Finite Group Databasebeta

← #29658832

Subgroups — #29658832

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2#21456960C211yesmaximal, characteristic, complemented
3#21455612C211yesmaximal, characteristic
4#21445087C214yesmaximal, complemented
5#21444990C214yesmaximal, complemented
6#21444490C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444209C211yesmaximal, characteristic
9C61×#3104C211yesmaximal, characteristic, complemented
10#16840485—22no—
11#16840485—22no—
12#16840485—22no—
13#16840485C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519765—21no—
17C2×#14519753—21no—
18C2×C4×Dic122—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21#16840158C2212yescomplemented
22#16840112C2212yescomplemented
23#16840090—22no—
24#16840059—22no—
25#16840049C2212yes—
26#16840047—22no—
27#16840039C2212yes—
28#16839815C2211yescharacteristic
29#16839805—21no—
30#16839786C2211yescharacteristic
31#16839766—21no—
32#16839507C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#477C2212yescomplemented
39C61×#462C2212yescomplemented
40C61×#451—21no—
41C4×Q16×C61—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C2×#11758194—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777D412yes—
50#14519777C2312yes—
51#14519772—28no—
52#14519772—28no—
53#14519765—24no—
54#14519765—24no—
55#14519753—24no—
56#14519753—24no—
57C4×Dic122—24no—
58C4×Dic122—24no—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244D412yes—
62C4×D244C2312yes—
63#14519735—24no—
64#14519735—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D412yes—
68C2×#11758196—22no—
69C2×#11758198—22no—
70C2×#11758198—22no—
71C2×#11758198—21no—
72C2×C4×Dic61—22no—
73#14519710—22no—
74#14519708—24no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519701—24no—
80#14519697—22no—
81#14519697—22no—
82C22⋊#92418—21no—
83C4×#92418C2312yes—
84C22×Q8×C61D411yescharacteristic
85C2×Q16×C61—22no—
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—
103C4⋊C8×C61C2312yes—
104#103×C61—22no—
105#103×C61—22no—
106C22⋊C8×C61—21no—
107C4×C8×C61—21no—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—22no—
122C2×D244—22no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—44no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—42no—
131C2×Dic122—42no—
132Q16×C61—44no—
133C2×C8×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—41no—
145C42×C61—22no—
146C42×C61—22no—
147C42×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2411yescharacteristic
152#11758198—28no—
153#11758198—28no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—21no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D414yes—
161C4⋊Dic61C2×D412yes—
162#11758196—28no—
163C4×Dic61—28no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61#12011yescharacteristic
168C22×D61—22no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—42no—
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—42no—
175Q8×C61—42no—
176Q8×C61—42no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—41no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#13512yes—
189C2×C4×C61#13712yes—
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×Dic61—22no—
194C2×Dic61—24no—
195C2×Dic61—42no—
196C2×Dic61—44no—
197D244—22no—
198D244—22no—
199D244—42no—
200C4×D61—44no—
201C4×D61—44no—
202Dic122—42no—
203Dic122—42no—
204Dic122—44no—
205C8×C61—42no—
206#92418—42no—
207#92418—42no—
208#18781—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61#45212yes—
212C22×C61C2×#12011yescharacteristic
213C2×D61—42no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—84no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61#45212yes—
225C4×C61C2×#13511yescharacteristic
226C4×C61C2×#13711yescharacteristic
227C4×C61#57512yes—
228Dic61—44no—
229Dic61—42no—
230C2×#556—611no—
231#3229—611no—
232#3104D6111yescharacteristic, complemented
233#3077—611no—
234#1652—614no—
235#1638—614no—
236#1546—611no—
237#1487—611no—
238#1478—611no—
239C2×C61—22no—
240C2×C61C2×#45211yescharacteristic
241C2×C61#300511yescharacteristic
242C2×C61#302011yescharacteristic
243D61—82no—
244#556—614no—
245#556—1222no—
246#556—1222no—
247#556—1222no—
248#533C2×D6111yescharacteristic
249C2×C4⋊Q8—611no—
250C2×#126—1222no—
251C2×#124—1221no—
252C2×C22⋊Q8—1221no—
253C2×C4×Q8—1221no—
254C2×C4×Q8C2×D6111yescharacteristic
255C2×C4×D4—611no—
256C2×#117—1221no—
257#498—612no—
258#487—612no—
259#484—21no—
260#483—1222no—
261#478—1222no—
262#477C2×D6112yescomplemented
263#476—612no—
264#475—1222no—
265#474—612no—
266#462C2×D6112yescomplemented
267#451—21no—
268C4×Q16—22no—
269#431—611no—
270#431C2×D6111yescharacteristic
271#428—1221no—
272#423—611no—
273#418—1221no—
274#332—614no—
275Q8⋊C8—614no—
276D4⋊C8—614no—
277C61#1878111yescharacteristic, complemented, Sylow
278C22×Q8—1221no—
279C22×Q8C22⋊D6111yescharacteristic
280C22×D4—611no—
281C23×C4—1221no—
282C2×Q16—22no—
283C2×M16—1221no—
284C4⋊Q8—612no—
285C4⋊Q8—612no—
286C4⋊Q8—1221no—
287C4⋊Q8—1221no—
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#126—1228no—
291#126—1228no—
292#125—21no—
293#124—1224no—
294#124—1224no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—1224no—
298C22⋊Q8—1224no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—1224no—
304C4×Q8—1224no—
305C4×Q8C22⋊D6112yes—
306C4×Q8C22×D6112yes—
307C4×D4—612no—
308C4×D4—612no—
309C4×D4—1221no—
310C4×D4—1221no—
311#117—21no—
312#117—1224no—
313#117—1224no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1221no—
323C2×C42—21no—
324C2×C42—1222no—
325C2×C42C22×D6111yescharacteristic
326#107—22no—
327#107—1222no—
328#106—1222no—
329#106—1224no—
330C4⋊C8—612no—
331C4⋊C8—1221no—
332C4⋊C8—1221no—
333C4⋊C8C22×D6112yes—
334#103—22no—
335#103—22no—
336#103—1224no—
337#102—1222no—
338#102—1222no—
339C22⋊C8—21no—
340C22⋊C8—1221no—
341C4×C8—21no—
342C4×C8—612no—
343C24—1221no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—42no—
348C2×Q8—1222no—
349C2×Q8—1222no—
350C2×Q8—2442no—
351C2×Q8—2442no—
352C2×Q8C2×C22⋊D6112yes—
353C2×D4—612no—
354C2×D4—1222no—
355C2×D4—1221no—
356C2×D4—1222no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—1221no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—2444no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367Q16—44no—
368M16—2442no—
369C2×C8—22no—
370C2×C8—1222no—
371C2×C8—1222no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—614no—
382C4⋊C4—612no—
383C4⋊C4—1228no—
384C4⋊C4—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4C2×C22⋊D6112yes—
389C22⋊C4—41no—
390C22⋊C4—1228no—
391C22⋊C4—1228no—
392C22⋊C4—1222no—
393C22⋊C4—1222no—
394C42—22no—
395C42—22no—
396C42—41no—
397C42—1228no—
398C42D4×D6111yescharacteristic
399C42D4×D6111yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—1222no—
403C23—2442no—
404C23—2442no—
405C23—2442no—
406C23#1451974511yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—2442no—
413Q8—2442no—
414Q8—2444no—
415D4—1222no—
416D4—1222no—
417D4—2442no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—1222no—
427C2×C4—1224no—
428C2×C4—2444no—
429C2×C4—2442no—
430C2×C4—2444no—
431C2×C4—2444no—
432C2×C4#1451974511yescharacteristic
433C2×C4#1451974511yescharacteristic
434C2×C4#1451974511yescharacteristic
435C2×C4#1451980012yes—
436C2×C4#1451980912yes—
437C2×C4C2×D4×D6111yescharacteristic
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C22×C22⋊D6111yescharacteristic
440C8—42no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—4884no—
446C22—2442no—
447C22—2442no—
448C22—2442no—
449C22#1683992312yes—
450C22C2×#1451974511yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4—2444no—
459C4—2442no—
460C4#1683992312yes—
461C4C2×#1451980011yescharacteristic
462C4C2×#1451980911yescharacteristic
463C4#1684056012yes—
464C2—22no—
465C2—4882no—
466C2C2×#1683992311yescharacteristic
467C2#2145505911yescharacteristic
468C2#2145518111yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3104C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3104D61
hypocenterC61#18781
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
2. #21456960C2semidirect
4. #21445087C2semidirect
5. #21444990C2semidirect
6. #21444490C2semidirect
7. #21444248C2semidirect
9. C61×#3104C2semidirect
21. #16840158C22semidirect
22. #16840112C22semidirect
32. #16839507C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#462C22semidirect
232. #3104D61semidirect
262. #477C2×D61semidirect
266. #462C2×D61semidirect
277. C61#18781semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214569602
#214556122
#214450872
#214449902
#214444902
#214442482
#214442092
C61×#31042
#1878161