Finite Group Databasebeta

← #29697191

Subgroups — #29697191

Order 15616.

1190 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519697C211yesmaximal, characteristic
2#21448696C211yesmaximal, characteristic, complemented
3#21448639C211yesmaximal, characteristic
4#21446503C212yesmaximal, complemented
5#21446444C212yesmaximal, complemented
6#21446081C212yesmaximal, complemented
7#21445842C211yesmaximal, characteristic, complemented
8#21445751C211yesmaximal, characteristic
9#21445633C212yesmaximal, complemented
10#21445593C211yesmaximal, characteristic, complemented
11C61×#1824C211yesmaximal, characteristic, complemented
12C23×D244C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519739C2212yes—
15C2×C22⋊C4×D61—21no—
16C2×#14519708C2212yescomplemented
17C2×#14519699C2212yes—
18C2×#14519699C2212yes—
19C2×#14519697C418yescomplemented
20C2×#14519697C2211yescharacteristic
21C2×#14519697C2211yescharacteristic
22C2×#14519697C2211yescharacteristic
23C2×#14519697C2211yescharacteristic
24C2×C8×Dic61C2212yescomplemented
25#16839663C2212yes—
26#16839661C2212yescomplemented
27#16839648—22no—
28#16839640—22no—
29#16839622—21no—
30#16839622C2211yescharacteristic
31#16839617—21no—
32C4⋊C4×Dic61C2212yescomplemented
33C4×C22⋊Dic61—21no—
34C4×C22⋊Dic61C2211yescharacteristic
35#16839524C2212yescomplemented
36#16839516C2212yescomplemented
37C22×C4⋊C4×C61C2211yescharacteristic
38C2×C22⋊C8×C61C2211yescharacteristic
39C61×#380C2211yescharacteristic
40C61×#340C2212yescomplemented
41C61×#340C2212yescomplemented
42C24×D61—21no—
43C22×D244—24no—
44C22×D244—21no—
45C22×D244—22no—
46C22×D244—42no—
47C22×D244C2×C414yes—
48C22×D244C2311yescharacteristic
49C22×D244C2311yescharacteristic
50C22×C4×D61—22no—
51C22×#92418—42no—
52C22×#92418C2×C414yescomplemented
53C22×#92418C2311yescharacteristic
54C22×#92418C2311yescharacteristic
55#14519739—24no—
56#14519739—42no—
57C2×C4⋊Dic61—22no—
58C2×C4⋊Dic61C2312yes—
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61D411yescharacteristic
61C2×C22⋊Dic61D411yescharacteristic
62C2×#11758198—22no—
63C2×#11758198—22no—
64C2×#11758198—22no—
65C22⋊C4×D61—44no—
66C2×C4×Dic61—22no—
67C2×C4×Dic61C2312yes—
68#14519708—42no—
69#14519699—24no—
70#14519699—42no—
71#14519697C2×C414yes—
72#14519697C2×C414yes—
73#14519697C2×C414yescomplemented
74#14519697C2×C414yescomplemented
75C8×Dic61—42no—
76#14519684—21no—
77#14519684—22no—
78#14519684—22no—
79#14519684D412yes—
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C22×C8×C61C2312yes—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61—22no—
85C2×C4⋊C4×C61C2×C414yescomplemented
86C2×C4⋊C4×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—21no—
90C22⋊C8×C61—22no—
91#95×C61—21no—
92C24×C61C2×D411yescharacteristic
93C23×D61—21no—
94C23×D61—21no—
95C23×D61—24no—
96C23×D61—44no—
97C23×D61—42no—
98C23×D61—42no—
99C23×D61—82no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—21no—
103C22×C4×C61—21no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—82no—
107C22×C4×C61C22×C412yes—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4112yes—
110C22×C4×C61C2411yescharacteristic
111C22×Dic61—22no—
112C22×Dic61C2×D412yes—
113C22×Dic61C2×D412yes—
114C2×D244—21no—
115C2×D244—22no—
116C2×D244—22no—
117C2×D244—22no—
118C2×D244—28no—
119C2×D244—24no—
120C2×D244—21no—
121C2×D244—44no—
122C2×D244—48no—
123C2×D244—44no—
124C2×D244C22×C412yes—
125C2×D244C22×C412yes—
126C2×D244#4114yes—
127C2×C4×D61—44no—
128C2×C4×D61—44no—
129C2×C4×D61—82no—
130C2×C8×C61—22no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61C22×C412yes—
139C4⋊C4×C61C22×C412yes—
140C22⋊C4×C61—42no—
141C22⋊Dic61—22no—
142C22⋊Dic61—42no—
143C22⋊Dic61SD1614yes—
144#11758198—44no—
145#11758198—44no—
146#11758198—82no—
147C4⋊Dic61—22no—
148C4⋊Dic61—22no—
149C4⋊Dic61—82no—
150C4×Dic61—22no—
151C4×Dic61—22no—
152C4×Dic61—82no—
153C2×#92418—22no—
154C2×#92418—22no—
155C2×#92418—44no—
156C2×#92418C22×C412yes—
157C2×#92418C22×C412yes—
158C23×C61—22no—
159C23×C61—22no—
160C23×C61C4×D412yes—
161C23×C61C2×SD1612yes—
162C23×C61#13512yes—
163C23×C61C22×D411yescharacteristic
164C22×D61—22no—
165C22×D61—24no—
166C22×D61—22no—
167C22×D61—48no—
168C22×D61—41no—
169C22×D61—41no—
170C22×D61—42no—
171C22×D61—42no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—42no—
175C22×D61—48no—
176C22×D61—84no—
177C22×D61—88no—
178C22×D61—88no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—84no—
191C2×C4×C61C4×D412yes—
192C2×C4×C61C23×C411yescharacteristic
193C2×C4×C61C2×#4112yes—
194C2×C4×C61C2×#4111yescharacteristic
195C2×C4×C61H(5,𝔽2)12yes—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—22no—
201C2×Dic61—42no—
202C2×Dic61—42no—
203C2×Dic61—42no—
204C2×Dic61C2×SD1614yes—
205D244—22no—
206D244—22no—
207D244—22no—
208D244—24no—
209D244—48no—
210D244—42no—
211D244#11714yes—
212C4×D61—88no—
213C8×C61—42no—
214#92418—22no—
215#92418—22no—
216#22574—611nomaximal, Sylow
217C22×C61—22no—
218C22×C61—22no—
219C22×C61—24no—
220C22×C61—24no—
221C22×C61C4×SD1614yes—
222C22×C61C2×C4×D411yescharacteristic
223C22×C61#54612yes—
224C22×C61C22×SD1611yescharacteristic
225C22×C61C2×#13511yescharacteristic
226C22×C61#57612yes—
227C2×D61—24no—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—44no—
231C2×D61—44no—
232C2×D61—42no—
233C2×D61—42no—
234C2×D61—42no—
235C2×D61—816no—
236C2×D61—82no—
237C2×D61—82no—
238C4×C61—22no—
239C4×C61—42no—
240C4×C61—42no—
241C4×C61—82no—
242C4×C61C2×#11711yescharacteristic
243C4×C61#51811yescharacteristic
244C4×C61#55112yes—
245Dic61—24no—
246Dic61—42no—
247Dic61—82no—
248C22×#102—611no—
249#2345—611no—
250D4×C22⋊C4—611no—
251#1934—612no—
252#1926—612no—
253#1864—612no—
254#1824D6111yescharacteristic, complemented
255#1807—611no—
256#1786—611no—
257C4×#102—612no—
258C8×C22⋊C4—611no—
259C2×C61—24no—
260C2×C61#234211yescharacteristic
261C2×C61C2×C4×SD1611yescharacteristic
262C2×C61#294811yescharacteristic
263C2×C61#329512yes—
264C2×C61#329712yes—
265D61—44no—
266D61—82no—
267C23×D4—611no—
268C23×C8—611no—
269C2×C4×D4—612no—
270C22×C4⋊C4C2×D6111yescharacteristic
271C22×C22⋊C4—1221no—
272C2×#106—612no—
273C2×#102—618no—
274C2×#102—611no—
275C2×#102—611no—
276C2×#102—611no—
277C2×#102—611no—
278C2×#102—612no—
279C2×#102—612no—
280C2×C22⋊C8C2×D6111yescharacteristic
281C2×C4×C8—612no—
282#390—612no—
283#389—612no—
284#387—1222no—
285#386—1222no—
286#380—611no—
287#380—1221no—
288#380C2×D6111yescharacteristic
289#379—1221no—
290C4×C4⋊C4—612no—
291C4×C22⋊C4—611no—
292C4×C22⋊C4—1221no—
293#340—612no—
294#340C2×D6112yescomplemented
295#340C2×D6112yescomplemented
296#336—612no—
297C61#2257411yescharacteristic, complemented, Sylow
298C25—1221no—
299C22×D4—614no—
300C22×D4—611no—
301C22×D4—611no—
302C22×D4—1224no—
303C22×D4—1221no—
304C22×D4—1222no—
305C22×D4—2442no—
306C23×C4—21no—
307C23×C4—1222no—
308C23×C4C22×D6111yescharacteristic
309C22×C8—611no—
310C22×C8—611no—
311C22×C8—614no—
312C22×C8—2442no—
313C22×C8C22×D6112yes—
314C4×D4—1224no—
315C4×D4—2442no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4C4×D6114yescomplemented
321C2×C4⋊C4C22×D6112yes—
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—611no—
326C2×C22⋊C4—611no—
327C2×C22⋊C4—1222no—
328C2×C22⋊C4—1222no—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—2444no—
332C2×C42—612no—
333C2×C42—1222no—
334#106—2442no—
335#102—614no—
336#102—614no—
337#102—614no—
338#102—614no—
339#102—1224no—
340#102—2442no—
341C22⋊C8—22no—
342C4×C8—2442no—
343#95—21no—
344#95—612no—
345#95—1222no—
346#95—1222no—
347#95—1221no—
348C24—1221no—
349C24—1224no—
350C24—1221no—
351C24—4882no—
352C24—2444no—
353C24—2442no—
354C24—2442no—
355C24D4×D6111yescharacteristic
356C2×D4—612no—
357C2×D4—612no—
358C2×D4—614no—
359C2×D4—1228no—
360C2×D4—1222no—
361C2×D4—1221no—
362C2×D4—1221no—
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—1224no—
366C2×D4—2444no—
367C2×D4—2448no—
368C2×D4—2444no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—82no—
376C22×C4—612no—
377C22×C4—612no—
378C22×C4—1222no—
379C22×C4—4882no—
380C22×C4—2444no—
381C22×C4—2444no—
382C22×C4C2×C4×D6112yes—
383C22×C4D4×D6111yescharacteristic
384C22×C4#1175822412yes—
385C22×C4C23×D6111yescharacteristic
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—612no—
390C2×C8—612no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—2444no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—1222no—
400C4⋊C4—1222no—
401C4⋊C4—4882no—
402C4⋊C4C2×C4×D6112yes—
403C4⋊C4C2×C4×D6112yes—
404C22⋊C4—42no—
405C22⋊C4—614no—
406C22⋊C4—1222no—
407C22⋊C4—4882no—
408C22⋊C4—2444no—
409C22⋊C4—2444no—
410C22⋊C4—2442no—
411C42—1222no—
412C42—1222no—
413C42—4882no—
414C23—22no—
415C23—22no—
416C23—1222no—
417C23—1224no—
418C23—1222no—
419C23—4888no—
420C23—4888no—
421C23—4884no—
422C23—2448no—
423C23—2441no—
424C23—2441no—
425C23—2442no—
426C23—2448no—
427C23—2442no—
428C23—2441no—
429C23—2441no—
430C23—2442no—
431C23#1451973712yes—
432C23SD16×D6112yes—
433C23#1451980012yes—
434C23C2×D4×D6111yescharacteristic
435D4—614no—
436D4—1222no—
437D4—1222no—
438D4—1222no—
439D4—1224no—
440D4—2448no—
441D4—2442no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—84no—
454C2×C4—614no—
455C2×C4—1222no—
456C2×C4—1222no—
457C2×C4—1222no—
458C2×C4—1222no—
459C2×C4—1222no—
460C2×C4—4888no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4#1451973712yes—
465C2×C4C22×C4×D6111yescharacteristic
466C2×C4#41×D6112yes—
467C2×C4C2×#1175822411yescharacteristic
468C2×C4#1451982312yes—
469C8—42no—
470C8—1222no—
471C8—1222no—
472C22—24no—
473C22—22no—
474C22—22no—
475C22—24no—
476C22—1224no—
477C22—48816no—
478C22—4882no—
479C22—4882no—
480C22—2442no—
481C22—2442no—
482C22—2442no—
483C22—2444no—
484C22—2442no—
485C22—2444no—
486C22—2442no—
487C22#1683984714yes—
488C22C2×#1451973711yescharacteristic
489C22#1684040412yes—
490C22C2×SD16×D6111yescharacteristic
491C22C2×#1451980011yescharacteristic
492C22#1684056612yes—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—1224no—
498C4—4882no—
499C4—2442no—
500C4#117×D6111yescharacteristic
501C4#1684023611yescharacteristic
502C4#1684043512yes—
503C2—24no—
504C2—4882no—
505C2—2444no—
506C2#2144866111yescharacteristic
507C2C2×#1683984711yescharacteristic
508C2#2145466711yescharacteristic
509C2#2145764112yes—
510C2#2145766712yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1824C2
socleC23×C61C22×D4
prime radicalC23×D244C22
cosocleC22×C4×C61C24
hypercenter#1824D61
hypocenterC61#22574
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1824

Complements

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

NH ≅ G/Ntype
2. #21448696C2semidirect
4. #21446503C2semidirect
5. #21446444C2semidirect
6. #21446081C2semidirect
7. #21445842C2semidirect
9. #21445633C2semidirect
10. #21445593C2semidirect
11. C61×#1824C2semidirect
16. C2×#14519708C22semidirect
19. C2×#14519697C4semidirect
24. C2×C8×Dic61C22semidirect
26. #16839661C22semidirect
32. C4⋊C4×Dic61C22semidirect
35. #16839524C22semidirect
36. #16839516C22semidirect
40. C61×#340C22semidirect
41. C61×#340C22semidirect
52. C22×#92418C2×C4semidirect
73. #14519697C2×C4semidirect
74. #14519697C2×C4semidirect
85. C2×C4⋊C4×C61C2×C4semidirect
254. #1824D61semidirect
294. #340C2×D61semidirect
295. #340C2×D61semidirect
297. C61#22574semidirect
320. C2×C4⋊C4C4×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145196972
#214486962
#214486392
#214465032
#214464442
#214460812
#214458422
#214457512
#214456332
#214455932
C61×#18242
#2257461