61—21no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C22⋊C4×C61—42no—
142C22⋊Dic61—22no—
143C22⋊Dic61—22no—
144#11758198—44no—
145#11758198—44no—
146#11758198—42no—
147C4⋊Dic61—24no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C2×D412yes—
153C4⋊Dic61C2×D412yes—
154#11758196—44no—
155C4×Dic61—22no—
156C4×Dic61—22no—
157C4×Dic61—44no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—44no—
162C23×C61—22no—
163C23×C61—22no—
164C23×C61#12112yes—
165C23×C61#13512yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C23×C61C22×D411yescharacteristic
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—42no—
173Q8×C61—24no—
174Q8×C61—24no—
175Q8×C61—48no—
176Q8×C61—42no—
177Q8×C61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—24no—
182C2×C4×C61—24no—
183C2×C4×C61—48no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61#12112yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4112yes—
192C2×C4×C61C2×#4111yescharacteristic
193C2×C4×C61H(5,𝔽2)12yes—
194C22⋊D61—84no—
195C2×Dic61—22no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—48no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211D244—41no—
212D244—41no—
213D244—42no—
214C8×C61—41no—
215C8×C61—41no—
216#92418—42no—
217#24481—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61#46314yes—
223C22×C61C2×#12111yescharacteristic
224C22×C61#54612yes—
225C22×C61C2×#13511yescharacteristic
226C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
227C22×C61#57812yes—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—82no—
232C2×D61—82no—
233C4×C61—22no—
234C4×C61—44no—
235C4×C61—82no—
236C4×C61#52511yescharacteristic
237C4×C61#53511yescharacteristic
238C4×C61#55112yes—
239Dic61—42no—
240Dic61—42no—
241Dic61—42no—
242Dic61—42no—
243Dic61—81no—
244C2×#480—611no—
245C22×#103—611no—
246#2436—611no—
247#2344—611no—
248#2068—612no—
249#2042—612no—
250#1983—612no—
251#1928—612no—
252#1893—611no—
253#1850—611no—
254#1787D6111yescharacteristic, complemented
255C2×C61—24no—
256C2×C61#241111yescharacteristic
257C2×C61C2×#46311yescharacteristic
258C2×C61#304411yescharacteristic
259C2×C61#330012yes—
260C2×C61#330812yes—
261D61—161no—
262C23×Q8C2×D6111yescharacteristic
263C23×C8—611no—
264C2×#124—612no—
265C2×#121—611no—
266C2×#117—1221no—
267C22×C4⋊C4—611no—
268#480—1224no—
269C2×#107—611no—
270C2×#107—611no—
271C2×C4⋊C8—612no—
272C2×#103—612no—
273C2×#103—612no—
274C2×#103—1224no—
275C2×#103C2×D6112yes—
276C2×#103C2×D6112yes—
277C2×#102—611no—
278C2×#102—611no—
279C2×#102—612no—
280C2×C22⋊C8C2×D6111yescharacteristic
281#393—1222no—
282#391—612no—
283#389—612no—
284#387—1222no—
285#386—1221no—
286#386—1221no—
287#381—1221no—
288#380C2×D6111yescharacteristic
289C4×C4⋊C4—612no—
290C4×C22⋊C4—611no—
291#340—612no—
292#336—612no—
293C61#2448111yescharacteristic, complemented, Sylow
294C22×Q8—21no—
295C22×Q8—24no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—42no—
299C22×Q8C22×D6112yes—
300C22×D4—611no—
301C22×D4—1221no—
302C23×C4—21no—
303C23×C4—1221no—
304C23×C4C22×D6111yescharacteristic
305C22×C8—612no—
306C22×C8—1222no—
307C22×C8—2442no—
308C22×C8C22×D6111yescharacteristic
309C22×C8C22×D6111yescharacteristic
310#124—1224no—
311#124—2442no—
312#121—1222no—
313#121—1222no—
314#117—2444no—
315C2×C4⋊C4—614no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4C22×D6112yes—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—611no—
325C2×C22⋊C4—611no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1221no—
329C2×C42—612no—
330C2×C42—1222no—
331#107—1222no—
332#107—1222no—
333C4⋊C8—2442no—
334#103—24no—
335#103—42no—
336#103—1224no—
337#103—1224no—
338#102—1222no—
339#102—1222no—
340#102—2442no—
341C22⋊C8—22no—
342#95—21no—
343#95—612no—
344#95—1222no—
345#95—1222no—
346#95—1222no—
347#95—1222no—
348C24—1221no—
349C24D4×D6111yescharacteristic
350C2×Q8—22no—
351C2×Q8—28no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—42no—
360C2×Q8#1175822414yes—
361C2×D4—1221no—
362C2×D4—1222no—
363C2×D4—1221no—
364C2×D4—1222no—
365C2×D4—2442no—
366C2×D4—2442no—
367C2×D4—2442no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—612no—
375C22×C4—612no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4—1224no—
379C22×C4—1221no—
380C22×C4—2442no—
381C22×C4—2442no—
382C22×C4#1175822212yes—
383C22×C4D4×D6112yes—
384C22×C4D4×D6111yescharacteristic
385C22×C4C23×D6111yescharacteristic
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—2444no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—612no—
397C4⋊C4—612no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1224no—
401C4⋊C4—1221no—
402C4⋊C4—1221no—
403C4⋊C4—2444no—
404C22⋊C4—42no—
405C22⋊C4—1222no—
406C22⋊C4—1222no—
407C22⋊C4—2444no—
408C22⋊C4—2444no—
409C22⋊C4—2442no—
410C42—1222no—
411C42—1222no—
412C42—2444no—
413C23—22no—
414C23—22no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23—2442no—
419C23—2442no—
420C23#1451975012yes—
421C23#1451980012yes—
422C23#1451980412yes—
423C23C2×D4×D6111yescharacteristic
424Q8—24no—
425Q8—24no—
426Q8—48no—
427Q8—42no—
428Q8—42no—
429D4—4884no—
430D4—2441no—
431D4—2441no—
432D4—2442no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1222no—
450C2×C4—1222no—
451C2×C4—1222no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2448no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4#1451975012yes—
461C2×C4C2×D4×D6111yescharacteristic
462C2×C4C2×#1175822412yes—
463C2×C4C2×#1175822211yescharacteristic
464C2×C4#1451982212yes—
465C8—41no—
466C8—41no—
467C8—2442no—
468C22—24no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—4882no—
473C22—4881no—
474C22—4882no—
475C22—4881no—
476C22—4881no—
477C22#1683998014yes—
478C22C2×#1451975011yescharacteristic
479C22#1684039912yes—
480C22C2×#1451980011yescharacteristic
481C22C2×#1451980411yescharacteristic
482C22#1684057812yes—
483C4—22no—
484C4—44no—
485C4—82no—
486C4—4881no—
487C4—2442no—
488C4—2442no—
489C4—2442no—
490C4—2442no—
491C4#1684026811yescharacteristic
492C4#1684032711yescharacteristic
493C4#1684043812yes—
494C2—24no—
495C2—9761no—
496C2#2144920611yescharacteristic
497C2C2×#1683998011yescharacteristic
498C2#2145534811yescharacteristic
499C2#2145771012yes—
500C2#2145781412yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1787D61
hypocenterC61#24481
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. #21448690C2semidirect
5. #21447212C2semidirect
6. #21447056C2semidirect
7. #21446748C2semidirect
8. #21446462C2semidirect
10. #21445996C2semidirect
11. C61×#1787C2semidirect
254. #1787D61semidirect
293. C61#24481semidirect

Maximal subgroups

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

subgroupindex
C2×#168400762
C22×#145197012
#214494732
#214486902
#214472122
#214470562
#214467482
#214464622
#214462452
#214459962
C61×#17872
#2448161