Finite Group Databasebeta

← #29734145

Subgroups — #29734145

Order 15616.

1076 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840497C211yesmaximal, characteristic
2C2×#16840128C211yesmaximal, characteristic
3#21451208C211yesmaximal, characteristic, complemented
4#21447056C212yesmaximal, complemented
5#21447056C212yesmaximal, complemented
6#21446758C211yesmaximal, characteristic, complemented
7#21446745C211yesmaximal, characteristic
8#21446077C212yesmaximal, complemented
9#21446063C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#479C211yesmaximal, characteristic, complemented
12#16840497—24no—
13#16840497—24no—
14C2×#14519774C2211yescharacteristic
15C2×#14519766C2212yes—
16C2×#14519765—21no—
17C2×#14519762—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244—21no—
20C2×#14519739C2212yes—
21#16840128—24no—
22C2×#14519701C2212yes—
23C2×#14519699C2212yes—
24C2×#14519699C2212yes—
25C2×#14519697C2212yes—
26C2×#14519697C2212yes—
27C2×#14519688C2211yescharacteristic
28#16839692—21no—
29#16839682—21no—
30#16839658C2212yes—
31#16839658C2212yes—
32#16839657—22no—
33C4×C4⋊Dic61C2211yescharacteristic
34#16839524C2212yes—
35#16839516C2212yes—
36#16839516C2212yes—
37C2×#125×C61C2211yescharacteristic
38C2×C4×Q8×C61C2211yescharacteristic
39C61×#479—24no—
40C2×#107×C61C2211yescharacteristic
41C2×#106×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×#103×C61C2211yescharacteristic
44C2×#103×C61C2211yescharacteristic
45C22×Dic122D411yescharacteristic
46C22×D244—21no—
47C22×D244—21no—
48C22×D244C2312yes—
49C22×C4×D61—22no—
50C22×#92418C2312yes—
51#14519774—22no—
52#14519774D414yes—
53#14519766—24no—
54#14519766—24no—
55#14519765—28no—
56#14519762—216no—
57C4×Dic122—22no—
58C4×Dic122—22no—
59C4×D244—28no—
60#14519739—24no—
61#14519739—24no—
62C2×C4⋊Dic61D411yescharacteristic
63C2×C4⋊Dic61C2312yes—
64C2×#11758196—22no—
65C2×#11758196—22no—
66C2×#11758196D412yes—
67C2×#11758198—22no—
68C2×#11758198—22no—
69C2×#11758198—22no—
70C2×C4×Dic61—21no—
71C2×C4×Dic61C2312yes—
72#14519701—24no—
73#14519699—24no—
74#14519699—24no—
75#14519697—24no—
76#14519697—24no—
77#14519688—22no—
78#14519684—21no—
79#14519684—22no—
80#14519684—22no—
81C22×Q8×C61C2311yescharacteristic
82C22×C8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#125×C61—22no—
85#125×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61D414yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—21no—
91C2×C4⋊C4×C61C2311yescharacteristic
92C2×C4⋊C4×C61C2311yescharacteristic
93C2×C4⋊C4×C61C2311yescharacteristic
94C2×C42×C61—21no—
95C2×C42×C61C2311yescharacteristic
96#107×C61—22no—
97#106×C61—22no—
98C4⋊C8×C61—22no—
99#103×C61—22no—
100#103×C61—22no—
101C23×D61—22no—
102C2×Q8×C61—21no—
103C2×Q8×C61—22no—
104C2×Q8×C61—22no—
105C2×Q8×C61C2×D412yes—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—21no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2411yescharacteristic
113C22×Dic61—22no—
114C22×Dic61C2×D412yes—
115C22×Dic61C2×D412yes—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—22no—
120C2×D244—24no—
121C2×D244—22no—
122C2×D244—24no—
123C2×D244—44no—
124C2×D244—81no—
125C2×D244—81no—
126C2×D244C2×D414yes—
127C2×C4×D61—28no—
128C2×C4×D61—82no—
129C2×C4×D61—82no—
130C2×Dic122—22no—
131C2×Dic122—42no—
132C2×C8×C61—21no—
133C2×C8×C61—21no—
134C2×C8×C61—22no—
135C2×C8×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—42no—
144C4⋊C4×C61C2×D412yes—
145C42×C61—24no—
146C42×C61C2×D412yes—
147C42×C61C2×D412yes—
148#11758198—28no—
149#11758198—28no—
150#11758198—28no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153C4⋊Dic61#4114yes—
154#11758196—24no—
155#11758196—44no—
156#11758196—82no—
157C4×Dic61—22no—
158C4×Dic61—22no—
159C4×Dic61—81no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61C22×D411yescharacteristic
164C22×D61—24no—
165C22×D61—42no—
166C22×D61—42no—
167C22×D61—42no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—42no—
171Q8×C61—22no—
172Q8×C61—22no—
173Q8×C61—42no—
174C2×C4×C61—22no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61#12012yes—
188C2×C4×C61#12012yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C2×#4112yes—
192C2×C4×C612−1+412yes—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—24no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—42no—
201C2×Dic61—42no—
202D244—24no—
203D244—24no—
204D244—48no—
205D244—42no—
206D244—42no—
207D244—84no—
208C4×D61—88no—
209Dic122—44no—
210C8×C61—41no—
211C8×C61—41no—
212#92418—42no—
213#25658—611nomaximal, Sylow
214C22×C61C2×#12011yescharacteristic
215C22×C61#54812yes—
216C22×C61#57512yes—
217C22×C61#57611yescharacteristic
218C22×C61#57711yescharacteristic
219C2×D61—44no—
220C2×D61—44no—
221C2×D61—44no—
222C2×D61—82no—
223C2×D61—82no—
224C2×D61—82no—
225C2×D61—82no—
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#53211yescharacteristic
233C4×C61#53611yescharacteristic
234C4×C61#54712yes—
235Dic61—44no—
236Dic61—42no—
237Dic61—42no—
238C2×#559—611no—
239C2×#490—611no—
240C2×#479D6111yescharacteristic, complemented
241#2584—611no—
242#2042—612no—
243#2042—612no—
244#1985—611no—
245#1982—611no—
246#1864—612no—
247#1862—612no—
248#1763—611no—
249C2×C61#243211yescharacteristic
250C2×C61#300511yescharacteristic
251C2×C61#301211yescharacteristic
252C2×C61#337012yes—
253C2×C61#337112yes—
254D61—84no—
255#559—1224no—
256#559—1224no—
257C2×#127—611no—
258C2×#125C2×D6111yescharacteristic
259C2×#124—612no—
260C2×#124—1221no—
261C2×#123—1222no—
262C2×C4×Q8—611no—
263C2×C4×Q8C2×D6111yescharacteristic
264C2×C4×D4—612no—
265C2×C4×D4—1221no—
266#490—1224no—
267#479—24no—
268C2×#107C2×D6111yescharacteristic
269C2×#106C2×D6111yescharacteristic
270C2×C4⋊C8C2×D6111yescharacteristic
271C2×#103—612no—
272C2×#103C2×D6111yescharacteristic
273C2×#103C2×D6111yescharacteristic
274C2×#102—612no—
275C2×#102—612no—
276C2×#102—612no—
277C2×#102—612no—
278C2×#97—611no—
279#398—1221no—
280#395—1221no—
281#389—612no—
282#389—612no—
283#389—1222no—
284C4×C4⋊C4—611no—
285#340—612no—
286#336—612no—
287#336—612no—
288C61#2565811yescharacteristic, complemented, Sylow
289C22×Q8—611no—
290C22×Q8C22×D6111yescharacteristic
291C22×D4—612no—
292C22×D4—1221no—
293C22×D4—1221no—
294C23×C4—1222no—
295C22×C8—612no—
296C22×C8C22×D6111yescharacteristic
297C22×C8C22×D6111yescharacteristic
298#127—614no—
299#127—1222no—
300#125—22no—
301#125—22no—
302#124—1228no—
303#124—1224no—
304#124—1224no—
305#123—12216no—
306C4×Q8—22no—
307C4×Q8—1222no—
308C4×Q8—1222no—
309C4×Q8C22⋊D6114yes—
310C4×D4—1228no—
311C4×D4—1224no—
312C4×D4—1224no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4C22×D6111yescharacteristic
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1222no—
326C2×C22⋊C4—1222no—
327C2×C42—21no—
328C2×C42—612no—
329C2×C42—1221no—
330C2×C42C22×D6111yescharacteristic
331#107—22no—
332#106—22no—
333C4⋊C8—22no—
334#103—22no—
335#103—22no—
336#103—1224no—
337#102—1224no—
338#102—1224no—
339#102—1224no—
340#102—1224no—
341#97—1222no—
342#95—1222no—
343#95—1222no—
344#95—1221no—
345C24—1222no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—1222no—
350C2×Q8—2442no—
351C2×Q8C2×C22⋊D6112yes—
352C2×D4—614no—
353C2×D4—1222no—
354C2×D4—1222no—
355C2×D4—1222no—
356C2×D4—1222no—
357C2×D4—1222no—
358C2×D4—1224no—
359C2×D4—1224no—
360C2×D4—4881no—
361C2×D4—4881no—
362C2×D4—2444no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—612no—
368C22×C4—612no—
369C22×C4—1228no—
370C22×C4—1222no—
371C22×C4—4882no—
372C22×C4—4882no—
373C22×C4D4×D6111yescharacteristic
374C22×C4D4×D6111yescharacteristic
375C22×C4C23×D6111yescharacteristic
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—21no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—614no—
392C4⋊C4—1222no—
393C4⋊C4—1224no—
394C4⋊C4—1222no—
395C4⋊C4—4882no—
396C4⋊C4—2444no—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—1228no—
399C22⋊C4—1228no—
400C22⋊C4—1228no—
401C42—24no—
402C42—1222no—
403C42—1222no—
404C42—4881no—
405C42D4×D6112yes—
406C42C2×C22⋊D6112yes—
407C23—1224no—
408C23—2442no—
409C23—2442no—
410C23—2442no—
411C23—2442no—
412C23—2442no—
413C23—2442no—
414C23C2×D4×D6111yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—2444no—
419D4—1224no—
420D4—1224no—
421D4—4884no—
422D4—2448no—
423D4—2442no—
424D4—2442no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1222no—
442C2×C4—1222no—
443C2×C4—1222no—
444C2×C4—1224no—
445C2×C4—4888no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4#1451974512yes—
449C2×C4#1451974512yes—
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4#41×D6112yes—
453C2×C4#1451982512yes—
454C8—41no—
455C8—41no—
456C8—2442no—
457C22—4882no—
458C22—4882no—
459C22—4882no—
460C22—4882no—
461C22—2444no—
462C22—2444no—
463C22—2444no—
464C22C2×#1451974511yescharacteristic
465C22#1684042212yes—
466C22#1684056112yes—
467C22#1684056411yescharacteristic
468C22#1684056911yescharacteristic
469C4—22no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—2442no—
476C4—2444no—
477C4—2442no—
478C4#1684030811yescharacteristic
479C4#1684033611yescharacteristic
480C4#1684041412yes—
481C2—4884no—
482C2#2144944811yescharacteristic
483C2#2145506011yescharacteristic
484C2#2145511911yescharacteristic
485C2#2145857512yes—
486C2#2145859012yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenterC2×#479D61
hypocenterC61#25658
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. #21451208C2semidirect
4. #21447056C2semidirect
5. #21447056C2semidirect
6. #21446758C2semidirect
8. #21446077C2semidirect
9. #21446063C2semidirect
10. #21445649C2semidirect
11. C2×C61×#479C2semidirect
240. C2×#479D61semidirect
288. C61#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#168404972
C2×#168401282
#214512082
#214470562
#214467582
#214467452
#214460772
#214460632
#214456492
C2×C61×#4792
#2565861