Finite Group Databasebeta

← #29658796

Subgroups — #29658796

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×D4×Dic122C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic, complemented
3#21455620C211yesmaximal, characteristic, complemented
4#21445080C214yesmaximal, complemented
5#21444983C214yesmaximal, complemented
6#21444490C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic
8#21444213C211yesmaximal, characteristic, complemented
9C61×#3101C211yesmaximal, 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#16840158C2212yescomplemented
22#16840112C2212yescomplemented
23#16840090—22no—
24#16840066—22no—
25#16840054C2212yescomplemented
26#16840047—22no—
27#16840044C2212yescomplemented
28#16839815C2211yescharacteristic
29#16839805—21no—
30#16839786C2211yescharacteristic
31#16839770—21no—
32#16839507C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#483—21no—
38C61×#476C2212yes—
39C61×#461C2212yes—
40C61×#450—21no—
41C4×SD16×C61—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122—24no—
45C22×Dic122—21no—
46C22×Dic122—24no—
47C22×Dic122—44no—
48C22×Dic122—44no—
49C22×Dic122D411yescharacteristic
50C23×Dic61—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—22no—
76#14519708—22no—
77#14519708—24no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80C4⋊#92418C2312yes—
81#14519701—24no—
82#14519700—22no—
83#14519700—22no—
84C22⋊#92418—21no—
85C4×#92418C2312yes—
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#106×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109C4×C8×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×Dic61—21no—
125C22×Dic61—28no—
126C22×Dic61—22no—
127C22×Dic61—21no—
128C22×Dic61—48no—
129C2×Dic122—21no—
130C2×Dic122—22no—
131C2×Dic122—28no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—48no—
135C2×Dic122—416no—
136C2×Dic122—48no—
137C2×Dic122—42no—
138C2×Dic122—42no—
139C2×Dic122—48no—
140C2×Dic122SD1614yescomplemented
141C2×Dic122C2×D412yes—
142SD16×C61—44no—
143C2×C8×C61—22no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C22⋊C4×C61—41no—
155C42×C61—41no—
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2411yescharacteristic
160C22⋊Dic61—28no—
161C4⋊Dic61—22no—
162C4⋊Dic61—21no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61C2×D414yes—
166C4⋊Dic61C2×D412yes—
167#11758196—28no—
168#11758196—22no—
169#11758196—22no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172#11758194—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—24no—
176C23×C61—22no—
177C23×C61—42no—
178C23×C61—42no—
179C23×C61#12011yescharacteristic
180Q8×C61—42no—
181Q8×C61—42no—
182D4×C61—24no—
183D4×C61—22no—
184D4×C61#13714yes—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C2×SD1612yes—
198C2×C4×C61AGL(1,ℤ/8)12yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—22no—
203C2×Dic61—24no—
204C2×Dic61—22no—
205C2×Dic61—24no—
206C2×Dic61—416no—
207C2×Dic61—42no—
208C2×Dic61—44no—
209Dic122—22no—
210Dic122—24no—
211Dic122—416no—
212Dic122—42no—
213Dic122—42no—
214Dic122—44no—
215Dic122C2×SD1614yes—
216C8×C61—42no—
217#92418—42no—
218#92418—42no—
219#18777—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61—24no—
223C22×C61—48no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61#45012yes—
227C22×C61C2×#12011yescharacteristic
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—42no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61#45012yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C61C2×#13712yes—
238Dic61—24no—
239Dic61—44no—
240Dic61—42no—
241C2×D4×Q8—611no—
242#3229—611no—
243#3101D6111yescharacteristic
244#3078—611no—
245#1651—614no—
246#1637—614no—
247#1546—611no—
248#1486—611no—
249#1479—611no—
250C2×C61—22no—
251C2×C61—44no—
252C2×C61C2×#45011yescharacteristic
253C2×C61#300411yescharacteristic
254C2×C61#302011yescharacteristic
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×Q8—611no—
266C2×C4×D4—1221no—
267C2×C4×D4C2×D6111yescharacteristic
268#498—612no—
269#487—612no—
270#483—21no—
271#483—1222no—
272#479—1222no—
273#477—612no—
274#476C2×D6112yes—
275#475—612no—
276#475—1222no—
277#461C2×D6112yes—
278#450—21no—
279C4×SD16—22no—
280#431—611no—
281#431C2×D6111yescharacteristic
282#428—1221no—
283#423—611no—
284#419—1221no—
285#332—614no—
286Q8⋊C8—614no—
287D4⋊C8—614no—
288C61#1877711yescharacteristic, complemented, Sylow
289C22×Q8—611no—
290C22×Q8—1224no—
291C22×Q8—1222no—
292C22×Q8—1224no—
293C22×Q8—1221no—
294C22×Q8—2444no—
295C22×Q8—2444no—
296C22×D4C22⋊D6111yescharacteristic
297C23×C4—21no—
298C23×C4—1222no—
299C2×SD16—22no—
300C2×M16—1221no—
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—1228no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—612no—
319C4×Q8—612no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—21no—
323C4×D4—21no—
324C4×D4—1224no—
325C4×D4—1224no—
326C4×D4C22⋊D6112yes—
327C4×D4C22×D6112yes—
328#117—21no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—612no—
331C2×C4⋊C4—1222no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22⋊D6111yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—1222no—
337C2×C42—1221no—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340#106—22no—
341#106—1222no—
342#106—1224no—
343C4⋊C8—612no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8C22×D6112yes—
347#103—22no—
348#103—1222no—
349#103—1222no—
350#103—1224no—
351#102—22no—
352C22⋊C8—21no—
353C22⋊C8—1221no—
354C4×C8—21no—
355C4×C8—612no—
356C24—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—612no—
360C2×Q8—614no—
361C2×Q8—1228no—
362C2×Q8—1222no—
363C2×Q8—1221no—
364C2×Q8—1222no—
365C2×Q8—1222no—
366C2×Q8—2448no—
367C2×Q8—2448no—
368C2×Q8—2448no—
369C2×Q8—24416no—
370C2×Q8—2442no—
371C2×Q8—2442no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4C2×C22⋊D6112yes—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—1221no—
381C22×C4—1228no—
382C22×C4—1221no—
383C22×C4—1222no—
384C22×C4—2448no—
385C22×C4D4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—44no—
389M16—2442no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—614no—
401C4⋊C4—612no—
402C4⋊C4—1228no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—1221no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4C2×C22⋊D6112yes—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—1228no—
414C42—41no—
415C42—1222no—
416C42—1222no—
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—24no—
422C23—22no—
423C23—42no—
424C23—42no—
425C23#1451974511yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—614no—
429Q8—1222no—
430Q8—1224no—
431Q8—24416no—
432Q8—2442no—
433Q8—2442no—
434Q8—2444no—
435D4—24no—
436D4—22no—
437D4#1451980714yes—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—1222no—
448C2×C4—1224no—
449C2×C4—1222no—
450C2×C4—1224no—
451C2×C4—24416no—
452C2×C4—2442no—
453C2×C4—2444no—
454C2×C4#1451974511yescharacteristic
455C2×C4#1451974511yescharacteristic
456C2×C4#1451974511yescharacteristic
457C2×C4SD16×D6112yes—
458C2×C4#1451980412yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—42no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—24no—
468C22—48no—
469C22—42no—
470C22—42no—
471C22#1683991112yes—
472C22C2×#1451974511yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—1224no—
480C4—2442no—
481C4—2444no—
482C4#1683991112yes—
483C4C2×SD16×D6111yescharacteristic
484C4C2×#1451980411yescharacteristic
485C4C2×#1451980712yes—
486C2—22no—
487C2—44no—
488C2C2×#1683991111yescharacteristic
489C2#2145505211yescharacteristic
490C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3101C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3101D61
hypocenterC61#18777
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3101

Complements

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

NH ≅ G/Ntype
2. #21456960C2semidirect
3. #21455620C2semidirect
4. #21445080C2semidirect
5. #21444983C2semidirect
6. #21444490C2semidirect
8. #21444213C2semidirect
21. #16840158C22semidirect
22. #16840112C22semidirect
25. #16840054C22semidirect
27. #16840044C22semidirect
32. #16839507C22semidirect
33. #16839492C22semidirect
140. C2×Dic122SD16semidirect
288. C61#18777semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1222
#214569602
#214556202
#214450802
#214449832
#214444902
#214442442
#214442132
C61×#31012
#1877761