Finite Group Databasebeta

← #29658543

Subgroups — #29658543

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2#21456764C211yesmaximal, characteristic, complemented
3#21455823C211yesmaximal, characteristic
4#21445025C214yesmaximal, complemented
5#21444952C214yesmaximal, complemented
6#21444501C211yesmaximal, characteristic, complemented
7#21444243C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic, complemented
9C61×#3078C211yesmaximal, characteristic, complemented
10#16840472—22no—
11#16840472—22no—
12#16840472—22no—
13#16840472C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×D244—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840141C2212yescomplemented
22#16840114C2212yescomplemented
23#16840090—21no—
24#16840049C2212yes—
25#16839972—22no—
26#16839964C2212yes—
27#16839917—22no—
28#16839912—21no—
29C4×#11758201—22no—
30#16839815C2211yescharacteristic
31#16839815C2211yescharacteristic
32#16839510C2214yescomplemented
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×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758202—24no—
48C2×#11758201—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777D412yes—
52#14519777C2312yes—
53#14519769—24no—
54#14519769—24no—
55#14519755—28no—
56#14519755—28no—
57#14519753—24no—
58#14519753—24no—
59C4×D244—24no—
60C4×D244—24no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74#14519708—22no—
75C4⋊#92418—21no—
76C4⋊#92418C2312yes—
77#14519701—22no—
78#14519700—24no—
79#14519697—22no—
80C22⋊#92418—21no—
81C22⋊#92418—21no—
82C4×#92418—21no—
83C4×#92418C2312yes—
84C22×Q8×C61D411yescharacteristic
85C2×M16×C61—21no—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103#106×C61—22no—
104C4⋊C8×C61—21no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#103×C61—22no—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61C2×D412yes—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×Dic61—22no—
121C22×Dic61—21no—
122C2×D244—21no—
123C2×D244—24no—
124C2×D244C2×D412yes—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—28no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×C4×D61—84no—
131C2×Dic122—22no—
132C2×Dic122—82no—
133C2×Dic122C2×D414yes—
134M16×C61—42no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—41no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61—41no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2411yescharacteristic
154#11758202—84no—
155#11758201—44no—
156#11758198—28no—
157#11758198—22no—
158#11758198—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61—22no—
161C4⋊Dic61—21no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61C2×D412yes—
165#11758196—28no—
166#11758196—28no—
167C2×#92418—22no—
168C2×#92418—22no—
169C23×C61#12011yescharacteristic
170C22×D61—24no—
171C22×D61—22no—
172C22×D61—42no—
173C22×D61—42no—
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—42no—
177Q8×C61—42no—
178Q8×C61—42no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—41no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61#13512yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×Dic61—22no—
196C2×Dic61—24no—
197C2×Dic61—44no—
198C2×Dic61—42no—
199D244—24no—
200D244—22no—
201D244#13714yes—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—44no—
205C4×D61—88no—
206Dic122—44no—
207Dic122—82no—
208C8×C61—42no—
209#92418—42no—
210#92418—42no—
211#18753—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61#45212yes—
215C22×C61C2×#12011yescharacteristic
216C2×D61—24no—
217C2×D61—48no—
218C2×D61—42no—
219C2×D61—42no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—81no—
226C4×C61—81no—
227C4×C61#45212yes—
228C4×C61C2×#13511yescharacteristic
229C4×C61C2×#13712yes—
230C4×C61C2×#13711yescharacteristic
231Dic61—42no—
232Dic61—44no—
233C2×#554—611no—
234#3209—611no—
235#3101—611no—
236#3078D6111yescharacteristic, complemented
237#1643—614no—
238#1632—614no—
239#1548—611no—
240#1486—611no—
241#1479—611no—
242C2×C61—22no—
243C2×C61C2×#45211yescharacteristic
244C2×C61#300411yescharacteristic
245C2×C61#301911yescharacteristic
246D61—44no—
247#554—614no—
248#554—1222no—
249#554—1222no—
250#554—1222no—
251#533C2×D6111yescharacteristic
252C2×C4⋊Q8—611no—
253C2×#125—1221no—
254C2×C22⋊Q8—1222no—
255C2×C22⋊Q8—1221no—
256C2×C4×Q8C2×D6111yescharacteristic
257C2×C4×D4—611no—
258C2×C4×D4—1221no—
259C22×C4⋊C4—1221no—
260#494—612no—
261#487—612no—
262#483—1221no—
263#479—22no—
264#477C2×D6112yescomplemented
265#476—612no—
266#475C2×D6112yescomplemented
267#462—1222no—
268#461—612no—
269#451—1222no—
270#450—1221no—
271C4×SD16—1222no—
272#431—611no—
273#431—611no—
274#428—21no—
275#423C2×D6111yescharacteristic
276#419—21no—
277#333—614no—
278Q8⋊C8—614no—
279D4⋊C8—614no—
280C61#1875311yescharacteristic, complemented, Sylow
281C22×Q8—612no—
282C22×Q8C22⋊D6111yescharacteristic
283C22×D4—611no—
284C23×C4—1222no—
285C23×C4—1221no—
286C2×Q16—1224no—
287C2×SD16—1222no—
288C2×M16—21no—
289C4⋊Q8—612no—
290C4⋊Q8—612no—
291C4⋊Q8—1221no—
292C4⋊Q8—1221no—
293C4⋊Q8C22×D6111yescharacteristic
294C4⋊Q8C22×D6111yescharacteristic
295#125—21no—
296#125—1224no—
297#125—1224no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—1228no—
301C22⋊Q8—1228no—
302C22⋊Q8—1224no—
303C22⋊Q8—1224no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×Q8C22⋊D6112yes—
309C4×Q8C22×D6112yes—
310C4×D4—612no—
311C4×D4—612no—
312C4×D4—1224no—
313C4×D4—1224no—
314C4×D4—1221no—
315C4×D4—1221no—
316#117—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—611no—
319C2×C4⋊C4—1224no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1222no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1224no—
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1221no—
328C2×C42—21no—
329C2×C42C22×D6111yescharacteristic
330#107—22no—
331#106—22no—
332#106—1222no—
333C4⋊C8—21no—
334C4⋊C8—612no—
335C4⋊C8—1221no—
336C4⋊C8C22×D6112yes—
337#103—22no—
338#103—22no—
339#103—1224no—
340#103—1222no—
341#102—1222no—
342C22⋊C8—1221no—
343C22⋊C8—1221no—
344C4×C8—612no—
345C4×C8—1221no—
346C24—1221no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—22no—
352C2×Q8—614no—
353C2×Q8—1222no—
354C2×Q8—4882no—
355C2×Q8C2×C22⋊D6112yes—
356C2×D4—612no—
357C2×D4—1224no—
358C2×D4—1221no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—1228no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—1222no—
365C22×C4—1222no—
366C22×C4—4884no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4D4×D6111yescharacteristic
370C22×C4D4×D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372Q16—4884no—
373SD16—2444no—
374M16—42no—
375C2×C8—22no—
376C2×C8—1222no—
377C2×C8—1222no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—612no—
388C4⋊C4—1228no—
389C4⋊C4—1228no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4—1221no—
393C4⋊C4—1222no—
394C4⋊C4—1222no—
395C4⋊C4C2×C22⋊D6112yes—
396C22⋊C4—41no—
397C22⋊C4—1228no—
398C22⋊C4—1222no—
399C22⋊C4—1222no—
400C42—22no—
401C42—22no—
402C42—41no—
403C42D4×D6111yescharacteristic
404C42D4×D6111yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—1224no—
408C23—1222no—
409C23—2442no—
410C23—2442no—
411C23#1451974511yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—4882no—
418Q8—2444no—
419D4—614no—
420D4—1224no—
421D4—1222no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—1222no—
431C2×C4—1224no—
432C2×C4—4888no—
433C2×C4—2444no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2444no—
438C2×C4#1451974511yescharacteristic
439C2×C4#1451974511yescharacteristic
440C2×C4#1451974511yescharacteristic
441C2×C4#1451979712yes—
442C2×C4#1451980812yes—
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4C2×D4×D6111yescharacteristic
445C2×C4C22×C22⋊D6111yescharacteristic
446C8—42no—
447C8—2442no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—1224no—
452C22—2448no—
453C22—2442no—
454C22—2442no—
455C22#1683992212yes—
456C22C2×#1451974511yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—42no—
462C4—81no—
463C4—81no—
464C4—2444no—
465C4—2442no—
466C4#1683992212yes—
467C4C2×#1451979711yescharacteristic
468C4#137×D6112yes—
469C4C2×#1451980811yescharacteristic
470C2—22no—
471C2—2444no—
472C2C2×#1683992211yescharacteristic
473C2#2145505111yescharacteristic
474C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3078C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3078D61
hypocenterC61#18753
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. #21456764C2semidirect
4. #21445025C2semidirect
5. #21444952C2semidirect
6. #21444501C2semidirect
8. #21444214C2semidirect
9. C61×#3078C2semidirect
21. #16840141C22semidirect
22. #16840114C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#475C22semidirect
236. #3078D61semidirect
264. #477C2×D61semidirect
266. #475C2×D61semidirect
280. C61#18753semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214567642
#214558232
#214450252
#214449522
#214445012
#214442432
#214442142
C61×#30782
#1875361