Finite Group Databasebeta

← #29652460

Subgroups — #29652460

Order 15616.

1450 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1450 conjugacy classes of subgroups — too many to draw. Only its 167 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
1C2×#16840343C211yesmaximal, characteristic, complemented
2#21455071C212yesmaximal, complemented
3#21444787C214yesmaximal, complemented
4#21444753C214yesmaximal, complemented
5#21444347C212yesmaximal, complemented
6#21444149C211yesmaximal, characteristic, complemented
7C61×#2406C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9C2×#14519802—22no—
10#16840343—24no—
11#16840343—22no—
12#16840343C2214yes—
13C2×#14519757—22no—
14C2×#14519752—22no—
15C2×#14519751—21no—
16C2×#14519751C2212yes—
17#16839907—24no—
18#16839901C2214yescomplemented
19#16839896C2214yescomplemented
20#16839763—22no—
21#16839726C2212yes—
22#16839495C2218yescomplemented
23#16839488C2214yescomplemented
24C22×#41×C61C2211yescharacteristic
25C2×#123×C61—21no—
26C22×C4⋊C4×C61C2211yescharacteristic
27C61×#395C2212yescomplemented
28C61×#394C2212yescomplemented
29C61×#386—21no—
30C61×#385—22no—
31C61×#381C2211yescharacteristic
32C2×#11758222—24no—
33C2×#11758222—24no—
34C2×#11758222—48no—
35C22×Dic122—21no—
36C22×D244—21no—
37C22×D244—21no—
38C22×D244D412yes—
39C22×C22⋊D61—22no—
40C22×C4×D61—22no—
41#14519802—48no—
42C2×#11758201—24no—
43C2×#11758199—24no—
44C2×#11758194—22no—
45#14519757—28no—
46#14519757—28no—
47#14519752—28no—
48#14519752—28no—
49#14519751—24no—
50#14519751—24no—
51#14519751—22no—
52#14519751—22no—
53#14519751D414yes—
54#14519751C2314yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61D412yes—
57C2×#11758198—22no—
58C2×#11758198—22no—
59#14519701—24no—
60#14519698—24no—
61C22⋊#92418—22no—
62C22⋊#92418C2314yes—
63C2×#41×C61—22no—
64C2×#41×C61—22no—
65C2×#41×C61—44no—
66C22×Q8×C61C2311yescharacteristic
67C22×D4×C61—21no—
68C22×D4×C61C2311yescharacteristic
69C23×C4×C61—21no—
70C23×C4×C61D411yescharacteristic
71C23×C4×C61D411yescharacteristic
72C23×C4×C61C2311yescharacteristic
73#123×C61—44no—
74C2×C4⋊C4×C61—22no—
75C2×C4⋊C4×C61—22no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61D412yes—
78C2×C4⋊C4×C61C2312yes—
79C2×C22⋊C4×C61—21no—
80C2×C22⋊C4×C61—22no—
81C2×C22⋊C4×C61—22no—
82#95×C61—21no—
83#95×C61—22no—
84#95×C61C2312yes—
85C24×C61—21no—
86C24×C61C2×D411yescharacteristic
87C23×D61—22no—
88#41×C61—28no—
89#41×C61—44no—
90#41×C61—44no—
91#41×C61—44no—
92C2×Q8×C61—22no—
93C2×Q8×C61—42no—
94C2×D4×C61—24no—
95C2×D4×C61—22no—
96C2×D4×C61—42no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C22×C4×C61—21no—
101C22×C4×C61—21no—
102C22×C4×C61—24no—
103C22×C4×C61—21no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D412yes—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2411yescharacteristic
118C2×C22⋊D61—24no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—48no—
121C22×Dic61—22no—
122#11758222—416no—
123#11758222—48no—
124#11758222—48no—
125C2×D244—24no—
126C2×D244—24no—
127C2×D244—22no—
128C2×D244—44no—
129C2×D244—44no—
130C2×D244—82no—
131C2×D244C2×D414yes—
132C2×C4×D61—28no—
133C2×C4×D61—48no—
134C2×Dic122—24no—
135C2×Dic122—44no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—24no—
138C4⋊C4×C61—44no—
139C22⋊C4×C61—44no—
140C22⋊C4×C61—42no—
141C22⋊C4×C61—44no—
142#11758201—48no—
143#11758199—48no—
144#11758198—28no—
145#11758198—24no—
146#11758198—24no—
147C4⋊Dic61—24no—
148C4⋊Dic61—22no—
149C4⋊Dic61C2×D414yes—
150#11758194—44no—
151C2×#92418—24no—
152C23×C61—22no—
153C23×C61—22no—
154C23×C61—22no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—42no—
158C23×C61—41no—
159C23×C61—42no—
160C23×C61#12011yescharacteristic
161C23×C61#12011yescharacteristic
162C23×C61C22×D411yescharacteristic
163C22×D61—24no—
164C22×D61—44no—
165C22×D61—44no—
166C22×D61—44no—
167Q8×C61—24no—
168Q8×C61—42no—
169D4×C61—24no—
170D4×C61—42no—
171D4×C61—44no—
172D4×C61—44no—
173C2×C4×C61—24no—
174C2×C4×C61—24no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—41no—
183C2×C4×C61—48no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—81no—
192C2×C4×C61#12012yes—
193C2×C4×C61AGL(1,ℤ/8)14yescomplemented
194C2×C4×C61C22×D412yes—
195C22⋊D61—48no—
196C22⋊D61—48no—
197C2×Dic61—24no—
198C2×Dic61—44no—
199D244—44no—
200D244—48no—
201D244—82no—
202D244—84no—
203C4×D61—416no—
204Dic122—48no—
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—42no—
217C22×C61—82no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61B(4,𝔽2)12yes—
221C22×C61C2×#12011yescharacteristic
222C22×C61C2×AGL(1,ℤ/8)12yes—
223C2×D61—44no—
224C2×D61—44no—
225C2×D61—44no—
226C2×D61—88no—
227C4×C61—24no—
228C4×C61—22no—
229C4×C61—22no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—82no—
233Dic61—44no—
234C2×#537—611no—
235#3006—612no—
236#2406D6111yescharacteristic, complemented
237#1604—614no—
238#1598—614no—
239#1510—612no—
240#1463—611no—
241C2×C61—22no—
242C2×C61—22no—
243C2×C61—41no—
244C2×C61—82no—
245C2×C61#300311yescharacteristic
246C2×C61#302011yescharacteristic
247C2×C61C2×B(4,𝔽2)11yescharacteristic
248D61—84no—
249C22×#41—1221no—
250C22×#41C2×D6111yescharacteristic
251C2×AGL(1,ℤ/8)—1222no—
252#537—614no—
253#537—1224no—
254#537—1222no—
255C2×#123—21no—
256C2×C22⋊Q8—1222no—
257C2×#121—612no—
258C2×#121—1222no—
259C2×#121—1221no—
260C22×C4⋊C4C2×D6111yescharacteristic
261#449—1224no—
262#448—614no—
263#447—614no—
264#417—1222no—
265#407—612no—
266#395C2×D6112yescomplemented
267#394C2×D6112yescomplemented
268#386—21no—
269#385—22no—
270#381C2×D6111yescharacteristic
271#327—618no—
272#324—614no—
273C61#1814211yescharacteristic, complemented, Sylow
274C2×#41—22no—
275C2×#41—22no—
276C2×#41—44no—
277C2×#41—1224no—
278C2×#41—1224no—
279C2×#41—2448no—
280C22×Q8—1221no—
281C22×Q8C22×D6111yescharacteristic
282C22×D4—21no—
283C22×D4—612no—
284C22×D4—1221no—
285C22×D4—1221no—
286C22×D4—1222no—
287C22×D4C22×D6111yescharacteristic
288C23×C4—21no—
289C23×C4—1222no—
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—44no—
298C22⋊Q8—1228no—
299C22⋊Q8—1228no—
300#121—614no—
301#121—614no—
302#121—1228no—
303#121—1228no—
304#121—1224no—
305#121—1222no—
306#121—1222no—
307#121—1224no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—612no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4C22⋊D6112yes—
314C2×C4⋊C4C22×D6112yes—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1222no—
320#103—1224no—
321#102—1224no—
322C22⋊C8—614no—
323C22⋊C8—1222no—
324#95—21no—
325#95—22no—
326#95C22×D6112yes—
327C24—21no—
328C24—1222no—
329C24C2×C22⋊D6111yescharacteristic
330#41—28no—
331#41—44no—
332#41—44no—
333#41—44no—
334#41—24416no—
335#41—2448no—
336#41—2448no—
337C2×Q8—22no—
338C2×Q8—42no—
339C2×Q8—1224no—
340C2×Q8—2444no—
341C2×D4—24no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—614no—
348C2×D4—1224no—
349C2×D4—1224no—
350C2×D4—1224no—
351C2×D4—1224no—
352C2×D4—1222no—
353C2×D4—4882no—
354C2×D4—2444no—
355C2×D4—2444no—
356C2×D4—2448no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—1228no—
370C22×C4—1222no—
371C22×C4—2448no—
372C22×C4D4×D6112yes—
373C22×C4D4×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—44no—
385C4⋊C4—614no—
386C4⋊C4—1224no—
387C4⋊C4—1222no—
388C22⋊C4—44no—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—1228no—
392C22⋊C4—1224no—
393C22⋊C4—1224no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—1224no—
403C23—2444no—
404C23—2444no—
405C23—2444no—
406C23#1451974511yescharacteristic
407C23#1451974511yescharacteristic
408C23C22×C22⋊D6111yescharacteristic
409Q8—24no—
410Q8—42no—
411Q8—2448no—
412D4—24no—
413D4—42no—
414D4—44no—
415D4—44no—
416D4—4882no—
417D4—4884no—
418D4—2444no—
419D4—2448no—
420D4—2448no—
421D4—2448no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—48no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—1224no—
442C2×C4—24416no—
443C2×C4—2444no—
444C2×C4#1451974512yes—
445C2×C4#1451980214yescomplemented
446C2×C4C2×D4×D6112yes—
447C8—2444no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—44no—
456C22—44no—
457C22—42no—
458C22—82no—
459C22—81no—
460C22—81no—
461C22—4888no—
462C22—2444no—
463C22—2444no—
464C22—2444no—
465C22#1683994812yes—
466C22C2×#1451974511yescharacteristic
467C22C2×#1451980212yes—
468C4—22no—
469C4—22no—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—82no—
474C4—2444no—
475C2—22no—
476C2—22no—
477C2—41no—
478C2—82no—
479C2—4884no—
480C2#2145504511yescharacteristic
481C2#2145518211yescharacteristic
482C2C2×#1683994811yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2406C2
socleC22×C61C2×#120
hypercenter#2406D61
hypocenterC61#18142
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. C2×#16840343C2semidirect
2. #21455071C2semidirect
3. #21444787C2semidirect
4. #21444753C2semidirect
5. #21444347C2semidirect
6. #21444149C2semidirect
7. C61×#2406C2semidirect
18. #16839901C22semidirect
19. #16839896C22semidirect
22. #16839495C22semidirect
23. #16839488C22semidirect
27. C61×#395C22semidirect
28. C61×#394C22semidirect
193. C2×C4×C61AGL(1,ℤ/8)semidirect
236. #2406D61semidirect
266. #395C2×D61semidirect
267. #394C2×D61semidirect
273. C61#18142semidirect
445. C2×C4#14519802semidirect

Maximal subgroups

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

subgroupindex
C2×#168403432
#214550712
#214447872
#214447532
#214443472
#214441492
C61×#24062
#1814261