Finite Group Databasebeta

← #29658795

Subgroups — #29658795

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×D244C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic, complemented
3#21455823C211yesmaximal, characteristic
4#21445081C214yesmaximal, complemented
5#21444982C214yesmaximal, complemented
6#21444492C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic, complemented
9C61×#3078C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11Q8×D244—22no—
12Q8×D244—22no—
13Q8×D244—22no—
14Q8×D244C2214yes—
15C2×#14519777—21no—
16C2×#14519777C2211yescharacteristic
17C2×#14519757—22no—
18C2×#14519757—21no—
19C2×C4×D244—21no—
20C2×C4×D244C2211yescharacteristic
21#16840158C2212yescomplemented
22#16840112C2212yescomplemented
23#16840090—22no—
24#16840090—21no—
25#16840049C2212yes—
26#16840047—22no—
27#16839964C2212yes—
28#16839912—21no—
29C4×#11758201—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839509C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#479—22no—
38C61×#477C2212yescomplemented
39C61×#475C2212yescomplemented
40C61×#428—21no—
41C61×#423C2211yescharacteristic
42C61×#419—21no—
43C22×Dic122—22no—
44C22×Dic122—21no—
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—44no—
48C2×Q8×D61—44no—
49C22×D244D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C2×#11758201—22no—
53#14519777—24no—
54#14519777—21no—
55#14519777—21no—
56#14519777—24no—
57#14519777D412yes—
58#14519777C2312yes—
59#14519757—24no—
60#14519757—28no—
61#14519757—24no—
62#14519757—28no—
63C4×D244—24no—
64C4×D244—24no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244D412yes—
68C4×D244C2312yes—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×C4⋊Dic61D412yes—
73C2×#11758198—22no—
74C2×#11758198—21no—
75#14519708—24no—
76#14519708—22no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519701—22no—
80#14519701—24no—
81#14519697—22no—
82C22⋊#92418—21no—
83C22⋊#92418—21no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C22×Q8×C61D411yescharacteristic
87C2×M16×C61—21no—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97C4×Q8×C61D412yes—
98C4×Q8×C61C2312yes—
99#117×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C42×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105#106×C61—22no—
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#103×C61—22no—
109#103×C61—22no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61SD1614yescomplemented
116C2×Q8×C61C2×D412yes—
117C22×C4×C61—21no—
118C22×C4×C61—21no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×Dic61—22no—
123C22×Dic61—21no—
124Q8×D61—48no—
125Q8×D61—416no—
126Q8×D61—48no—
127C2×D244—21no—
128C2×D244—24no—
129C2×D244C2×D412yes—
130C2×C4×D61—28no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—48no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×Dic122—28no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—48no—
140C2×Dic122—42no—
141C2×Dic122—42no—
142M16×C61—42no—
143C2×C8×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61—41no—
153C4⋊C4×C61C2×D412yes—
154C22⋊C4×C61—41no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61—41no—
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2×D411yescharacteristic
161C42×C61C2411yescharacteristic
162#11758201—44no—
163#11758198—28no—
164#11758198—22no—
165#11758198—22no—
166C4⋊Dic61—28no—
167C4⋊Dic61—22no—
168C4⋊Dic61—21no—
169C4⋊Dic61—22no—
170C4⋊Dic61—22no—
171C4⋊Dic61C2×D412yes—
172C4⋊Dic61C2×D414yes—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61#12011yescharacteristic
176C22×D61—24no—
177C22×D61—22no—
178C22×D61—42no—
179C22×D61—42no—
180Q8×C61—22no—
181Q8×C61—24no—
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61C2×SD1614yes—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—41no—
190C2×C4×C61—42no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61C2×SD1612yes—
197C2×C4×C61AGL(1,ℤ/8)12yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×Dic61—22no—
202C2×Dic61—24no—
203C2×Dic61—42no—
204C2×Dic61—44no—
205D244—24no—
206D244—22no—
207D244#13714yes—
208C4×D61—416no—
209C4×D61—42no—
210C4×D61—42no—
211C4×D61—44no—
212Dic122—416no—
213Dic122—42no—
214Dic122—42no—
215Dic122—44no—
216C8×C61—42no—
217#92418—42no—
218#92418—42no—
219#18777—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#45012yes—
223C22×C61C2×#12011yescharacteristic
224C2×D61—24no—
225C2×D61—48no—
226C2×D61—42no—
227C2×D61—42no—
228C4×C61—24no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—81no—
234C4×C61—81no—
235C4×C61#45012yes—
236C4×C61C22×SD1611yescharacteristic
237C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C61C2×#13712yes—
239Dic61—42no—
240Dic61—44no—
241C2×D4×Q8—611no—
242#3229—611no—
243#3101—611no—
244#3078D6111yescharacteristic, complemented
245#1651—614no—
246#1637—614no—
247#1546—611no—
248#1486—611no—
249#1479—611no—
250C2×C61—22no—
251C2×C61C2×#45011yescharacteristic
252C2×C61#300411yescharacteristic
253C2×C61#302011yescharacteristic
254D61—44no—
255C23×Q8—1221no—
256D4×Q8—614no—
257D4×Q8—1222no—
258D4×Q8—1222no—
259D4×Q8—1222no—
260#533C2×D6111yescharacteristic
261C2×C4⋊Q8—611no—
262C2×C4⋊Q8—1221no—
263C2×C22⋊Q8—1222no—
264C2×C22⋊Q8—1221no—
265C2×C4×Q8C2×D6111yescharacteristic
266C2×C4×D4—611no—
267C2×C4×D4—1221no—
268#498—612no—
269#487—612no—
270#483—1221no—
271#483—1222no—
272#479—22no—
273#477C2×D6112yescomplemented
274#476—612no—
275#475—1222no—
276#475C2×D6112yescomplemented
277#461—612no—
278#450—1221no—
279C4×SD16—1222no—
280#431—611no—
281#431—611no—
282#428—21no—
283#423C2×D6111yescharacteristic
284#419—21no—
285#332—614no—
286Q8⋊C8—614no—
287D4⋊C8—614no—
288C61#1877711yescharacteristic, complemented, Sylow
289C22×Q8—1222no—
290C22×Q8—1224no—
291C22×Q8—1224no—
292C22×Q8—1221no—
293C22×Q8—2444no—
294C22×Q8—2444no—
295C22×Q8C22⋊D6111yescharacteristic
296C22×D4—611no—
297C23×C4—1222no—
298C23×C4—1221no—
299C2×SD16—1222no—
300C2×M16—21no—
301C4⋊Q8—612no—
302C4⋊Q8—612no—
303C4⋊Q8—1224no—
304C4⋊Q8—1224no—
305C4⋊Q8—1221no—
306C4⋊Q8—1221no—
307C4⋊Q8C22×D6111yescharacteristic
308C4⋊Q8C22×D6111yescharacteristic
309#125—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—1228no—
313C22⋊Q8—1224no—
314C22⋊Q8—1228no—
315C22⋊Q8—1224no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—21no—
319C4×Q8—21no—
320C4×Q8C22⋊D6112yes—
321C4×Q8C22×D6112yes—
322C4×D4—612no—
323C4×D4—612no—
324C4×D4—1224no—
325C4×D4—1224no—
326C4×D4—1221no—
327C4×D4—1221no—
328#117—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—612no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C42—21no—
338C2×C42C22×D6111yescharacteristic
339#107—22no—
340#106—22no—
341#106—1222no—
342#106—1224no—
343C4⋊C8—21no—
344C4⋊C8—612no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6112yes—
347#103—22no—
348#103—22no—
349#103—1222no—
350#103—1224no—
351#102—1222no—
352C22⋊C8—1221no—
353C22⋊C8—1221no—
354C4×C8—612no—
355C4×C8—1221no—
356C24—1221no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—1228no—
362C2×Q8—1222no—
363C2×Q8—1222no—
364C2×Q8—2448no—
365C2×Q8—2448no—
366C2×Q8—24416no—
367C2×Q8—2448no—
368C2×Q8—2442no—
369C2×Q8—2442no—
370C2×Q8#1175820114yescomplemented
371C2×Q8C2×C22⋊D6112yes—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1221no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—1228no—
378C22×C4—1222no—
379C22×C4—1221no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—2448no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4D4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—2444no—
389M16—42no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—612no—
403C4⋊C4—614no—
404C4⋊C4—1228no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4C2×C22⋊D6112yes—
410C22⋊C4—41no—
411C22⋊C4—1228no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C42—22no—
415C42—22no—
416C42—41no—
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1224no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23#1451974511yescharacteristic
426Q8—22no—
427Q8—24no—
428Q8—42no—
429Q8—42no—
430Q8—24416no—
431Q8—2442no—
432Q8—2442no—
433Q8—2444no—
434Q8C2×#1175820114yes—
435D4—614no—
436D4—1224no—
437D4—1222no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—1224no—
447C2×C4—1222no—
448C2×C4—24416no—
449C2×C4—2444no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2444no—
454C2×C4#1451974511yescharacteristic
455C2×C4#1451974511yescharacteristic
456C2×C4#1451974511yescharacteristic
457C2×C4C2×#1175820112yes—
458C2×C4#1451980212yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—42no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—1224no—
468C22—2448no—
469C22—2442no—
470C22—2442no—
471C22#1683991212yes—
472C22C2×#1451974511yescharacteristic
473C4—21no—
474C4—21no—
475C4—24no—
476C4—21no—
477C4—42no—
478C4—81no—
479C4—81no—
480C4—2444no—
481C4—2442no—
482C4#1683991212yes—
483C4C22×#1175820111yescharacteristic
484C4C2×#1451980211yescharacteristic
485C4#137×D6112yes—
486C2—22no—
487C2—2444no—
488C2C2×#1683991211yescharacteristic
489C2#2145505111yescharacteristic
490C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3078C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3078D61
hypocenterC61#18777
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
2. #21456960C2semidirect
4. #21445081C2semidirect
5. #21444982C2semidirect
6. #21444492C2semidirect
8. #21444214C2semidirect
9. C61×#3078C2semidirect
21. #16840158C22semidirect
22. #16840112C22semidirect
32. #16839509C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#475C22semidirect
115. C2×Q8×C61SD16semidirect
244. #3078D61semidirect
273. #477C2×D61semidirect
276. #475C2×D61semidirect
288. C61#18777semidirect
370. C2×Q8#11758201semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D2442
#214569602
#214558232
#214450812
#214449822
#214444922
#214442432
#214442142
C61×#30782
#1877761