Finite Group Databasebeta

← #29652461

Subgroups — #29652461

Order 15616.

1450 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 183 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
1#21455069C212yesmaximal, complemented
2#21449154C211yesmaximal, characteristic, complemented
3#21444788C214yesmaximal, complemented
4#21444754C214yesmaximal, complemented
5#21444348C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C2×C61×#537C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519803—22no—
10C2×#14519760—21no—
11#16839905—24no—
12#16839900C2214yescomplemented
13#16839895C2214yescomplemented
14#16839761—22no—
15#16839726C2212yes—
16#16839682C2212yescomplemented
17#16839679C2212yescomplemented
18#16839641—21no—
19#16839633—22no—
20#16839625C2211yescharacteristic
21#16839496C2218yescomplemented
22#16839488C2214yescomplemented
23C22×#41×C61—21no—
24C61×#537—24no—
25C61×#537—22no—
26C61×#537C2214yes—
27C2×C22⋊Q8×C61—22no—
28C2×#121×C61—22no—
29C2×#121×C61—21no—
30C2×#121×C61C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C2×#11758222—22no—
33C2×#11758222—22no—
34C2×#11758222—44no—
35C22×Dic122C2311yescharacteristic
36C22×D244C2311yescharacteristic
37C22×C22⋊D61—21no—
38C22×C4×D61—21no—
39#14519803—48no—
40C2×#11758200—24no—
41C2×#11758199—24no—
42C2×#11758194—22no—
43#14519760—44no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—21no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48#14519700—24no—
49#14519697—24no—
50C22⋊#92418—22no—
51C22⋊#92418C2314yes—
52#14519684—22no—
53#14519684—21no—
54#14519684C2312yes—
55C2×#41×C61—24no—
56C2×#41×C61—24no—
57C2×#41×C61—48no—
58C22×Q8×C61—21no—
59C22×D4×C61—21no—
60C22×D4×C61—22no—
61C22×D4×C61—21no—
62C22×D4×C61D412yes—
63C23×C4×C61—22no—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C22⋊Q8×C61—28no—
68C22⋊Q8×C61—28no—
69#121×C61—24no—
70#121×C61—28no—
71#121×C61—28no—
72#121×C61—22no—
73#121×C61—22no—
74#121×C61—24no—
75#121×C61D414yes—
76#121×C61C2314yes—
77C2×C4⋊C4×C61—21no—
78C2×C4⋊C4×C61—22no—
79C2×C4⋊C4×C61—22no—
80C2×C4⋊C4×C61D412yes—
81C2×C4⋊C4×C61D412yes—
82C2×C4⋊C4×C61C2312yes—
83C2×C22⋊C4×C61—22no—
84C2×C22⋊C4×C61—22no—
85C24×C61—22no—
86C24×C61C2×D411yescharacteristic
87C23×D61—21no—
88#41×C61—416no—
89#41×C61—48no—
90#41×C61—48no—
91C2×Q8×C61—24no—
92C2×Q8×C61—44no—
93C2×D4×C61—24no—
94C2×D4×C61—24no—
95C2×D4×C61—24no—
96C2×D4×C61—24no—
97C2×D4×C61—22no—
98C2×D4×C61—44no—
99C2×D4×C61—44no—
100C2×D4×C61—48no—
101C2×D4×C61—82no—
102C2×D4×C61C2×D414yes—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—24no—
106C22×C4×C61—28no—
107C22×C4×C61—21no—
108C22×C4×C61—22no—
109C22×C4×C61—21no—
110C22×C4×C61—42no—
111C22×C4×C61—48no—
112C22×C4×C61—42no—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D412yes—
118C22×C4×C61C2411yescharacteristic
119C2×C22⋊D61—24no—
120C2×C22⋊D61—42no—
121C2×C22⋊D61—41no—
122C2×C22⋊D61—41no—
123C22×Dic61—21no—
124C22×Dic61—22no—
125#11758222—28no—
126#11758222—44no—
127#11758222—44no—
128#11758222—44no—
129C2×D244—22no—
130C2×D244—42no—
131C2×C4×D61—42no—
132C2×C4×D61—42no—
133C2×C4×D61—42no—
134C2×Dic122—22no—
135C2×Dic122—42no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61C2×D414yes—
141C22⋊C4×C61—28no—
142C22⋊C4×C61—24no—
143C22⋊C4×C61—24no—
144C22⋊Dic61—42no—
145#11758200—48no—
146#11758199—48no—
147#11758198—44no—
148#11758198—44no—
149#11758196—44no—
150#11758194—44no—
151C2×#92418—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—24no—
155C23×C61—41no—
156C23×C61—44no—
157C23×C61—44no—
158C23×C61—44no—
159C23×C61#12011yescharacteristic
160C23×C61#12011yescharacteristic
161C23×C61C22×D411yescharacteristic
162C22×D61—22no—
163C22×D61—41no—
164C22×D61—42no—
165C22×D61—41no—
166C22×D61—42no—
167Q8×C61—48no—
168D4×C61—44no—
169D4×C61—48no—
170D4×C61—48no—
171D4×C61—48no—
172D4×C61—82no—
173D4×C61—84no—
174C2×C4×C61—24no—
175C2×C4×C61—24no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—24no—
179C2×C4×C61—22no—
180C2×C4×C61—24no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—416no—
185C2×C4×C61—44no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—81no—
189C2×C4×C61#12012yes—
190C2×C4×C61AGL(1,ℤ/8)14yescomplemented
191C2×C4×C61C22×D412yes—
192C22⋊D61—44no—
193C22⋊D61—44no—
194C2×Dic61—22no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200D244—24no—
201D244—42no—
202C4×D61—48no—
203Dic122—24no—
204Dic122—42no—
205#92418—44no—
206#18142—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—44no—
215C22×C61—44no—
216C22×C61—44no—
217C22×C61—44no—
218C22×C61—88no—
219C22×C61B(4,𝔽2)12yes—
220C22×C61C2×#12011yescharacteristic
221C22×C61C2×AGL(1,ℤ/8)12yes—
222C2×D61—44no—
223C2×D61—42no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—24no—
228C4×C61—22no—
229C4×C61—22no—
230C4×C61—44no—
231C4×C61—42no—
232Dic61—42no—
233Dic61—82no—
234C2×#537D6111yescharacteristic, complemented
235#3006—612no—
236#2406—611no—
237#1604—614no—
238#1598—614no—
239#1510—612no—
240#1463—611no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—41no—
244C2×C61—84no—
245C2×C61#300311yescharacteristic
246C2×C61#302011yescharacteristic
247C2×C61C2×B(4,𝔽2)11yescharacteristic
248D61—82no—
249C22×#41—21no—
250C22×#41—611no—
251C2×AGL(1,ℤ/8)—1222no—
252#537—24no—
253#537—22no—
254#537C2×D6114yes—
255C2×#123—1221no—
256C2×C22⋊Q8—22no—
257C2×#121—22no—
258C2×#121—21no—
259C2×#121C2×D6112yes—
260C22×C4⋊C4C2×D6111yescharacteristic
261#449—1224no—
262#448—614no—
263#447—614no—
264#417—1222no—
265#407—612no—
266#395—612no—
267#394—612no—
268#386—1221no—
269#385—1222no—
270#381—611no—
271#327—618no—
272#324—614no—
273C61#1814211yescharacteristic, complemented, Sylow
274C2×#41—24no—
275C2×#41—24no—
276C2×#41—48no—
277C2×#41—1222no—
278C2×#41—1222no—
279C2×#41—2444no—
280C22×Q8—21no—
281C22×Q8—611no—
282C22×D4—21no—
283C22×D4—22no—
284C22×D4—21no—
285C22×D4—611no—
286C22×D4—1221no—
287C22×D4C22⋊D6112yes—
288C23×C4—22no—
289C23×C4—1221no—
290C23×C4C22⋊D6111yescharacteristic
291C23×C4C22⋊D6111yescharacteristic
292C23×C4C22×D6111yescharacteristic
293AGL(1,ℤ/8)—2448no—
294C2×SD16—1224no—
295C2×D8—1224no—
296C2×M16—1222no—
297#123—2444no—
298C22⋊Q8—28no—
299C22⋊Q8—28no—
300#121—24no—
301#121—28no—
302#121—28no—
303#121—24no—
304#121—22no—
305#121—22no—
306#121C22⋊D6114yes—
307#121C22×D6114yes—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—1222no—
312C2×C4⋊C4C22⋊D6112yes—
313C2×C4⋊C4C22⋊D6112yes—
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1222no—
320#103—1224no—
321#102—1224no—
322C22⋊C8—614no—
323C22⋊C8—1222no—
324#95—612no—
325#95—1221no—
326#95—1222no—
327C24—22no—
328C24—1221no—
329C24C2×C22⋊D6111yescharacteristic
330#41—416no—
331#41—48no—
332#41—48no—
333#41—1228no—
334#41—2444no—
335#41—2444no—
336#41—2444no—
337C2×Q8—24no—
338C2×Q8—44no—
339C2×Q8—1222no—
340C2×Q8—2442no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—24no—
345C2×D4—22no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—82no—
350C2×D4—1224no—
351C2×D4—1222no—
352C2×D4—2442no—
353C2×D4—2442no—
354C2×D4—2441no—
355C2×D4—2441no—
356C2×D4C2×C22⋊D6114yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—42no—
365C22×C4—48no—
366C22×C4—42no—
367C22×C4—1221no—
368C22×C4—1222no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4C2×C22⋊D6112yes—
373C22×C4C2×C22⋊D6112yes—
374C22×C4C2×C22⋊D6112yes—
375C22×C4C2×C22⋊D6112yes—
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378SD16—2448no—
379D8—2448no—
380M16—2444no—
381C2×C8—1224no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—2444no—
387C4⋊C4C2×C22⋊D6114yes—
388C22⋊C4—28no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—2444no—
392C22⋊C4—2442no—
393C22⋊C4—2444no—
394C23—22no—
395C23—22no—
396C23—24no—
397C23—41no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—1222no—
402C23—2441no—
403C23—2442no—
404C23—2441no—
405C23—2442no—
406C23#1451974611yescharacteristic
407C23#1451974611yescharacteristic
408C23C22×C22⋊D6111yescharacteristic
409Q8—48no—
410Q8—1224no—
411Q8—2442no—
412D4—44no—
413D4—48no—
414D4—48no—
415D4—48no—
416D4—82no—
417D4—84no—
418D4—1224no—
419D4—2442no—
420D4—2444no—
421D4—2444no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—416no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—81no—
437C2×C4—1222no—
438C2×C4—2448no—
439C2×C4—2442no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4#1451974612yes—
445C2×C4#1451980314yescomplemented
446C2×C4C22×C22⋊D6112yes—
447C8—2444no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—44no—
456C22—44no—
457C22—44no—
458C22—44no—
459C22—88no—
460C22—4882no—
461C22—4881no—
462C22—4881no—
463C22—2444no—
464C22—2442no—
465C22#1683994912yes—
466C22C2×#1451974611yescharacteristic
467C22C2×#1451980312yes—
468C4—22no—
469C4—22no—
470C4—24no—
471C4—42no—
472C4—44no—
473C4—4882no—
474C4—2442no—
475C2—22no—
476C2—22no—
477C2—41no—
478C2—84no—
479C2—4882no—
480C2#2145504611yescharacteristic
481C2#2145518311yescharacteristic
482C2C2×#1683994911yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#537C2
socleC22×C61C2×#120
hypercenterC2×#537D61
hypocenterC61#18142
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21455069C2semidirect
2. #21449154C2semidirect
3. #21444788C2semidirect
4. #21444754C2semidirect
5. #21444348C2semidirect
6. #21444149C2semidirect
7. C2×C61×#537C2semidirect
12. #16839900C22semidirect
13. #16839895C22semidirect
16. #16839682C22semidirect
17. #16839679C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
190. C2×C4×C61AGL(1,ℤ/8)semidirect
234. C2×#537D61semidirect
273. C61#18142semidirect
445. C2×C4#14519803semidirect

Maximal subgroups

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

subgroupindex
#214550692
#214491542
#214447882
#214447542
#214443482
#214441492
C2×C61×#5372
#1814261