Finite Group Databasebeta

← #29718427

Subgroups — #29718427

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×#16840078C211yesmaximal, characteristic
2C22×#14519701C211yesmaximal, characteristic, complemented
3#21448694C211yesmaximal, characteristic, complemented
4#21447200C212yesmaximal, complemented
5#21447055C212yesmaximal, complemented
6#21446752C212yesmaximal, complemented
7#21446450C212yesmaximal, complemented
8#21446248C211yesmaximal, characteristic
9#21445999C211yesmaximal, characteristic, complemented
10#21445754C211yesmaximal, characteristic, complemented
11C61×#2436C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C22×C4⋊Dic61C2211yescharacteristic
14#16840078—24no—
15C2×#14519708C2211yescharacteristic
16C2×#14519708C2211yescharacteristic
17C2×C4⋊#92418C2212yes—
18C2×#14519701—24no—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×#14519701C2212yes—
22C2×#14519701C2212yes—
23C2×#14519698C2211yescharacteristic
24C2×#14519698C2211yescharacteristic
25C2×#14519698C2212yes—
26C2×C22⋊#92418C2211yescharacteristic
27#16839660C2212yes—
28#16839645—22no—
29#16839622C2211yescharacteristic
30#16839620—21no—
31C4×C4⋊Dic61C2212yes—
32C22⋊C4×Dic61—21no—
33#16839524C2212yes—
34#16839517C2212yes—
35C23×Q8×C61C2211yescharacteristic
36C2×#124×C61C2212yes—
37C2×#121×C61C2211yescharacteristic
38C61×#393—22no—
39C61×#391C2212yes—
40C61×#386—21no—
41C61×#386—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C23×Dic61—21no—
44C22×#92418—22no—
45C22×#92418—42no—
46C22×#92418C2311yescharacteristic
47C22×#92418C2311yescharacteristic
48C22×#92418C2312yes—
49C2×C4⋊Dic61—21no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61D414yes—
52C2×C4⋊Dic61C2312yes—
53C2×C4⋊Dic61C2311yescharacteristic
54C2×C4⋊Dic61C2311yescharacteristic
55C2×#11758196—22no—
56C2×C22⋊Dic61—21no—
57C2×C4×Dic61—22no—
58#14519708—22no—
59#14519708—22no—
60C4⋊#92418—42no—
61#14519701—24no—
62#14519701—24no—
63#14519701—24no—
64#14519701—42no—
65#14519698—22no—
66#14519698—22no—
67#14519698—42no—
68C22⋊#92418—22no—
69#14519684—22no—
70#14519684—22no—
71#14519684—21no—
72C22×Q8×C61—21no—
73C22×Q8×C61—24no—
74C22×Q8×C61—22no—
75C22×Q8×C61—22no—
76C22×Q8×C61—42no—
77C22×Q8×C61C2312yes—
78C22×D4×C61—21no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82#124×C61—24no—
83#124×C61—42no—
84#121×C61—22no—
85#121×C61—22no—
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—22no—
88C2×C22⋊C4×C61—22no—
89C2×C22⋊C4×C61—21no—
90C2×C22⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61D411yescharacteristic
92C2×C42×C61C2312yes—
93#95×C61—22no—
94#95×C61—22no—
95#95×C61D412yes—
96C24×C61—21no—
97C24×C61C2×D411yescharacteristic
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—
109C2×D4×C61—21no—
110C2×D4×C61—21no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—42no—
114C2×D4×C61—42no—
115C2×D4×C61—42no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—44no—
121C22×C4×C61—42no—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61#4112yes—
127C22×C4×C61C2411yescharacteristic
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61—82no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—44no—
138C22⋊C4×C61—44no—
139C22⋊C4×C61—42no—
140C42×C61—22no—
141C42×C61—22no—
142C22⋊Dic61—42no—
143C4⋊Dic61—24no—
144C4⋊Dic61—21no—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61C2×D412yes—
149C4⋊Dic61C2×D412yes—
150#11758196—82no—
151C4×Dic61—82no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—22no—
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
171Q8×C61—24no—
172Q8×C61—24no—
173Q8×C61—48no—
174Q8×C61—42no—
175Q8×C61—42no—
176D4×C61—41no—
177D4×C61—41no—
178D4×C61—42no—
179D4×C61—84no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—24no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—48no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61#12112yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C2×#4112yes—
200C2×C4×C61C2×#4111yescharacteristic
201C2×C4×C61H(5,𝔽2)12yes—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—42no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—84no—
210#92418—41no—
211#92418—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—44no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61#52511yescharacteristic
235C4×C61#53511yescharacteristic
236C4×C61#55112yes—
237Dic61—42no—
238Dic61—42no—
239Dic61—82no—
240C2×#481—611no—
241C22×#103—611no—
242#2436D6111yescharacteristic
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×Q8C2×D6111yescharacteristic
259C23×C8—611no—
260C2×#124C2×D6112yes—
261C2×#121C2×D6111yescharacteristic
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—612no—
270C2×#103—612no—
271C2×#103—1224no—
272C2×#102—611no—
273C2×#102—611no—
274C2×#102—612no—
275C2×C22⋊C8—611no—
276#393—22no—
277#391C2×D6112yes—
278#389—612no—
279#387—1222no—
280#386—21no—
281#386—21no—
282#380—611no—
283#380—1221no—
284C4×C4⋊C4—612no—
285C4×C22⋊C4—1221no—
286C4×C22⋊C4C2×D6111yescharacteristic
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—21no—
297C22×D4C22×D6111yescharacteristic
298C23×C4—21no—
299C23×C4—1221no—
300C23×C4C22×D6111yescharacteristic
301C22×C8—612no—
302C22×C8—611no—
303C22×C8—611no—
304C22×C8—1222no—
305C22×C8—2442no—
306#124—24no—
307#124—42no—
308#121—22no—
309#121—22no—
310C2×C4⋊C4—614no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—612no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4—1222no—
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—1221no—
322C2×C22⋊C4C22⋊D6111yescharacteristic
323C2×C22⋊C4C22⋊D6111yescharacteristic
324C2×C42—1222no—
325C2×C42C22×D6112yes—
326#106—1222no—
327#106—1222no—
328C4⋊C8—2442no—
329#103—1224no—
330#103—1224no—
331#103—1224no—
332#103—2442no—
333#102—1222no—
334#102—1222no—
335#102—2442no—
336C22⋊C8—1222no—
337#95—22no—
338#95—22no—
339#95—1222no—
340#95—1222no—
341#95—1221no—
342#95C22⋊D6112yes—
343C24—21no—
344C24C2×C22⋊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—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—44no—
368C22×C4—42no—
369C22×C4—1224no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1222no—
373C22×C4—4882no—
374C22×C4D4×D6112yes—
375C22×C4#1175822412yes—
376C22×C4C2×C22⋊D6112yes—
377C22×C4C2×C22⋊D6112yes—
378C22×C4C2×C22⋊D6111yescharacteristic
379C22×C4C23×D6111yescharacteristic
380C2×C8—1221no—
381C2×C8—1221no—
382C2×C8—1222no—
383C2×C8—1222no—
384C2×C8—1222no—
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—2444no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—612no—
391C4⋊C4—612no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1224no—
395C4⋊C4—1222no—
396C4⋊C4—1222no—
397C4⋊C4—4882no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—42no—
403C22⋊C4—2442no—
404C42—22no—
405C42—22no—
406C42—4882no—
407C23—22no—
408C23—22no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23#1451974812yes—
415C23#1451979712yes—
416C23#1451980712yes—
417C23C22×C22⋊D6111yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—48no—
421Q8—42no—
422Q8—42no—
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—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—1222no—
445C2×C4—1222no—
446C2×C4—4884no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451974812yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C2×#1175822412yes—
455C2×C4C2×#1175822411yescharacteristic
456C2×C4#1451982212yes—
457C8—2441no—
458C8—2441no—
459C8—2442no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—82no—
465C22—82no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#1683998114yes—
470C22C2×#1451974811yescharacteristic
471C22#1684040612yes—
472C22C2×#1451979711yescharacteristic
473C22C2×#1451980711yescharacteristic
474C22#1684057012yes—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—44no—
479C4—81no—
480C4—4882no—
481C4—2442no—
482C4—2442no—
483C4#1684026811yescharacteristic
484C4#1684032711yescharacteristic
485C4#1684044312yes—
486C2—24no—
487C2—161no—
488C2#2144920011yescharacteristic
489C2C2×#1683998111yescharacteristic
490C2#2145532711yescharacteristic
491C2#2145772112yes—
492C2#2145793012yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2436C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenter#2436D61
hypocenterC61#24483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2436

Complements

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

NH ≅ G/Ntype
2. C22×#14519701C2semidirect
3. #21448694C2semidirect
4. #21447200C2semidirect
5. #21447055C2semidirect
6. #21446752C2semidirect
7. #21446450C2semidirect
9. #21445999C2semidirect
10. #21445754C2semidirect
289. C61#24483semidirect

Maximal subgroups

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

subgroupindex
C2×#168400782
C22×#145197012
#214486942
#214472002
#214470552
#214467522
#214464502
#214462482
#214459992
#214457542
C61×#24362
#2448361