Finite Group Databasebeta

← #29688950

Subgroups — #29688950

Order 15616.

3090 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 3090 conjugacy classes of subgroups — too many to draw. Only its 259 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519747C211yesmaximal, characteristic
2#379×D61C212yesmaximal, complemented
3#21446271C218yesmaximal, complemented
4#21445893C211yesmaximal, characteristic, complemented
5#21445733C212yesmaximal, complemented
6C61×#1819C211yesmaximal, characteristic, complemented
7C25×D61C2211yescharacteristic
8C23×D244C412yescomplemented
9C23×C22⋊D61C2211yescharacteristic
10C2×#14519747—24no—
11C2×#14519747—24no—
12C2×#14519747C414yescomplemented
13C2×#14519747C2214yes—
14C22×#11758198C412yescomplemented
15C2×C22⋊C4×D61—24no—
16C2×C22⋊C4×D61C2214yescomplemented
17C2×#14519691C2218yescomplemented
18#16839618C2212yes—
19#16839617—24no—
20#16839617C2214yescomplemented
21#16839529C2214yescomplemented
22C22×C22⋊C4×C61C2211yescharacteristic
23C61×#379C2212yes—
24C61×#342C2214yescomplemented
25C24×D61—48no—
26C24×D61—44no—
27C24×D61C2×C412yes—
28C24×D61C2×C412yes—
29C24×D61C2×C411yescharacteristic, complemented
30C24×D61D414yes—
31C24×D61D414yes—
32C24×D61C2312yes—
33C22×D244—24no—
34C22×D244—22no—
35C22×D244—22no—
36C22×D244—48no—
37C22×D244C2×C418yescomplemented
38C22×C22⋊D61—24no—
39C22×C22⋊D61—44no—
40C22×C22⋊D61C2×C412yescomplemented
41C22×C22⋊D61C2312yes—
42C23×Dic61C2×C411yescharacteristic, complemented
43C22×C4×D61—24no—
44#14519747—24no—
45#14519747—28no—
46#14519747—24no—
47#14519747—24no—
48#14519747—28no—
49#14519747—24no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61C2314yes—
52C2×#11758198—24no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×#11758198—28no—
56C2×#11758198—24no—
57C2×#11758198C2×C418yescomplemented
58C22⋊C4×D61—48no—
59C22⋊C4×D61—416no—
60#14519691—48no—
61C25×C61C2311yescharacteristic
62C23×C4×C61C2×C412yescomplemented
63C2×C22⋊C4×C61—24no—
64C2×C22⋊C4×C61—24no—
65C2×C22⋊C4×C61C2×C412yescomplemented
66C2×C22⋊C4×C61C2314yes—
67C2×C22⋊C4×C61C2312yes—
68C24×C61—24no—
69C24×C61—44no—
70C24×C61C22×C411yescharacteristic
71C24×C61C22×C411yescharacteristic
72C24×C61C2×D414yes—
73C24×C61C2411yescharacteristic
74C23×D61—28no—
75C23×D61—28no—
76C23×D61—216no—
77C23×D61—22no—
78C23×D61—48no—
79C23×D61—48no—
80C23×D61—48no—
81C23×D61—48no—
82C23×D61—44no—
83C23×D61—44no—
84C23×D61—44no—
85C23×D61—48no—
86C23×D61—48no—
87C23×D61—48no—
88C23×D61—42no—
89C23×D61—464no—
90C23×D61—44no—
91C23×D61C22⋊C414yes—
92C23×D61C22⋊C412yes—
93C23×D61C22⋊C412yes—
94C23×D61C22⋊C414yes—
95C23×D61C22⋊C412yes—
96C23×D61C22⋊C412yes—
97C23×D61C22×C414yes—
98C23×D61C22×C412yes—
99C23×D61C2×D412yes—
100C23×D61C2×D414yes—
101C23×D61C2×D412yes—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—24no—
105C22×C4×C61—44no—
106C22×C4×C61C22×C414yes—
107C2×C22⋊D61—24no—
108C2×C22⋊D61—24no—
109C2×C22⋊D61—28no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—416no—
112C22×Dic61—22no—
113C22×Dic61—24no—
114C22×Dic61—42no—
115C22×Dic61C22×C412yes—
116C2×D244—28no—
117C2×D244—28no—
118C2×D244—48no—
119C2×D244—416no—
120C2×D244—48no—
121C2×D244—416no—
122C2×C4×D61—416no—
123C2×C4×D61—48no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—28no—
127C22⋊C4×C61—48no—
128C22⋊Dic61—28no—
129C22⋊Dic61—48no—
130#11758198—28no—
131#11758198—28no—
132#11758198—416no—
133#11758198—48no—
134C23×C61—22no—
135C23×C61—24no—
136C23×C61—22no—
137C23×C61—28no—
138C23×C61—28no—
139C23×C61—42no—
140C23×C61—44no—
141C23×C61—42no—
142C23×C61—416no—
143C23×C61C2×C22⋊C412yes—
144C23×C61C2×C22⋊C412yes—
145C23×C61C23×C411yescharacteristic
146C23×C61C22×D412yes—
147C23×C61H(5,𝔽2)14yes—
148C22×D61—24no—
149C22×D61—28no—
150C22×D61—28no—
151C22×D61—24no—
152C22×D61—28no—
153C22×D61—24no—
154C22×D61—28no—
155C22×D61—28no—
156C22×D61—28no—
157C22×D61—28no—
158C22×D61—28no—
159C22×D61—28no—
160C22×D61—22no—
161C22×D61—24no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61—44no—
165C22×D61—44no—
166C22×D61—42no—
167C22×D61—44no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—44no—
171C22×D61—42no—
172C22×D61—44no—
173C22×D61—44no—
174C22×D61—44no—
175C22×D61—464no—
176C22×D61—432no—
177C22×D61—432no—
178C22×D61—432no—
179C22×D61—464no—
180C22×D61—44no—
181C22×D61—44no—
182C22×D61—44no—
183C22×D61—44no—
184C22×D61—88no—
185C22×D61#9918yescomplemented
186C22×D61C2×C22⋊C414yes—
187C22×D61C2×C22⋊C414yes—
188C2×C4×C61—24no—
189C2×C4×C61—44no—
190C2×C4×C61—44no—
191C2×C4×C61—44no—
192C2×C4×C61—48no—
193C2×C4×C61—44no—
194C22⋊D61—48no—
195C22⋊D61—48no—
196C2×Dic61—22no—
197C2×Dic61—24no—
198C2×Dic61—44no—
199C2×Dic61—48no—
200C2×Dic61—44no—
201D244—416no—
202D244—416no—
203C4×D61—816no—
204#21746—611nomaximal, Sylow
205C22×C61—28no—
206C22×C61—22no—
207C22×C61—24no—
208C22×C61—22no—
209C22×C61—22no—
210C22×C61—22no—
211C22×C61—24no—
212C22×C61—44no—
213C22×C61—416no—
214C22×C61—44no—
215C22×C61C2×#9914yes—
216C22×C61C22×C22⋊C411yescharacteristic
217C22×C61#51812yes—
218C22×C61#53414yes—
219C2×D61—216no—
220C2×D61—24no—
221C2×D61—24no—
222C2×D61—24no—
223C2×D61—24no—
224C2×D61—24no—
225C2×D61—48no—
226C2×D61—48no—
227C2×D61—416no—
228C2×D61—416no—
229C2×D61—416no—
230C2×D61—416no—
231C2×D61—48no—
232C2×D61—48no—
233C2×D61—48no—
234C2×D61—42no—
235C2×D61—44no—
236C2×D61—88no—
237C2×D61—84no—
238C2×D61C2×#9918yes—
239C4×C61—44no—
240C4×C61—84no—
241Dic61—42no—
242Dic61—84no—
243C22×#120—611no—
244C2×#379—612no—
245#1898—618no—
246#1819—611no—
247#1819D6111yescharacteristic, complemented
248#1781—612no—
249C2×C61—24no—
250C2×C61—22no—
251C2×C61—24no—
252C2×C61—42no—
253C2×C61#230811yescharacteristic
254C2×C61C22×#9912yes—
255D61—28no—
256D61—44no—
257D61—82no—
258C26—611no—
259C23×D4—611no—
260C23×D4—612no—
261C2×#120—614no—
262C2×#120—614no—
263C2×#120—1224no—
264C2×#120—1224no—
265C22×C22⋊C4—612no—
266C22×C22⋊C4—614no—
267C22×C22⋊C4—1224no—
268C22×C22⋊C4C2×D6111yescharacteristic
269C2×#99—618no—
270#379—612no—
271#379—614no—
272#379—1224no—
273#379C2×D6112yes—
274#342—614no—
275#342C2×D6114yescomplemented
276C61#2174611yescharacteristic, complemented, Sylow
277C25—612no—
278C25—614no—
279C25—612no—
280C25—612no—
281C25—614no—
282C25—611no—
283C25—2444no—
284C25—2448no—
285C25C22×D6111yescharacteristic
286C22×D4—612no—
287C22×D4—612no—
288C22×D4—618no—
289C22×D4—1222no—
290C22×D4—1224no—
291C22×D4—1222no—
292C22×D4—1224no—
293C22×D4—2448no—
294C22×D4—2444no—
295C23×C4—611no—
296C23×C4—1224no—
297C23×C4C4×D6112yescomplemented
298#120—1224no—
299#120—1228no—
300#120—1224no—
301#120—1224no—
302#120—1228no—
303#120—1224no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—618no—
307C2×C22⋊C4—614no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1224no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1224no—
312C2×C22⋊C4—1228no—
313C2×C22⋊C4—1224no—
314C2×C22⋊C4—2448no—
315C2×C22⋊C4—24416no—
316C2×C22⋊C4C4×D6112yescomplemented
317C2×C22⋊C4C22×D6114yes—
318C2×C22⋊C4C22×D6112yes—
319#99—2448no—
320C24—24no—
321C24—44no—
322C24—612no—
323C24—612no—
324C24—614no—
325C24—614no—
326C24—614no—
327C24—612no—
328C24—612no—
329C24—612no—
330C24—614no—
331C24—612no—
332C24—612no—
333C24—12216no—
334C24—1228no—
335C24—1228no—
336C24—1222no—
337C24—2442no—
338C24—2444no—
339C24—2444no—
340C24—2444no—
341C24—2448no—
342C24—2448no—
343C24—2448no—
344C24—2444no—
345C24—2448no—
346C24—2448no—
347C24—2448no—
348C24—2448no—
349C24—24464no—
350C24C2×C4×D6111yescharacteristic
351C24C2×C4×D6111yescharacteristic
352C24D4×D6114yes—
353C24C23×D6111yescharacteristic
354C2×D4—1228no—
355C2×D4—1228no—
356C2×D4—1224no—
357C2×D4—1228no—
358C2×D4—1224no—
359C2×D4—2448no—
360C2×D4—24416no—
361C2×D4—2448no—
362C2×D4—24416no—
363C2×D4—2444no—
364C2×D4—24416no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—612no—
370C22×C4—1222no—
371C22×C4—1224no—
372C22×C4—24416no—
373C22×C4—2448no—
374C22×C4—2442no—
375C22×C4C2×C4×D6114yes—
376C22⋊C4—24no—
377C22⋊C4—24no—
378C22⋊C4—28no—
379C22⋊C4—48no—
380C22⋊C4—1228no—
381C22⋊C4—1228no—
382C22⋊C4—1228no—
383C22⋊C4—24416no—
384C22⋊C4—2448no—
385C22⋊C4—2448no—
386C23—22no—
387C23—24no—
388C23—22no—
389C23—28no—
390C23—28no—
391C23—42no—
392C23—44no—
393C23—42no—
394C23—416no—
395C23—614no—
396C23—618no—
397C23—614no—
398C23—1228no—
399C23—1228no—
400C23—1228no—
401C23—1228no—
402C23—1228no—
403C23—1228no—
404C23—1224no—
405C23—1228no—
406C23—1224no—
407C23—1228no—
408C23—1224no—
409C23—1228no—
410C23—1224no—
411C23—1222no—
412C23—4888no—
413C23—2444no—
414C23—2444no—
415C23—2444no—
416C23—2442no—
417C23—2442no—
418C23—2442no—
419C23—2444no—
420C23—2442no—
421C23—2444no—
422C23—2444no—
423C23—2442no—
424C23—2444no—
425C23—2442no—
426C23—2444no—
427C23—2444no—
428C23—2444no—
429C23—24432no—
430C23—24432no—
431C23—24464no—
432C23—24464no—
433C23—24432no—
434C23—2444no—
435C23C22⋊C4×D6112yes—
436C23C22⋊C4×D6112yes—
437C23C22×C4×D6111yescharacteristic
438C23C2×D4×D6112yes—
439C23#1451982314yes—
440D4—24416no—
441D4—24416no—
442D4—2448no—
443D4—2448no—
444C2×C4—24no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—48no—
449C2×C4—44no—
450C2×C4—1222no—
451C2×C4—1224no—
452C2×C4—48816no—
453C2×C4—2448no—
454C2×C4—2444no—
455C2×C4—2444no—
456C22—28no—
457C22—24no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—22no—
463C22—44no—
464C22—416no—
465C22—44no—
466C22—618no—
467C22—12216no—
468C22—1224no—
469C22—1224no—
470C22—1224no—
471C22—1224no—
472C22—1224no—
473C22—4884no—
474C22—4888no—
475C22—2448no—
476C22—24416no—
477C22—24416no—
478C22—24416no—
479C22—2448no—
480C22—2448no—
481C22—2448no—
482C22—24416no—
483C22—2448no—
484C22—2444no—
485C22—2442no—
486C22#99×D6114yes—
487C22C2×C22⋊C4×D6111yescharacteristic
488C22#1684023612yes—
489C22#1684032314yes—
490C4—44no—
491C4—84no—
492C4—4884no—
493C4—2442no—
494C2—24no—
495C2—22no—
496C2—24no—
497C2—42no—
498C2—1228no—
499C2—4882no—
500C2—2444no—
501C2#2144839611yescharacteristic
502C2C2×#99×D6112yes—
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC24C23×D61
FittingC61×#1819C2
socleC22×C61C22×C22⋊C4
prime radicalC22×#14519747C2
cosocleC24×C61C24
hypercenter#1819D61
hypocenterC61#21746
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
2. #379×D61C2semidirect
3. #21446271C2semidirect
4. #21445893C2semidirect
5. #21445733C2semidirect
6. C61×#1819C2semidirect
8. C23×D244C4semidirect
12. C2×#14519747C4semidirect
14. C22×#11758198C4semidirect
16. C2×C22⋊C4×D61C22semidirect
17. C2×#14519691C22semidirect
20. #16839617C22semidirect
21. #16839529C22semidirect
24. C61×#342C22semidirect
29. C24×D61C2×C4semidirect
37. C22×D244C2×C4semidirect
40. C22×C22⋊D61C2×C4semidirect
42. C23×Dic61C2×C4semidirect
57. C2×#11758198C2×C4semidirect
62. C23×C4×C61C2×C4semidirect
65. C2×C22⋊C4×C61C2×C4semidirect
185. C22×D61#99semidirect
247. #1819D61semidirect
275. #342C2×D61semidirect
276. C61#21746semidirect
297. C23×C4C4×D61semidirect
316. C2×C22⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197472
#379×D612
#214462712
#214458932
#214457332
C61×#18192
#2174661