Finite Group Databasebeta

← #29656216

Subgroups — #29656216

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic122C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3#21455945C211yesmaximal, characteristic, complemented
4#21445080C214yesmaximal, complemented
5#21445067C214yesmaximal, complemented
6#21444535C211yesmaximal, characteristic, complemented
7#21444265C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic
9C61×#3110C211yesmaximal, characteristic
10C23×Dic122—21no—
11D4×Dic122—22no—
12D4×Dic122—22no—
13D4×Dic122—22no—
14D4×Dic122C2214yes—
15C2×#14519777C2211yescharacteristic
16C2×#14519776—21no—
17C2×#14519756—21no—
18C2×#14519754—22no—
19C2×C4×Dic122C2211yescharacteristic
20C2×D4×Dic61—21no—
21#16840172—22no—
22#16840165C2212yescomplemented
23#16840158C2212yescomplemented
24#16840090—21no—
25#16840044C2212yescomplemented
26#16839969C2212yescomplemented
27#16839917—21no—
28#16839867—22no—
29#16839819C2211yescharacteristic
30#16839805—22no—
31C4×#11758194C2211yescharacteristic
32#16839507C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#474C2212yes—
39C61×#461C2212yes—
40C61×#450—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—21no—
46C22×Dic122—24no—
47C22×Dic122—48no—
48C22×Dic122D411yescharacteristic
49C23×Dic61—22no—
50C2×#11758202—22no—
51C2×#11758194—21no—
52#14519777—21no—
53#14519777—21no—
54#14519777D412yes—
55#14519777C2312yes—
56#14519776—24no—
57#14519776—24no—
58#14519756—24no—
59#14519756—24no—
60#14519754—28no—
61#14519754—28no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122D412yes—
65C4×Dic122C2312yes—
66D4×Dic61—24no—
67D4×Dic61—24no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D412yes—
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×C22⋊Dic61—22no—
74C2×C4×Dic61—21no—
75#14519710—24no—
76#14519708—22no—
77#14519708—24no—
78C4⋊#92418C2312yes—
79#14519701—22no—
80#14519700—22no—
81C22⋊#92418—21no—
82#14519688—21no—
83#14519688—21no—
84C4×#92418C2312yes—
85C22×D4×C61D411yescharacteristic
86C23×C4×C61—21no—
87C2×SD16×C61—22no—
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×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99#117×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C22⋊C4×C61—21no—
102C2×C42×C61C2311yescharacteristic
103#107×C61—22no—
104C4⋊C8×C61C2312yes—
105#103×C61—22no—
106#102×C61—22no—
107C22⋊C8×C61—21no—
108#97×C61—21no—
109C24×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×D4×C61—24no—
113C2×D4×C61—21no—
114C2×D4×C61C2×D412yes—
115C22×C4×C61—21no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—44no—
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×Dic61—21no—
123C22×Dic61—28no—
124C22×Dic61—22no—
125C22×Dic61—21no—
126C22×Dic61—48no—
127C2×Dic122—21no—
128C2×Dic122—24no—
129C2×Dic122—28no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—416no—
133C2×Dic122—48no—
134C2×Dic122—48no—
135C2×Dic122—44no—
136C2×Dic122—48no—
137C2×Dic122C2×D412yes—
138SD16×C61—44no—
139C2×C8×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—41no—
147C4⋊C4×C61C2×D412yes—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156C22⋊Dic61—28no—
157#11758202—44no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61C2×D412yes—
163C4⋊Dic61C2×D414yes—
164#11758196—28no—
165#11758196—22no—
166#11758196—22no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169#11758194—24no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—24no—
173C23×C61—22no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61#12011yescharacteristic
177Q8×C61—42no—
178Q8×C61—42no—
179D4×C61—24no—
180D4×C61—22no—
181D4×C61#13714yes—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×Dic61—24no—
198C2×Dic61—22no—
199C2×Dic61—22no—
200C2×Dic61—24no—
201C2×Dic61—416no—
202C2×Dic61—44no—
203C2×Dic61—42no—
204Dic122—24no—
205Dic122—22no—
206Dic122—416no—
207Dic122—44no—
208Dic122—44no—
209Dic122AGL(1,ℤ/8)14yes—
210C8×C61—42no—
211#92418—24no—
212#92418—42no—
213#18540—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—24no—
217C22×C61—48no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61#45612yes—
221C22×C61C2×#12011yescharacteristic
222C4×C61—21no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61C2×AGL(1,ℤ/8)12yes—
230C4×C61C2×#13712yes—
231Dic61—24no—
232Dic61—44no—
233Dic61—42no—
234C2×D4×Q8—611no—
235#3165—611no—
236#3113—611no—
237#3110D6111yescharacteristic
238#1651—614no—
239#1649—614no—
240#1556—611no—
241#1491—611no—
242#1490—611no—
243C2×C61—22no—
244C2×C61—44no—
245C2×C61#300311yescharacteristic
246C2×C61#300411yescharacteristic
247C2×C61C2×#45611yescharacteristic
248C23×Q8—1221no—
249D4×Q8—614no—
250D4×Q8—1222no—
251D4×Q8—1222no—
252D4×Q8—1222no—
253#533C2×D6111yescharacteristic
254C2×C4⋊Q8—611no—
255C2×C4⋊Q8—1221no—
256C2×C22⋊Q8—1222no—
257C2×C22⋊Q8—1221no—
258C2×C4×Q8—611no—
259C2×C4×D4—1221no—
260C2×C4×D4C2×D6111yescharacteristic
261#501—1222no—
262#500—612no—
263#498—612no—
264#484—21no—
265#483—1221no—
266#475—612no—
267#474C2×D6112yes—
268#462—612no—
269#461C2×D6112yes—
270#451—1221no—
271#450—21no—
272#441—1222no—
273#440—22no—
274#432—611no—
275#432C2×D6111yescharacteristic
276#428—1222no—
277C4×M16—611no—
278#332—614no—
279Q8⋊C8—614no—
280D4⋊C8—614no—
281C61#1854011yescharacteristic, complemented, Sylow
282C22×Q8—611no—
283C22×Q8—1224no—
284C22×Q8—1222no—
285C22×Q8—1224no—
286C22×Q8—1221no—
287C22×Q8—2448no—
288C22×D4C22⋊D6111yescharacteristic
289C23×C4—21no—
290C23×C4—1222no—
291C2×Q16—1222no—
292C2×SD16—22no—
293C2×M16—1221no—
294C4⋊Q8—612no—
295C4⋊Q8—612no—
296C4⋊Q8—1224no—
297C4⋊Q8—1224no—
298C4⋊Q8—1221no—
299C4⋊Q8—1221no—
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6111yescharacteristic
302#125—21no—
303C22⋊Q8—21no—
304C22⋊Q8—21no—
305C22⋊Q8—1228no—
306C22⋊Q8—1228no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—612no—
312C4×Q8—612no—
313C4×Q8—1221no—
314C4×Q8—1221no—
315C4×D4—21no—
316C4×D4—21no—
317C4×D4—1224no—
318C4×D4—1224no—
319C4×D4C22⋊D6112yes—
320C4×D4C22×D6112yes—
321#117—21no—
322C2×C4⋊C4—611no—
323C2×C4⋊C4—612no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4—1221no—
327C2×C4⋊C4C22⋊D6111yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1222no—
330C2×C42—1221no—
331C2×C42C22×D6111yescharacteristic
332#107—22no—
333#107—1224no—
334#106—1224no—
335#106—1222no—
336C4⋊C8—612no—
337C4⋊C8C22×D6112yes—
338#103—22no—
339#103—1222no—
340#103—1222no—
341#102—22no—
342C22⋊C8—21no—
343C22⋊C8—1221no—
344#97—21no—
345#97—1221no—
346#97—1221no—
347C4×C8—612no—
348C24—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—612no—
352C2×Q8—1224no—
353C2×Q8—1228no—
354C2×Q8—1222no—
355C2×Q8—1221no—
356C2×Q8—1222no—
357C2×Q8—2444no—
358C2×Q8—2448no—
359C2×Q8—2448no—
360C2×Q8—24416no—
361C2×Q8—2448no—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4C2×C22⋊D6112yes—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—44no—
369C22×C4—1221no—
370C22×C4—1228no—
371C22×C4—1221no—
372C22×C4—1222no—
373C22×C4—2448no—
374C22×C4D4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377Q16—2444no—
378SD16—44no—
379M16—1224no—
380C2×C8—22no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—612no—
391C4⋊C4—614no—
392C4⋊C4—1228no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4—1221no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C4⋊C4C2×C22⋊D6112yes—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—41no—
403C22⋊C4—1228no—
404C42—41no—
405C42—1222no—
406C42—1222no—
407C42D4×D6111yescharacteristic
408C42D4×D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C42C23×D6111yescharacteristic
411C23—24no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23#1451974511yescharacteristic
416Q8—42no—
417Q8—42no—
418Q8—614no—
419Q8—1224no—
420Q8—1222no—
421Q8—2444no—
422Q8—24416no—
423Q8—2444no—
424D4—24no—
425D4—22no—
426D4#1451980714yes—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—1224no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—1224no—
440C2×C4—24416no—
441C2×C4—2444no—
442C2×C4—2442no—
443C2×C4#1451974511yescharacteristic
444C2×C4#1451974511yescharacteristic
445C2×C4#1451974511yescharacteristic
446C2×C4C2×D4×D6111yescharacteristic
447C2×C4C2×D4×D6111yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C8—42no—
450C8—1224no—
451C8—2442no—
452C22—21no—
453C22—21no—
454C22—24no—
455C22—48no—
456C22—42no—
457C22—42no—
458C22#1683994312yes—
459C22C2×#1451974511yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—42no—
465C4—81no—
466C4—81no—
467C4—1224no—
468C4—2442no—
469C4—2444no—
470C4AGL(1,ℤ/8)×D6112yes—
471C4C2×#1451980712yes—
472C2—22no—
473C2—44no—
474C2#2145504411yescharacteristic
475C2#2145505211yescharacteristic
476C2C2×#1683994311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3110D61
hypocenterC61#18540
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. #21455945C2semidirect
4. #21445080C2semidirect
5. #21445067C2semidirect
6. #21444535C2semidirect
7. #21444265C2semidirect
22. #16840165C22semidirect
23. #16840158C22semidirect
25. #16840044C22semidirect
26. #16839969C22semidirect
32. #16839507C22semidirect
33. #16839492C22semidirect
281. C61#18540semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214563662
#214559452
#214450802
#214450672
#214445352
#214442652
#214442622
C61×#31102
#1854061