Finite Group Databasebeta

← #29734138

Subgroups — #29734138

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×#16840126C211yesmaximal, characteristic
2C2×#16840069C211yesmaximal, characteristic, complemented
3#21451207C211yesmaximal, characteristic, complemented
4#21447055C212yesmaximal, complemented
5#21447055C212yesmaximal, complemented
6#21446760C211yesmaximal, characteristic, complemented
7#21446747C211yesmaximal, characteristic
8#21446080C212yesmaximal, complemented
9#21446066C212yesmaximal, complemented
10#21445649C211yesmaximal, characteristic, complemented
11C2×C61×#559C211yesmaximal, characteristic
12C2×#14519769C2211yescharacteristic
13#16840126—24no—
14#16840069—24no—
15C2×#14519710C2211yescharacteristic
16C2×#14519708C2211yescharacteristic
17C2×C4⋊#92418C2211yescharacteristic
18C2×#14519701C2212yes—
19C2×#14519701C2211yescharacteristic
20C2×#14519701C2211yescharacteristic
21C2×#14519698C2212yes—
22C2×#14519698C2212yes—
23C2×#14519698C2212yes—
24C2×#14519698C2212yes—
25C2×#14519688C2211yescharacteristic
26#16839693—21no—
27#16839681—21no—
28#16839660C2212yes—
29#16839660C2212yes—
30#16839657—22no—
31C4×C4⋊Dic61C2211yescharacteristic
32#16839524C2212yes—
33#16839517C2212yes—
34#16839517C2212yes—
35C61×#559—24no—
36C61×#559—24no—
37C2×#127×C61C2211yescharacteristic
38C2×#124×C61—21no—
39C2×#124×C61C2212yes—
40C2×#123×C61—22no—
41C2×C4×Q8×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C2×C4×D4×C61—21no—
44C2×C4×D4×C61C2212yes—
45C22×#92418C2312yes—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48#14519769—22no—
49#14519769—22no—
50C2×C4⋊Dic61C2311yescharacteristic
51C2×C4⋊Dic61C2311yescharacteristic
52C2×C4⋊Dic61C2312yes—
53C2×#11758196—21no—
54C2×#11758196—22no—
55C2×#11758196—21no—
56C2×#11758196—22no—
57C2×C4×Dic61—21no—
58#14519710—22no—
59#14519708—22no—
60C4⋊#92418—22no—
61#14519701—22no—
62#14519701—22no—
63#14519701—24no—
64#14519698—24no—
65#14519698—24no—
66#14519698—24no—
67#14519698—24no—
68#14519688—22no—
69#14519684—21no—
70#14519684—22no—
71#14519684—22no—
72C22×Q8×C61D411yescharacteristic
73C22×Q8×C61C2311yescharacteristic
74C22×D4×C61—21no—
75C22×D4×C61—21no—
76C22×D4×C61C2312yes—
77C23×C4×C61—22no—
78#127×C61—22no—
79#127×C61D414yes—
80#124×C61—28no—
81#124×C61—24no—
82#124×C61—24no—
83#123×C61—216no—
84C4×Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61D414yes—
88C4×D4×C61—28no—
89C4×D4×C61—24no—
90C4×D4×C61—24no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C4⋊C4×C61C2311yescharacteristic
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C22⋊C4×C61—22no—
98C2×C42×C61—21no—
99C2×C42×C61C2312yes—
100C2×C42×C61C2311yescharacteristic
101C24×C61—22no—
102C2×Q8×C61—22no—
103C2×Q8×C61—21no—
104C2×Q8×C61—22no—
105C2×Q8×C61—22no—
106C2×Q8×C61—42no—
107C2×Q8×C61C2×D412yes—
108C2×D4×C61—24no—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—44no—
116C2×D4×C61—81no—
117C2×D4×C61—81no—
118C2×D4×C61C2×D414yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—28no—
122C22×C4×C61—82no—
123C22×C4×C61—82no—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2411yescharacteristic
129C22×Dic61—22no—
130C22×Dic61—21no—
131C22×Dic61—21no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—24no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61C2×D412yes—
137C22⋊C4×C61—28no—
138C22⋊C4×C61—28no—
139C22⋊C4×C61—28no—
140C42×C61—24no—
141C42×C61—22no—
142C42×C61—22no—
143C42×C61C2×D412yes—
144C42×C61C2×D412yes—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61#4114yes—
151#11758196—44no—
152#11758196—42no—
153#11758196—42no—
154#11758196—82no—
155C4×Dic61—81no—
156C2×#92418—22no—
157C2×#92418—21no—
158C2×#92418—21no—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C2×#92418—22no—
163C23×C61—24no—
164C23×C61—42no—
165C23×C61—42no—
166C23×C61—42no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61C22×D411yescharacteristic
171Q8×C61—22no—
172Q8×C61—22no—
173Q8×C61—44no—
174Q8×C61—42no—
175D4×C61—24no—
176D4×C61—24no—
177D4×C61—48no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—84no—
181C2×C4×C61—22no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—42no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—88no—
196C2×C4×C61#12012yes—
197C2×C4×C61#12012yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C2×#4112yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—24no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210#92418—41no—
211#92418—41no—
212#92418—42no—
213#25658—611nomaximal, Sylow
214C22×C61—44no—
215C22×C61—44no—
216C22×C61—44no—
217C22×C61—82no—
218C22×C61—82no—
219C22×C61—82no—
220C22×C61—82no—
221C22×C61C2×#12011yescharacteristic
222C22×C61#54812yes—
223C22×C61#57512yes—
224C22×C61#57611yescharacteristic
225C22×C61#57711yescharacteristic
226C4×C61—22no—
227C4×C61—22no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61#53211yescharacteristic
233C4×C61#53611yescharacteristic
234C4×C61#54712yes—
235Dic61—44no—
236Dic61—81no—
237Dic61—81no—
238C2×#559D6111yescharacteristic
239C2×#490—611no—
240C2×#479—611no—
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—84no—
250C2×C61#243211yescharacteristic
251C2×C61#300511yescharacteristic
252C2×C61#301211yescharacteristic
253C2×C61#337012yes—
254C2×C61#337112yes—
255#559—24no—
256#559—24no—
257C2×#127C2×D6111yescharacteristic
258C2×#125—611no—
259C2×#124—21no—
260C2×#124C2×D6112yes—
261C2×#123—22no—
262C2×C4×Q8C2×D6111yescharacteristic
263C2×C4×Q8C2×D6111yescharacteristic
264C2×C4×D4—21no—
265C2×C4×D4C2×D6112yes—
266#490—1224no—
267#479—1224no—
268C2×#107—611no—
269C2×#106—611no—
270C2×C4⋊C8—611no—
271C2×#103—611no—
272C2×#103—611no—
273C2×#103—612no—
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×Q8C22⋊D6111yescharacteristic
290C22×Q8C22×D6111yescharacteristic
291C22×D4—21no—
292C22×D4—21no—
293C22×D4C22×D6112yes—
294C23×C4—22no—
295C22×C8—612no—
296C22×C8—611no—
297C22×C8—611no—
298#127—22no—
299#127C22⋊D6114yes—
300#125—1222no—
301#125—1222no—
302#124—28no—
303#124—24no—
304#124—24no—
305#123—216no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8C22⋊D6114yes—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—24no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—612no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4C22⋊D6111yescharacteristic
322C2×C4⋊C4C22⋊D6112yes—
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—22no—
327C2×C42—21no—
328C2×C42—1221no—
329C2×C42C22×D6112yes—
330C2×C42C22×D6111yescharacteristic
331#107—1222no—
332#106—1222no—
333C4⋊C8—1222no—
334#103—1224no—
335#103—1222no—
336#103—1222no—
337#102—1224no—
338#102—1224no—
339#102—1224no—
340#102—1224no—
341#97—1222no—
342#95—1221no—
343#95—1222no—
344#95—1222no—
345C24—22no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8C2×C22⋊D6112yes—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—44no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4C2×C22⋊D6114yes—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—28no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—1222no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4C2×C22⋊D6112yes—
372C22×C4C2×C22⋊D6112yes—
373C22×C4C2×C22⋊D6111yescharacteristic
374C22×C4C2×C22⋊D6111yescharacteristic
375C22×C4C23×D6111yescharacteristic
376C2×C8—1222no—
377C2×C8—1221no—
378C2×C8—1221no—
379C2×C8—1222no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—21no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—24no—
387C4⋊C4—614no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—4882no—
394C4⋊C4—2444no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C42—22no—
402C42—22no—
403C42—24no—
404C42—4881no—
405C42C2×C22⋊D6112yes—
406C42C2×C22⋊D6112yes—
407C23—24no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23C22×C22⋊D6111yescharacteristic
415Q8—22no—
416Q8—22no—
417Q8—44no—
418Q8—42no—
419D4—24no—
420D4—24no—
421D4—48no—
422D4—42no—
423D4—42no—
424D4—84no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—88no—
440C2×C4—1224no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4#1451974612yes—
449C2×C4#1451974612yes—
450C2×C4C22×C22⋊D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C2×C4#41×D6112yes—
453C2×C4#1451982412yes—
454C8—2441no—
455C8—2441no—
456C8—2442no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22C2×#1451974611yescharacteristic
465C22#1684041912yes—
466C22#1684056012yes—
467C22#1684056511yescharacteristic
468C22#1684057311yescharacteristic
469C4—22no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—4881no—
476C4—4881no—
477C4—2444no—
478C4#1684030811yescharacteristic
479C4#1684033311yescharacteristic
480C4#1684041112yes—
481C2—84no—
482C2#2144944311yescharacteristic
483C2#2145506111yescharacteristic
484C2#2145512311yescharacteristic
485C2#2145856612yes—
486C2#2145858112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#559C2
socleC23×C61C22×D4
prime radicalC2×#127×C61C22
hypercenterC2×#559D61
hypocenterC61#25658
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#16840069C2semidirect
3. #21451207C2semidirect
4. #21447055C2semidirect
5. #21447055C2semidirect
6. #21446760C2semidirect
8. #21446080C2semidirect
9. #21446066C2semidirect
10. #21445649C2semidirect
288. C61#25658semidirect

Maximal subgroups

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

subgroupindex
C2×#168401262
C2×#168400692
#214512072
#214470552
#214467602
#214467472
#214460802
#214460662
#214456492
C2×C61×#5592
#2565861