Finite Group Databasebeta

← #29718425

Subgroups — #29718425

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×#16840080C211yesmaximal, characteristic
2C22×#14519700C211yesmaximal, characteristic, complemented
3#21449477C211yesmaximal, characteristic
4#21447197C212yesmaximal, complemented
5#21447060C212yesmaximal, complemented
6#21446751C212yesmaximal, complemented
7#21446453C212yesmaximal, complemented
8#21446245C211yesmaximal, characteristic
9#21445999C211yesmaximal, characteristic, complemented
10#21445753C211yesmaximal, characteristic, complemented
11C61×#2345C211yesmaximal, characteristic, complemented
12C23×Dic122C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519765C2212yes—
15C2×#14519751C2211yescharacteristic
16#16840080—24no—
17C2×#14519708C2211yescharacteristic
18C2×#14519708C2211yescharacteristic
19C2×C4⋊#92418C2212yes—
20C2×#14519700—24no—
21C2×#14519700C2212yes—
22C2×#14519700C2212yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519697C2212yes—
28C2×C22⋊#92418C2211yescharacteristic
29#16839675—22no—
30#16839668C2212yes—
31#16839639—21no—
32#16839639—21no—
33#16839524C2212yes—
34#16839517C2212yes—
35C22×C4⋊C4×C61C2211yescharacteristic
36C61×#389C2212yes—
37C61×#387—22no—
38C61×#380—21no—
39C61×#380C2211yescharacteristic
40C4×C4⋊C4×C61C2212yes—
41C4×C22⋊C4×C61—21no—
42C4×C22⋊C4×C61C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—22no—
46C22×Dic122—21no—
47C22×Dic122—42no—
48C22×Dic122C2312yes—
49C22×D244C2311yescharacteristic
50C22×C22⋊D61—21no—
51C23×Dic61—21no—
52C22×#92418—22no—
53C22×#92418—42no—
54C22×#92418C2311yescharacteristic
55C22×#92418C2311yescharacteristic
56C22×#92418C2312yes—
57#14519765—24no—
58#14519765—42no—
59#14519751—22no—
60#14519751—22no—
61C2×C4⋊Dic61C2311yescharacteristic
62C2×#11758198—21no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65#14519708—22no—
66#14519708—22no—
67C4⋊#92418—42no—
68#14519700—24no—
69#14519700—24no—
70#14519700—24no—
71#14519700—42no—
72#14519697—22no—
73#14519697—22no—
74#14519697—42no—
75C22⋊#92418—22no—
76#14519684—22no—
77#14519684—22no—
78C23×C4×C61—21no—
79C23×C4×C61C2311yescharacteristic
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61D414yes—
84C2×C4⋊C4×C61C2312yes—
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C4⋊C4×C61C2311yescharacteristic
87C2×C22⋊C4×C61—21no—
88C2×C22⋊C4×C61D411yescharacteristic
89C2×C22⋊C4×C61D411yescharacteristic
90C2×C42×C61—22no—
91C2×C42×C61C2312yes—
92#95×C61—22no—
93#95×C61—22no—
94#95×C61—21no—
95#95×C61D412yes—
96C24×C61C2×D411yescharacteristic
97C23×D61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—24no—
103C22×C4×C61—22no—
104C22×C4×C61—82no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61#4112yes—
110C22×C4×C61C2411yescharacteristic
111C2×C22⋊D61—42no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—42no—
114C22×Dic61—22no—
115C22×Dic61—21no—
116C22×Dic61—44no—
117C22×Dic61—42no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—22no—
121C2×D244—22no—
122C2×Dic122—28no—
123C2×Dic122—22no—
124C2×Dic122—22no—
125C2×Dic122—22no—
126C2×Dic122—22no—
127C2×Dic122—22no—
128C2×Dic122—48no—
129C2×Dic122—44no—
130C2×Dic122—44no—
131C2×Dic122—42no—
132C2×Dic122#4114yes—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—82no—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D412yes—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—42no—
144C42×C61—22no—
145C42×C61—22no—
146C42×C61—82no—
147#11758198—44no—
148#11758198—44no—
149#11758198—42no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
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#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—
171C2×C4×C61—22no—
172C2×C4×C61—22no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—84no—
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—24no—
196C2×Dic61—48no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202D244—41no—
203D244—41no—
204D244—42no—
205Dic122—24no—
206Dic122—24no—
207Dic122—48no—
208Dic122—42no—
209Dic122—42no—
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#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—42no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—82no—
235C4×C61#52511yescharacteristic
236C4×C61#53511yescharacteristic
237C4×C61#55112yes—
238Dic61—44no—
239Dic61—81no—
240C2×#481—611no—
241C22×#103—611no—
242#2436—611no—
243#2345D6111yescharacteristic, complemented
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#241111yescharacteristic
253C2×C61C2×#46411yescharacteristic
254C2×C61#304211yescharacteristic
255C2×C61#330112yes—
256C2×C61#331612yes—
257D61—161no—
258C23×Q8—611no—
259C23×C8—611no—
260C2×#124—612no—
261C2×#121—611no—
262C22×C4⋊C4C2×D6111yescharacteristic
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—1222no—
277#391—612no—
278#389C2×D6112yes—
279#387—22no—
280#386—1221no—
281#386—1221no—
282#380—21no—
283#380C2×D6111yescharacteristic
284C4×C4⋊C4C2×D6112yes—
285C4×C22⋊C4—21no—
286C4×C22⋊C4C2×D6111yescharacteristic
287#340—612no—
288#336—612no—
289C61#2448311yescharacteristic, complemented, Sylow
290C22×Q8—612no—
291C22×Q8—1224no—
292C22×Q8—1221no—
293C22×Q8—1222no—
294C22×Q8—1222no—
295C22×Q8—2442no—
296C22×D4—611no—
297C22×D4—1221no—
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—1224no—
307#124—2442no—
308#121—1222no—
309#121—1222no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—611no—
314C2×C4⋊C4D24414yes—
315C2×C4⋊C4C22×D6112yes—
316C2×C4⋊C4C22×D6111yescharacteristic
317C2×C4⋊C4C22×D6111yescharacteristic
318C2×C22⋊C4—21no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4C22⋊D6111yescharacteristic
323C2×C22⋊C4C22⋊D6111yescharacteristic
324C2×C42—22no—
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—21no—
340#95—1222no—
341#95—1222no—
342#95C22⋊D6112yes—
343C24—1221no—
344C24C2×C22⋊D6111yescharacteristic
345C2×Q8—614no—
346C2×Q8—1228no—
347C2×Q8—1222no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—1222no—
351C2×Q8—1222no—
352C2×Q8—2444no—
353C2×Q8—2448no—
354C2×Q8—2444no—
355C2×Q8—2442no—
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—21no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—82no—
370C22×C4—1222no—
371C22×C4—1221no—
372C22×C4—2444no—
373C22×C4—2442no—
374C22×C4C2×D24412yes—
375C22×C4#1175822212yes—
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—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—82no—
394C4⋊C4—1221no—
395C4⋊C4—1221no—
396C4⋊C4C2×D24412yes—
397C4⋊C4C2×D24412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—42no—
401C22⋊C4—2444no—
402C22⋊C4—2444no—
403C22⋊C4—2442no—
404C42—22no—
405C42—22no—
406C42—82no—
407C23—22no—
408C23—22no—
409C23—2441no—
410C23—2441no—
411C23—2441no—
412C23—2442no—
413C23—2442no—
414C23#1451975112yes—
415C23#1451979712yes—
416C23#1451980812yes—
417C23C22×C22⋊D6111yescharacteristic
418Q8—1224no—
419Q8—1224no—
420Q8—2448no—
421Q8—2442no—
422Q8—2442no—
423D4—4884no—
424D4—2441no—
425D4—2441no—
426D4—2442no—
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—22no—
438C2×C4—22no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—84no—
445C2×C4—1224no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4—2448no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4#1451975112yes—
453C2×C4C22×D24411yescharacteristic
454C2×C4#41×D6112yes—
455C2×C4C2×#1175822211yescharacteristic
456C2×C4#1451982312yes—
457C8—2441no—
458C8—2441no—
459C8—2442no—
460C22—24no—
461C22—24no—
462C22—22no—
463C22—22no—
464C22—4882no—
465C22—4882no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22#1683998414yes—
470C22C2×#1451975111yescharacteristic
471C22#1684040012yes—
472C22C2×#1451979711yescharacteristic
473C22C2×#1451980811yescharacteristic
474C22#1684057312yes—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—82no—
481C4—4881no—
482C4—2444no—
483C4#1684027411yescharacteristic
484C4#1684033111yescharacteristic
485C4#1684043612yes—
486C2—24no—
487C2—9761no—
488C2#2144919811yescharacteristic
489C2C2×#1683998411yescharacteristic
490C2#2145533111yescharacteristic
491C2#2145771512yes—
492C2#2145792412yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2345C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#2345D61
hypocenterC61#24483
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2345

Complements

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

NH ≅ G/Ntype
2. C22×#14519700C2semidirect
4. #21447197C2semidirect
5. #21447060C2semidirect
6. #21446751C2semidirect
7. #21446453C2semidirect
9. #21445999C2semidirect
10. #21445753C2semidirect
11. C61×#2345C2semidirect
243. #2345D61semidirect
289. C61#24483semidirect

Maximal subgroups

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

subgroupindex
C2×#168400802
C22×#145197002
#214494772
#214471972
#214470602
#214467512
#214464532
#214462452
#214459992
#214457532
C61×#23452
#2448361