Finite Group Databasebeta

← #29658507

Subgroups — #29658507

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840469C211yesmaximal, characteristic
2#21456764C211yesmaximal, characteristic, complemented
3#21455854C211yesmaximal, characteristic, complemented
4#21445032C214yesmaximal, complemented
5#21444951C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444247C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic
9C61×#3077C211yesmaximal, characteristic
10#16840469—22no—
11#16840469—22no—
12#16840469—22no—
13#16840469C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519768—21no—
16C2×#14519756—21no—
17C2×#14519754—22no—
18C2×C4×Dic122C2211yescharacteristic
19C2×D4×Dic61—21no—
20C22×C4⋊Dic61—21no—
21#16840141C2212yescomplemented
22#16840114C2212yescomplemented
23#16840095—21no—
24#16840054C2212yescomplemented
25#16839972—22no—
26#16839969C2212yescomplemented
27#16839917—22no—
28#16839917—21no—
29C4×#11758202—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839510C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#478—22no—
38C61×#476C2212yes—
39C61×#474C2212yes—
40C61×#428—21no—
41C61×#423C2211yescharacteristic
42C61×#418—21no—
43C22×Dic122D411yescharacteristic
44C22×Dic122D412yes—
45C23×Dic61—22no—
46C2×#11758202—22no—
47C2×#11758202—24no—
48#14519777—21no—
49#14519777—21no—
50#14519777D412yes—
51#14519777C2312yes—
52#14519768—24no—
53#14519768—24no—
54#14519756—24no—
55#14519756—24no—
56#14519754—28no—
57#14519754—28no—
58C4×Dic122—21no—
59C4×Dic122—21no—
60C4×Dic122D412yes—
61C4×Dic122C2312yes—
62D4×Dic61—24no—
63D4×Dic61—24no—
64C2×C4⋊Dic61—24no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—24no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×#11758196—22no—
71C2×#11758196—21no—
72C2×C22⋊Dic61—22no—
73C2×C4×Dic61—21no—
74#14519710—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519700—24no—
79#14519700—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C2×M16×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#106×C61—22no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106#102×C61—22no—
107#102×C61—22no—
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—22no—
113C2×D4×C61Q1614yescomplemented
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—22no—
124C22×Dic61—21no—
125C22×Dic61—28no—
126C22×Dic61—84no—
127C2×Dic122—21no—
128C2×Dic122—22no—
129C2×Dic122—42no—
130C2×Dic122—82no—
131C2×Dic122C2×D412yes—
132C2×Dic122C2×D414yes—
133M16×C61—42no—
134C2×C8×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61C2×D412yes—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—41no—
146C42×C61—41no—
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2411yescharacteristic
151C22⋊Dic61—28no—
152#11758202—44no—
153#11758202—84no—
154C4⋊Dic61—28no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D412yes—
161#11758196—28no—
162#11758196—22no—
163#11758196—22no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61#12011yescharacteristic
173Q8×C61—42no—
174Q8×C61—42no—
175D4×C61—22no—
176D4×C61—24no—
177D4×C61C2×Q1614yes—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—44no—
182C2×C4×C61—42no—
183C2×C4×C61—44no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61C2×Q1612yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—24no—
197C2×Dic61—42no—
198C2×Dic61—44no—
199C2×Dic61—42no—
200C2×Dic61—88no—
201Dic122—22no—
202Dic122—22no—
203Dic122—42no—
204Dic122—44no—
205Dic122—82no—
206C8×C61—42no—
207#92418—42no—
208#92418—42no—
209#18749—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—24no—
213C22×C61—48no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61#45112yes—
217C22×C61C2×#12011yescharacteristic
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61#45112yes—
225C4×C61C22×Q1611yescharacteristic
226C4×C61C2×#13711yescharacteristic
227C4×C61#57512yes—
228Dic61—42no—
229Dic61—44no—
230Dic61—42no—
231C2×#554—611no—
232#3209—611no—
233#3104—611no—
234#3077D6111yescharacteristic
235#1644—614no—
236#1632—614no—
237#1548—611no—
238#1487—611no—
239#1478—611no—
240C2×C61—22no—
241C2×C61—44no—
242C2×C61C2×#45111yescharacteristic
243C2×C61#300511yescharacteristic
244C2×C61#301911yescharacteristic
245#554—614no—
246#554—1222no—
247#554—1222no—
248#554—1222no—
249#533C2×D6111yescharacteristic
250C2×C4⋊Q8—611no—
251C2×#125—1221no—
252C2×C22⋊Q8—1222no—
253C2×C22⋊Q8—1221no—
254C2×C4×Q8—611no—
255C2×C4×D4—1221no—
256C2×C4×D4C2×D6111yescharacteristic
257C22×C4⋊C4—1221no—
258#494—612no—
259#487—612no—
260#484—1221no—
261#478—22no—
262#477—612no—
263#476C2×D6112yes—
264#474C2×D6112yes—
265#462—612no—
266#462—1222no—
267#451—1221no—
268#451—1222no—
269C4×Q16—1222no—
270#431—611no—
271#431—611no—
272#428—21no—
273#423C2×D6111yescharacteristic
274#418—21no—
275#333—614no—
276Q8⋊C8—614no—
277D4⋊C8—614no—
278C61#1874911yescharacteristic, complemented, Sylow
279C22×Q8—612no—
280C22×Q8—611no—
281C22×D4C22⋊D6111yescharacteristic
282C23×C4—21no—
283C23×C4—1222no—
284C2×Q16—1224no—
285C2×Q16—1222no—
286C2×M16—21no—
287C4⋊Q8—612no—
288C4⋊Q8—612no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8C22×D6111yescharacteristic
292C4⋊Q8C22×D6111yescharacteristic
293#125—21no—
294#125—1224no—
295#125—1224no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—1228no—
299C22⋊Q8—1228no—
300C22⋊Q8—1224no—
301C22⋊Q8—1224no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—612no—
305C4×Q8—612no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—1224no—
311C4×D4—1224no—
312C4×D4C22⋊D6112yes—
313C4×D4C22×D6112yes—
314#117—21no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1224no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1224no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4C22⋊D6111yescharacteristic
324C2×C22⋊C4—21no—
325C2×C22⋊C4—1222no—
326C2×C42—1221no—
327C2×C42C22×D6111yescharacteristic
328#107—22no—
329#107—1222no—
330#106—22no—
331C4⋊C8—21no—
332C4⋊C8—612no—
333C4⋊C8—1221no—
334C4⋊C8C22×D6112yes—
335#103—1222no—
336#103—1224no—
337#103—1222no—
338#102—22no—
339#102—22no—
340C22⋊C8—1221no—
341C22⋊C8—1221no—
342C4×C8—612no—
343C4×C8—1221no—
344C24—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—612no—
348C2×Q8—614no—
349C2×Q8—1222no—
350C2×Q8—1221no—
351C2×Q8—4882no—
352C2×Q8—2442no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4#1175820214yescomplemented
356C2×D4C2×C22⋊D6112yes—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—1228no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1222no—
365C22×C4—4884no—
366C22×C4D4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369Q16—4884no—
370Q16—2444no—
371M16—42no—
372C2×C8—22no—
373C2×C8—1222no—
374C2×C8—1222no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—612no—
383C4⋊C4—1228no—
384C4⋊C4—1228no—
385C4⋊C4—1222no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4—1221no—
389C4⋊C4—1222no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4C2×C22⋊D6112yes—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—41no—
396C22⋊C4—1228no—
397C42—41no—
398C42—1222no—
399C42—1222no—
400C42D4×D6111yescharacteristic
401C42D4×D6111yescharacteristic
402C42C2×C22⋊D6111yescharacteristic
403C42C23×D6111yescharacteristic
404C23—22no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23#1451974511yescharacteristic
409Q8—42no—
410Q8—42no—
411Q8—1222no—
412Q8—1222no—
413Q8—4882no—
414Q8—2444no—
415Q8—2442no—
416D4—22no—
417D4—24no—
418D4C2×#1175820214yes—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—1222no—
428C2×C4—1222no—
429C2×C4—1224no—
430C2×C4—4888no—
431C2×C4—2442no—
432C2×C4—2444no—
433C2×C4—2442no—
434C2×C4#1451974511yescharacteristic
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4C2×#1175820212yes—
438C2×C4#1451980712yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C2×D4×D6111yescharacteristic
441C2×C4C22×C22⋊D6111yescharacteristic
442C8—42no—
443C8—2442no—
444C8—2442no—
445C22—21no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—42no—
450C22—42no—
451C22#1683991712yes—
452C22C2×#1451974511yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—81no—
458C4—81no—
459C4—2442no—
460C4—2444no—
461C4—2442no—
462C4#1683991712yes—
463C4C22×#1175820211yescharacteristic
464C4C2×#1451980711yescharacteristic
465C4#1684055812yes—
466C2—22no—
467C2—44no—
468C2C2×#1683991711yescharacteristic
469C2#2145505811yescharacteristic
470C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3077C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3077D61
hypocenterC61#18749
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3077

Complements

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

NH ≅ G/Ntype
2. #21456764C2semidirect
3. #21455854C2semidirect
4. #21445032C2semidirect
5. #21444951C2semidirect
6. #21444501C2semidirect
7. #21444247C2semidirect
21. #16840141C22semidirect
22. #16840114C22semidirect
24. #16840054C22semidirect
26. #16839969C22semidirect
32. #16839510C22semidirect
33. #16839492C22semidirect
113. C2×D4×C61Q16semidirect
278. C61#18749semidirect
355. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
C2×#168404692
#214567642
#214558542
#214450322
#214449512
#214445012
#214442472
#214442102
C61×#30772
#1874961